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Mass and geometric measurements of binary radio pulsars Fonseca, Emmanuel 2016

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Mass and Geometric Measurementsof Binary Radio PulsarsbyEmmanuel FonsecaB. Sc., Physics, the Pennsylvania State University, 2010M. Sc., Astronomy, the University of British Columbia, 2012A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Physics & Astronomy)The University of British Columbia(Vancouver)October 2016c©Emmanuel Fonseca, 2016AbstractOne of the primary, long-term goals in high-energy astrophysics is the mea-surement of macroscopic parameters that constrain the equation of statefor compact stellar objects. For neutron stars, known to be composed ofthe densest matter in the Universe, measurements of their masses and sizesare of considerable importance due to the poorly understood processes thatgovern their interiors. Measurements of relativistic “post-Keplerian” effectsin binary systems can be used to significantly constrain viable equations ofstate, test modern theories of gravitation, verify binary-evolution models thatpredict correlations between certain binary parameters, and determine theGalactic neutron-star mass distribution that is expected to reflect differentsupernovae mechanisms and evolutionary paths.In this thesis, we use established pulsar-timing techniques to analyze sig-nals from radio pulsars in 25 binary systems, as well as from one pulsar ina hierarchical triple system, in order to detect perturbations from Keplerianmotion of the bodies. We characterize observed relativistic Shapiro timingdelays to derive estimates of the component masses and inclination angles in14 pulsar-binary systems, and measure a large number of secular variationsdue to kinematic, relativistic and/or third-body effects in the majority ofbinary systems studied here. We find a wide range of pulsar masses (mp),with values as low as mp = 1.18+0.10−0.09 M for PSR J1918−0642 and as highas mp = 1.928+0.017−0.017 M for PSR J1614−2230 (both 68.3% credibility), andmake new detections of the Shapiro-delay signal in four binary systems foriithe first time. In the relativistic PSR B1534+12 binary system, we derivean accurate and precise rate of geodetic precession of the pulsar-spin axis –due to secular variations of electromagnetic pulse structure – that is consis-tent with the prediction from general relativity. In the PSR B1620−26 triplesystem, we discuss ongoing efforts to simultaneously model both “inner”and “outer” orbits and tentatively measure secular variations of all “inner-orbital” elements; we show that these variations are likely due to third-bodyinteractions between the smaller orbit and outer companion, which can even-tually be used to constrain orientation angles and possibly the pulsar massin the near future.iiiPrefaceAll text, figures, and results presented in this dissertation are original prod-ucts of the author, E. Fonseca, under the supervision of I. H. Stairs. Nonethe-less, much of the pulsar data presented in this thesis were obtained in col-laboration with many observational astronomers:• In Chapter 2, the successful proposal for the ongoing NANOGrav “P2945”observing program at the Arecibo Observatory was written by E. Fon-seca, using simulated results provided by J. A. Ellis, X. Siemens, J.Cordes, and D. R. Madison. The P2945 data presented in Figure 2.2were collected by over 15 pulsar astronomers, including E. Fonseca. Alist of the observers for the Arecibo and GBT timing data is given inthe “Author Contributions” section of the study published by Arzou-manian et al. (2015b).• In Chapter 3, all text, results and figures were generated by E. Fonseca– the principal investigator of the project – and accepted for publica-tion in the Astrophysical Journal in September 2016 (see Fonseca et al.,2016). The interpretation of observed secular variations and analysisof Shapiro timing delays were performed by E. Fonseca, with guidanceand insight provided by I. H. Stairs, D. J. Nice, T. T. Pennucci, J. A.Ellis, S. M. Ransom, and P. B. Demorest. The PAL2 Bayesian soft-ware discussed and used in this chapter was developed by J. A. Ellis.A portion of the data set was obtained for a targeted observing pro-gram devised and described by Pennucci (2015). The data presentedivin this chapter were collected by over 15 pulsar astronomers, includingE. Fonseca, as described in the “Author Contributions” section of thesubmitted study conducted by Fonseca et al. (2016). We are gratefulfor useful comments on the manuscript sent by C. Bassa, and for usefuldiscussion with C. Ng and P. C. C. Freire.• In Chapter 4, all GUPPI data collected after the start of the 2012year were obtained through successful proposals led by E. Fonseca,and collected by E. Fonseca, I. H. Stairs and Z. Arzoumanian. Earlierobservations with the Arecibo and NRAO facilities were performed byS. E. Thorsett and F. Camilo. Observations collected with the Effels-berg and Lovell telescopes were conducted by a large number of peopleover many years: the observations from the Effelsberg telescope weremaintained, processed and provided by N. Caballero and M. Kramer;and data collected at Jodrell Bank were maintained, processed andprovided by A. Lyne and B. Stappers.• In Chapter 5, the majority of results and text in Sections 5.2 and5.3 were first presented by Fonseca et al. (2014), which was based on aM. Sc. thesis written by E. Fonseca. However, the PUPPI observationspresented in Section 5.4 – successfully proposed for by E. Fonseca, I .H.Stairs and S. E. Thorsett – are processed and analyzed for the first timein this dissertation.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xvDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The Physics and Phenomenology of Pulsars . . . . . . . . . . 31.1.1 Pulsars are Neutron Stars . . . . . . . . . . . . . . . . 31.1.2 A “Lighthouse” in the (Cosmic) Darkness . . . . . . . 51.1.3 Observed Characteristics . . . . . . . . . . . . . . . . . 71.1.4 Binary Millisecond Pulsars . . . . . . . . . . . . . . . . 91.2 Data Acquisition and Instrumentation . . . . . . . . . . . . . 121.2.1 Dispersion from the Interstellar Medium . . . . . . . . 131.2.2 Incoherent De-Dispersion . . . . . . . . . . . . . . . . . 141.2.3 Coherent De-Dispersion . . . . . . . . . . . . . . . . . 151.3 Pulsar Timing, in a Nutshell . . . . . . . . . . . . . . . . . . . 16vi1.3.1 TOA Estimation . . . . . . . . . . . . . . . . . . . . . 171.3.2 The Timing Model for Isolated Pulsars . . . . . . . . . 181.4 Timing Delays from Binary Motion . . . . . . . . . . . . . . . 251.4.1 Orbits with Significant Eccentricity . . . . . . . . . . . 271.4.2 Orbits with No Significant Eccentricity . . . . . . . . . 301.5 Variations in the Orbital Elements . . . . . . . . . . . . . . . 331.5.1 Strong-Field Gravitation . . . . . . . . . . . . . . . . . 341.5.2 Kinematic Bias from Proper Motion . . . . . . . . . . 361.5.3 Kinematic Bias from Acceleration . . . . . . . . . . . . 371.5.4 Periodic Variations and Annual Orbital Parallax . . . . 381.6 This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 The North American Nanohertz Observatory for Gravita-tional Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.1 Data Acquisition and Analysis . . . . . . . . . . . . . . . . . . 422.2 Proposals for Observations: P2945 . . . . . . . . . . . . . . . 432.3 Contributions to NANOGrav Projects . . . . . . . . . . . . . 463 The NANOGrav Nine-Year Data Set: Mass and GeometricMeasurements of Binary Millisecond Pulsars . . . . . . . . . 483.1 Observations & Reduction . . . . . . . . . . . . . . . . . . . . 493.2 Binary Timing Models . . . . . . . . . . . . . . . . . . . . . . 533.2.1 Parametrizations of the Shapiro Delay . . . . . . . . . 533.3 Analyses of Mass & Geometric Parameters . . . . . . . . . . . 553.3.1 Bayesian Analyses of Shapiro-Delay Signals . . . . . . 573.3.2 Limits on Inclination from x˙ and the Absence of ShapiroDelay . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.4 Results & Discussion . . . . . . . . . . . . . . . . . . . . . . . 753.4.1 PSR J0613−0200 . . . . . . . . . . . . . . . . . . . . . 753.4.2 PSR J1455−3330 . . . . . . . . . . . . . . . . . . . . . 763.4.3 PSR J1600−3053 . . . . . . . . . . . . . . . . . . . . . 77vii3.4.4 PSR J1614−2230 . . . . . . . . . . . . . . . . . . . . . 783.4.5 PSR J1640+2224 . . . . . . . . . . . . . . . . . . . . . 793.4.6 PSR J1738+0333 . . . . . . . . . . . . . . . . . . . . . 813.4.7 PSR J1741+1351 . . . . . . . . . . . . . . . . . . . . . 823.4.8 PSR B1855+09 . . . . . . . . . . . . . . . . . . . . . . 833.4.9 PSR J1903+0327 . . . . . . . . . . . . . . . . . . . . . 843.4.10 PSR J1909−3744 . . . . . . . . . . . . . . . . . . . . . 853.4.11 PSR J1918−0642 . . . . . . . . . . . . . . . . . . . . . 863.4.12 PSR J1949+3106 . . . . . . . . . . . . . . . . . . . . . 873.4.13 PSR J2017+0603 . . . . . . . . . . . . . . . . . . . . . 883.4.14 PSR J2043+1711 . . . . . . . . . . . . . . . . . . . . . 893.4.15 PSR J2145−0750 . . . . . . . . . . . . . . . . . . . . . 903.4.16 PSR J2302+4442 . . . . . . . . . . . . . . . . . . . . . 913.4.17 PSR J2317+1439 . . . . . . . . . . . . . . . . . . . . . 923.5 Conclusions & Summary . . . . . . . . . . . . . . . . . . . . . 1094 Long-Term Timing of the PSR B1620−26 Triple System inMessier 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1144.1 Observations & Reduction . . . . . . . . . . . . . . . . . . . . 1174.2 Methods for Timing Analysis . . . . . . . . . . . . . . . . . . 1204.3 Timing Update for PSR B1620−26 . . . . . . . . . . . . . . . 1224.3.1 Analysis of “Pre-Gap” Data . . . . . . . . . . . . . . . 1244.3.2 Analysis of “Post-Gap” GASP and GUPPI Data . . . . 1334.3.3 Changes of Inner-Orbital Parameters over Time . . . . 1364.3.4 Global Analysis of All TOA Data . . . . . . . . . . . . 1374.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1434.4.1 Is PSR B1620−26 a Triple System? . . . . . . . . . . . 1434.4.2 Complications from Moons and Pulsar Glitches . . . . 1454.4.3 Bias in ν˙s from Cluster Acceleration . . . . . . . . . . . 1464.4.4 Bias in ν¨s from Cluster Jerks . . . . . . . . . . . . . . . 1484.4.5 Variations of the Inner-Orbital Elements . . . . . . . . 148viii4.5 Further Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 1505 Long-Term Observations of the Relativistic PSR B1534+12Binary System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1525.1 Observations & Reduction . . . . . . . . . . . . . . . . . . . . 1555.2 DM Variations in PSR B1534+12 . . . . . . . . . . . . . . . . 1575.3 Geodetic Precession and Secular Evolution in Pulse Structure 1625.3.1 Methodologies . . . . . . . . . . . . . . . . . . . . . . . 1635.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1675.4 Timing Observations with PUPPI . . . . . . . . . . . . . . . . 1705.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1746 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . 1786.1 Projections of Future PK Measurements . . . . . . . . . . . . 1826.2 The Era of Neutron-Star Mass Measurements . . . . . . . . . 188Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191A Probabilistic Analysis of Shapiro-Delay Parameters . . . . . 212A.1 Bayesian Interpretation of χ2-grid Analysis . . . . . . . . . . . 213A.2 Derivation of Uniform Distribution for Randomly OrientedOrbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215A.3 Translations of Probability Densities . . . . . . . . . . . . . . 216A.3.1 Probability in Orthometric Space . . . . . . . . . . . . 219A.4 Confidence Intervals . . . . . . . . . . . . . . . . . . . . . . . 220B Spin-Derivative Model of Long-Period Orbits . . . . . . . . . 222B.1 Derivation of Orbit-Induced Spin Derivatives . . . . . . . . . . 223B.2 Determination of Parameters . . . . . . . . . . . . . . . . . . . 226B.3 Complications from Spin-down . . . . . . . . . . . . . . . . . . 227B.4 Derived Quantities of the Orbit . . . . . . . . . . . . . . . . . 230ixList of Tables2.1 NANOGrav MSPs for the P2945 Observing Program. . . . . . 443.1 Keplerian Elements of the NANOGrav Binary MSPs. . . . . . 523.2 Summary of Secular Variations and Shapiro-Delay Parameters. 563.3 Upper Limits on Inclinations for MSPs with No ∆S. . . . . . . 633.4 χ2-grid Estimates of Component Masses and Inclinations forMSPs with Significant ∆S. . . . . . . . . . . . . . . . . . . . . 743.5 Precise Estimates of mc for Low-Mass He White Dwarfs . . . . 1104.1 Frequency-Derivative timing Models for Pre-Gap, Post-GapSubsets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1264.2 Solutions of the Joshi & Rasio Method and Outer-Orbital El-ements for PSR B1620−26. . . . . . . . . . . . . . . . . . . . 1284.3 Two-Orbit Timing Solution for B1620−26 for Pre-Gap DataSubset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1324.4 Spin-Frequency Timing Model for Entire B1620−26 Data Set. 1385.1 MCMC Results of Geodetic-Precession Parameters for PSRB1534+12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1685.2 PUPPI Data Parameters for B1534+12. . . . . . . . . . . . . 171xList of Figures1.1 P˙s Versus Ps for All Known Radio Pulsars. . . . . . . . . . . . 101.2 Schematic of Orientation for an Eccentic Binary System. . . . 261.3 ∆S for PSR J1918−0642. . . . . . . . . . . . . . . . . . . . . . 332.1 TOA residuals for P2945 MSPs. . . . . . . . . . . . . . . . . . 463.1 PAL2 posterior PDFs of Shapiro-delay parameters for PSRJ2043+1711. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.2 (mc, cos i) Probability Map for PSR J0023+0923. . . . . . . . 653.3 (mc, cos i) Probability Map for PSR J1012+5307. . . . . . . . 663.4 (mc, cos i) Probability Map for PSR J1455−3330. . . . . . . . 673.5 (mc, cos i) Probability Map for PSR J1643−1224. . . . . . . . 683.6 (mc, cos i) Probability Map for PSR J1738+0333. . . . . . . . 693.7 (mc, cos i) Probability Map for PSR J1853+1303. . . . . . . . 703.8 (mc, cos i) Probability Map for PSR J1910+1256. . . . . . . . 713.9 (mc, cos i) Probability Map for PSR J2145−0750. . . . . . . . 723.10 (mc, cos i) Probability Map for PSR J2214+3000. . . . . . . . 733.11 Probability Maps and Posterior PDFs of mp, mc and cos i forPSR J0613−0200. . . . . . . . . . . . . . . . . . . . . . . . . . 943.12 Probability Maps and Posterior PDFs of mp, mc and cos i forPSR J1600−3053. . . . . . . . . . . . . . . . . . . . . . . . . . 953.13 Probability Maps and Posterior PDFs of mp, mc and cos i forPSR J1614−2230. . . . . . . . . . . . . . . . . . . . . . . . . . 96xi3.14 Probability Maps and Posterior PDFs of mp, mc and cos i forPSR J1640+2224 . . . . . . . . . . . . . . . . . . . . . . . . . 973.15 Probability Maps and Posterior PDFs of mp, mc and cos i forPSR J1741+1351. . . . . . . . . . . . . . . . . . . . . . . . . . 983.16 Probability Maps and Posterior PDFs of mp, mc and cos i forPSR B1855+09. . . . . . . . . . . . . . . . . . . . . . . . . . . 993.17 Probability Maps and Posterior PDFs of mp, mc and cos i forPSR J1903+0327. . . . . . . . . . . . . . . . . . . . . . . . . . 1003.18 Probability Maps and Posterior PDFs of mp, mc and cos i forPSR J1909−3744. . . . . . . . . . . . . . . . . . . . . . . . . . 1013.19 Probability Maps and Posterior PDFs of mp, mc and cos i forPSR J1918−0642. . . . . . . . . . . . . . . . . . . . . . . . . . 1023.20 Probability Maps and Posterior PDFs of mp, mc and cos i forPSR J1949+3106. . . . . . . . . . . . . . . . . . . . . . . . . . 1033.21 Probability Maps and Posterior PDFs of mp, mc and cos i forPSR J2017+0603. . . . . . . . . . . . . . . . . . . . . . . . . . 1043.22 Probability Maps and Posterior PDFs of mp, mc and cos i forPSR J2043+1711. . . . . . . . . . . . . . . . . . . . . . . . . . 1053.23 Probability Maps and Posterior PDFs of mp, mc and cos i forPSR J2145−0750. . . . . . . . . . . . . . . . . . . . . . . . . . 1063.24 Probability Maps and Posterior PDFs of mp, mc and cos i forPSR J2302+4442. . . . . . . . . . . . . . . . . . . . . . . . . . 1073.25 Probability Maps and Posterior PDFs of mp, mc and cos i forPSR J2317+1439. . . . . . . . . . . . . . . . . . . . . . . . . . 1083.26 Pb Versus mc for NANOGrav MSP-Binary Systems with He-WD Companions. . . . . . . . . . . . . . . . . . . . . . . . . . 1124.1 TOA Residuals for PSR B1620−26, All Data with Fixed Two-Orbit Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.2 TOA Residuals for B1620−26, Pre-Gap Data Subset. . . . . . 127xii4.3 MCMC Estimates of Spin-Frequency Derivatives for B1620−26Pre-Gap Data Subset. . . . . . . . . . . . . . . . . . . . . . . 1304.4 Posterior Distributions of Outer-Orbital Elements for Pre-GapData Subset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1314.5 TOA Residuals for B1620−26, Post-Gap Data Subset. . . . . . 1354.6 Solutions of the Joshi & Rasio (1997) Method for the Post-GapData Subset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1364.7 Posterior Distributions of Outer-Orbital Elements for Post-Gap Data Subset. . . . . . . . . . . . . . . . . . . . . . . . . . 1394.8 TOA Residuals for B1620−26, Sll data with Frequency-DerivativeModel of Outer Orbit. . . . . . . . . . . . . . . . . . . . . . . 1404.9 Ps Versus Time for B1620−26. . . . . . . . . . . . . . . . . . . 1424.10 Zoomed TOA Residuals for Frequency-Derivative Model ofOuter Orbit for All Data. . . . . . . . . . . . . . . . . . . . . 1505.1 DM Variations for PSR B1534+12. . . . . . . . . . . . . . . . 1595.2 Structure Function of DM Variations for B1534+12. . . . . . . 1615.3 Difference in 2005 Mark IV and ASP Profiles. . . . . . . . . . 1645.4 Polarization Data of Full-Summed 2003 Campaign Profile forB1534+12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1665.5 MCMC Posterior Distribution and Chain for Ωspin1 . . . . . . . 1695.6 β Versus Time for B1534+12. . . . . . . . . . . . . . . . . . . 1705.7 Current TOA Residuals for B1534+12, Including PUPPI Data. 1725.8 1400-MHZ Observation of B1534+12 on MJD 56326. . . . . . 1735.9 DM Variations in PUPPI Era for B1534+12. . . . . . . . . . . 1745.10 Evaluation of Six PK Effects Measured for B1534+12. . . . . . 1766.1 Summary of Pulsar Masses Measured in This Thesis. . . . . . 1806.2 Future Times of Detection for Different Secular Variations, forB1534+12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187xiii6.3 Neutron-Star Mass-Radius Relations and Observational Con-straints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190xivAcknowledgementsThose who get to know me quickly find out that I’m a very nostalgic person.In the process of writing this dissertation, and especially this section, manymemories have returned to me in numerous and funny ways. I rememberthe first time I saw Comet Hale-Bopp, around the age of 10, in the northernsky from a frigid night in Malden, Massachusetts. I also remember manydays of my time at the “Zen House”, a Kitsilano home filled with fifteen ofthe greatest people to have ever roam the Earth. I especially remember thefirst time I met Ingrid H. Stairs, my Ph. D. supervisor, and her enthusiasticwillingness to take me on as a graduate student.First and foremost, I truly thank Ingrid for everything. Her expertise,patience, teachings and her trust in my judgement have instilled within me aconfidence that I never thought I could obtain. The numerous opportunitiesprovided by Ingrid allowed me to travel near and far, meeting with leading ex-perts and inspiring students (both graduate and undergraduate), all of whomare great people. I feel truly grateful to be a part of the pulsar community,a comparatively small and yet powerful group of researchers. In particular,I thank the following pulsar astronomers and mentors for their collaborationand help throughout my time as a graduate student: Steve Thorsett, Za-ven Arzoumanian, Tim Pennucci, Maura McLaughlin, Scott Ransom, PaulDemorest, David Nice, and Jim Cordes.At UBC, I am deeply thankful to my Ph. D. committee – Jeremy Heyl,Brett Gladman, Gary Hinshaw, and Bill Unruh – who provided useful com-xvmentary that undoubtedly improved the quality of the work presented below.I also thank the UBC Department of Physics and Astronomy, particularlyits secretaries and staff, for their constant and prompt support during theeasy and hard times. I am definitely thankful to Theresa Liao and DaveNg for allowing me to contribute towards their science outreach and public-engagement efforts, and wish to acknowledge here the good work that theydo. I am grateful for the camaraderie shared with the graduate students andpostdocs at UBC astronomy, as well all of those times spent together outsideof our offices and instead within nearby pubs.At Penn State, my alma mater, I thank the astronomy faculty and partic-ularly Donald Schneider for teaching one of the most difficult courses I’ve evertaken, and inspiring me to continue onward down the path of an astronomer.I also thank the folks at the Swift Gamma-Burst Mission, especially ScottKoch, Erik Hoversten, Peter Roming and Stephen Holland, all of whom gaveme my first shot at research. Steve recommended UBC to me as a potentialgraduate-school option, and the rest is history.This work, carried out over six wonderful and complex years, would nothave been possible without the selfless support of family and friends. Tothe roommates of the “Zen” and “King Ed” houses: thank you for yourfriendship, your openness, the happiness and hilarity that filled our timetogether. To the Birds and their lovely family: thank you for the chats andexile at W. Ninth and Discovery, in the peaceful Point Grey. To (Dra.) AnuGutierrez: thank you for sharing your love and beauty with me. To my oldfriends from Malden, wherever you are now: thank you for the catchups,coffee and Piza Pizza; we will do it all again soon.My family – the Fonseca’s, the Bernal’s and the Trahan’s – is filled withmany, many lovely people. All of them cared for me, worried about me, andbelieved in me. To my parents and siblings – Martha (ne´e Bernal), Gustavo,Leo, and Denise – who have been there the whole way: los amo. May youand your names be remembered forever.xviFor mi familia, mis amigos, and mis amores:this is written for youand by you.xviiChapter 1IntroductionThis dissertation is set to be completed nearly fifty years after the discov-ery of radio pulsars made by Jocelyn Bell-Burnell (and first summarized byHewish et al., 1968). The discovery itself was a decisive moment in thehistory of physics and astronomy as it confirmed the existence of neutronstars – tiny, compact stars comprised mostly of neutrons that are supportedagainst gravitational collapse by quantum-mechanical degeneracy pressureand nuclear interactions – and demonstrated that rotating neutron stars withbeamed radiation along their magnetic poles could be observed at radio fre-quencies as pulsars. Over 2,500 pulsars are currently known to reside in theGalaxy, according to the catalog maintained by the Australia Telescope Na-tional Facility (ATNF; Manchester et al., 2005)1; the number continues togrow as large-scale surveys search for them with increasing sensitivity (e.g.Manchester et al., 2001; Coenen et al., 2014). In the five decades since theirdiscovery, pulsars have repeatedly been used to directly address open prob-lems in physics and astronomy; they continue to yield game-changing resultswith exquisitely high precision. Indeed, two Nobel Prizes in physics have sofar been awarded for results obtained by studying pulsars: one Prize2 for the1http://www.atnf.csiro.au/people/pulsar/psrcat/2http://www.nobelprize.org/nobel_prizes/physics/laureates/1974/1discovery of pulsars; and a second Prize3 for the confirmation that gravita-tional waves exist in Nature using pulsars in relativistic binary systems.As shown in all subsequent chapters of this thesis, radio pulsars in gravit-ationally-bound orbital systems serve as probes of gravitation and binary-formation mechanisms. Many aspects regarding the evolution of their pro-genitor orbits can be inferred from precise measurements of the five basicKeplerian orbital parameters and the observed spin properties (see Lorimer,2008, for a review). Pulsars within relativistic binary systems further ex-hibit a variety of “post-Keplerian” (PK) effects that can be used to measureadditional parameters of each system, such as the binary-component massesand system orientation (Damour & Deruelle, 1985, 1986). PK measurementsoffer uniquely powerful constraints on the internal structure of ultra-compactobjects (e.g. Lattimer & Prakash, 2004) and the inferred mass distributionof the neutron-star population (Thorsett & Chakrabarty, 1999; O¨zel et al.,2012; Kiziltan et al., 2013). Binary radio pulsars provide a desirable environ-ment to test gravitational theory and understand late-stage stellar evolutionwith high precision.The purpose of this doctoral thesis is to present results obtained fromdetailed analyses of radio pulsars in binary systems. In this introductorychapter, we present an overview of the current physical paradigm of pul-sars, the phenomena that affect radio-frequency observations of pulsars, andgeneral analysis techniques that are employed in all projects discussed insubsequent chapters. This context is necessary for the interpretation of re-sults presented throughout the dissertation. In Section 1.1, we provide abrief summary of the current understanding of pulsars as neutron stars, gen-eral pulsar properties, and a brief overview of the evolution of pulsars inbinary systems. In Section 1.3, we outline the procedure for modeling pul-sar data that is used for all radio pulsars studied below. In Section 1.4, weexplicitly discuss the models used to describe binary motion for eccentric3http://www.nobelprize.org/nobel_prizes/physics/laureates/1993/2and nearly-circular pulsar-binary systems. In Section 1.2, we briefly discussthe observing systems used over the years to collect the pulsar data we usein the subsequent chapters of this dissertation. In Section 1.5, we discussthe relativistic and geometric phenomena that can be measured using binarypulsars and the information that can be learned from modeling these effects.In Section 1.6, we summarize the information presented in this chapter andbriefly outline the content of the following chapters presented in this thesis.1.1 The Physics and Phenomenology of Pul-sarsAll known pulsars in our Galaxy collectively span a wide range of pulseperiods, with the smallest spin periods on the order of 1 ms. Observations ofindividual radio pulsars yield unique parameters that describe each neutronstar’s spin properties, the tenuous interstellar medium along the observer’sline of sight to the pulsar, and any kinematic terms associated with secularor orbital motion. However, an evolving census of these parameters for theGalactic pulsar population has allowed for a qualitative picture of a pulsarto be formed with wide acceptance in the astrophysical community. In thissection, we describe the general picture of a pulsar, expected spin propertiesof radio pulsars that will be exploited throughout this thesis, and an overviewof pulsars in binary systems. We reserve discussion of the quantitative modelsthat describe the various physical effects observable using pulsars for Section1.3.1.1.1 Pulsars are Neutron StarsThe theoretical discovery of neutron stars was first made by Baade & Zwicky(1934) shortly after the discovery of the neutron in 1932 by James Chadwick.Walter Baade and Fritz Zwitcky argued that supernova explosions generally3represent the transition from a fusion-powered star to a degenerate objectthat is mostly composed of neutrons. Pulsars therefore represent the rem-nants of old, massive stars that have aged through the branches of stellarevolution and ultimately suffered extensive (but not complete) gravitationalcollapse. The association of pulsars with neutron stars was first made inthe discovery study undertaken by Hewish et al. (1968), though radially-pulsating white dwarfs could have also explained their observations at thetime. Shortly after this discovery, the neutron-star picture was solidified withsubsequent discoveries of radio pulsars at the heart of the Crab and Vela neb-ulae (Staelin & Reifenstein, 1968; Large et al., 1968), which both have pulseperiods far smaller than the physically allowed pulsation rates for compactobjects. The mechanisms proposed by Pacini (1967) and Gold (1968), whichaccounted for observed high-energy radiation at the center of supernova rem-nants as well as the range and regularity of pulse periods known at the time,quickly established pulsars are rotating neutron stars.Initial calculations of the internal structure for neutron stars were firstperformed by Oppenheimer & Volkoff (1939), who considered a cold Fermi-Dirac gas while neglecting thermal and nuclear sources of pressure; theypredicted that such objects must have masses around 0.7 M and be verysmall in size, with radii R ∼ 10 km. Modern calculations that use numericalsupercomputing and account for repulsive nuclear forces predict that neutronstars can have a maximum mass somewhat less than 3 M, depending onthe non-nucleonic composition under consideration, before undergoing com-plete gravitational collapse into a black hole (e.g. Kiziltan et al., 2013). Themaximum mass for neutron stars therefore depends on the equation of state(EOS) that governs their internal structure (e.g. Lattimer & Prakash, 2015).Indeed, one of the driving motivations for studying pulsars in binary systemsis to determine the masses of the observed neutron stars, which can be usedto place constraints on viable EOSs (e.g. O¨zel & Freire, 2016).41.1.2 A “Lighthouse” in the (Cosmic) DarknessThe conventional model of a pulsar illustrates a highly-magnetized neutronstar that emits cone-shaped beams of radiation at radio wavelengths alongits magnetic poles. A small region centered on each polar cap is threadedby open magnetic-field lines which accelerates charged plasma away fromthe surface of the pulsar; the charged plasma produces photons during thisacceleration into a conic area. The extreme electromagnetic fields aroundthe vicinity of a neutron star produce a nebulous sphere of charged par-ticles that co-rotates with the compact object, generally referred to as a“magnetosphere” (Goldreich & Julian, 1969; Sieber & Wielebinski, 1973).Despite nearly five decades of pulsar research, there is still no theoreticalframework that fully describes the broad-band, frequency-dependent, lumi-nous radio emission, though the strong external magnetic fields are expectedto play an integral role in their production (e.g. Rankin, 2015, and referencestherein). Modern research in magnetosphere structure seeks to obtain solu-tions for non-idealized electromagnetic fields and radiation mechanisms (e.g.Kalapotharakos et al., 2012), while accounting for the implied changes ofmagnetospheric properties observed in “intermittent” pulsars (e.g. Krameret al., 2006a; Lorimer et al., 2012).Radio pulsars are observed to be highly-polarized sources, with large de-grees of linearly-polarized intensity (L) and comparatively weaker circularly-polarized emission (V ). Along with the Stokes Q and U polarization param-eters of the observed radio emission, pulsar signals can be generally repre-sented by the Stokes vector ~S = (I,Q, U, V ), where I ≥ √Q2 + U2 + V 2is the total intensity of the pulsar signal and L =√Q2 + U2. If the entirepolarization state is well measured, the polarization position angle (Ψ) canbe computed to beΨ =12arctan(UQ). (1.1)5and is a function of the phase of pulsar rotation (φ). Radhakrishnan & Cooke(1969) developed a model that explains smooth, swing-like variations in Ψ asa swing of the plane of linear polarization that is tied to the magnetic fieldlines, assuming that the magnetic field is dipolar. This model of polarizationorientation is generally referred to as a rotating vector model (RVM), and theassumption of dipolar geometry relates Ψ to intrinsic geometric parametersthat describe the relative orientation of the spin and magnetic axes of thepulsar:tan(Ψ−Ψ0) = sinα sin(φ− φ0)sin(α + β) cosα− cos(α + β) sinα cos(φ− φ0) , (1.2)where α is the misalignment angle between the spin and magnetic axes, βis the minimum angle between the magnetic axis and the line of sight, and(φ0,Ψ0) are fiducial values of φ and Ψ, respectively. In principle, polarizationdata can be used to constrain the geometry of pulsars; in practice, however,few radio pulsars exhibit values of Ψ as a function of φ that are well-modeledby Equation 1.2.It is important to note that Equations 1.1 and 1.2 are derived under theassumption that Ψ is measured in the clockwise direction on the plane ofthe sky. This is inconsistent with the general convention, maintained bythe International Astronomical Union (IAU), that position angles definedwithin the plane of the sky are measured in the counter-clockwise directionfrom celestial North (Everett & Weisberg, 2001). We nonetheless use theseequations in this dissertation, while noting that any results obtained by usingEquations 1.1 and 1.2 can be converted to the convention-standard values byapplying the appropriate change in angular basis:6αIAU = pi − α (1.3)βIAU = −β (1.4)In summary, the “cosmic lighthouse” model describes the observed po-larized radiation from pulsars as a periodic “slice” of the radio-emission conethat occurs once per rotation. Naturally, a distant observer will see a radiopulse so long as the emission cone contains the observer’s line of sight. Thelighthouse model makes no assumptions about conic structure of the radiobeam and, indeed, many different pulsars have complex (i.e. non-Gaussian,multiple-component) pulse profiles. In order to classify a spinning neutronstar as a pulsar, the model does assume that the vector of spin angular mo-mentum is misaligned with the axis of magnetic poles, i.e. α > 0, which isnecessary in order for the beamed radiation to appear as pulses to the distantobserver. However, values of α can be quite small (e.g. Stairs et al., 1999).1.1.3 Observed CharacteristicsThere are several important observational properties of radio pulsars thatare considered to be general among the population and are well understood,despite the underlying uncertainty in neutron-star structure and the natureof the radio emission mechanism. For instance, the rotation of pulsars isobserved to be remarkably stable; the regularity in pulsar rotation is main-tained by the neutron star’s large moment of inertia Irot ∼ 1045 g cm2. Suchrotational stability is a defining characteristic of radio pulsars and was im-mediately seen in PSR B1919+214, the first pulsar that was discovered by4Radio pulsars, designated with “PSR” that stands for pulsating source of radio emis-sion, are formally given names based on their equatorial coordinates as measured at somereference epoch. Thus, PSR B1534+12 has a right ascension of 15h34s and a declinationof 12 degrees North of the celestial equator, relative to the B1950 reference epoch. TheJ2000 name for the same pulsar is PSR J1537+1155.7Jocelyn Bell-Burnell, as a steady period of elapsed time between radio pulses(Ps) that corresponds to the period of pulsar rotation. Current instrumenta-tion at radio telescopes allows for precise measurements of each pulse’s timeof arrival (TOA), such that TOA uncertainties for comparatively bright pul-sars are at the microsecond level. A classic example of such precision canbe seen in PSR J0437−4715, the closest and brightest radio pulsar known,where a recent analysis by Reardon et al. (2016) used 5065 TOAs collectedof 15 years and determined the spin frequency νs = 1/Ps to beνs = 173.6879458121843± 0.0000000000005 s−1.The precision in current measurements of pulsar rotation is comparable tothe atomic-transition clocks used for maintaining terrestrial timescales (e.g.Matsakis et al., 1997). In fact, there has been recent work undertaken toexploit the rotational stability of the most stable pulsars to develop andmaintain an independent timescale based on pulsar-TOA measurements (e.g.Hobbs et al., 2012).Radio pulsars are also observed to spin with intrinsically increasing spinperiods over time. This physically corresponds to a loss of rotational ki-netic energy (Erot) in pulsar rotation; if the pulsar has an angular rotationfrequency Ωs = 2pi/Ps, thenE˙rot =dErotdt=ddt(12IrotΩ2s)= IrotΩsΩ˙s = 4pi2IrotP˙sP 3s. (1.5)The loss of rotational energy in spinning neutron stars, generally referredto as “spin-down” in the literature, supplies power to their immediate en-vironments and produces high-energy radiation. This was first proposed byPacini (1967) as a mechanism for the observed high-energy radiation in theCrab nebula, within a year before the observational discovery of neutron starsmade by Jocelyn Bell-Burnell. If one assumes that the spin-down is purelydue to magnetic dipole radiation, then the strength of the magnetic field at8the neutron-star surface can be determined up to a factor of sinα. For aradio pulsar with radius R = 10 km and α = pi/2, the surface field strengthisB(r = R) =√3c38pi2IrotR6 sin2 αPsP˙s (1.6)= 3.29× 1019√PsP˙s Gauss. (1.7)Figure 1.1 shows a plot of all known pulsars with observed Ps and P˙svalues in the ATNF catalog. This figure is comparable to the Hertzsprung-Russell diagram for cluster stars, in that observed groupings of pulsars reflectunderlying physical processes that contributed to their current states. “Nor-mal” pulsars typically have rotation periods Ps ∼ 1 s and spin-down ratesP˙s ∼ 10−15, and are referred to as “normal” because they are generally ob-served to be isolated objects with no binary companions. A small number ofexceptions exist, such as PSR J1740-3052 (e.g. Madsen et al., 2012), whereslow-spinning pulsars are observed to orbit companions whose stellar evolu-tion has had little or no impact on the spin evolution of the pulsar. As such,the rotation of normal pulsars is understood to have only been affected byspin-down through magnetic dipole radiation after the formative supernovaevent. Recent studies have shown that the magnetic field of normal pulsarscan decay with timescales as short as 105 years (Igoshev & Popov, 2015),though this remains a controversial subject of study.1.1.4 Binary Millisecond PulsarsIn stark contrast to the normal pulsars, millisecond pulsars (MSPs) – withexceptionally stable rotation periods Ps < 20 ms and spin-down rates P˙s <10−17 – are understood to be the end products of prolonged, stable masstransfer onto a neutron star from an evolving (sub)giant progenitor com-9Figure 1.1: P˙s versus Ps for all known radio pulsars with available spin andspin-down measurements. The blue lines correspond to values of both pa-rameters that yield the denoted surface magnetic field (B) for a neutron starwith Irot = 1045 g cm2. Red points denote 24 binary pulsars that are studiedin Chapter 3 of this thesis, while the cyan and green points represent PSRsB1620−26 and B1534+12 that are studied in Chapters 4 and 5, respectively.As we discuss in Section 4.4, there are several different physical mechanisms(other than pure spin-down) that bias the observed value of P˙s for B1620−26.panion. This long-term “recycling” process due to Roche-lobe overflow ofthe companion’s outermost layer increases the neutron star’s spin frequencywhile circularizing its orbit and reducing the magnetic-field strength overthe course of accretion (e.g. Alpar et al., 1982). The post-accretion spinparameters are therefore changed from what they were when the neutronstar was born; in Figure 1.1, this is generally seen as a migration from thelarger population of normal pulsars to the smaller MSP population. Theresultant companion object will likely be a low-mass white dwarf (WD), but,in principle, it is possible for the companion to be fully evaporated by the10high-energy radiation from the spun-up neutron star (Ruderman et al., 1989).Furthermore, recent discoveries have implied that current eccentric MSP bi-naries may have once been a part of progenitor triple systems that have sinceundergone disruption to form their current states (e.g. Freire et al., 2011).Even with their binary origin, there are a number of pulsars within the MSPpopulation that are observed to be isolated and believed to have undergonegravitational disruption of their progenitor orbits (Lorimer, 2008).If there are no external perturbations from nearby stars, the dissipativetidal interactions due to stable mass transfer between components will gov-ern the dynamical evolution of the orbit up to the termination of transfer(e.g. Phinney, 1992; Tauris & Savonije, 1999). Therefore, the post-accretionorbital elements will likely depend on several accretion-related factors. Anotable prediction is a correlation between the resultant mass of the WDcompanion (mc) and post-accretion orbital period (Pb) for “wide” binarysystems (with Pb > 1 day; e.g. Tauris & Savonije, 1999), where numericalsimulations of mass transfer showed thatmc =(Pbb)1/a+ c (1.8)where the values of (a, b, c) weakly depend on the metallicity of the pro-genitor companion. One can therefore compute an expected value of mc ifthe companion is known or expected to currently be a low-mass WD. Evo-lutionary models can therefore be used in conjunction with pulsar-timingmeasurements to constrain additional parameters of interest, such as thepulsar mass (mp) and the inclination of the orbit relative to the plane of thesky (i).111.2 Data Acquisition and InstrumentationIn its raw form, radio-frequency radiation from pulsars is collected with adirectional antenna and processed through a series of filters, amplifiers andmixers to produce a usable signal stream (e.g. Chapter 5 of Lorimer &Kramer, 2005). Hewish et al. (1968) used an array of dipole antennas anda pen-chart recorder to make the first (low-frequency, narrow-band) obser-vations of PSR B1919+21. Modern observations of radio pulsars use largesingle-dish telescopes (or an array of small single-dish telescopes), receiverswith large bandwidths and sophisticated computer hardware to make mea-surements with comparatively greater sensitivity and localization of the radiosources. The most common radio-pulsar observations use receivers centeredon frequencies that collectively span the range 0.1-10 GHz.Upon reception and initial processing of the radiation into voltage, theproceeding steps in real-time signal processing depend on the desired type ofobservation. For blind-search observations, where parts of the sky with noknown pulsars are searched to potentially discover new sources, the mixedand amplified stream is processed through spectrometers and decomposedinto “channelized” data contained within finite, contiguous frequency chan-nels. The sub-banded signal is detected and recorded for offline search of pul-sations using software such as the PRESTO suite.5 Pulsar-searching obser-vations typically occur at low frequencies since pulsars are typically brighterand telescope beams are wider at low receiver frequencies. For observationsof known pulsars, it is standard practice to further process the data streamin real time to remove the dispersive effect of the interstellar medium fromthe broadband signal and average successive, low-S/N pulses together, givena sufficiently accurate measure of the pulsar’s apparent spin period at thetime of observation. The resultant data are higher-S/N pulses within eachfrequency channel across the receiver band; each of these folded profiles,5http://www.cv.nrao.edu/~sransom/presto/12comprised of a large number of individual radio pulses detected within eachchannel and within some chosen sub-integration timescale, is then recordedfor subsequent offline analysis. In what follows below, we refer to the pro-cess of dispersion removal prior to the recording and integration of data asde-dispersion.Any given TOA analyzed in this dissertation was obtained through oneof two broadband de-dispersion techniques, depending on the type of pulsarsignal processor (or backend) used at the time of observation. For recentgenerations of pulsar backends, the determination the full Stokes polarizationvector is done in software using two input channels with orthogonal sense ofpolarization, regardless of the de-dispersion technique.6 In this section, webriefly describe the two de-dispersion methods used throughout this work.1.2.1 Dispersion from the Interstellar MediumThe Galaxy is filled with a cold, generally tenuous collection of dust, ions,neutral gas and free electrons that make up the interstellar medium (ISM)between stars, with a higher concentration of material within the Galacticdisk. A broadband electromagnetic signal that propagates through such amedium with free-electron number density ne will experience a dispersion inthe wave packet that culminates in a nonzero lag between signal componentsat different frequencies (e.g. Jackson, 1962).Radio telescopes typically use a variety of receivers with different band-widths that are tuned to different central frequencies; therefore, TOAs forthe same pulse observed at different channel frequencies will typically have alag between them due to the ISM along the line of sight between the observerand the pulsar. The time delay for ISM dispersion between signals recordedat two different receiver-channel frequencies f1 and f2 is given as ∆DM =6The earliest generations of pulsar backends required the use of additional hardware,such as adding and multiplying polarimeters (e.g. von Hoensbroech & Xilouris, 1996),in order to preserve signal-phase information and derive the components of the Stokespolarization vector.13C(f−21 −f−22 )×DM, where C = (4.148808±0.000003)×103 MHz2 pc−1 cm3s is a collection of physical constants7 and DM =∫ d0ne(l)dl is the pulsar’sdispersion measure, an electron column-density parameter. In practice, DMis usually measured relative to an infinite frequency, f2 =∞, so that the DMof a TOA measured at frequency f = f2 can be rewritten to yield∆DM =(C ×DM)f 2. (1.9)If left uncorrected, pulse profiles from a pulsar with nonzero DM that areobtained at different observing frequencies will produce a smeared profilewhen they are averaged together. The DM is therefore another definingcharacteristic of radio pulsars that directly affects the precision to whichTOAs can be measured.1.2.2 Incoherent De-DispersionThe oldest data that are presented in this dissertation, most of which werecollected in the 1990s, were obtained using pulsar backends that employed theincoherent de-dispersion technique for correcting ISM-related delays betweenreceiver channels (e.g. Lorimer & Kramer, 2005). This “brute-force” methodof de-dispersion was typically employed using analogue filter bank spectrom-eters which first decomposed the mixed voltage signal into a number of spec-tral channels across the observed bandwidth. The incoherent de-dispersiontechnique uses the general form of Equation 1.9, ∆t = C×DM×(f−2ref −f−2chan),where fref is taken to be the central frequency of the observed receiver bandand fchan is the frequency of the channel under consideration, to computeand directly apply the predicted timing delay. While this technique is effec-tive and simple to implement, one of the major disadvantages of incoherentde-dispersion is the imperfect removal of DM timing delays within each indi-7The uncertainty in C reflects the experimental uncertainties in the electron’s chargeand mass that are used in the computation of the dispersion constant.14vidual channel of finite width; a residual smearing of the pulse will occur ineach channel and will therefore limit TOA precision depending on the chan-nel width. The Mark III pulsar backend (Stinebring et al., 1992) is a classicexample of a signal processor that used the incoherent de-dispersion methodfor DM-delay removal with channels widths of 10 MHz.1.2.3 Coherent De-DispersionModern pulsar instrumentation uses improved floating-point precision andbit-sampling technology to fully de-disperse the incoming radio signal in soft-ware, and in real time. This technique is generally referred to as coherent de-dispersion. Hankins & Rickett (1975) showed that, in the frequency domain,the raw complex voltage detected by the radio telescope (Vf ) is proportionalto the intrinsic complex voltage produced at the point of emission from thepulsar (Vf,int); they are related to one another through the transfer function(H), which acts as a type of filter and fully characterizes the ISM-dispersioneffect on the received broadband signal:Vf (f + fcen) = H(f + fcen)Vf,int(f + fcen), (1.10)whereH(f + fcen) = exp[i2piCf 2(f + fcen)f 2cenDM]. (1.11)The inverse transfer function, H−1, can therefore be computed and appliedto Vf given some nominal value of DM to recover the original form of thepulsar signal prior to dispersion. Software filter banks are then applied todecompose the signal into spectral channels with the smearing of each pulsedue to ISM dispersion completely removed.The majority of data for all pulsars studied in this dissertation was col-lected using pulsar backends that employed (and still currently employ) thecoherent de-dispersion method. The specific coherent-de-dispersion signal15processors used are discussed in the following chapters.1.3 Pulsar Timing, in a NutshellRadio pulsars are observed as faint, periodic flashes of electromagnetic ra-diation at radio frequencies. So long as the radio beam sweeps across theirline of sight, an observer will see pulses from the rotating neutron star. Af-ter many subsequent observations of this (isolated) pulsar are made over aseveral-month timescale, the observer will eventually note that the pulsar’sspin frequency (νs) is decreasing over time due to spin-down. Since spin-down rates are typically very small, such an observer can construct a general“timing model” that describes the spin frequency of the isolated pulsar as aTaylor-expanded function of time (e.g. Lorimer & Kramer, 2005),νs(t) = νs(t0) + ν˙s,0(t− t0) + 12ν¨s,0(t− t0)2 + ... (1.12)where the time derivatives are evaluated at the reference epoch t0. In thisidealized case of an isolated neutron star, the observer can gradually ex-tend their data set and make refined measurements on the spin parameters,including higher-order time derivatives in their model when needed.In practice, however, this process of modeling the observed spin behavioris nontrivial and ultimately more rewarding. For instance, single-dish radiotelescopes on Earth do not reside in an inertial reference frame with respectto a given pulsar; the Earth is in orbit about the Solar System Barycentre(SSB) and also spins about its own precessing rotation axis at the decreasingsidereal rate. Also, a given pulsar is likely to have non-zero secular mo-tion relative to the SSB due to asymmetries in the supernova explosion thatformed the neutron star. Furthermore, pulsars in binary/triple systems willundergo orbital motion that induces regular Doppler shifts in the observedspin frequency. Finally, observations of pulsars made at different telescope-receiver frequencies will not record identical TOAs for the same pulses, but16rather a lag that changes with the receiver frequency due to ISM dispersion.The robust construction of a timing model8 for a radio pulsar thereforerequires an explicit account of all relevant physical processes that affect thepulsar’s observed spin behavior. This procedure, referred to as pulsar timingthroughout pulsar literature and in this thesis, is the foundational method forstudying radio pulsars. As discussed in this section and Section 1.4, pulsartiming yields direct measurements of parameters that describe spin prop-erties, astrometry (i.e., position, proper motion, and parallax), frequency-dependent effects associated with the tenuous interstellar medium (ISM), andorbital motion. An accurate pulsar-timing model can also be used to fold (i.e.shift to a common phase and average) observed pulses obtained over manydays or years to form an integrated, high signal-to-noise pulse profile, whichis useful for obtaining high-precision TOAs and resolving polarization prop-erties. In this section, we briefly discuss the procedure for measuring TOAsas well as the theoretical models that describe many of the effects observ-able with pulsar timing. We reserve a full and separate discussion of binarytiming models for Section 1.4, since the analysis of binary/triple motion ofpulsars is the central theme of this dissertation.1.3.1 TOA EstimationRadio telescopes use hydrogen-maser atomic clocks at each site location inorder to assign a time stamp at the beginning of each recorded data stream.For pulsars with pre-existing timing solutions, obtained from iterative anal-yses of the first TOAs measured after their discovery, a series of consecutivepulses are folded together modulo the computed pulse period in order toform a high-S/N pulse; its time stamp is the average of all time stamps ofthe middle of each individual sub-integration that formed the folded profile.TOAs are then determined from a cross correlation between the observed8Timing models are also referred to as “timing solutions” and “pulsar ephemerides”,and we use these terms interchangeably throughout the text.17pulse profile (P ) and a template profile (SP). A template profile can bedetermined from the folding of many previously-recorded pulses, or can becreated as a multi-component model made with an appropriate number ofGaussian distributions (e.g. Kramer, 1994). In general, P and SP are linearlyrelated to each other,P (t) = a+ bSP(t− τ) + n(t) (1.13)where 0 < t < Ps, a is a baseline offset, b is a scale factor, τ is the phaseshift between P and SP, in units of the local value of Ps, and n(t) is arandom, time-dependent term that quantifies observed noise in the recordeddata stream. The n(t) term is typically negligible for high S/N profiles, and sothe cross correlation can be done in a straightforward manner to determineτ . The TOA is then defined to be the sum of the mid-point time of theobservation with τ multiplied by the local value of Ps. In practice, the crosscorrelation is performed in the frequency domain using Fourier transformsfor high-precision determination of TOAs (Taylor, 1992).1.3.2 The Timing Model for Isolated PulsarsTOAs are locally measured quantities, collectively referred to as topocentricTOAs (ttop), that can be recorded with several telescopes located acrossthe Earth. In order for Equation 1.12 to be applied successfully, the ttopmeasurements must be transformed to the equivalent times measured relativeto the SSB. For isolated pulsars, this amounts to: applying appropriate clockcorrections; accounting for orbital motion within the Solar System; modelingfor transverse motion of the pulsar; and correcting delays between TOAscollected at different receiver frequencies. We discuss each of these timedelays and their respective models in this subsection.The following principles, corrections, physical mechanisms and fitting pro-cedures described above are embodied in two standard pulsar-timing software18packages: TEMPO9 and TEMPO210 (Hobbs et al., 2006). We use both toolsas indicated throughout this dissertation.Local Clock TransformationsSince hydrogen-maser clocks at different telescopes maintain local observa-tory time, biases in parameter estimate will occur should TOAs collectedfrom different observatories be combined without the appropriate clock cor-rections. TOAs are first corrected to the Coordinated Universal Time (UTC)maintained by the Global Positioning System (GPS), and then corrected fur-ther to Terrestrial Time, or TT(BIPM)11, that is based on a number of main-tained atomic clocks across the globe. Clock corrections (generally referredto as ∆C below) also account for the non-uniform rotation of the Earth andinclude leap seconds when needed.Dynamical Effects in the Solar SystemTime delays associated with orbital motion within the Solar System can beseparated into three distinct components. The most prominent of these or-bital effects is the Ro¨mer timing delay (∆R), which describes the classical(i.e. non-relativistic) propagation delay experienced by an observer at differ-ent points in space over the course of the orbit. The Shapiro timing delayin the Solar System (∆S), one of the four classic tests of GR first proposedby Shapiro (1964), accounts for general-relativistic propagation delays fromvarying spacetime curvature due to the presence of Solar System bodies nearthe observer’s line of sight. The Einstein timing delay (∆E) is a cumulativeeffect of general-relativistic gravitational redshift of the pulsar signal due tothe Solar System bodies, as well as the special-relativistic time dilation dueto Earth’s motion. The functional form for these effects are given as (e.g.9http://tempo.sourceforge.net/10http://sourceforge.net/projects/tempo2/11http://www.bipm.org/en/bipm-services/timescales/19Lorimer & Kramer, 2005),∆R = −1c(~rSSB + ~robs) · sˆ (1.14)∆S = −2∑iGmic3ln[sˆ · ~ri,⊕ + ri,⊕sˆ · ~ri,PSR + ri,PSR](1.15)∆E =1c2∑i∫ (Gmiri,⊕ + 12v2⊕)dt (1.16)and the various terms are summarized as follows. The sˆ unit points fromthe SSB in the direction of the pulsar. The ~rSSB vector points from the SSBto the center of mass of Earth, while the ~robs vector points from the Earth’scenter of mass to the location of the observatory where TOAs were recorded.The ~ri terms denote relative vectors between the observatory on Earth andthe i-th Solar System body (~ri,⊕), or between the i-th body and the pulsar(~ri,PSR). The mi terms are the gravitational masses for the i-th body, ~v⊕ isthe orbital velocity of the Earth, G is Newton’s gravitational constant and cis the speed of light.The body-specific terms in Equations 1.14-1.16 are independently com-puted using a Solar-system planetary ephemeris, maintained by the NASAJet Propulsion Laboratory12, that specifies the masses and past/future three-dimensional locations for the Sun, Moon, and all eight planets. We used theDE421 or DE430 ephemerides as specified in each chapter. The masses andcomponents of the various ~r vectors are held fixed during the construction ofa timing model, so that the only quantities left to determine are the compo-nents of the sˆ vector, which yield to the two-dimensional coordinates of thepulsar on the sky. Any significant transverse motion of the pulsar relative tothe SSB will lead to a secular change in sˆ, and so the proper-motion termscan be directly measured should sˆ change over time.12ssd.jpl.nasa.gov/eph_info.html20For sufficiently nearby pulsars, with distances d < 1 kpc, a radio-timingsignature of the astrometric parallax ($ = d−1) can be measured in TOAdata. The periodic timing delay introduced by such a signature has thefollowing functional form (Backer & Hellings, 1986),∆$ = −$2c[(~rSSB + ~robs)× sˆ]2 (1.17)where the components of ~rSSB and ~robs are determined from the same So-lar System ephemeris described above, and sˆ is gradually resolved from anapplication of ∆R (Equation 1.14).Evolution in the ISMThe ISM is a naturally dynamic and turbulent environment. This evolutionwas first seen as measurable variations in DM for the Crab and Vela pulsars(Rankin & Roberts, 1971; Isaacman & Rankin, 1977), and is readily seenin TOAs collected for isolated and binary pulsars with modern broadband-oriented instrumentation (e.g. Keith et al., 2013; Lam et al., 2015). In thepresence of such variations, a single value of DM will not sufficiently char-acterize the frequency dependence of TOA data and will introduce biases inother timing-model parameters. The TEMPO software package contains abuilt-in feature, called “DMX”, for measuring DM at some prescribed timeinterval, which can be as low as one day for a per-epoch determination ofDM. While this type of modeling introduces more degrees of freedom withinthe timing model, it nonetheless allows for more robust determination of thetime dependence in DM over the span of the data set.Intrinsic Evolution of the Pulse ProfileFor many pulsars with different DMs and timing-model parameters, the in-tegrated, high-S/N pulse profile has been shown to yield slightly differentshapes when observing with different receivers, or when using a single receiver21with a large bandwidth. This frequency-dependent evolution is believed tobe an intrinsic property of the poorly-understood mechanism for radio emis-sion, and can be interpreted as the emission at different frequencies occurringat different heights from the polar cap (Komesaroff, 1970).As with the general radio-emission mechanism, there is no sufficient frame-work that can fully account for the observed variations in profile shape as afunction of frequency for all pulsars. Moreover, recent work has shown thatthe degree of profile evolution can vary substantially between radio pulsars,with several pulsars shown no significant shape variations over a wide rangeof receiver frequencies (e.g. Arzoumanian et al., 2015b). For pulsars withsignificant profile evolution, the cross-correlation of broadband data with asingle profile template with produce TOA lags as a function of frequencythat generally differs from the f−2 dependence of timing delays due to DM.When needed, we use the heuristic model developed by Arzoumanianet al. (2015b) that modeled the frequency-dependent (FD) evolution of theprofile shape in the following manner:∆tFD =n∑i=1ci log(f1 GHz)i(1.18)where f is the channel frequency of the observed TOA, n is the user-definednumber of log-polynomial terms to be used, and ci is the i-th coefficient,such that i = 1, 2, . . . n. The ci terms are the allowed free parameters inTEMPO and TEMPO2. In practice, pulsars that show no significant changein pulse structure across observing frequency do not require an applicationof ∆tFD, whereas pulsars with comparatively large changes require up to fivelog-polynomial terms when applying Equation 1.18.Determination of the Model and Noise ProcessesA timing solution of an isolated pulsar can be obtained so long as it explicitlymodels the aforementioned physical effects and applies the appropriate clock22corrections. This amounts to transforming the observed topocentric TOAsto an inertial reference frame, taken to be the SSB13,t = ttop + ∆C + ∆R + ∆S + ∆E + ∆$ + ∆DM + ∆FDand using these corrected/transformed TOAs to model the spin behaviorthrough Equation 1.12. The timing model therefore consists of free param-eters that describe the spin frequency and its derivatives, astrometry, thepulsar’s DM, and any significant variations in DM. The parameters thatbest describe the observed TOAs are determined using a least-squares fit-ting procedure, where the best-fit parameters minimize the χ2 goodness-of-fitstatistic,χ2 =NTOA∑i((νs,mod(t)− νs)2σ2i). (1.19)In Equation 1.19, NTOA is the number of TOAs being modeled and σi is theuncertainty in the i-th TOA.The use of Equation 1.19 and the corresponding covariance matrix oftiming-model parameters assumes that the timing residuals – the differencebetween measured TOAs and those predicted from a given timing model– are uncorrelated between data subsets collected with different receiversand backends. In other words, the best-fit residuals are assumed to form anormal distribution with a mean of zero that reflects a white-noise randomprocess. However, older-generation TOA data have been shown to harborsystematic errors associated with limits in instrumentation from incoherentde-dispersion and low-bit resolution, despite the appearance of “flat” best-fitresiduals that indicate a good fit of the timing model.13The acceleration due to gravity from a solar-mass star at a distance d = 1 pc isGm/d2 ∼ 10−13 m s−2, while the acceleration due to gravity from the Galactic centerat the SSB is ∼ 10−11 m s−2. Both accelerations are considerably smaller in order ofmagnitude when compared to the acceleration due to gravity of the Sun at Earth, which is∼ 10−8 m s−2. Therefore, the SSB is a comparatively more “inertial” frame of reference.23A historic, ad hoc solution to this problem is the slight alteration of TOAmeasurement uncertainties for data subsets to values that ultimately producethe desired best-fit statistic, which is usually taken to be χ2red = χ2/NDOF ≈ 1,where NDOF is the number of degrees of freedom. Additional uncertainty canbe added in quadrature (by an amount σq) and/or as a multiplicative factor(σf), such thatσ = σf√σ2o + σ2q (1.20)and σo is the original TOA uncertainty. For TOAs collected with modern,coherent-de-dispersion backends, σf ≈ 1 and σq ∼ 1 µs. The correspondingvalues of σf and σq for historic processors can be larger than the modern-backend values by an order of magnitude.Several bright, nearby pulsars show “red” noise (e.g. for PSR B1937+21;Kaspi et al., 1994; Verbiest et al., 2009) that appear as non-random structurein TOA residuals and can produce biases in timing-model parameters whenleft unaccounted, even after modeling all seemingly relevant physical effects.Red noise – also referred to as timing noise in the literature – is believed toreflect real instabilities in pulsar rotation due to torque fluctuations associ-ated with the superfluid interior and/or the co-rotating magnetosphere (e.g.Shannon & Cordes, 2010). Historically, a practical solution towards model-ing red-noise signatures in timing residuals is the use of higher-order timederivatives in νs (e.g. Arzoumanian et al., 1994).A more sophisticated method for addressing red noise in TOA resid-uals was recently proposed by Coles et al. (2011), where a “generalized”least-squares (GLS) method accounts for correlated noise in TOA residualsthrough a linear transformation of the covariance matrix that whitens TOAresiduals; the transformed data can then be used with Equation 1.19 to yieldaccurate, unbiased timing parameters.241.4 Timing Delays from Binary MotionIn principle, the timing delays for radial displacement from binary motionof a pulsar have a similar, additive form to those that describe the Earth’smotion in the solar system (Equations 1.14-1.16). The general timing formulafor TOA correction and transformation is finally given as (e.g. Lorimer &Kramer, 2005)t = ttop + ∆C + ∆R + ∆S + ∆E + ∆$ + ∆DM + ∆R + ∆S + ∆E (1.21)where ∆R, ∆S, and ∆E are the Ro¨mer, Shapiro and Einstein timing delays ofthe pulsar-binary system. However, the parameters that describe a pulsar-binary system are not known beforehand. Indeed, a pulsar must be re-observed several times upon discovery in order to measure significant changesin the observed spin period due to radial motion in an inclined orbit; theobserved shifts can be used to constrain and make initial estimates of thebinary parameters, though an explicit timing model is eventually requiredto precisely determine the orbital elements. Once a binary timing modelis successfully applied to TOA data, future observations can be made torefine ∆R,S,E and the binary parameters that describe them. If relevant tothe system, future TOAs can eventually yield estimates of variations in theorbital elements over time due to one or more effects described below and insubsequent chapters.The binary systems studied in this thesis already had long-term timingsolutions that explicitly modeled their respective orbits prior to their studyfor this work; we update these timing solutions in an effort to measure orimprove prior estimates of masses and geometry, and derive tests of grav-itational theory whenever possible. In this section, we present two binarytiming models that describe pulsar-binary systems and long-term changes inthe orbital elements. These two general binary models exist due to compli-25EastTo ObserverNorthABCplane of skyplane of orbitΩωiuFigure 1.2: A schematic of an eccentric binary orbit – shown as a solid-blackellipse, with the center of mass at the origin of the drawn coordinate system– and the various angles of pulsar/binary orientation defined in Section 1.4.The portion of the orbit shown in gray lies on the opposite side of the tan-gential plane of the sky. The pulsar is shown as a black dot and its directionof motion along the orbit is given by the black arrow.cations in applying an elliptical-orbit model to a highly circular orbit, wherecertain timing parameters become ill-defined and numerically unstable asfree parameters in a least-squares model fit. Both binary timing models areused extensively in this dissertation.Throughout this dissertation, we adopt the IAU convention for character-izing the three-dimensional orientation of orbital planes for nearly all pulsar-binary systems.14 An example of an eccentric orbit and the angles that define14The only exception to the angle convention in this dissertation is PSR B1534+12 (thesubject of Chapter 5), which uses the convention defined by Damour & Taylor (1992): i ismeasured such that i = 0◦ corresponds to the orbital-angular-momentum vector pointing26its orientation is shown in Figure 1.2. In this figure, points labeled “A”, “B”and “C”, as well as the gray lines that connect them to the origin of the coor-dinate system, are drawn to help indicate how several angles are measured.The relevant angles are defined as follows: the longitude of the ascendingnode (Ω) is measured from the direction of celestial North to the line con-necting the origin and A (which contains the point along the orbit wherethe pulsar pierces the tangential plane of the sky in the direction away fromobservers on Earth) in the plane of the sky and in the direction of celestialEast, such that 0◦ < Ω < 360◦; the argument of periastron (ω) is measuredbetween the ascending node and the line between the origin and B, in theplane of the orbit, such that 0◦ < ω < 360◦; and the true anomaly (u) ismeasured between periastron and the line connecting the origin and C, inthe plane of the orbit, such that 0◦ < u < 360◦. The system inclination (i) ismeasured between the planes of the orbit and the sky, such that 0◦ < i < 180◦and i = 0◦ corresponds to the vector of orbital angular momentum pointingin the direction towards Earth.1.4.1 Orbits with Significant EccentricityA general binary orbit will have an elliptical shape that is characterized byits eccentricity (e), where a circular orbit corresponds to e = 0 and 0 < e < 1for gravitationally bound systems. An orbit with a statistically significanteccentricity has a well-defined ω (see Figure 1.2) that corresponds to the pointof closest relative approach and maximum orbital speed. The Ro¨mer timingdelay for a pulsar (∆R) in an eccentric binary system with orbital period Pband semi-major axis ap is dependent on the parameters that directly affectthe radial motion of the pulsar along the line of sight (Blandford & Teukolsky,1976):away from the Earth; and Ω is measured with the opposite sense than the one used in theIAU convention, from celestial North in the direction of celestial West.27∆R = x[(cosE − e) sinω + sinE√1− e2 cosω](1.22)where x = ap sin i/c is the semi-major axis projected along the line of sightdue to an inclination (i) of the orbital plane, and where c is the speed of light.The “eccentric anomaly” (E) is computed using a set of “Kepler’s equations”that are modified to include first-order perturbations in the orbital periodand periastron argument (Damour & Deruelle, 1986):E − e sinE = nb[(t− T0) + 12P˙bPb(t− T0)2](1.23)u(E) = 2 arctan[√1 + e1− e tan(E2)](1.24)ω = ω0 +ω˙nbu(E) (1.25)where u(E) is the true anomaly of the pulsar (shown in Figure 1.2) at sometime t in an inertial reference frame relative to the binary system (e.g. a co-ordinate system centered at the SSB), T0 is the epoch of periastron passage,ω0 is the value of ω measured at a time T0, and nb = 2pi/Pb is the angu-lar orbital frequency. The Keplerian timing parameters are five parametersthat describe the basic properties of every eccentric binary system: {x, Pb,e, ω, T0}. With a measurement of ∆R alone, ap and sin i cannot be sepa-rately measured. Furthermore, the time derivatives in Pb and ω, as well asvariations in x, are only measured in certain cases where there are intrinsicand/or kinematic effects that produce apparent changes in these three pa-rameters over time. The physical effects that produce observable variationsare discussed in Section 1.5 below.In theory, the Ro¨mer timing delay can be written in a more general formto account for radial and tangential corrections of the eccentricity that areunique to the general-relativistic formulation of the eccentric two-body prob-28lem (Damour & Deruelle, 1985):∆R = x[(cosE − er) sinω + sinE√1− e2θ cosω](1.26)where er = e(1+δr), eθ = e(1+δθ), and {δr, δθ} are the radial and tangentialdeformation parameters of the orbit’s eccentricity, respectively. However,Damour & Deruelle (1986) showed that δr cannot be separately measuredfrom parameters related to pulsar rotation, and that δθ can be measured solong as ω has changed significantly over time through relativistic precessionsand a large data span (on the order of decades) has been obtained. Wetherefore ignored these terms in all analyses presented in this dissertation(i.e. we hold their values fixed at δr = δθ = 0) and use the Ro¨mer delayshown in Equation 1.22 to model the timing delay of eccentric orbits. Weconsider the likelihood of measuring δθ in the relativistic PSR B1534+12system (Chapter 5) within the coming years in Section 6.1.The Shapiro timing delay experienced by a pulsar (∆S) is also an impor-tant effect that can be present in a pulsar-binary system (Damour & Deru-elle, 1985, 1986). As with the analogous Solar-System effect, the relativisticShapiro timing delay is a measure of the change in spacetime curvature thatis traced by the pulsar signal as it propagates from different points of theorbit towards an observer on Earth. The Shapiro delay is most prominentfor pulsar-binary systems that appear “edge-on”, i.e. when i → pi/2, wherethe signal will propagate closest to the companion star at superior conjunc-tion. For an eccentric orbit, the Shapiro timing delay is given as (Damour &Deruelle, 1986)∆S = −2r ln[1−e cosE−s((cosE−e) sinω+sinE√1− e2 cosω)](1.27)where r and s are the “range” and “shape” parameters of the Shapiro tim-ing delay, respectively. In most theories of gravitation, s = sin i, whereas29Einstein’s theory of general relativity (GR) requires that r = Tmc, whereT = GM/c3 = 4.925490947 µs. A significant measurement of ∆S is there-fore useful as it yields estimates of both mc and sin i simultaneously.The Einstein timing delay (∆E) is typically only measurable for highlyrelativistic binary systems with e ∼ 0.1 or greater, where periastron advanceis significant and both components are degenerate, compact objects withorbital periods on the order of hours, or even a few days for the most ec-centric systems (Damour & Deruelle, 1985, 1986). ∆E is a measure of bothrelativistic time dilation and gravitational redshift, and is given as∆E = γ sinE (1.28)where γ is the amplitude of the effect with unit of time. The full pulsar-binarytiming model, defined as the sum of the three delays given in Equations 1.22,1.27 and 1.28, are referred to as the “DD” binary model in TEMPO andTEMPO2.The forms of ∆R,S,E shown in Equations 1.22-1.28, do not depend on aparticular theory of strong-field gravitation. In other words, the {P˙b, ω˙,r, s, γ, δθ} parameters can be measured in a theory-independent manner,without the need to assume general relativity or any other valid gravitationaltheory. The DD model therefore allows for unambiguous tests of gravitationby comparing the measured PK parameters with their values as predicted bythe theory in question, and we discuss this type of analysis in Section 1.5.1.4.2 Orbits with No Significant EccentricityIn practice, TOAs collected over several years and many orbits are neededin order to measure the eccentricity of a low-e pulsar-binary system. Untila sufficient time span is reached, ω and T0 lose their meaning in describingthe near-circular orbit and are numerically unstable parameters in the least-squares fit of the timing model. A second binary timing model, first presented30by Lange et al. (2001), was developed in order to re-parametrize the Ro¨merand Shapiro timing delay for circular orbits in terms of a first-order expansionin e. This near-circular binary timing model, referred to in TEMPO andTEMPO2 as “ELL1”, introduces a different set of orbital parameters:∆R = x(sin Φ +κ2sin 2Φ− η2cos 2Φ)(1.29)∆S = −2r ln(1− s sin Φ) (1.30)where Φ = nb(t − Tasc) is the celestial argument of the pulsar at the pulseemission time t, Tasc is the mean epoch of ascending-node passage15, andboth κ and η are collectively referred to as the Laplace-Lagrange eccentricityparameters. The low-e form of the Ro¨mer timing delay (Equation 1.29) doesnot include a constant additive term of −3xe/2 since pulsar timing onlyseparately measures binary terms that vary periodically with time.The ELL1 eccentricity parameters (κ, η, and Tasc) are related to the DDeccentricity parameters (e, ω, and T0) using the following relations (Langeet al., 2001):η = e sinω (1.31)κ = e cosω (1.32)Tasc = T0 − ω/nb. (1.33)The full ELL1 timing-model fit produces estimates of η, κ and Tasc withlittle numerical correlation, even if the {η, κ} parameters are not statisticallysignificant. Secular variations in the orbital elements are measured as Taylorexpansions in those parameters about Tasc.15As noted by Lange et al. (2001), the true time of ascending-node passage is Tasc +2η/nb.31The low-e expansion used for the development of the ELL1 model pro-duces a slight degeneracy when attempting to measure the Shapiro timingdelay for low-inclination, low-e systems. This degeneracy is best seen bycomputing the Fourier expansion of Equation 1.30 in terms of the orbitalperiod of the binary system (Lange et al., 2001):∆S = 2r(a0 + b1 sin Φ− a2 cos 2Φ + . . .) (1.34)where the a and b are the even and odd harmonic amplitudes of the Fourierbasis, respectively, that depend on the inclination of the system. In the caseof low inclination, only the first one or two harmonic terms in Equation 1.34will be significant, and higher order terms will be negligible; the sum ∆R+∆Stherefore yields an expression that is identical in form to Equation 1.29 withthe exception that two of the ELL1 timing parameters are modified by theharmonic amplitudes:xobs = x+ 2rb1 (1.35)ηobs = η +4ra2x(1.36)For low-inclination systems, one must therefore compute the “intrinsic” ELL1parameters (x, η) using the observed parameters in order to determine thetrue values of the Keplerian elements using Equations 1.31-1.33 (see Langeet al., 2001; Freire & Wex, 2010).Furthermore, one can use the presence of orbital harmonic structure inTOA residuals as a confirmation of the Shapiro timing delay. Figure 1.3shows best-fit TOA residuals for PSR J1918−0642, a low-e binary systemthat is studied in Chapter 2 and uses the ELL1 timing model, as a functionof the orbital phase. We see in the middle panel of Figure 1.3 that the Shapirotiming delay is not fully absorbed when only fitting for ∆R, which is a visualindication that the system is highly inclined; this is because the higher-order32Figure 1.3: The Shapiro timing delay observed in PSR J1918−0642 (a highly-inclined binary MSP that is studied in Chapter 3) when using the ELL1binary timing model. The blue and green points denote TOAs collected at430 and 1400 MHz, respectively. The top panel shows the “full”, intrinsiceffect that ∆S (Equation 1.27) has on TOA residuals, assuming that theintrinsic binary parameters are known and holding ∆R fixed at these values(while not fitting for ∆S). The middle panel shows the remaining harmonicstructure of ∆S after absorption of its first two Fourier harmonic coefficientswhen fitting only for ∆R. The bottom panels shows the best timing-modelfit, obtained when fitting for both ∆R and ∆S simultaneously.harmonic terms in Equation 1.34, jointly represented as the “...” term, arestatistically significant and essentially represent the separately-measurablepart of ∆S.1.5 Variations in the Orbital ElementsIn the absence of observed secular variations or relativistic phenomena, themass function (fm) of the pulsar-binary system can be used as a measure of33the mass and inclination of the Keplerian system; it is computed using theobserved Keplerian elements that describe the pulsar’s orbit:fm =n2bx3TM =(mc sin i)3(mp +mc)2. (1.37)For strictly Keplerian motion, and without any independent knowledge ofthe binary companion (e.g. through optical spectroscopy and radial-velocitymeasurements), the intrinsic parameters of the binary system (mp, mc, i,Ω) cannot be uniquely determined. However, estimates of the companionmass can be made using Equation 1.37 since the pulsar mass is restrictedby theory to be no larger than 3 M. A nominal value of mp = 1.35 M(Thorsett & Chakrabarty, 1999) can be assumed to yield estimates of mc fora given value of sin i; and a minimum companion mass can be computed forthe case where sin i = 1. Conversely, one can assume the mc − Pb relationfor a MSP-binary system suspected to undergone significant mass transfer tocompute a value of mc using Equation 1.8, and then derive values of mp fordifferent inclination angles.Many pulsar-binary systems examined in this dissertation exhibit one ormore variations in their orbital elements and PK corrections of the observedorbital motion. The PK effects are particularly interesting since any con-tending gravitational theory must yield correct predictions of the observedPK motion as functions of fundamental quantities that uniquely describe thetheory. However, several other effects can give rise to secular variations that,if left unaccounted for, will bias the secular PK variations and their interpre-tation. In this section, we outline several of the most dominant mechanismsthat give rise to observed variations in the orbital elements.1.5.1 Strong-Field GravitationPulsar-binary systems in tight orbits with WDs or other neutron stars typ-ically exhibit PK effects that are observed as secular changes in the orbital34elements. We assumed GR to be valid throughout this dissertation in or-der to explicitly interpret these effects; we refer to the secular PK variationsin this chapter as (P˙b)GR, (ω˙)GR, (x˙)GR and (e˙)GR, where the dots indicatederivatives in time. According to GR, each of the PK quantities (includingthe Shapiro r and s parameters) are functions of at least one of the twocomponent masses (Damour & Taylor, 1992). The Shapiro-delay parametersand first-order PK variations are given as(P˙b)GR = −192pi5(nbT)5/3(1 +7324e2 +3796e4)× (1− e2)−7/2[mpmc(mp +mc)1/3](1.38)(ω˙)GR = 3(n5bT2)1/3(1− e2)−1(mp +mc)2/3 (1.39)γ =(T2nb)1/3[mc(mp + 2mc)(mp +mc)4/3]e (1.40)s =(Tn2b)1/3[(mp +mc)2/3mc]x = sin i (1.41)r = Tmc (1.42)where the values of mp and mc are assumed to be in units of M in eachof the above expressions. Throughout this thesis, we ignore the (x˙)GR =(a˙p)GR sin i/c and (e˙)GR terms since these particular rates of change are im-measurable on the timescales spanned by the data sets analyzed below.16Measurements of r (Equation 1.42) and s (Equation 1.41) from an ob-served ∆S alone therefore yields estimates of mp and mc, as well as a measureof i since s = sin i. Even without an observed Shapiro timing delay, estimates16Kepler’s third law requires that (x˙)GR/x = 2(P˙b)GR/(3Pb). In the case of relativisticpulsar-binary systems, Pb ∼ hours and P˙b due to GR is measurable after nearly a decadeof observation. However, typical values of x ∼ seconds and so the value of x˙ due to GRis comparatively much smaller than the rate of change in period, as required by Kepler’slaw.35of mp and mc can be obtained so long as two of the other corrections due toGR are measured. A classic example of such a scenario is the Hulse-Taylorpulsar, PSR B1913+16, for which the P˙b− ω˙−γ combination were measuredand used to derive high-precision estimates of both masses (Weisberg et al.,2010; Weisberg & Huang, 2016). Three or more observed PK effects leadto an overdetermined system of equations that can be solved to obtaineda high-precision estimate of the component masses, as well as one or more“tests” of GR when using different combinations of three PK parameters tocheck for self-consistency with measurement uncertainties.1.5.2 Kinematic Bias from Proper MotionBesides the intrinsic changes within orbits from PK effects, apparent secularvariations in the orbital elements will also be induced from significant rela-tive motion between the pulsar-binary and SSB reference frames (Kopeikin,1996). The secular variations in x and ω from proper motion (µ) – to whichwe refer in this study as (x˙)µ and (ω˙)µ – arise from a long-term change incertain elements of orientation as the binary system moves across the sky.The kinematic terms for x˙ and ω˙ are described as trigonometric functions ofi and Ω:(x˙)µ = xµ cot i sin(Θµ − Ω), (1.43)(ω˙)µ = µ csc i cos(Θµ − Ω). (1.44)where Θµ is the position angle of proper motion, computed internally withinTEMPO2.361.5.3 Kinematic Bias from AccelerationA separate kinematic bias that produces observed secular variations in or-bital elements can arise from several forms of relative acceleration betweenthe pulsar-binary and SSB systems (e.g. Damour & Taylor, 1992; Nice &Taylor, 1995), the most prominent of which are: differential rotation in theGalactic disk; acceleration in the Galactic gravitational potential vertical tothe disk (e.g. Kuijken & Gilmore, 1989); and apparent acceleration due tosignificant proper motion (Shklovskii, 1970). The kinematic bias from rela-tive acceleration produces a rate of change in the Doppler shift (D) in, forexample, Pb, such that(P˙bPb)D=D˙D=azc− cos b(Θ20cR0)(cos l +βsin2 l + β2)+µ2dc(1.45)where the terms in Equation 1.45 are summarized as follows: (l, b) are theGalactocentric coordinates of the MSP; R0 is the distance between the Sunand the Galactic center; Θ0 is the circular Galactic-rotation speed of theSun; µ is the proper motion of the pulsar-binary system; d is the distanceto the binary system; az is the component of acceleration in the Galacticpotential that is vertical to the Galactic disk; and β = (d/R0) cos b − cos l.Throughout this dissertation, we use the az model developed by Kuijken &Gilmore (1989), who used photometric and spectroscopic data of K-dwarfstars with known distances to measure az out to d = 3 kpc and found thatazc= 1.08× 10−19[1.25z(z2 + 0.0324)1/2+ 0.58z]s−1, (1.46)where z = d sin b is distance from the Galactic plane. The changing Dopplershifts ultimately produce an apparent variation in x and Pb, though the effectis negligibly small for the former parameter. We refer to the component ofthe secular variation due to the acceleration bias as (P˙b)D = Pb(D˙/D).37Several other pulsar-timing parameters (such as x and Ps) are similarly af-fected by the change in Doppler factors due to relative acceleration. In otherwords, (x˙)D = x(D˙/D) and (P˙s)D = Ps(D˙/D). However, the Doppler com-ponent of x˙ is generally considered to be negligibly small since x ∼ seconds,and the smallest known values of Pb ∼ hours. The Doppler component of P˙sis inseparable from the spin-down term if an accurate measure of distance isnot known.1.5.4 Periodic Variations and Annual Orbital ParallaxFor sufficiently nearby pulsar-binary systems, the observed system orienta-tion will change periodically as the Earth and the MSP orbit their respectivebarycenters and at their respective orbital periods. The “mixed” periodicvariations in x and ω, collectively referred to as the “annual orbital paral-lax” (Kopeikin, 1995), depend on i, Ω, and the observed parallax ($) of thepulsar-binary system:xobs = x[1−$ cot i(∆~Io sin Ω−∆ ~J0 cos Ω)](1.47)ωobs = ω −$ csc i[∆~Io cos Ω + ∆ ~J0 sin Ω)](1.48)where $ is the annual astrometric parallax, and ∆~Io = ~rSSB · ~I0 and ∆ ~J0 =~rSSB · ~J0 are the time-varying projections of the SSB position vector of theEarth, defined in Equation 1.14, onto two of the orthogonal basis vectors(~I0, ~J0) centered on the pulsar-binary system that define the celestial eastand north directions, respectively. Annual orbital parallax has been detectedfor PSRs J0437-4715 (e.g. Verbiest et al., 2008) and J1713+0747 (e.g. Zhuet al., 2015) as significant improvements in timing-model fits when explicitlymodeling the effect. In practice, annual orbital parallax does not signifi-cantly improve the estimates of timing-model parameters due to its small38signature on TOA residuals. However, annual orbital parallax can be usedin conjunction with measurements of the Shapiro timing delay, annual as-trometric parallax and x˙ due to kinematic bias (Equation 1.43) to uniquelysolve for the three-dimensional geometry of the pulsar-binary system whereall of these effects are measured.1.6 This ThesisThe major goal of this dissertation is to measure and/or constrain the massand geometric parameters of binary pulsars through the analysis of pulsar-timing data and interpretation of observed variations in the orbital elements.The following chapters can be described and summarized as follows:• in Chapter 2, we discuss the contributions that author made as a mem-ber of the North American Nanohertz Observatory for GravitationalWaves (NANOGrav) during his graduate career;• in Chapter 3, we present detailed analyses of 24 binary pulsars that arecurrently being observed as part of the NANOGrav program;• in Chapter 4, we present a long-term and ongoing analysis of PSRB1620-26, a 11-ms pulsar within a gravitationally-bound triple systemthat resides within the Messier 4 globular cluster;• in Chapter 5, we present updates of the long-term analysis of PSRB1534+12, a 37.9-ms pulsar in a 10-hr, relativistic orbit with anotherneutron star;• in Chapter 6, we summarize the results obtained for this thesis, per-form simulations to determine when other secular variations could beobservable in PSR B1534+12, and describe possible avenues for furtherwork.39Chapter 2The North AmericanNanohertz Observatory forGravitational WavesThe recent, direct detection of gravitational waves (GWs) at kHz frequenciesusing the Laser Interferometer Gravitational-Wave Observatory has officiallyheralded the era of observational GW astronomy (Abbott et al., 2016). Alongwith LIGO, several major, international efforts are currently underway inorder to directly detect and characterize GWs across different parts of theGW spectrum. Several expected sources of GWs are merging black holes,merging neutron stars (e.g. Hulse & Taylor, 1975; Kramer et al., 2006b;Fonseca et al., 2014), primordial relics from the inflation era of the earlyUniverse (e.g. Grishchuk, 2005), and cosmic strings (Vilenkin & Shellard,1994).NANOGrav1 is one of three “pulsar timing array” (PTA) collaborations,comprised of faculty, researchers and students, that regularly monitor an in-creasing number of MSPs with the primary goal of directly detecting GWsat nanohertz frequencies. The NANOGrav collaboration members are affili-1http://nanograv.org/40ated with universities and research institutes within Canada and the UnitedStates. The other two PTAs are comprised of institutions and universi-ties across Europe, which are collectively referred to as the European Pul-sar Timing Array (EPTA), and groups across Australian universities thatare collectively referred to in literature as the Parkes Pulsar Timing Array(PPTA). The union of NANOGrav, the EPTA and PPTA is referred as theInternational PTA (IPTA; Manchester & IPTA, 2013). A PTA essentiallyserves as a Galactic-scale detector for perturbations of the spacetime metricgenerated by, most prominently, the expected merger of binary supermas-sive black holes (SMBHs) scattered across the Universe (Sazhin, 1978; De-tweiler, 1979); the superposition of GW signals from SMBH binaries forms astochastic background of nanohertz-frequency GWs. Recent work has shownthat PTAs can eventually also become sensitive to individual, localizablesources of nanohertz-frequency GWs generated from nearby galaxy clusters(e.g. Sesana et al., 2009), as well as permanent post-merger deformations ofspacetime referred to as GW “memory” (e.g. Madison et al., 2014).PTA collaborations make high-precision timing observations of an arrayof the brightest and most stable MSPs on a regular basis using the sametechniques and analysis methods discussed in Chapter 1. For NANOGrav,these measurements are made with the 305-m Arecibo Observatory in PuertoRico and the 100-m Robert C. Byrd Green Bank Telescope (GBT) in GreenBank, West Virginia (USA). The collective goal of NANOGrav, along withthe EPTA and PPTA, is the detection and characterization of nanohertz-frequency GWs within the next decade. The foundational method for de-tection of the stochastic background with PTAs was outlined by Hellings& Downs (1983), who proposed the cross-correlation of TOA residuals forall PTA MSPs at different sky locations to search for correlated structuredue to passing GWs. The effectiveness of the cross-correlation method nat-urally depends on our ability to accurately measure and model the observedTOA variations with precision on the order of 10 ns, where GW structure is41expected to be observable (e.g. Jenet et al., 2006; Sesana et al., 2008).The author formally joined NANOGrav as a (graduate) student memberin October 2012, shortly after the start of his Ph. D. program, though beganassisting with timing observations around June 2012. After several years ofundertaking data acquisition and constructing timing solutions for variousNANOGrav MSPs, he became a full member of the collaboration in Novem-ber 2014. During his Ph. D. career, he has attended weekly online meetingsfor the timing and observing groups within NANOGrav, and has presentedvarious projects at in-person NANOGrav meetings across North Americanas well as two IPTA meetings in Krabi, Thailand and Banff, Canada. In thischapter, we briefly summarize the contributions made by E. Fonseca for thebenefit of the NANOGrav collaboration.2.1 Data Acquisition and AnalysisThe nominal NANOGrav observing program, which formally began in 2004,observes all PTA MSPs every ∼3 weeks using the Arecibo Telescope and/orthe GBT.2 In order to eventually produce high-quality pulsar data and timingsolutions, observations are typically carried out using two widely-separatedtelescope receivers and span 20-30 minutes per receiver, per source. Thereare currently ∼50 MSPs undergoing observations for the NANOGrav PTA,and the PTA increase in size by ∼3 MSPs per year, and so observations cancollectively take up ∼50 hours per month of the observing year. Capable andexperienced observers are therefore essential for the acquisition and model-construction of NANOGrav MSPs.The author has regularly collected NANOGrav TOA data using boththe Arecibo telescope and GBT during his Ph. D. career. Between Novem-ber 2013 and January 2016, the author collected 180 hours’ worth of pulsar2Two NANOGrav MSPs – PSRs J1713+0747 and B1937+21 – are observed using bothobservatories. All other MSPs are observed using only one of the two radio telescopes.42TOAs (Nice, 2016). This total amount of observing includes supplementaryNANOGrav observations conducted for a proposal led by the author duringthe 2015 observing year, which is discussed in Section 2.2.During his graduate career, the author also contributed to the clean-ing and initial analysis of TOA data for five NANOGrav MSPs: PSRsJ1643−1224; J1853+1303; J1910+1256; J1949+3106; and B1953+29, usingthe procedures discussed in Sections 1.3 and 1.4 for the inclusion of relevanttiming parameters. The data acquisition and timing analyses conducted bythe author, along with all full and active members of NANOGrav, culmi-nated in the publication of the NANOGrav nine-year data set (Arzoumanianet al., 2015b).2.2 Proposals for Observations: P2945The success of the IPTA experiment depends on many observational andanalytical factors that are currently being addressed by PTA collaborations.Standard “pulsar timing” techniques, discussed in Sections 1.3 and 1.4 above,provide the crucial means for understanding the environment and spin be-havior of each MSP. However, recent studies have demonstrated a need tocompensate for intrinsic “timing noise” of varying strength and propertieswithin several NANOGrav MSPs (e.g. Shannon & Cordes, 2012; Perrodinet al., 2013; Arzoumanian et al., 2015b). Additional complications arisedue to temporal variations in dispersive properties and the frequency de-pendence of pulse structure. However, the dedicated work of NANOGravfaculty, post-doctoral and student members has so far yielded analysis tech-niques and pipelines that address these issues and allow for current upperlimits of the GW-signal strength. While the NANOGrav PTA is consistentlybecoming a more powerful tool for the detection and study of the stochasticGW background (e.g. Siemens et al., 2013), more work is needed to improveits sensitivity to individual, localizable sources of GWs.43Source Ps (ms) DM (pc cm−3) LST (rise-set) RMS residual (µs)J0030+0451 4.87 4.33 23:31-01:29 0.265J1640+2224 3.16 18.43 15:18-18:03 0.189J1713+0747 4.57 15.99 16:05-18:22 0.065J2043+1711 2.38 20.70 19:21-22:06 0.136J2317+1439 3.45 21.90 21:56-00:38 0.337Table 2.1: Observing parameters for the five MSPs observed that are partof the P2945 program at the Arecibo Observatory. Note that the “rise” andset” times constitute times when the source enters in and exits out of theArecibo field of view, respectively.Arzoumanian et al. (2014) provided upper limits on the signal strengthof individual-source GWs using the NANOGrav five-year data set (Demor-est et al., 2013) and found that the detectability of individual-source GWsstrongly depends on two factors. One factor is the angular separation of sev-eral MSPs relative to these bright GW sources, which are likely to be nearbygalaxy clusters such as the Virgo and Fornax clusters (e.g. Simon et al., 2014).Localized GW signals will require nearby “detectors”, so suitable MSPs withsmall angular separations to these individual GW sources will be necessaryin order to make such detections with confidence. As mentioned above, thispoint is continually addressed by including several newly discovered, brightMSPs each year that are ideally distributed randomly across the sky. The sec-ond, more limiting factor involves the timing precision of MSPs close to thesepotential GW sources of interest. Arzoumanian et al. (2014) found that thebest-timed NANOGrav pulsar – PSR J1713+0747 (e.g. Splaver et al., 2005)– vastly dominates in S/N contribution to individual-source GW detectionwhen compared to the other 16 MSPs in their study. In principle, then, theoptimal observing program would be high cadence observations of the singlebest-timed pulsar. However, this would not be a robust experiment, sincean apparent GW signal in the timing of that single pulsar could potentiallyalso be explained by pulsar rotation irregularities or ISM effects. In order tomake a reliable detection, the GW signal needs to be seen in timing data of44more than one pulsar. Thus the optimal experimental design involves high-cadence observations, on timescales shorter than the nominal NANOGravobserving program, of a small number of precisely timed pulsars.In August 2014, the author served as principal investigator and wrotea telescope proposal for an additional NANOGrav observing program thatis supplementary to the nominal program discussed in Section 2.1. Thetelescope proposal, submitted to the Arecibo Observatory for the September2014 deadline, requested weekly observations of five NANOGrav MSPs whosetiming data are shown in Table 2.1. The scientific justification in the firstproposal, given the designation “P2945” by the Arecibo Observatory3, waswritten by the author using results based on simulated data generated by sev-eral NANOGrav collaborators. The proposal was accepted in December 2014with the full amount of requested observing time granted to NANOGrav, fora total of 260 hours collected for all five MSPs. Observations for P2945formally began on 1 January 2015.In August 2015, the author and several NANOGrav members re-submitteda telescope proposal for the continuation of the P2945 observing program.The scientific justification for the second submission was similar in form tothe original version submitted in the first P2945 proposal, but included aninitial timing analysis of P2945 data collected during January through Au-gust of the 2015 observing year that was performed by the author. Thesedata are shown in Figure 2.2. The second P2945 proposal was successfullyaccepted and will continue throughout the 2016 observing year. These datawill be published and be made publicly available in a forthcoming exten-sion of the NANOGrav data set (Arzoumanian et al., 2016), and will likelybe a part of several studies characterizing noise and DM variations usinghigh-cadence data.3http://www.naic.edu/vscience/schedule/2015Spring/FonsecatagP2945.pdf45Figure 2.1: TOA residuals for the five NANOGrav MSPs observed underP2945. Red and blue points represent 430 MHz and 1400 MHz data col-lected using the PUPPI pulsar processor currently used at Arecibo; greenand magenta points represent 430 and 1400 MHz data collected with thepredecessor (ASP) machine. All other points were collected with the GreenBank Telescope in West Virginia, USA. (Note: there are NANOGrav datathat have been collected during the apparent gap in data around 2014-2015as part of the nominal observing program, but they are excluded in this figureto highlight the amount of data collected under P2945.)2.3 Contributions to NANOGrav ProjectsAt its core, NANOGrav is a collaborative effort that requires dedicated timeand productivity from observers, data analysts and theorists alike. The re-cent publication of the NANOGrav nine-year data set has so far spurred amultitude of studies that probe a wide variety of scientific questions regard-ing:• limits on the strength of the stochastic, nanohertz-frequency GW signal46(Arzoumanian et al., 2016),• analysis of astrometric parameters observed in NANOGrav MSPs (Matthewset al., 2016),• impact of interstellar scintillation in TOA precision for NANOGravMSPs (Levin et al., 2016),• assessment of PTA times to detection of the stochastic GW background(Taylor et al., 2016),• assessment of noise budget in NANOGrav TOA data (Lam et al., 2016),and• analysis of secular and PK variations in orbital parameters of NANOGravbinary MSPs.The last of the studies listed above is the subject of Chapter 3.The author contributed text and analysis to the publication of the NANO-Grav nine-year data set (see “Author Contributions” section of Arzoumanianet al., 2015b), and the astrometry study performed by Matthews et al. (2016).For the latter study, Matthews et al. (2016) used secular variations in Pbobserved in two NANOGrav binary-MSP systems – PSRs J1614−2230 andJ1909−3744 – to place formidable constraints on the distance to both bi-naries, using the methodology discussed in Section 1.5. These two systemsand their orbital variations are discussed in Sections 3.4.4 and 3.4.10 below,respectively.47Chapter 3The NANOGrav Nine-YearData Set: Mass and GeometricMeasurements of BinaryMillisecond Pulsars1The NANOGrav nine-year data set (Arzoumanian et al., 2015b) containsTOAs collected for 37 MSPs, 25 of which reside in binary systems of differentshapes, sizes an orbital periods. The data span for each binary MSP variesbetween ∼ 2 to 9 years, depending on when the MSP was discovered and/orincluded into the NANOGrav PTA. The regular, monthly cadence and dual-receiver strategy of the NANOGrav observing program collectively yield anideal data set for tracking long-term changes in orientation and/or relativisticphenomena over time. Moreover, long data sets with a large number of TOAscollected at different times (and different phases of each orbit) are ideal forresolving the Shapiro timing delay. The theory introduced in Sections 1.4and 1.5 show that measurements of geometric and/or relativistic phenomena1This study was recently accepted by the Astrophysical Journal for publication, and isavailable online in the arXiv repository (Fonseca et al., 2016).48can be related to intrinsic properties of the components and orientation ofthe binary system, which are otherwise not uniquely accessible through strictKeplerian timing. In this chapter, we investigate the various effects observedin NANOGrav MSP-binary systems to determine the relevant mechanismswithin each system, as well as extract mass and/or geometric informationwhenever possible.In Section 3.1, we provide details regarding the general NANOGrav ob-serving program as well as targeted observations that were obtained specifi-cally for the detection of possible Shapiro timing delays in several NANOGravMSP-binary systems. In Section 3.2, we describe the timing models and an-alytical methods used to derive the orbital elements, as well as theoreticalconstraints that can be placed on the component masses and system orien-tation from observed variations in the orbital elements. In Section 3.3, wediscuss the methods used to characterize the physical parameters of interest,and in particular the component masses and system geometries. In Section3.4, we discuss results obtained for select individual MSP-binary systems.Finally, in Section 3.5, we summarize the main findings of our study andprovide a broader context for the implications these measurements have onunderstanding stellar-binary evolution and the overall mass distribution ofbinary MSPs.3.1 Observations & ReductionThe full details regarding data collection, calibration, pulse arrival-time de-termination and noise modeling for the NANOGrav nine-year data set areprovided in Arzoumanian et al. (2015b). Here we provide a brief summaryof this information. The data are publicly available for download online.2All 37 NANOGrav MSPs were observed on a monthly basis using eitherthe 300-m William E. Gordon Arecibo Telescope in Puerto Rico or the 100-2http://data.nanograv.org49m Robert C. Byrd Green Bank Telescope (GBT) in West Virginia, USA,as early as 2004 until late 2013. In the cases of PSRs J1713+0747 andB1937+21, both telescopes were used to monitor these MSPs. In additionto the monthly-cadence program, concentrated observing campaigns of 12MSPs were made at specific orbital phases and were designed to maximizesensitivity to the Shapiro timing delay (Pennucci, 2015).For the monthly observations at both telescopes, as well as the targetedShapiro-delay campaigns at Arecibo, each MSP was observed using two ra-dio receivers at widely separated frequencies in order to accurately measurethe pulsar’s line-of-sight dispersion properties on monthly timescales, andto account for any evolution in these frequency-dependent properties overtime. The dual-receiver observations at Arecibo were performed contigu-ously during each observing session. The same measurements at the GBTwere typically performed within several days of one another due to a needfor retraction and extension of the prime-focus boom when switching be-tween receivers. For the targeted Shapiro-delay observations at the GBT,only one receiver was used due to time constraints. The receivers used forthe NANOGrav observations reported here were centered near: 327 MHz (atArecibo only); 430 MHz (at Arecibo only); 820 MHz (at GBT only); 1400MHz; and 2030 MHz (at Arecibo only).In order to calibrate each MSP signal, pulsed broadband signals from anoise diode were recorded for several continuum radio sources of known fluxdensity that were observed once every month during each observing year.The quasar J1413+1509 was used as the continuum source at Arecibo, whilethe quasar B1442+101 was similarly used at the GBT. For each receiver,two calibration scans of the same continuum source were obtained: one wascentered on the continuum source, and another was obtained typically 1degree offset from the central position. The difference in “on” and “off”calibration signals yields the conversion factor from units of machine countsto flux density. A similar noise-diode signal was obtained for each pulsar at50its position during every observing session in order to convert raw voltages toflux densities using the conversion factors determined from the continuum-calibration observations.Two generations of pulsar backend processors were used at each tele-scope for real-time coherent de-dispersion and folding of the signal usingpre-determined ephemerides of each MSP based on early timing solutions.The identical ASP and GASP pulsar machines (Demorest, 2007; Ferdman,2008) were used from the start of the NANOGrav observing program in 2004until their decommissioning in 2011-2012. These backends decomposed theincoming signal into contiguous 4-MHz channels that spanned 20-64 MHzin usable bandwidth, depending on the receiver used and radio-frequency-interference environment. The PUPPI and GUPPI machines (DuPlain et al.,2008; Ford et al., 2010), currently in use at both telescopes, can process upto 800 MHz in bandwidth using smaller, 1.5625-MHz channels. Both sets ofmachines generated folded pulse profiles resolved into 2048 bins across thepulsar’s spin period.Arzoumanian et al. (2015b) used the standard cross-correlation methodfor the determination of each folded profile’s time of arrival (TOA), where asingle, de-noised profile template is matched in the Fourier domain with allprofiles obtained at some observing frequency and bandwidth (Taylor, 1992).Prior to correlation, we averaged data both over time (20-30 min or 2% of aMSP-binary orbit per TOA, whichever was shorter) and over a small fractionof the available bandwidth.51PSR x (lt-s) Pb (days) e ω (deg) T0 (MJD) η κ Tasc (MJD)J0023+0923 0.03484105(11) 0.13879914244(4) 0.000025(5) 82.0(12.0) 56179.082(5) 0.000024(5) 0.000003(5) 56179.08248997(8)J0613−0200 1.0914422(5) 1.198512556680(13) 0.00000443(17) 35.0(3.0) 54890.089(10) 0.0000026(2) 0.00000362(8) 54889.991808565(12)J1012+5307 0.5818176(6) 0.60467271380(6) 0.0000013(17) 75.0(75.0) 54901.95(13) 0.0000012(16) 0.0000003(16) 54901.95231605(11)J1455−3330 32.3622120(3) 76.174567474(14) 0.00016965(2) 223.458(6) 55531.1454(14) . . . . . . . . .J1600−3053 8.8016526(10) 14.348468(3) 0.000173741(11) 181.854(16) 55419.1115(6) . . . . . . . . .J1614−2230 11.29119744(7) 8.68661942171(9) 0.000001333(8) 175.9(4) 55662.053(10) 0.000000096(9) −0.000001330(7) 55658.145347857(6)J1640+2224 55.329717(4) 175.460597(13) 0.00079725(2) 50.7313(15) 54784.4707(7) . . . . . . . . .J1643−1224 25.0725904(3) 147.01739554(4) 0.000505752(18) 321.849(2) 54870.5948(8) . . . . . . . . .J1713+0747 32.34242188(14) 67.82513826930(19) 0.0000749402(6) 176.1963(16) 53761.0327(3) . . . . . . . . .J1738+0333 0.3434297(2) 0.35479073425(6) 0.0000004(10) 252.0(140.0) 55598.94(14) −0.0000004(10) −0.0000001(9) 55598.93613993(12)J1741+1351 11.0033168(4) 16.3353478266(6) 0.00000998(2) 204.00(17) 55812.321(8) −0.00000406(3) −0.00000912(2) 55819.25468493(3)J1853+1303 40.76952255(13) 115.653786432(6) 0.000023700(6) 346.656(11) 56128.563(3) . . . . . . . . .B1855+09 9.2307805(2) 12.32717119133(19) 0.00002163(2) 276.54(5) 54975.5129(19) . . . . . . . . .J1903+0327 105.593463(3) 95.17411738(8) 0.43667843(2) 141.6536021(15) 55776.9743424(3) . . . . . . . . .J1909−3744 1.89799095(4) 1.533449451246(8) 0.000000092(13) 179.0(13.0) 54514.49(6) 0.00000000(2) −0.000000092(12) 54513.989936084(3)J1910+1256 21.1291025(2) 58.466742058(5) 0.00023020(2) 106.013(6) 54956.3186(11) . . . . . . . . .J1918−0642 8.3504665(2) 10.9131775801(2) 0.000020340(18) 219.38(6) 54893.7305(17) −0.00001291(2) −0.000015721(13) 54897.63652454(2)J1949+3106 7.288647(7) 1.9495344177(8) 0.0000429(3) 208.0(6) 56365.552(3) −0.0000201(5) −0.0000379(2) 56365.97423581(3)B1953+29 31.4126915(2) 117.349097292(19) 0.000330230(15) 29.483(2) 55265.7096(7) . . . . . . . . .J2017+0603 2.1929208(9) 2.1984811364(4) 0.00000685(15) 177.0(3.0) 56201.626(15) 0.0000004(3) −0.00000684(15) 56200.64259488(3)J2043+1711 1.6239584(2) 1.48229078649(14) 0.00000489(13) 240.4(1.2) 56173.974(5) −0.00000425(13) −0.00000242(9) 56174.306240718(10)J2145−0750 10.16410849(17) 6.83890250963(11) 0.000019295(19) 200.91(5) 54902.6174(9) . . . . . . . . .J2214+3000 0.0590817(3) 0.4166329463(9) 0.000009(11) 345.0(72.0) 56221.96(8) −0.000002(10) 0.000008(11) 56221.9632381(4)J2302+4442 51.4299676(5) 125.93529697(13) 0.000503021(17) 207.8925(18) 56302.6599(6) . . . . . . . . .J2317+1439 2.313943(4) 2.45933146519(2) 0.0000007(5) 101.0(42.0) 54976.1(3) −0.0000007(5) 0.00000015(6) 54976.609358785(14)Table 3.1: Values in parentheses denote the 1σ uncertainty in the preceding digit(s), as determined fromTEMPO2. For MSPs with both DD and ELL1 parameters listed in this table, we used the ELL1 modelto describe the Keplerian orbit in the TEMPO2 fit, and then derived the corresponding DD values; the 1σuncertainties for the derived DD parameters were computed by propagating 1σ uncertainties in the fittedELL1 parameters. The values for PSR J1713+0747 were taken from Zhu et al. (2015).523.2 Binary Timing ModelsWe used the TEMPO2 pulsar-timing software package for the analysis oftopocentric TOAs collected for all NANOGrav binary MSPs, based on so-lutions made publicly available by Arzoumanian et al. (2015b) that wereobtained using GLS fitting. Each timing model includes parameters that de-scribe the given pulsar’s spin and spin-down rates, astrometry (i.e. ecliptic-coordinate position, proper motion and annual timing parallax), DM evalu-ated at monthly intervals, binary motion, and evolution in pulse-profile struc-ture as a function of observing frequency. As discussed in Chapter 2, severalNANOGrav collaborators have led studies that focus on different subsets oftiming parameters among the NANOGrav MSPs in order to maximize theamount of astrophysical information derivable from them. For this project,we only directly examine the measurements relevant to binary motion andany observed variations in the orbital elements.For each binary system, five Keplerian parameters were included in thetiming model. We also included timing parameters that describe secularvariations in the orbital elements, and/or the Shapiro timing delay, if theleast-squares fit in TEMPO2 was significantly improved, such that the F-test significance value was at least 0.0027 (i.e. each parameter is at least 3σsignificant). Finally, we chose to fit for secular variations in the projectedsemi-major axis (x) for PSRs J1600−3053 and J1909−3744, and a secularvariation in Pb of PSR J1614−2230, despite their lack of 3σ significance; thereasons for these additions are discussed in Section 3.4 below.3.2.1 Parametrizations of the Shapiro DelayThe timing solutions developed by Arzoumanian et al. (2015b) used the “tra-ditional” parametrization of the Shapiro timing delay discussed in Chapter1, where ∆S is a function of r and s (Equation 1.27 or 1.30 for DD or ELL1models, respectively). For this detailed study of MSP-binary systems, we53also created timing solutions that used the “orthometric” parametrization ofthe Shapiro timing delay (Freire & Wex, 2010). The orthometric frameworkexpresses the observed ∆S as a Fourier expansion across each system’s or-bital period and uses two different PK parameters that are derived from theharmonics of the Shapiro-delay signal to describe the relativistic effect. Itis a generalized framework based on the Fourier expansion of ∆S introducedin Section 1.4 for the entire range of system inclination. In the orthomet-ric framework, the PK parameters are either the third and fourth harmonicamplitudes of ∆S (referred to as h3 and h4, respectively), or h3 and the ortho-metric ratio ς = h4/h3. In practice, the choice of (h3, h4) as PK parametersis most appropriate for low-e systems with i < 60◦, while (h3, ς) is used foreccentric systems and low-e systems with i > 60◦.While no new physical information is made available by its PK parame-ters, the orthometric parametrization reduces statistical correlation betweenthe Shapiro-delay parameters. The orthometric model therefore provides amore numerically stable solution to the timing of binary pulsars with signif-icant Shapiro-delay signals, particularly in low-e systems where ∆S is moredifficult to measure. The available orthometric PK parameters are related tothe traditional PK parameters as nonlinear functions:ς =√1− cos i1 + cos i(3.1)h3 = rς3 (3.2)h4 = h3ς. (3.3)As shown by Freire & Wex (2010), the statistical significance of h3 reflects thedegree to which ∆S is measurable and can therefore be used as a straightfor-ward indicator for the detection of the Shapiro timing delay in a pulsar-binarysystem. In this study, we considered the Shapiro delay to be measurable ifthe estimate of h3 was statistically significant to at least 3σ. For all systems54with significant ∆S, as well as systems with statistically significant eccentric-ities that did not pass the h3 significance test, we used the (h3, ς) parametersto describe the Shapiro timing delay. For low-e systems with no significant∆S, we instead parameterized ∆S using the (h3, h4) combination.Given the relations between the (mc, sin i) and (h3, ς) parameters inEquations 3.1 and 3.2, physical arguments require that h3 > 0 and 0 < ς < 1.Equation 3.3 subsequently requires that h4 be positive, as well. However,TEMPO2 does not impose any theoretically-motivated constraints on theShapiro-delay parameters (traditional or orthometric) during a timing-modelfit; it is therefore mathematically allowed for the Shapiro-delay terms topossess values that exceed their physical limits. Such limit discrepancies arenot expected to be an issue for significant ∆S signals, but may occur for non-detections of the Shapiro delay due to large statistical correlation betweenparameters when the ∆S signal is weak.3.3 Analyses of Mass & Geometric Parame-tersWe measured the Shapiro timing delay in fifteen NANOGrav binary-MSPsystems, as well as many secular and PK variations in several orbital ele-ments, based on the F-test significance criterion used by Arzoumanian et al.(2015b). In this work, we analyzed only fourteen of the fifteen MSPs with sig-nificant ∆S since PSR J1713+0747 was recently studied by Zhu et al. (2015)using NANOGrav and archival data sets. The other fourteen NANOGravbinary MSPs with significant ∆S also passed the 3σ-significance test of h3,as described in Section 3.2.1. The secular/PK measurements are shown inTable 3.2.55PSR ω˙ (deg yr−1) x˙ (10−12) P˙b (10−12) h3 (µs) h4 (µs) ς Detection of ∆S? Span (yr)J0023+0923 . . . . . . . . . 0.06(5) −0.00(6) . . . N 2.3J0613−0200 . . . . . . . . . 0.28(3) . . . 0.74(8) Y 8.6J1012+5307 . . . . . . . . . −0.00(9) 0.05(10) . . . N 9.2J1455−3330 . . . −0.021(5) . . . 0.3(2) . . . 0.7(4) N 9.2J1600−3053 0.007(2) −0.0017(9) . . . 0.39(3) . . . 0.62(6) Y 6.0J1614−2230 . . . . . . 1.3(7) 2.329(11) . . . 0.9859(2) Y 5.1J1640+2224 −0.00028(5) 0.0145(10) . . . 0.57(6) . . . 0.61(8) Y 8.9J1643−1224 . . . −0.047(3) . . . −0.09(13) . . . 1.2(8) N 9.0J1713+0747 . . . 0.00645(11) . . . 0.54(3) . . . 0.73(1) Y 8.8J1738+0333 . . . . . . . . . 0.02(12) 0.06(13) . . . N 4.0J1741+1351 . . . −0.0094(18) . . . 0.46(6) . . . 0.85(10) Y 4.2J1853+1303 . . . 0.0147(19) . . . 0.11(11) . . . 0.5(1.2) N 5.6B1855+09 . . . . . . . . . 1.04(4) . . . 0.969(5) Y 8.9J1903+0327 0.0002410(13) . . . . . . 2.0(3) . . . 0.70(8) Y 4.0J1909−3744 . . . −0.00044(16) 0.509(9) 0.868(7) . . . 0.9381(16) Y 9.1J1910+1256 . . . −0.017(2) . . . 0.3(2) . . . 0.7(7) N 8.8J1918−0642 . . . . . . . . . 0.83(3) . . . 0.918(8) Y 9.0J1949+3106 . . . . . . . . . 2.5(5) . . . 0.77(10) Y 1.2B1953+29 . . . 0.011(3) . . . −0.1(6) . . . 0.8(5) N 7.2J2017+0603 . . . . . . . . . 0.31(7) . . . 0.72(8) Y 1.7J2043+1711 . . . . . . . . . 0.60(3) . . . 0.890(13) Y 2.3J2145−0750 . . . 0.0098(19) . . . 0.10(5) . . . 0.94(17) N 9.1J2214+3000 . . . . . . . . . −0.3(2) −0.1(3) . . . N 2.1J2302+4442 . . . . . . . . . 1.5(3) . . . 0.55(15) Y 1.7J2317+1439 . . . . . . . . . 0.33(6) . . . 0.49(14) Y 8.9Table 3.2: Values in parentheses denote the 1σ uncertainty in the preceding digit(s), as determined fromTEMPO2.563.3.1 Bayesian Analyses of Shapiro-Delay SignalsWe used the procedure outlined in Appendix A to perform a statistically rig-orous analysis of the fourteen MSPs in the nine-year data set with significantShapiro-delay measurements and obtain robust estimates of mp, mc, and i.For each of the fourteen MSPs, we first created a uniform, two-dimensionaln × n grid of χ2 values for different combinations of mc = r/T (Equation1.42) and cos i, where n = 200 or greater in order to minimize artifacts frominterpolation. With the exception of the noise parameters, all other timing-model parameters were allowed to vary freely when estimating the χ2 at eachgrid coordinate; the noise terms were held fixed at their maximum-likelihoodvalues as determined by Arzoumanian et al. (2015b). We used cos i insteadof sin i as a grid coordinate since a collection of randomly-oriented binarysystems possesses a uniform distribution in cos i; this assertion is proven inSection A.2 below.Each χ2 map was then converted to a two-dimensional probability distri-bution function (PDF) by using a Bayesian likelihood density of the followingform,p(data|mc, cos i) ∝ e−(χ2−χ20)/2 (3.4)where χ20 is the minimum value of the χ2 distribution defined on the two-dimensional grid. Bayes’ theorem subsequently yields the two-dimensionalposterior PDF, p(mc, cos i|data), when using the joint-uniform prior distri-bution of the two Shapiro-delay parameters. We then marginalized (i.e. in-tegrated) the two-dimensional PDF over cos i to obtain the one-dimensionalPDF in mc, and marginalized over mc to obtain the one-dimensional PDF incos i. In order to obtain a PDF in mp, we transformed the two-dimensional(mc, cos i) probability grid to one in the (mp, cos i) space by applying thetransformation rule for PDFs of random variables,57p(mp, cos i|data) = p(mc, cos i|data)∂mc∂mp,where the partial derivative is evaluated by using the mass function (Equation1.37, for a fixed value of cos i). The expressions for map translation areprovided in Section A.3.For the two-dimensional grids, we computed χ2 values over 0 < cos i < 1and, unless otherwise noted, 0 < mc < 1.4 M. The latter upper limitapproximately corresponds to the Chandrasekhar limit for a non-rotatingwhite dwarf. (Three exceptions to this cut-off limit are PSRs J1903+0327,J1949+3106, and J2302+4442, which are discussed individually in Section3.4 below.) The arbitrary upper limit on the companion mass does notaffect the most significant ∆S measurements, where all non-zero probabilityis typically enclosed in a small, elliptical region of the (mc, cos i) space. Thecut-off value only biases estimates made for statistically weak Shapiro-delaymeasurements, where non-zero probability can extend to large values of mcand low values of i; this bias is discussed in Section 3.3.2 below.We applied the same set of χ2-grid and marginalization procedures de-scribed above for the fourteen timing models with significant ∆S that usedthe (h3, ς) orthometric parametrization. However, we first created a χ2 gridin uniform steps of the (h3, ς) parameters, and afterwards converted the resul-tant likelihood density to the (mc, cos i) probability map by using Equations3.1 and 3.2 when applying the PDF-transformation rule.The choice in parametrization of ∆S amounts to a difference in prior prob-abilities on the physical parameters (mc, sin i) when performing the MCMCor χ2-grid analysis described above, due to the nonlinear relation between thephysical and orthometric parameters (Equations 3.1-3.3). Our first choice ofprior, in (mc, cos i), is motivated by the expected distribution of randomlyoriented binary systems – uniform in cos i – though the choice of uniform mcis arbitrary. On the other hand, Freire & Wex (2010) argue that a statisticalanalysis of the orthometric parameters is preferable since h3 and ς are re-58lated to the Fourier harmonics of ∆S and make no immediate assumption onthe probability distributions of physical parameters. Simulations by Freire &Wex show that the one-dimensional posterior PDFs of the physical parame-ters will be affected in cases of low inclination, where ∆S is typically weakerand the posterior density is heavily influenced by the choice of prior infor-mation. For cases in which there is a highly-significant measurement of ∆S,such that the posterior density spans a small range of parameter space, thetwo choices of priors give essentially the same results. We present the resultsobtained from both sets of priors to demonstrate the effects such choices haveon our mass measurements.MCMC Analysis of Shapiro-delay ParametersAs a check on the χ2-grid procedure described above, we evaluated the pa-rameters of each binary system using a Bayesian Markov Chain Monte Carlo(MCMC; e.g. Gregory, 2005a) analysis of all timing-model parameters. Inthe MCMC analysis, where we used the PAL2 Bayesian inference suite3, thejoint likelihood density includes all spin, astrometric, binary and noise termsas parameters to be sampled. The Bayesian analysis uses the traditional(mc, cos i) parameterization for the Shapiro delay, along with uniform priorson these and all other timing model parameters. We analytically marginal-ized the joint posterior over the DM, profile-evolution, and backend-offsetparameters in order to reduce computational needs.In principle, the MCMC analysis therefore provides a more robust explo-ration of the parameter space and timing-model behavior than the χ2-gridanalysis, since the MCMC method samples the noise parameters, while theχ2-grid holds the noise parameters fixed. Moreover, for the MCMC analysis,the computation of mp accounts for the small uncertainty in the mass func-tion, as it uses the posterior distributions for the Shapiro-delay and Keplerianparameters.3https://github.com/jellis18/PAL259Figure 3.1: Normalized posterior PDFs of mp, mc and cos i for PSRJ2043+1711. The red-solid curves were obtained from a χ2-grid analysis,and the blue-dashed curves were generated from an MCMC analysis of alltiming-model parameters (including terms that characterize red- and white-noise processes) when drawing 106 samples and using a thinning factor of10 to reduce autocorrelation. The χ2-grid and MCMC methods yield nearlyidentical estimates of the posterior PDFs.60Figure 3.1 shows the normalized posterior PDFs of the Shapiro-delayparameters for PSR J2043+1711 (see Section 3.4.14) estimated from boththe χ2-grid and MCMC analyses. It is clear that the χ2-grid and MCMCanalyses yield nearly identical estimates of the posterior distributions of thecomponent masses and cos i. This consistency between methods is seen for all14 MSPs with significant ∆S. Thus the χ2-grid method is a reliable methodfor estimating posterior PDFs when using an adequate (fixed) noise model.All estimates reported below were obtained from the χ2-grid method andverified using PAL2.Constraints from (ω˙)GR on Shapiro-delay ParametersBoth PSRs J1600−3053 and J1903+0327 exhibit statistically significant mea-surements of ω˙ and ∆S. As discussed in Sections 3.4.3 and 3.4.9 below, theω˙ measurements in these two systems are likely due to GR. We thereforegenerated additional χ2 grids of the two Shapiro-delay parameters for PSRsJ1600−3053 and J1903+0327 that used the statistical significance of ω˙ toimprove our estimates of the Shapiro-delay parameters in the following man-ner:• for each (mc, cos i) coordinate on the χ2 grid, we computed a value ofmp using the mass function for the given system; for the orthometricgrids, we first used Equations 3.1 and 3.2 to compute mc and cos iat each (h3, ς) grid coordinate, and then used the mass function tocompute mp;• we then used the values of mp and mc, along with the Keplerian ele-ments of the given system, to compute (ω˙)GR using Equation 1.39 atthe (mc, cos i) or (h3, ς) grid points;• we then held the ω˙ parameter fixed in the timing solution at the valuegiven by (ω˙)GR, along with the Shapiro-delay parameters, and usedTEMPO2 to obtain a constrained χ2 value.61We then used Equation 3.4 and the marginalization procedures discussedabove to obtain constrained PDFs of mp, mc and i from both parametriza-tions of ∆S.Constraints from geometric variations on Shapiro-delay Parame-tersPSRs J1640+2224 and J1741+1351 have significant measurements of ∆Sand secular variations in x that are likely due to proper motion, a biasingeffect discussed in Section 1.5.2. However, PSR J1640+2224 also exhibitsa significant ω˙ that is currently not well understood in terms of the variousmechanisms outlined in Section 1.5 above (see Section 3.4.5 for a discussion).We therefore only analyze the observed geometric variation in x for PSRJ1741+1351.In the case of PSR J1741+1351, we generated χ2 grids that explicitlymodeled the observed x˙ in terms of system geometry at each grid point usingEquation 1.43. The observed Shapiro delay yields a measure of sin i, and soan estimate of Ω can be made using the observed x˙.4 We used the T2 binarytiming model in TEMPO2, a general binary framework that uses the DD orELL1 models when appropriate but also allows for i and Ω to be used asfit parameters; the T2 timing model computes both the secular and periodicvariations in x (Equations 1.43 and 1.47) and ω (Equations 1.44 and 1.48)given the two geometric parameters.The explicit modeling of orbital variations due to changes in geometryintroduces Ω as an a priori unknown parameter; we therefore generated three-dimensional χ2 grids in the uniform (mc, cos i, Ω) and (h3, ς, Ω) phase spacesfor PSR J1741+1351, using Equation 3.4 as the likelihood density at each gridpoint in the three-dimensional phase space. We then appropriately translated4The sign ambiguity of cos i as well as the functional form of (x˙)µ results in four possiblecombinations of (i, Ω). We discuss the possibility of breaking the degeneracy in the lastparagraph of Section 3.3.1.62PSR iSD (deg) ix˙ (deg)J0023+0923 < 56 . . .J1012+5307 < 66 . . .J1455−3330 < 85 < 77J1643−1224 < 73 < 37J1738+0333 < 70 . . .J1853+1303 < 74 < 63J1910+1256 < 63 < 63B1953+29 < 80 < 77J2145−0750 < 80 < 73J2214+3000 < 75 . . .Table 3.3: Upper limits of the system inclination for MSPs that do nothave significant measurements of ∆S; if available, we use the measured x˙to compute a second, independent constraint. All upper limits are at 95%confidence.and marginalized the three-dimensional probability maps in order to obtainone-dimensional posterior PDFs of mp, mc, i and Ω.If the Shapiro timing delay and only x˙ are measured, then the (mc, cos i,Ω) and (h3, ς, Ω) grid analyses will introduce a sign ambiguity in Ω dueto the fact that the variation depends both cot i and sin(Θµ − ω). In thiscase, the ambiguity in both cos i and Ω results in a four-fold degeneracy inthe system orientation (i, Ω) of the orbit. However, if two or more secularand/or periodic variations are measured, then the four-fold degeneracy canbe broken to determine a unique orientation of the MSP-binary orbit. Weconsider the relevance of annual orbital parallax for PSRs J1640+2224 andJ1741+1351 below.3.3.2 Limits on Inclination from x˙ and the Absence ofShapiro DelayA constraint on the system inclination angle can still be placed using the x˙measurements listed in Table 3.2 (e.g. Nice et al., 2001) for cases where the63Shapiro timing delay is not detected. This is possible since the trigonometricterm for Ω in Equation 1.43 cannot exceed unity, i.e. sin(Θµ−Ω) ≤ 1, wherethe equality corresponds to an alignment between the proper-motion vectorand the projection of the orbital angular moment vector on the plane of thesky. The “magnitude” of the effect can therefore be written as |x˙|µ,max =µx| cot i|, and an upper limit on the system inclination can be calculated asi < arctan[xµ|x˙|obs]. (3.5)We computed a 95.4%-credibility upper limit on the system inclination usingEquation 3.5 and the 2σ lower limit of the x˙ measurements reported in Table3.3 for systems with no detected Shapiro delay.Another constraint on i can be placed by using a non-detection of theShapiro timing delay. The Shapiro-delay χ2 grids of pulsar-binary systemswith no measurable ∆S contain zero probability in regions of the (mc, cos i)space that correspond to large companion masses and high inclinations.These regions can be excluded based on statistically poor timing-model fitsto the NANOGrav nine-year data sets.A complication arises from the cut-off value in mc when generating theχ2 grids as discussed in Section 3.3.1: the cut-off value disregards regionsof the (mc, cos i) phase space with non-zero probability density. We believethat the cut-off value in mc is nonetheless justified since the only MSP with asuspected main-sequence-star companion is PSR J1903+0327.5 The inclusionof more probability density in non-detection χ2 grids would shift the upperlimit on i to lower values, so the upper limits on i we report in this study areconsidered to be conservative. Figures 3.2-3.10 show the χ2-grid and upper-limit results for all binary MSPs with no significant detection of ∆S, and theestimates of upper limits on i for these systems are provided in Table 3.3.5While we extended the upper limit on mc for PSRs J1949+3106 and J2302+4442 to5 M, we believe that the detections of ∆S in their TOA residuals warrant more stringentanalysis of the probability density.64Figure 3.2: Top A (mc, cos i) probability map for PSR J0023+0923. Thesignificance of h3 in this system is less than 3σ, so we only compute upperlimits on i. The inner, middle and outer red contours encapsulate 68.3%,95.4% and 99.7% of the total probability. Bottom. Posterior PDF of thederived inclination angle, obtained from the (mc, cos i) grid shown in thetop panel. The shaded blue region under the PDF contains 95% of the totalprobability.65Figure 3.3: Top. A (mc, cos i) probability map for PSR J1012+5307. Thesignificance of h3 in this system is less than 3σ, so we only compute upperlimits on i. The inner, middle and outer red contours encapsulate 68.3%,95.4% and 99.7% of the total probability. Bottom. Posterior PDF of thederived inclination angle, obtained from the (mc, cos i) grid shown on the top.The shaded blue region under the PDF contains 95% of the total probability.66Figure 3.4: Top. A (mc, cos i) probability map for PSR J1455−3330. Thesignificance of h3 in this system is less than 3σ, so we only compute upperlimits on i. The inner, middle and outer red contours encapsulate 68.3%,95.4% and 99.7% of the total probability. Bottom. Posterior PDF of thederived inclination angle, obtained from the (mc, cos i) grid shown on the top.The shaded blue region under the PDF contains 95% of the total probability.67Figure 3.5: Top. A (mc, cos i) probability map for PSR J1643−1224. Thesignificance of h3 in this system is less than 3σ, so we only compute upperlimits on i. The inner, middle and outer red contours encapsulate 68.3%,95.4% and 99.7% of the total probability. Bottom. Posterior PDF of thederived inclination angle, obtained from the (mc, cos i) grid shown on the top.The shaded blue region under the PDF contains 95% of the total probability.68Figure 3.6: Top. A (mc, cos i) probability map for PSR J1738+0333. Thesignificance of h3 in this system is less than 3σ, so we only compute upperlimits on i. The inner, middle and outer red contours encapsulate 68.3%,95.4% and 99.7% of the total probability. Bottom. Posterior PDF of thederived inclination angle, obtained from the (mc, cos i) grid shown on the top.The shaded blue region under the PDF contains 95% of the total probability.69Figure 3.7: Top. A (mc, cos i) probability map for PSR J1853+1303. Thesignificance of h3 in this system is less than 3σ, so we only compute upperlimits on i. The inner, middle and outer red contours encapsulate 68.3%,95.4% and 99.7% of the total probability. Bottom. Posterior PDF of thederived inclination angle, obtained from the (mc, cos i) grid shown on the top.The shaded blue region under the PDF contains 95% of the total probability.70Figure 3.8: Top. A (mc, cos i) probability map for PSR J1910+1256. Thesignificance of h3 in this system is less than 3σ, so we only compute upperlimits on i. The inner, middle and outer red contours encapsulate 68.3%,95.4% and 99.7% of the total probability. Bottom. Posterior PDF of thederived inclination angle, obtained from the (mc, cos i) grid shown on the top.The shaded blue region under the PDF contains 95% of the total probability.71Figure 3.9: Top. A (mc, cos i) probability map for PSR J2145−0750. Thesignificance of h3 in this system is less than 3σ, so we only compute upperlimits on i. The inner, middle and outer red contours encapsulate 68.3%,95.4% and 99.7% of the total probability. Bottom. Posterior PDF of thederived inclination angle, obtained from the (mc, cos i) grid shown on the top.The shaded blue region under the PDF contains 95% of the total probability.72Figure 3.10: Top. A (mc, cos i) probability map for PSR J2214+3000. Thesignificance of h3 in this system is less than 3σ, so we only compute upperlimits on i. The inner, middle and outer red contours encapsulate 68.3%,95.4% and 99.7% of the total probability. Bottom. Posterior PDF of thederived inclination angle, obtained from the (mc, cos i) grid shown on the top.The shaded blue region under the PDF contains 95% of the total probability.73PSR Pulsar Mass (M) Companion Mass (M) System Inclination (deg)Trad Ortho Trad Ortho Trad OrthoJ0613−0200 2.3+2.7−1.1 2.1+2.1−1.0 0.21+0.23−0.10 0.19+0.15−0.07 66+8−12 68+7−10J1600−3053 2.4+1.5−0.9 2.4+1.3−0.8 0.33+0.14−0.10 0.33+0.13−0.08 63+5−5 64+4−5J1614−2230 1.928+0.017−0.017 1.928+0.017−0.017 0.493+0.003−0.003 0.493+0.003−0.003 89.189+0.014−0.014 89.188+0.014−0.014J1640+2224 4.4+2.9−2.0 5.2+2.6−2.0 0.6+0.4−0.2 0.7+0.3−0.2 60+6−6 58+6−6J1713+0747 1.31+0.11−0.11 1.31+0.11−0.11 0.286+0.012−0.012 0.286+0.012−0.012 71.9+0.7−0.7 71.9+0.7−0.7J1741+1351 1.87+1.26−0.69 1.781.08−0.63 0.32+0.15−0.09 0.31+0.13−0.08 66+5−6 66+5−6B1855+09 1.30+0.11−0.10 1.31+0.12−0.10 0.236+0.013−0.011 0.238+0.013−0.012 88.0+0.3−0.4 88.0+0.3−0.4J1903+0327 1.65+0.02−0.02 1.65+0.02−0.03 1.06+0.02−0.02 1.06+0.02−0.02 72+2−3 72+2−3J1909−3744 1.55+0.03−0.03 1.55+0.03−0.03 0.214+0.003−0.003 0.214+0.003−0.003 86.33+0.09−0.10 86.33+0.09−0.10J1918−0642 1.18+0.10−0.09 1.19+0.10−0.09 0.219+0.012−0.011 0.219+0.012−0.011 85.0+0.5−0.5 85.0+0.5−0.5J1949+3106 4.0+3.6−2.5 4.0+3.4−2.3 2.1+1.6−1.0 1.9+1.5−0.9 67+9−8 68+8−8J2017+0603 2.4+3.4−1.4 2.0+2.8−1.1 0.32+0.44−0.16 0.27+0.30−0.12 62+9−12 65+7−11J2043+1711 1.41+0.20−0.18 1.43+0.21−0.18 0.175+0.016−0.015 0.177+0.017−0.015 83.2+0.8−0.9 83.1+0.8−0.9J2302+4442 5.3+3.2−3.6 5.5+3.0−3.2 2.3+1.7−1.3 1.8+1.6−1.0 54+12−7 57+11−9J2317+1439 4.7+3.4−2.8 4.1+3.5−2.4 0.7+0.5−0.4 0.5+0.5−0.3 47+10−7 51+1010Table 3.4: Estimate of mp, mc and i for NANOGrav Binary MSPs with significasnt Shapiro-delay signals.All uncertainties reflect 68.3% confidence intervals. “Trad” refers to estimates made with the traditional(mc, sin i) Shapiro-delay model, while “Ortho” refers to those made with the orthometric (h3, ς) model.Difference in median values and confidence intervals reflect the consequence in choosing uniform Bayesianpriors on the (mc, sin i) or (h3, ς) parameters for weak measurements of ∆S. Observed secular variationsused as constraints for PSRs J1600−3053, J1741+1351, and J1903+0327. The values for PSR J1713+0747were taken from Zhu et al. (2015).743.4 Results & DiscussionThe traditional and orthometric parameterizations of the Shapiro timing de-lay yield consistent measurements of the component masses, i, and Ω (whenthe latter angle is measurable) in the fourteen NANOGrav MSP-binary sys-tems with significant ∆S that we analyze here. We report estimates thatwere made using both Shapiro-delay models for each of these 14 MSPs inTable 3.4. Any differences in the estimates and confidence intervals derivedfrom the traditional (mc, sin i) or orthometric (h3, ς) probability grids reflectdifferent Bayesian priors on those PK parameters; the most highly-inclinedsystems produced essentially identical estimates. These features are consis-tent with the expectations discussed in Section 3.3.1.Unless otherwise specified, all numerical values with uncertainties pre-sented below reflect 68.3% equal-tailed credible intervals; that is, we computethe credible interval by numerically integrating each (normalized) posteriorPDF to values of the parameter that contain 15.9% (lower bound), 50% (me-dian), and 84.1% (upper bound) of all probability (see Appendix A.4 fordetails).3.4.1 PSR J0613−0200PSR J0613−0200 is a 3.1-ms pulsar in a 1.2-day orbit that was discoveredin a survey of the Galactic disk using the Parkes radio telescope (Lorimeret al., 1995). A previous long-term timing study of this MSP by Hotanet al. (2006) used the lack of a Shapiro-delay detection to place constraintson the companion mass and system inclination, such that 0.13 < mc/M <0.15 and 59◦ < i < 68◦ if mp = 1.3 M. Two recent, independent TOAanalyses of PSR J0613-0200 were performed by Reardon et al. (2016) andDesvignes et al. (2016). Reardon et al. used an 11-yr data set collectedfor the Parkes Pulsar Timing Array (PPTA) and did not report any secularvariations or PK effects. Desvignes et al. used a 16-yr data set collected75for the European Pulsar Timing Array (EPTA) to be measure a significantP˙b = 4.8(1.1)×10−14. Neither study reports a detection of the Shapiro timingdelay. A recent optical-spectroscopy study did not detect the companion toPSR J0613−0200, and placed a 5σ-detection lower limit on the photometricR-band magnitude to be R > 23.8 (Bassa et al., 2015).For the first time, we report the detection of the Shapiro timing delayin the PSR J0613−0200 system using the NANOGrav nine-year data set.It is likely that the Shapiro-delay signal in PSR J0613−0200 went unde-tected by Reardon et al. (2016) and Desvignes et al. (2016) because of thebetter sensitivity achieved with the GBT and GUPPI backend, as reflectedby the factor of 2-3 improvement in TOA root-mean-square (RMS) residu-als between the NANOGrav and PPTA/EPTA data sets. The χ2 grids andmarginalized PDFs for PSR J0613−0200 are shown in Figure 3.11. Our cur-rent estimates of mc = 0.18+0.15−0.07 M and i = 68+7−10 degrees are consistentwith the predictions made by Hotan et al. (2006), though our derived es-timate of mp = 2.3+2.7−1.1 M is not yet precise enough to yield a meaningfulconstraint on the pulsar mass.3.4.2 PSR J1455−3330PSR J1455−3330 is a 7.9-ms pulsar in a 76-day orbit and was discoveredin a survey of the Galactic disk using the Parkes radio telescope (Lorimeret al., 1995). The long spin period of this MSP, along with its large orbitand anomalously large characteristic age, indicates potential disk instabilityduring the transfer phase that ultimately dontated little mass to the neutronstar (Li et al., 1998). A recent radio-timing analysis by Desvignes et al.(2016) reported a significant x˙ = −1.7(4)× 10−14.We measured a significant x˙ = −2.1(5) × 10−14 in the PSR J1455−3330system using the NANOGrav nine-year data set. Our estimate of x˙ is con-sistent with the one made by Desvignes et al. (2016) using an independentdata set. We did not detect a Shapiro timing delay, as indicated by the76insignificance of h3 and unconstrained estimate of ς listed in Table 3.2.3.4.3 PSR J1600−3053PSR J1600−3053 is a 3.6-ms pulsar in a 14.3-day orbit that was discovered ina survey of high Galactic latitudes using the Parkes radio telescope (Jacobyet al., 2007). A recent analysis of the PSR J1600−3053 system by Reardonet al. (2016) used PPTA data to make significant measurements of x˙ and theShapiro timing delay: mp = 2.4(1.7) M, mc = 0.34(15) M, sin i = 0.87(6),and x˙ = −4.2(7) × 10−15. Another recent and independent study by Desvi-gnes et al. (2016) used EPTA data to measure the orthometric parametersh3 = 0.33(2) µs and ς = 0.68(5), consistent with the component masses andinclination measured by Reardon et al., as well as x˙ = −2.8(5)× 10−15.We measured a significant ω˙ for the first time, as well as a Shapiro timingdelay in the PSR J1600-3053 system. We do not yet measure a 3σ significantx˙, likely because the NANOGrav data span for PSR J1600−3053 is ∼6 yr,several years shorter than the EPTA and PPTA data sets. Nevertheless, wedo make a tentative, ∼2σ detection of x˙ = −1.7(9)× 10−15 and have electedto include it as a free parameter in our timing solution. Our estimates of x˙and the orthometric parameters, h3 = 0.39(3) and ς = 0.62(6), are consistentwith those made by Desvignes et al. (2016).Our measurement of ω˙ = 7(2) × 10−3 deg yr−1 in the PSR J1600−3053system could, in principle, be due to a combination of physical effects dis-cussed in Section 1.5. The maximum amplitude of (ω˙)µ for PSR J1600−3053,computed from Equation 1.44, is (ω˙)µ,max = µ| csc i| ∼ 10−6 deg yr−1, whichis two orders of magnitude smaller than the uncertainty level for the observedω˙ in this MSP-binary listed in Table 3.2. Therefore, the observed ω˙ in thePSR J1600−3053 system cannot be due to secular variations from propermotion at the current level of precision.The predicted GR component of ω˙ of PSR J1600−3053 is on the order of10−3 deg yr−1 given the Keplerian parameters of the system shown in Table773.1, the same order of magnitude as our measured value. We therefore usedthe method described in Section 3.3.1 to include both ω˙ and the Shapiro-delay parameters when generating the two-dimensional χ2 grid. The χ2 gridsand marginalized PDFs for PSR J1600−3053 are shown in Figure 3.12; theconstrained estimates of the component masses and inclination are: mp =2.4+1.5−0.9 M; mc = 0.33+0.14−0.10 M; and i = 63(5) degrees. Our constrainedestimates of the Shapiro delay parameters are consistent with the estimatesmade by Reardon et al. (2016) and Desvignes et al. (2016).3.4.4 PSR J1614−2230PSR J1614−2230 is a 3.2-ms pulsar in a 8.7-day orbit with a massive WDcompanion; this MSP was discovered in a mid-latitude radio search of uniden-tified EGRET gamma-ray sources using the Parkes radio telescope (Hesselset al., 2005; Crawford et al., 2006). The PSR J1614−2230 system containsone of the most massive neutron stars known, mp = 1.97(4) M, as de-termined by a strategic set of observations that were made and used byDemorest et al. (2010) to measure the Shapiro timing delay in this highly-inclined binary system. Demorest et al. were able to rule out nearly allmodels for plausible neutron-star equations of state that invoke significantamounts of exotic matter. Moreover, the PSR J1614−2230 system providedearly evidence for relatively high “birth masses” of neutron stars after theirformation, and before the onset of mass transfer (Tauris et al., 2011).We made an improved measurement of the Shapiro timing delay in PSRJ1614−2230 when using the NANOGrav nine-year data set, which includesa subset of the GUPPI data used by Demorest et al. (2010). The χ2 gridsand marginalized PDFs for PSR J0613−0200 are shown in Figure 3.13. Theuncertainties in both mc = 0.493(3) M and i = 89.189(14) degrees havedecreased such that the uncertainty in mp = 1.928(17) M is a factor of ∼3less than that made by Demorest et al. (2010).Although there was not a formally significant measurement of orbital de-78cay, we nevertheless explored fitting for it. We measured (P˙b)obs = 1.3(7)×10−12. This is much larger than the component expected from general-relativistic orbital decay (Equation 1.38), (P˙b)GR = −0.00042 × 10−12. In-stead, it is attributable to the change in the Doppler shift due to the pulsarmotion, as discussed in Section 1.5; Equation 1.45 predicts that (P˙b)D =Pb(D˙/D) = 1.36× 10−12 when using the pulsar distance and proper motionmeasured in the NANOGrav nine-year timing model, and which is consistentwith the direct measurement we make here. Matthews et al. (2016) used theagreement between (P˙b)D and the observed value as a confirmation of theparallax distance to the pulsar. The precision of (P˙b)obs can be improvedby extending the observing span backwards using pre-GUPPI archival datapublished by Demorest et al. (2010) and forwards (through future observa-tions); this will eventually provide the most precise means for measuring thedistance to this pulsar.3.4.5 PSR J1640+2224PSR J1640+2224 is a 3.1-ms pulsar in a 175-day orbit that was discoveredin a Arecibo survey of high Galactic latitudes (Foster et al., 1995a,b). Thecompanion star in this system was observed using the Palomar 5.1-m opticaltelescope to have an effective temperature that is consistent with an old HeWD (Lundgren et al., 1995). The first dedicated radio-timing study of thePSR J1640+2224 system reported a tentative detection of the Shapiro timingdelay, with mc = 0.15+0.08−0.05 M and cos i = 0.11+0.09−0.07 (Lo¨hmer et al., 2005).However, Lo¨hmer et al. did not derive a statistically significant constrainton mp. A subsequent TOA analysis of the NANOGrav five-year data set(Demorest et al., 2013) used Markov chain fitting methods and noted issueswith the numerical stability of the observed Shapiro timing delay (Vigeland& Vallisneri, 2014). The most recent radio-timing study by Desvignes et al.(2016) used EPTA data to measure a significant x˙ = 1.07(16) × 10−14, butdid not measure a significant Shapiro delay.79We measured the Shapiro timing delay, x˙ = 1.45(10) × 10−14 and ω˙ =−2.8(5) × 10−4 deg yr−1 using the NANOGrav nine-year data set for PSRJ1640+2224. The χ2 grids and marginalized PDFs of the Shapiro-delay pa-rameters measured for this MSP are shown in Figure 3.14. Based on theShapiro timing delay alone, we estimated that mc = 0.6+0.4−0.2 M and i = 60(6)degrees with the corresponding mp = 4.4+2.9−2.0 M. The highly-significant x˙,consistent with the estimate made by Desvignes et al. (2016) at the 2σ un-certainty level, is most likely due to a secular change in the inclination ofthe wide binary system induced by proper motion; the current data set isnot sensitive to annual orbital parallax since the annual astrometric paral-lax was not found to be significant for PSR J1640+2224 (Matthews et al.,2016). However, we could not reconcile the 6σ-significant value of ω˙ withthe physical mechanisms outlined in Section 1.5. In what follows below inthis subsection, we explicitly discuss and reject the possibilities that wereconsidered to explain the ω˙ measurement.The general-relativistic component of ω˙ (Equation 1.39) cannot be thedominant term since our observed value is negative. We also rule out asignificant detection of (ω˙)GR since, given the fitted Keplerian elements listedin Table 3.1, its predicted value for large assumed component masses is onthe order of 10−6 deg yr−1. Furthermore, we reject the possibility of thismeasurement arising from secular orbital variations due to proper motion,since the predicted magnitude of (ω˙)µ (Equation 1.44) is also on the order of10−6 deg yr−1.In principle, a nonzero value of ω˙ can arise from a spin-induced quadrupoleterm in the companion’s gravitational potential due to classical spin-orbitcoupling (Wex, 1998); this effect has been observed in pulsar-binary sys-tems with main-sequence companions (e.g. Wex et al., 1998), and can alsobe observed in pulsar-WD systems in the case where a quadrupole termis induced from rapid rotation of the WD companion. This scenario wasfirst considered in early studies of the relativistic PSR J1141-6545 system80by Kaspi et al. (2000), where they noted that classical spin-orbital couplingwould cause a time derivative in the system inclination angle, di/dt, that iscomparable in order of magnitude to the component of ω˙ due to spin-orbitcoupling. We used the x˙ measured in the PSR J1640+2224 system, the factthat x˙ = d(ap sin i)/dt ≈ (ap cos i) di/dt, and the Shapiro-delay estimate ofsin i to compute the time rate of change in the system inclination, and foundthat di/dt ∼ 10−6 deg yr−1. This estimate of di/dt is two orders of magni-tude smaller than the observed ω˙, and we therefore reject the significance ofclassical spin-orbit coupling in our measurement of ω˙ in the PSR J1640+2224system.While third-body effects can give rise to measurable perturbations of thepulsar-binary’s Keplerian elements (e.g. Rasio, 1994), such interactions withanother massive component would first be observed as large variations inνs (see Chapter 4 of this thesis for an analysis of PSR B1620−26, one oftwo pulsar-triple systems). The NANOGrav 9-year timing solution for PSRJ1640+2224 does not show such variations in spin frequency, and so thereis no evidence that J1640+2224 is a triple system. Future observations ofJ1640+2224, along with historical data used by Lo¨hmer et al. (2005) andthe EPTA data set, will permit for even more stringent estimates of binary-parameter variations evaluated over a larger number of orbits, and ultimatelyyield a more robust timing solution.3.4.6 PSR J1738+0333PSR J1738+0333 is a 5.8-ms pulsar in a 8.5-hr orbit with a low-mass WDcompanion that was discovered in the Swinburne Intermediate Latitude Pul-sar Survey (Jacoby, 2004). Optical spectroscopy of the WD companionyielded a significant mass ration q = mp/mc = 8.1(2) andmc = 0.182+0.007−0.005 M,as well as consistent measures of the companion radius from both spec-troscopy and photometry (Antoniadis et al., 2012). A radio-timing studyreported the measurement of orbital decay that, after applying the correc-81tion for kinematic bias discussed in Section 1.5, is consistent with the com-ponent due to GR, yielding one of the most stringent tests on tensor-scalartheories of gravitation (Freire et al., 2012). The combination of these radioand optical analyses produced a derived estimate of mp = 1.47+0.07−0.06 M, aswell as an estimate of i = 32.6(1.0) that was computed using fm. Recentphotometric observations identified optical variability of the WD companionthat is consistent with pulsations of low-mass WDs (Kilic et al., 2015).We do not measure any significant Shapiro delay or secular variations inthe orbital elements. The (mc, cos i) χ2 grid and upper limit on i are shownin Figure 3.6, which is consistent with the derived estimate of i made byAntoniadis et al. (2012) and Freire et al. (2012).3.4.7 PSR J1741+1351PSR J1741+1351 is a 3.7-ms pulsar in a 16.3-day orbit that was discovered ina survey of high Galactic latitudes using the Parkes radio telescope (Jacobyet al., 2007). The Shapiro delay was initially detected in this system by Freireet al. (2006).We detected the Shapiro timing delay in the NANOGrav nine-year dataset for PSR J1741+1351, as well as a highly significant measurement of x˙that we report for the first time. The annual orbital parallax is not signifi-cant for this MSP since the annual astrometric parallax was not significantlymeasured (Matthews et al., 2016). As discussed in Section 3.3.1 above, wenonetheless generated a three-dimensional χ2 grid for different values of thetwo Shapiro-delay parameters and Ω, in order to constrain the system geom-etry using both measurements. Figure 3.15 shows the χ2-grid results for PSRJ1741+1351 when first generating a three-dimensional, uniform grid in the(mc, cos i, Ω) parameters. The two-dimensional (cos i, Ω) probability grid,obtained by marginalizing over mc, illustrates a highly non-elliptical covari-ance between the two parameters. The constrained estimates of the Shapiro-delay parameters are mp = 1.87+1.26−0.69 M, mc = 0.32+0.15−0.09 M, i = 66+5−682degrees, and Ω = 317(35) degrees.For comparison, we over-plotted the posterior PDFs obtained from a stan-dard two-dimensional χ2 grid over the traditional (mc, cos i) parameters,while allowing x˙ and all other parameters to vary freely in each timing-modelfit, as the grey lines in Figure 3.15. There are clear and significant differencesbetween the posterior PDFs, which strongly suggest correlation between x˙and one or both of the Shapiro delay parameters. The three-dimensional χ2-grid results indicate that explicit modeling of the highly-significant kinematicterm reduces correlation between the Shapiro-delay parameters and x˙, andproduces more sensible posterior PDFs of the component masses and systeminclination that are consistent with initial results presented by Freire et al.(2006).3.4.8 PSR B1855+09PSR B1855+09 is a 5.4-ms pulsar in a 12.3-day orbit with a WD compan-ion, and is also one of the earliest MSP discoveries made using the AreciboObservatory (Segelstein et al., 1986). This MSP-binary system was thefirst to yield a significant measurement of the Shapiro timing delay frompulsar-timing measurements (Ryba & Taylor, 1991). The most recent long-term radio timing study determined the pulsar mass to lie within the range1.4 < mp < 1.8 M (95% confidence; Nice et al., 2004). Optical follow-upobservations of the companion yielded a WD-cooling timescale of ∼10 Gyr,which is twice as long as the characteristic age of the MSP (van Kerkwijket al., 2000).We made a highly significant measurement of the Shapiro timing delaywhen using the NANOGrav nine-year data set for PSR B1855+09. Theχ2 grids and marginalized PDFs for PSR B1855+09 are shown in Figure3.16. Our estimates of the component masses and inclination angle – mp =1.30+0.11−0.10 M, mc = 0.236+0.013−0.011 M, and i = 88.0+0.3−0.4 degrees – are consistentwith, and more precise than, those previously made by Kaspi et al. (1994),83Nice et al. (2004) and Reardon et al. (2016).3.4.9 PSR J1903+0327PSR J1903+0327 is a 2.1-ms pulsar in an eccentric, 95-day orbit with amain-sequence companion (Champion et al., 2008). This binary system, lo-cated within the Galactic disk, posed a significant challenge to the standardview of MSP formation since tidal interactions are expected to produce low-eccentricity orbits with WD companions, as is observed for all other diskMSP-binary systems. Freire et al. (2011) performed the most recent pulsar-timing analysis of PSR J1903+0327 and argued that both binary componentswere once members of a progenitor triple system where the main-sequencecompanion was in an outer orbit about an inner MSP-WD binary; this sys-tem was subsequently disrupted and produced the binary currently observed,either by a chaotic third-body interaction or full dissipation of the innerWD companion. They combined their Shapiro-delay measurement for thissystem with a significant measurement of ω˙, which they argue is due toGR, to determine the component masses and inclination with high preci-sion: mp = 1.667(21) M; mc = 1.029(8) M; and 77.47(15) degrees (all99.7% confidence). Freire et al. also measured an x˙ = 0.020(3) × 10−12that they attributed to proper-motion bias. A recent optical analysis ofradial-velocity measurements estimated the mass ratio of this system to beq = mp/mc = 1.56(15) (68.3% confidence; Khargharia et al., 2012), consis-tent with the radio-timing estimate of q = 1.62(3) made by Freire et al.We also independently measure a significant ω˙ = 2.410(13)×10−4 deg yr−1in the PSR J1903+0327 system, as well as the Shapiro timing delay indicatedby the significance of h3 listed in Table 3.2. We do not measure a significant x˙.The observed ω˙ from our data set is consistent with the measurement madeby Freire et al. (2011), and so we used the methodology discussed in Section3.3.1 to constrain the Shapiro-delay parameters assuming that GR describesthe observed periastron shift. The constrained χ2 grids for PSR J1903+032784are shown in Figure 3.17. From these grids, we estimated the componentmasses and inclination to be: mp = 1.65(2) M; mc = 1.06(2) M; andi = 72+2−3 deg yr−1. The estimate of mp agrees with the Freire et al. measure-ment at the 68.3% credibility level, while mc and i are consistent at aboutthe 95.4% credibility level. We do not adjust the uncertainty in our measure-ment of ω˙ for the maximum uncertainty in (ω˙)µ, which Freire et al. do whenderiving their estimates. Our derived estimate of q = 1.56(3) also agreeswith the optical measurement and Freire et al. estimate mentioned above.3.4.10 PSR J1909−3744PSR J1909-3744 is a 2.9-ms pulsar in a 1.5-day orbit with a WD companion(Jacoby et al., 2005). The Shapiro timing delay has previously been ob-served in this system with high precision, leading to the first precise massmeasurement for an MSP (Jacoby et al., 2005; Hotan et al., 2006; Verbiest,2009). Two recent, independent TOA analyses of this pulsar were per-formed by Reardon et al. (2016) and Desvignes et al. (2016). Reardon etal. used the PPTA data set and reported significant Shapiro-delay param-eters, apparent orbital decay, and geometric variations the PSR J1909-3744system with the following measured and derived results: mp = 1.47(3) M;mc = 0.2067(19) M; i = 93.52(9)◦; and P˙b = 0.503(6) × 10−12. Desvi-gnes et al. analyzed the EPTA data set and also reported estimates ofthe Shapiro-delay parameters, apparent orbital decay, and geometric vari-ations: mp = 1.54(3) M; mc = 0.213(2) M; sin i = 0.99771(13); andP˙b = 0.503(5)× 10−12.We independently measure both Shapiro-delay parameters and P˙b withhigh significance when using the NANOGrav nine-year data set. We alsomake a marginal detection of x˙ = −4.4(1.6) × 10−16 when incorporating itas a free parameter, but it does not pass the F-test criterion.The component masses that we derived from the probability maps forJ1909−3744 shown in Figure 3.18, mp = 1.55(3) M and mc = 0.214(3) M,85agree with the estimates made by Reardon et al. (2016) and Desvignes et al.(2016). Our estimate of i = 86.33(10) degrees possesses a sign ambiguityin cos i, so i = 93.67(10) is an allowed solution for our analysis; the latterestimate agrees with the Reardon et al. and Desvignes et al. measurement.Given our measurements of the Keplerian and Shapiro-delay parameters,the expected orbital decay in this system from quadrupole gravitational-waveemission is (P˙b)GR = −0.00294 × 10−12, which is significantly less than ourmeasurement of P˙b. This low estimate of (P˙b)GR implies that P˙b = 0.509(9)×10−12 ≈ (P˙b)D, which agrees with the measurement and assessment made byReardon et al. (2016) and Desvignes et al. (2016). We therefore attribute theapparent orbital decay in PSR J1909−3744 system to biases from significantacceleration between the MSP-binary and SSB reference frames. Matthewset al. (2016) used our P˙b measurement to find the distance to PSR J1909-3744 to be 1.11(2) kpc, in agreement with their timing-parallax distance of1.07+0.04−0.03 kpc.3.4.11 PSR J1918−0642PSR J1918−0642 is a 7.6-ms pulsar in a 10.9-day orbit with a likely WDcompanion that was discovered by Edwards & Bailes (2001) in a multi-beamsurvey of intermediate Galactic latitudes using the Parkes Radio Telescope.An optical search for the companion of PSR J1918−0642 was unsuccessful(van Kerkwijk et al., 2005), requiring that the apparent R-band magnitude ofthe WD be R > 24. A long-term timing study of this MSP was carried out byJanssen et al. (2010) using the Westerbork, Nanc¸ay and Jodrell Bank radioobservatories at 1400 MHz for a combined timespan of 7.4 years. While onlyKeplerian parameters were measured, Janssen et al. (2010) combined theirdistance estimate to PSR J1918−0642 – based on their dispersion-measureestimate for this pulsar and the Cordes & Lazio (2001) electron-density modelfor the Galaxy – with the R > 24 limit, and the assumption that the white-dwarf cooling and pulsar spin-down are coeval, to further constrain the com-86panion to be a He or CO white dwarf with a thin hydrogen atmosphere. Theyused the mass function of the system, as well as an assumedmp = 1.35 M, tocompute a minimum companion mass of mc,min = 0.24 M. A recent radio-timing analysis by Desvignes et al. (2016) used the EPTA data to measurethe Shapiro delay in this system, with mp = 1.3+0.6−0.4 M, mc = 0.23(7) M,and cos i = 0.09+0.05−0.04.We measured a highly-significant Shapiro timing delay in the PSR J1918−0642binary system using the NANOGrav nine-year data set. The probabilitymaps computed from χ2 grids for PSR J1918−0642 are shown in Figure3.19. The significance of h3 in the PSR J1918−0642 system exceeds 27σ, afactor of ∼ 4 better than the h3 estimate made by Desvignes et al. (2016)when using their EPTA data set. Our precise measurements of the WDmass and inclination from the Shapiro timing delay are mc = 0.219+0.012−0.011 Mand i = 85.0(5) degrees regardless of choice in the parameterization of ∆S.The derived estimate of the pulsar mass is the first precise estimate for thissystem, and is suggestive of a low-mass neutron star: mp = 1.18+0.10−0.09 M.3.4.12 PSR J1949+3106PSR J1949+3106 is a 13.1-ms pulsar in a 1.9-day orbit with a massive com-panion that was discovered by the ongoing PALFA survey of the Galac-tic plane using the Arecibo telescope (Deneva et al., 2012). The initialradio-timing study by Deneva et al. used TOAs collected with the Arecibo,Green Bank, Nanc¸ay and Jodrell Bank telescopes over a four-year periodto make a significant detection of the Shapiro timing delay in this sys-tem. They reported significant measurements of the orthometric parameters,h3 = 2.4(1) µs and ς = 0.84(2), as well as derived estimates of componentmasses and system inclination: mp = 1.47+0.43−0.31 M; mc = 0.85+0.14−0.11 M; andi = 79.9+1.6−1.9 degrees.We independently measured a Shapiro timing delay in the PSR J1949+3106using the NANOGrav nine-year data set. The probability maps computed87from χ2 grids for PSR J1949+3106 are shown in Figure 3.20; we set mc,max =5 M when computing the χ2 grids since the peak-probability value is nearlyequal to our usual upper limit of mc,max = 1.4 M. Our measurements of theorthometric parameters, h3 = 2.5(5) µs and ς = 0.77(10), are consistent withthose made by (Deneva et al., 2012) at the 68.3% credibility level. The uncer-tainties in our measurements are comparatively larger due to the shorter timespan of our data set and, therefore, less TOA coverage across the orbit. Ourderived estimates of the component masses and inclination are subsequentlymuch less stringent than those made by Deneva et al.: mp = 4.0+3.6−2.5 M;mc = 2.1+1.6−1.0 M; and i = 67+9−8 degrees.3.4.13 PSR J2017+0603PSR J2017+0603 is a 2.9-ms pulsar in a 2.2-day orbit that was initially foundusing the Fermi Large Area Telescope (LAT) as a gamma-ray source withno known associations; radio pulsations were discovered and subsequentlytimed from this source using the Nancay Radio Telescope and Jodrell BankObservatory for nearly two years by Cognard et al. (2011). They used themass function of the PSR J2017+0603 system, along with an assumed mp =1.35 M, to compute a minimum companion mass of mc,min = 0.18 M.For the first time, we detect a Shapiro timing delay in the PSR J2017+0603system using the NANOGrav nine-year data set, with mc = 0.32+0.44−0.16 M andi = 62+9−12 degrees. The probability maps computed from χ2 grids for PSRJ2017+0603 are shown in Figure 3.21. The observed Shapiro delay in thissystem is currently weak since the marginalized, one-dimensional PDF ofmp = 2.4+3.4−1.4 M extends to large values of the neutron-star mass. However,we were able to make a significant detection using a comparatively small,1.7-yr data set that includes targeted observations at select orbital phasesdiscussed in Section 3.1; our measurement will improve with the inclusion offuture TOAs collected at different points in the orbit.883.4.14 PSR J2043+1711PSR J2043+1711 is a 2.4-ms pulsar in a 1.5-day orbit that was initiallyfound using the Fermi LAT as a gamma-ray source with no previously knownassociations. The radio counterpart was discovered using the Nancay andGreen Bank Telescopes; the Shapiro delay was detected in this MSP-binarysystem using a timing model derived from TOAs collected with the Nancay,Westerbork and Arecibo observatories over a three-year period (Guillemotet al., 2012). At the time of the initial study performed by Guillemot etal., the Shapiro timing delay was not significant enough to yield statisticallymeaningful estimates of the component masses and inclination angle. Theyplaced limits on the companion mass by assuming the validity of the mc-Pbrelation, and derived a preferred range of 0.20 < mc < 0.22 M; with thisconstraint, Guillemot et al. found the pulsar mass and inclination to be1.7 < mp < 2.0 M and i = 81.3(1.0) degrees, respectively.The NANOGrav nine-year data set on PSR J2043+1711, which includesthe targeted Shapiro-delay observations discussed in Section 3.1, yields asignificantly improved measurement of the component masses and systeminclination as shown in Table 3.4; the impact of the targeted observationson the significance of ∆S in the PSR J2043+1711 system was discussedby Pennucci (2015). The probability maps computed from χ2 grids forPSR J2043+1711 are shown in Figure 3.22. Our improved measurementsof mc = 0.175+0.016−0.015 M and i = 83.2+0.8−0.9 degrees are consistent with the ini-tial estimates made by Guillemot et al. (2012), though mc is moderatelylower than the range determined from the mc-Pb relation. Our derivedmp = 1.41+0.21−0.18 M is therefore slightly below the mp range determined byGuillemot et al. when assuming the validity of the mc − Pb relation.893.4.15 PSR J2145−0750PSR J2145−0750 is a 16-ms pulsar in a 6.8-day orbit with a white-dwarfcompanion and was discovered in a Parkes Telescope survey (Bailes et al.,1994). Both Phinney & Kulkarni (1994) and van den Heuvel (1994) arguedthat the J2145−0750 system likely experienced unstable mass transfer from“common-envelope” evolution, where the pulsar gradually expelled the outerlayers of the donor, in order to explain its unusually long pulsar-spin pe-riod and massive companion compared to other binary-MSP systems. Earlyoptical observations of the WD companion noted the difficulty in obtainingaccurate photometry due to the use of a dispersion-based distance estimateand the presence of a coincident field star (Lundgren et al., 1995). However,a recent study performed by Deller et al. (2016) combined improved opti-cal imaging with a precise VLBI distance of d = 613+16−14 pc to estimate acompanion mass of mc ≈ 0.85 M. Deller et al. also detected the orbitalreflex motion of J2145−0750 through their VLBI measurements, and inferredestimates of i = 21+7−4 degrees and Ω = 230(12) degrees.6We measured x˙ = 0.0098(19) × 10−12, consistent with estimates madeby Reardon et al. (2016). Our estimate of h3 = 0.10(5) µs does not passthe h3-significance test, and so we do not formally measure a significantShapiro timing delay from the radio-timing data alone. However, we usedthe estimate of mc = 0.83+0.06−0.06 M made by Deller et al. (2016) as a priordistribution when computing the posterior maps for PSR J2145−0750. Theresulting constraints on cos i and mp are shown in Figure 3.23, which yieldmp = 1.3+0.4−0.5 M and i = 34+5−7 degrees; these estimates are consistent withthose made by Deller et al., and with the upper limits on i we derive inSection 3.3.2, shown in Table 3.3.6Deller et al. (2016) report their estimate of Ω using a convention that measures Ω fromcelestial East through North. This convention is inconsistent with the North-through-Eastconvention we use in this work. We report their estimate of Ω relative to our convention.903.4.16 PSR J2302+4442PSR J2302+4442 is a 5.2-ms pulsar in a 126-day orbit that, along with PSRJ2017+0603 (Section 3.4.13) was initially found using the Fermi LAT as agamma-ray source with no known associations and observed in the radiousing the Nanc¸ay Radio Telescope and Jodrell Bank Observatory for nearlytwo years by Cognard et al. (2011). They used the mass function of the PSRJ2302+4442 system, along with an assumed mp = 1.35 M, to compute aminimum companion mass of mc,min = 0.3 M.For the first time, we tentatively detect a Shapiro timing delay in the PSRJ2302+4442 system using the NANOGrav nine-year data set. The probabil-ity maps computed from χ2 grids for PSR J2302+4442 are shown in Figure3.24. Due to the weak detection of ∆S and large correlation between r ands, the timing solution published by Arzoumanian et al. (2015b) used a fixedvalue of mc = 0.355 M that was computed from the mc-Pb relation whenfitting for all other timing parameters, including the Shapiro s parameter.In this study, we developed timing solutions using both the traditional andorthometric parameterizations of ∆S that allowed both PK parameters to befitted for. The value of h3 in the PSR J2302+4442 system exceeds 5σ andtherefore passes the h3 significance test for detection of ∆S.Our estimates of the companion mass and inclination aremc = 2.3+1.7−1.3 Mand i = 54+12−7 degrees, and the corresponding pulsar mass ismp = 5.3+3.2−3.6 M.We computed χ2 grids with mc,max = 5 M since the peak-probability valueof mc exceeds the usual upper limit of mc,max = 1.4 M. While the posteriorPDFs of the component masses span a large range of mass values, the sig-nificant estimates of s and ς indicate a measurable constraint on the systeminclination. The measurement of ∆S will improve in significance over timesince the current data set for PSR J2302+4442 only spans about 1.7 years –or ∼5 orbits, given the long Pb of this MSP-binary system – and so a verysmall fraction of the Shapiro-delay signal has been sampled. Furthermore,given the large orbit and modest inclination, we expect to see a measurable91secular variation in x within the next few years.3.4.17 PSR J2317+1439PSR J2317+1439 is a 3.4-ms pulsar in a 2.5-day orbit that was discovered in asurvey of high Galactic latitudes using the Arecibo Obsveratory and possessesone of the smallest eccentricities known (Camilo et al., 1993, 1996; Hobbset al., 2004). The most recent radio-timing analysis of PSR J2317+1439performed by Desvignes et al. (2016) did not yield any secular variationsin orbital parameters or a significant measurement of the Shapiro timingdelay when using their 17.3-yr EPTA data set. However, a Bayesian-timinganalysis performed by Vigeland & Vallisneri (2014) used the NANOGrav five-year data set (Demorest et al., 2013) to measure several secular variationsin the binary parameters: P˙b = 6.4(9) × 10−12; η˙ = −2(4) × 10−15; andκ˙ = 2.0(7)×10−14. Vigeland and Vallisneri noted that many of the posteriordistributions for binary parameters of J2317+1439 changed slightly whenusing different priors for the astrometric timing parallax.The original NANOGrav nine-year timing model for PSR J2317+1439contains parameters that describe secular variations in x and the Laplace-Lagrange eccentricity parameters, with η˙ = 5.0(9) × 10−15 s−1, all of whichpass the F-test criterion. We found that P˙b did not pass the F-test, so it wasnot fitted in the original NANOGrav nine-year timing solution. Moreover,both the F-test and the h3-significance test indicated that the Shapiro delaywas not significant, and so we also did not initially incorporate the Shapiro-delay parameters.Despite the statistical significance of η˙, we do not believe that the PSRJ2317+1439 system is experiencing physical processes that produce a chang-ing eccentricity. For instance, if mass transfer between components werecurrently taking place, we would expect to observe a spin-up phase; instead,we observe seemingly “normal” spin-down properties and stable rotation thatis typical of MSPs. The presence of a third massive body in a bound, hierar-92chical orbit about the pulsar-companion binary system would induce higher-order derivatives in spin frequency as well as additional third-body effectson the shape, size and period of the inner binary (e.g. Joshi & Rasio, 1997),most of which we do not see in the NANOGrav nine-year data set. Finally,the timescale for the observed change in η is estimated to be η/η˙ ≈ 0.7 years,which is implausibly short.Because the observed η˙ is physically implausible, and because covariancesbetween it and several other parameters distort the timing solution, we choseto hold both η˙ and κ˙ fixed to a value of zero (i.e. no change in the eccentricityparameters of the system) while re-fitting the nine-year timing model. In thiscase, we found that the significance of h3 exceeded 3σ and therefore includedthe Shapiro-delay parameters. We found that x˙ did not pass the F-test, andso did not fit for it in our modified solution. The new timing model forPSR J2317+1439 fits the data well (reduced χ2 = 1.0053 for 2531 degrees offreedom), though the original model published by Arzoumanian et al. (2015b)that fits for η˙ and κ˙ better fits the TOA data (reduced χ2 = 0.9966 for 2531degrees of freedom).We generated two-dimensional χ2 grids for the traditional and orthome-tric Shapiro-delay parameters. The probability maps and the marginalizedPDFs of the component masses and system inclination are shown in Figure3.25. Given the new binary timing model of PSR J2317+1439, we have madea weak detection of the Shapiro timing delay in this system since the two-dimensional probability density extends to large mc for low inclinations, andso the system inclination angle is not as well constrained as for the otherstronger detections. Our current estimates of the component masses andinclinations are mp = 4.7+3.4−2.8 M, mc = 0.7+0.5−0.4 M, and i = 47+10−7 degrees.93Figure 3.11: Probability maps and posterior PDFs of the traditional Shapiro-delay parameters measuredfor PSR J0613−0200. The inner, middle and outer red contours encapsulate 68.3%, 95.4% and 99.7% ofthe total probability defined on each two-dimensional map, respectively. In all slimmer panels, the bluesolid lines represent posterior PDFs obtained from marginalizing the appropriate two-dimensional map, thevertical red-dashed lines are bounds of the 68.3% confidence interval, and the red-solid line is the medianvalue.94Figure 3.12: Probability maps and posterior PDFs of the traditional Shapiro-delay parameters measuredfor PSR J0613−0200. The maps and PDFs for J1600−3053 were constrained assuming that the observed ω˙is due to GR (see Section 3.4.3). The inner, middle and outer red contours encapsulate 68.3%, 95.4% and99.7% of the total probability defined on each two-dimensional map, respectively. In all slimmer panels,the blue solid lines represent posterior PDFs obtained from marginalizing the appropriate two-dimensionalmap, the vertical red-dashed lines are bounds of the 68.3% confidence interval, and the red-solid line is themedian value.95Figure 3.13: Probability maps and posterior PDFs of the traditional Shapiro-delay parameters measuredfor PSR J1614−2230. The inner, middle and outer red contours encapsulate 68.3%, 95.4% and 99.7% ofthe total probability defined on each two-dimensional map, respectively. In all slimmer panels, the bluesolid lines represent posterior PDFs obtained from marginalizing the appropriate two-dimensional map, thevertical red-dashed lines are bounds of the 68.3% confidence interval, and the red-solid line is the medianvalue.96Figure 3.14: Probability maps and posterior PDFs of the traditional Shapiro-delay parameters measuredfor PSR J1640+2224. The inner, middle and outer red contours encapsulate 68.3%, 95.4% and 99.7% ofthe total probability defined on each two-dimensional map, respectively. In all slimmer panels, the bluesolid lines represent posterior PDFs obtained from marginalizing the appropriate two-dimensional map, thevertical red-dashed lines are bounds of the 68.3% confidence interval, and the red-solid line is the medianvalue.97Figure 3.15: Probability maps and posterior PDFs of the traditional Shapiro-delay parameters measuredfor PSR J1741+1351, including the PDF for Ω determined from a three-dimensional χ2 grid. The inner,middle and outer red contours encapsulate 68.3%, 95.4% and 99.7% of the total probability defined on eachtwo-dimensional map, respectively. In all slimmer panels, the blue solid lines represent posterior PDFsobtained from marginalizing the appropriate two-dimensional map, the vertical red-dashed lines are boundsof the 68.3% confidence interval, and the red-solid line is the median value. Shown for comparison, thegrey curves in the slimmer panels of PSR J1741+1351 are marginalized PDFs obtained from computing aseparate, two-dimensional χ2 grid over the (mc, cos i) parameters while letting x˙ be a free parameter in eachTEMPO2 fit. See Section 3.4.7 for a discussion on the visible differences in PDFs.98Figure 3.16: Probability maps and posterior PDFs of the traditional Shapiro-delay parameters measured forPSR B1855+09. The inner, middle and outer red contours encapsulate 68.3%, 95.4% and 99.7% of the totalprobability defined on each two-dimensional map, respectively. In all slimmer panels, the blue solid linesrepresent posterior PDFs obtained from marginalizing the appropriate two-dimensional map, the verticalred-dashed lines are bounds of the 68.3% confidence interval, and the red-solid line is the median value.99Figure 3.17: Probability maps and posterior PDFs of the traditional Shapiro-delay parameters measuredfor PSR J1903+0327, using statistical significance of ω˙ and the assumption of GR to constrain the proba-bility density. The inner, middle and outer red contours encapsulate 68.3%, 95.4% and 99.7% of the totalprobability defined on each two-dimensional map, respectively. In all slimmer panels, the blue solid linesrepresent posterior PDFs obtained from marginalizing the appropriate two-dimensional map, the verticalred-dashed lines are bounds of the 68.3% confidence interval, and the red-solid line is the median value.100Figure 3.18: Probability maps and posterior PDFs of the traditional Shapiro-delay parameters measuredfor PSR J1909−3744. The inner, middle and outer red contours encapsulate 68.3%, 95.4% and 99.7% ofthe total probability defined on each two-dimensional map, respectively. In all slimmer panels, the bluesolid lines represent posterior PDFs obtained from marginalizing the appropriate two-dimensional map, thevertical red-dashed lines are bounds of the 68.3% confidence interval, and the red-solid line is the medianvalue.101Figure 3.19: Probability maps and posterior PDFs of the traditional Shapiro-delay parameters measuredfor PSR J1918−0642. The inner, middle and outer red contours encapsulate 68.3%, 95.4% and 99.7% ofthe total probability defined on each two-dimensional map, respectively. In all slimmer panels, the bluesolid lines represent posterior PDFs obtained from marginalizing the appropriate two-dimensional map, thevertical red-dashed lines are bounds of the 68.3% confidence interval, and the red-solid line is the medianvalue.102Figure 3.20: Probability maps and posterior PDFs of the traditional Shapiro-delay parameters measuredfor PSR J1949+3106. The inner, middle and outer red contours encapsulate 68.3%, 95.4% and 99.7% ofthe total probability defined on each two-dimensional map, respectively. In all slimmer panels, the bluesolid lines represent posterior PDFs obtained from marginalizing the appropriate two-dimensional map, thevertical red-dashed lines are bounds of the 68.3% confidence interval, and the red-solid line is the medianvalue.103Figure 3.21: Probability maps and posterior PDFs of the traditional Shapiro-delay parameters measuredfor PSR J2017+0603. The inner, middle and outer red contours encapsulate 68.3%, 95.4% and 99.7% ofthe total probability defined on each two-dimensional map, respectively. In all slimmer panels, the bluesolid lines represent posterior PDFs obtained from marginalizing the appropriate two-dimensional map, thevertical red-dashed lines are bounds of the 68.3% confidence interval, and the red-solid line is the medianvalue.104Figure 3.22: Probability maps and posterior PDFs of the traditional Shapiro-delay parameters measuredfor PSR J2043+1711. The inner, middle and outer red contours encapsulate 68.3%, 95.4% and 99.7% ofthe total probability defined on each two-dimensional map, respectively. In all slimmer panels, the bluesolid lines represent posterior PDFs obtained from marginalizing the appropriate two-dimensional map, thevertical red-dashed lines are bounds of the 68.3% confidence interval, and the red-solid line is the medianvalue.105Figure 3.23: Probability maps and posterior PDFs of the traditional Shapiro-delay parameters measuredfor PSR J2145−0750. The inner, middle and outer red contours encapsulate 68.3%, 95.4% and 99.7% ofthe total probability defined on each two-dimensional map, respectively. In all slimmer panels, the bluesolid lines represent posterior PDFs obtained from marginalizing the appropriate two-dimensional map, thevertical red-dashed lines are bounds of the 68.3% confidence interval, and the red-solid line is the medianvalue.106Figure 3.24: Probability maps and posterior PDFs of the traditional Shapiro-delay parameters measuredfor PSR J2302+4442. The inner, middle and outer red contours encapsulate 68.3%, 95.4% and 99.7% ofthe total probability defined on each two-dimensional map, respectively. In all slimmer panels, the bluesolid lines represent posterior PDFs obtained from marginalizing the appropriate two-dimensional map, thevertical red-dashed lines are bounds of the 68.3% confidence interval, and the red-solid line is the medianvalue.107Figure 3.25: Probability maps and posterior PDFs of the traditional Shapiro-delay parameters measuredfor PSR J2317+1439. The inner, middle and outer red contours encapsulate 68.3%, 95.4% and 99.7% ofthe total probability defined on each two-dimensional map, respectively. In all slimmer panels, the bluesolid lines represent posterior PDFs obtained from marginalizing the appropriate two-dimensional map, thevertical red-dashed lines are bounds of the 68.3% confidence interval, and the red-solid line is the medianvalue.1083.5 Conclusions & SummaryWe have derived estimates of binary component masses and inclination anglesfor fourteen NANOGrav MSP-binary systems with significant measurementsof the Shapiro timing delay. Four of these fifteen Shapiro-delay signals –in PSRs J0613−0200, J2017+0603, J2302+4442, and J2317+1439 – havebeen measured for the first time. From the Shapiro timing delay alone, wewere able to measure high-precision neutron star masses as low as mp =1.18+0.10−0.09 M for PSR J1918−0642 and as high as mp = 1.928+0.017−0.017 M forPSR J1614−2230. Measurements of previously observed ∆S signals in theJ1918−0642 and J2043+1711 systems have been significantly improved uponin this work, with the pulsar mass for PSR J2043+1711 mp = 1.41+0.21−0.18 Mbeing measured significantly for the first time. For the fourteen MSPs withsignificant ∆S, we performed a rigorous analysis of the χ2 space for the twoShapiro-delay parameters, using priors uniform in the traditional (mc, sin i)and orthometric (h3, ς) parametrizations of the Shapiro timing delay, in orderto determine robust credible intervals of the physical parameters.Most of the NANOGrav binary MSPs exhibit significant changes in oneor more of their orbital elements over time. Whenever possible, we used thestatistical significance of the observed orbital variations to further constrainthe parameters of the observed Shapiro timing delay when performing theχ2-grid analysis. Assuming the validity of GR, we further constrained thecomponent masses in the PSR J1600−3053 and PSR J1903+0327 systems,which both experience significant periastron advance due to strong-field grav-itation; the precision of our ω˙ measurement for PSR J1903+0327 contributedto a highly constrained estimate of mp = 1.65+0.02−0.02 M that is consistent withprevious timing studies of this MSP using an independent data set. We alsoused the highly-significant x˙ measurement in the PSR J1741+1351 systemin combination with the Shapiro timing delay observed in this system, whichallowed for an estimation of Ω, albeit with a large uncertainty.109System Name mc (M) Pb (days) References (Mass Measurement, Identification)PSR J0348+0432 0.172(3) 0.102 Antoniadis et al. (2013)PSR J0751−1807 0.16(1) 0.263 Bassa et al. (2006b), Desvignes et al. (2016)PSR J1738+0333 0.181+0.007−0.005 0.354 Antoniadis et al. (2012)PSR J1012+5307 0.16(2) 0.604 van Kerkwijk et al. (1996), van Kerkwijk et al. (2005)J0247−25B 0.186(2) 0.667 Maxted et al. (2013)PSR J1910−5959A 0.180(18) 0.837 Bassa et al. (2006a), Corongiu et al. (2012)PSR J0337+1715i 0.19751(15) 1.629 Ransom et al. (2014), Kaplan et al. (2014)KOI 1224 0.22(2) 2.698 Breton et al. (2012)KOI 74 0.22(3) 5.189 van Kerkwijk et al. (2010)PSR J0437−4715 0.224(7) 5.741 Durant et al. (2012), Reardon et al. (2016)RRLYR 02792 0.260(15) 15.243 Pietrzyn´ski et al. (2012)PSR J0337+1715o 0.4101(3) 327.257 Ransom et al. (2014)Table 3.5: Uncertainties in Pb are suppressed due to the high precision to which they are measured. Valuesin parentheses denote the 1σ uncertainty in the preceding digit(s).110The relativistic Shapiro timing delay provides a direct measurement ofthe companion mass that is independent of the given system’s evolutionaryhistory, and that therefore can be used to test the plausibility of availablebinary-evolution paradigms. Figure 3.26 illustrates the Pb-vs-mc estimatesfor the NANOGrav MSP-binary systems that are known or suspected tohave He-WD companions, as well as a blue-shaded region that correspondsto the theoretical mc-Pb correlation (Equation 1.8) as predicted by Tauris &Savonije (1999). PSR J1903+0327 is excluded since its companion is likely amain sequence star, while PSR J1614−2230 is excluded since its companionis a carbon-oxygen WD and is believed to have evolved through a differentformation channel (Tauris et al., 2011). Figure 3.26 is recreated from the onepresented by Tauris & van den Heuvel (2014). Black points denote precisemeasurements of mWD in WD-binary systems examined in previous works;values and references for these data are provided in Table 3.5. The widthof the shaded region represents possible correlated values of Pb and mWDfor progenitor donor stars with different chemical compositions, particularlywith metallicities (Z) in the range 0.001 < Z < 0.02. While our mc estimatesgenerally agree with the predicted correlation, additional measurements athigher companion masses are needed in order to perform a robust explorationof the correlation parameters and their credible intervals.At its current level of precision, the low mass of PSR J1918−0642 isinteresting since this MSP possesses spin parameters that are indicative ofan old neutron star that experienced significant mass transfer and a sub-stantial spin-up phase. The implication of a low “birth mass” for neu-tron stars is consistent with early estimates of the initial-mass function(e.g. Timmes et al., 1996), though suggests that the neutron-star progeni-tor to J1918−0642 may have undergone an electron-capture supernova event(e.g. Schwab et al., 2010) which produces comparatively less-massive neutronstars. Similar conclusions have been drawn for the lighter neutron stars inthe J0737−3039A/B (Ferdman et al., 2013) and J1756−2251 (Ferdman et al.,111Figure 3.26: Pb versus mc for binary systems with He-WD companions. Redpoints are our new measurements (see Figure 9). Black points are WD-massmeasurements made for systems listed in Table 3.5. The shaded blue region isthe expected correlation between mc and Pb, computed by Tauris & Savonije(1999), for post-transfer He-WD binary systems with progenitor companionsthat have metallicities within the range 0.001 < Z < 0.02.1122014) double-neutron-star binary systems, though the evolutionary historyof these systems (with lesser degrees of mass transfer) are understood to bedifferent than that expected for PSR J1918−0642.Extending the data sets of these MSPs will refine observed secular varia-tions due to PK and/or kinematic-bias effects within the next few years. Fur-thermore, extending TOA coverage in orbital phase for PSRs J0613−0200,J1949+3106, J2017+0603, J2302+4442, and J2317+1439 will improve thesignificance of the Shapiro timing delay that we report in this study. In par-ticular, additional TOAs collected for PSRs J1640+2224 and J2317+1439will help in the assessment of their complex orbital behavior as seen inthe NANOGrav nine-year data set for these systems. The combination ofNANOGrav high-precision TOAs with archival data published in previousstudies will provide more accurate timing models and a complete picture ofthe physical processes that affect the NANOGrav MSP orbits.113Chapter 4Long-Term Timing of the PSRB1620−26 Triple System inMessier 4The proto-stellar environments that eventually form binary systems can alsoform dynamically complex systems with three or more massive components.Hydrodynamic simulations of gravitationally-driven collapse in proto-stellargas clouds have shown that such “multi-core” systems can be readily formedthrough substantial fragmentation (e.g. Boss, 1991). For the purposes oflong-term study, the most stable multi-core systems are those in a hierar-chical formation, where an “inner” system is orbited by an “outer” objectwith a large distance between their respective centers of mass; for these hi-erarchical systems, the mutual tidal perturbations are expected to influencethe dynamical evolution at a rate that keeps the total system gravitationallybound on long timescales.1 Recent observational studies have suggested thathierarchical triple systems, with three components, are indeed common in1For hierarchical triple systems, we refer to the orbital parameters of the inner andouter companions with subscripts “i” and “o”, respectively. The details regarding theouter-orbital parameters are summarized in Section 4.2 and discussed in greater detail inAppendix B.114nature and constitute ∼ 20% of all small-period binaries within the Galaxy(e.g. Rappaport et al., 2013).At this point in time, only two radio pulsars are known to reside in hi-erarchical triple systems. The first of these pulsars to be discovered, PSRB1620−26, was found in a search for radio pulsations in the Messier 4 (M4)globular cluster (Lyne, 1988) and was immediately noted to belong in a bi-nary system with a WD companion with (mc)i ≈ 0.3 M (McKenna & Lyne,1988). However, subsequent timing of PSR B1620−26 showed an unusuallylarge eccentricity of the 191-day pulsar-WD orbit, ei ≈ 0.025, that contra-dicts the standard formation model discussed in Section 1.1.4, as well as alarge time derivatives in νs that is unlike those seen in other stable binarypulsars (Thorsett, 1991).It was Backer et al. (1993) who first proposed the existence of a secondcompanion of substellar mass in the PSR B1620−26 system to explain thesediscrepancies, since gravitational acceleration from binary (or triple) motioncan produce large variations in νs from orbit-induced Doppler shifts. Using adata set that spanned nearly a decade in time, Thorsett et al. (1999) providedthe first timing solution for a hierarchical pulsar-triple system that describedthe observed spin-frequency derivatives in terms of Doppler shifts due totwo non-interacting Keplerian orbits. However, their analysis indicated thatthe outer-orbital companion mass (mc)o ∼ 0.01 M and outer-orbital pe-riod (Pb)o ∼ 100 years; the orbital elements that comprise the outer-orbitalRo¨mer delay, (∆R)o, were found to be covariant as fit parameters and not wellconstrained, and so Thorsett et al. provided estimates of the outer-orbitalelements for different fixed values of the outer-orbital eccentricity (eo).The study of orbital variations due to hierarchical three-body interactionsis expected to yield intrinsic properties of the complex dynamical system andsystem components, such as the component masses and mutual inclination ofthe inner and outer orbits (e.g. Ford et al., 2000b; Kopeikin & Vlasov, 2004).These variations can also be used as constraints to infer the evolution of115three-body systems to their current states. A notable example is the recentdiscovery and analysis of the PSR J0337+1715 system – the other hierarchicalpulsar-triple system – with two low-mass white dwarfs in short-period orbits,which used a newly developed three-body integration technique that allowsfor direct measurement of “interaction” parameters that quantify three-bodydynamical terms tied to mutual geometry and ratios of component masses(Ransom et al., 2014). Ransom et al. also showed that a double-Kepleriantiming model of the whole system that models inner-orbit perturbations asTaylor expansions does not adequately describe the complex timing behaviorfor three-body systems, which illustrates the need for explicit, simultaneousmodeling of both orbits and their mutual interactions.In the unique case of PSR B1620−26, the sub-stellar nature of the outercompanion yields interesting implications for planet formation in the early,metal-poor Universe. Thorsett et al. (1999) determined that (mc)o < 0.036 M(95% confidence), significantly smaller than the minimum mass needed forhydrogen fusion (∼ 0.08 M), and argued that the outer companion is likelya Jupiter-mass planet. The presence of planets in globular clusters wasfirst considered by Sigurdsson (1992), shortly after the discovery of the PSRB1257+12 planetary system (Wolszczan & Frail, 1992) and prior to the triple-system association for PSR B1620−26 by Backer et al. (1993); Sigurdssonnoted that the timescale for planet formation in globular clusters, which areamong the oldest collections of stars and metal-poor environments, may notbe sufficient for their creation due to the need for significant dust coagula-tion (Weidenschilling, 1980). However, Sigurdsson et al. (2003) used opticalphotometric data obtained with the Hubble Space Telescope to determine aninner-binary white dwarf age of 480(140)×106 years through isochrone fittingof their observed color-magnitude diagram of the system. The white-dwarfage is considerably smaller than the cluster age of 12.7(4)×109 years (Hansenet al., 2002), and supports the evolutionary scenario that the triple systemformed through a recent exchange interaction between an old NS-WD sys-116tem and a main-sequence/planet system in the cluster core, where the orig-inal WD companion was ejected and the newly formed neutron-star/main-sequence/planet triple was jettisoned out of the core. Subsequent accre-tion the inner neutron-star/main-sequence binary would eventually producea low-mass (∼0.3 M) white dwarf, which is consistent with the measurementof (mc)i = 0.34(4) M made by Sigurdsson et al. (2003).The presence of the planet in M4 therefore suggest that coagulation canindeed take place on timescales shorter than the average age of globularclusters, and that dynamical disruption from nearby stars cannot entirelyprevent a planet population to form within such dense stellar environments.Recent photometric measurements of globular clusters indicate that multiplestellar populations are actually common and suggest that a non-negligibleamount of heavy elements is available in these astrophysical relics (Nardielloet al., 2015; Piotto et al., 2015).In this chapter, we present current results obtained from an ongoing anal-ysis of the hierarchical PSR B1620−26 triple system. We analyze nearly 30years of TOA data collected with several premier radio facilities, estimate theelements of the outer orbit, and resolve secular variations of several inner-orbital parameters that are likely to be (in part) due to tidal interactionsfrom third-body effects.4.1 Observations & ReductionWe used the TOA data set constructed by Thorsett et al. (1999) as well asTOAs collected for an additional 16+ years after their publication. In thissection, we briefly summarize the observatories and pulsar backends used tocollect the TOAs we analyze in the subsection section of this chapter.The observations undertaken by Thorsett et al. (1999) were performedthree different radio facilities. A portion of the Thorsett et al. data set wascollected using the Very Large Array (VLA) near Socorro, New Mexico, USA,117over the course of one or two days per most of the months between Decem-ber 1990 to October 1998. A Princeton Mark III machine (Stinebring et al.,1992) was used to incoherently de-disperse data collected across a 50-MHzbandpass centered at 1660 MHz into a single, folded profile with a 5-minuteintegration time, which ultimately produced 486 TOAs. Another portion ofthe Thorsett et al. data set was collected with the 76-m Lovell Telescope atJodrell Bank, England, using a 64-MHz bandpass for low observing frequen-cies (400, 600 MHz) and a 32-MHz bandpass for higher frequencies (1400,1600 MHz) using analogue filter banks to incoherently de-disperse the datastream in hardware. For this dissertation, we use additional Jodrell-BankTOAs that were collected after the Thorsett et al. study was published upto late-March 2003; in total, 608 TOAs were collected with the Lovell Tele-scope at Jodrell Bank. The third portion of TOAs was collected using the43-m radio telescope at the NRAO in Green Bank, West Virginia (USA).For each observing epoch, the Spectral Processor – a fast-Fourier-transformspectrometer – was used to digitally sample 512 frequency channels acrossa 40-MHz bandpass centered on two observing frequencies (430 MHz and1400 MHz) using a 5-minute integration time; before February 1991, only256 channels across a 20-MHz bandpass were recorded. Each observationsession at Green Bank consisted of recording pulsar signals using two out offour available frequency receivers; these multi-frequency observations wereperformed contiguously.We extended the Thorsett et al. (1999) data set to incorporate high-precision TOAs with four different pulsar backends using the GBT, and ex-clusively using the 1400-MHz receiver. We used the same Spectral Proces-sor discussed above for initial GBT observations that began in November2001 and ended in August 2004, which yielded 137 TOAs. A portion of ourextended data set was collected with the Berkeley-Caltech Pulsar Machine(BCPM; Backer et al., 1997). The BCPM was used between July 2004 andMay 2009, and employed a digital filter bank to incoherently de-disperse an118incoming signal across 100 MHz in bandwidth with 4-bit sampling and auser-specified number of frequency channels. A total of 307 TOAs were col-lected with the BCPM backend. We used the GASP coherent-dedispersionbackend (first discussed in Section 3.1) between August 2005 and October2011 to collect a total of 717 TOAs. We began using the wide-band GUPPIcoherent-dedispersion backend since February 2011 and continue to use itfor ongoing observations, having so far collected 440 TOAs across the full800-MHz bandwidth. For all four GBT backends, we generally integratedsuccessive pulsars over a 3-minute timescale and across all available band-width to form high-S/N, averaged TOAs.It is important to note that the early and late portions of the GASP dataoverlap with segments of the BCPM and GUPPI data, respectively. Sincethese data are collected with the same telescope, and with the same receiver,the overlapping data essentially lead to a redundancy in a small fraction of theTOA measurements and a slight overweighting of pulsar-timing analyses tothese data. However, the overlap in data is needed for accurately determininginstrumental offsets between the various backends. We therefore currentlyretain all available TOAs for the analyses presented below.For the first time, we also incorporated TOAs collected with the 100-m Effelsberg Telescope in Bad Mu¨nstereifel, North Rhine-Westphalia, Ger-many. The Effelsberg data were processed using the coherent-dedispersionEffelsberg-Berkeley Pulsar Processor (EBPP; Backer et al., 1997). A singleTOA was obtained per observing epoch using the 1410-MHz receiver, afteraveraging 30-minutes of successive radio pulsars across the entire ∼ 64-MHzbandwidth. In total, 47 TOAs were collected with the Effelsberg Telescopeusing the EBPP backend between July 1997 and March 2009.1194.2 Methods for Timing AnalysisA grand total of 3,515 TOAs collected for PSR B1620−26 are analyzed in thisstudy. We first determined instrumental offsets between data collected withdifferent backends and/or telescopes by using TEMPO and a fixed timingsolution to fit for arbitrary time offsets between short overlapping segmentsof data. These offsets were then held fixed when performing the varioustiming analyses discussed in Section 4.3 below.We used the TEMPO pulsar-timing package for modeling TOAs collectedfor PSR B1620−26. In all analyses presented in Section 4.3 below, we directlymodeled the following physical effects described in Sections 1.3 and 1.4:• the J2000 positions and proper motion terms;• the timing parallax, fixed at a value of $ = 0.59” determined fromoptical color and magnitude measurements made by Peterson et al.(1995);• the five Keplerian elements that describe the Ro¨mer timing delay dueto the inner orbit, (∆R)i, along with first-order derivatives in one ormore elements;• the DM and a first-order rate of change in DM, which we hold fixedin our timing model using values first determined by Thorsett et al.(1999);• and the spin frequency, along with one or more spin-frequency timederivatives.It is important to note that, due to the complex nature of the triple sys-tem and its timing model, we do not yet perform a full analysis of TOAuncertainties and their slight adjustment to produce a best-fit χ2red ≈ 1.0.While this does not affect the robustness of the timing model in fitting our120data set, it will likely affect the interpretation of measurements that pos-sess marginal statistical significance. As discussed in Section 4.3, the lackof TOA-uncertainty analysis primarily affects the measurements of severalinner-orbital secular variations. However, we show in Section 4.4 that theorders of magnitude of the observed inner-orbital secular variations are con-sistent with those expected by theory and likely indicate real changes inseveral elements.In principle, the effects of the hierarchical outer orbit on pulsar TOAs canbe modeled with an additional Ro¨mer timing delay and its outer-orbital Ke-plerian elements.2 In this case, which we refer to as the “two-orbit” solution,the total Ro¨mer timing delay of the pulsar ephemeris is ∆R = (∆R)i + (∆R)oand two separate sets of Keplerian elements can be obtained using the ap-propriate eccentric or nearly-circular forms presented in Section 1.4. Thesimultaneous fit of both orbits, along with the other effects mentioned in theabove list, allows for an unambiguous determination of the spin frequencyand its first derivative due to spin-down and other kinematic effects discussedin Section 1.5.However, if the time spanned by a given TOA data set is significantlyless than the given binary’s orbital period, then a fit for ∆R and its orbitalparameters will not be well constrained and likely fail. These issues were firstnoted by Thorsett et al. (1999) when applying a two-orbit solution to theirdata set for PSR B1620−26, where the outer-orbital elements could not beuniquely obtained with sufficient numerical stability in the timing-model fit.Thorsett et al. instead fitted for (∆R)o for different fixed values of eo andfound a large variation in outer-orbital elements as a function of eo.As pointed out by Joshi & Rasio (1997), significant spin-frequency deriva-tives arise from the periodic Doppler shift due to unmodeled orbital motion.For long-period orbits, one can use the observed spin-frequency derivatives2Due to the wide, hierarchical orbital period of the outer companion, third-body pertur-bations of the inner-orbital elements can approximately be modeled as Taylor expansionsof those parameters over time.121to derive the orbital elements. We used the framework first developed byJoshi & Rasio (1997) that we derived, discussed and extended in AppendixB of this dissertation, in order to relate the spin-frequency derivatives to theouter-orbital parameters measured relative to the inner binary’s center ofmass. The use of the Joshi & Rasio (1997) method, as well as the assump-tion that mp + (mc)i + (mc)o ≈ mp + mWD ≈ 1.65 M, allows one to derivean estimate of the “full” semi-major axis of the outer orbit (ao), as well asthe true anomaly (u) and its time derivative (u˙). It is important to note thatthe Joshi & Rasio (1997) method yields direct estimates of the outer-orbitalorientation angles measured in the plane of the orbit (i.e. uo and ωo), thoughdoes not yield any information of orbital nodes; one can compute the corre-sponding orientation angles of the inner binary by noting that ωi = ωo + pi,and ui = uo = u.This indirect method for fitting the outer orbit requires at least five spin-frequency derivatives in order to uniquely solve for the five outer-orbital Ke-plerian parameters. In the analyses presented below, we measure five or moretime-derivatives with using all or subsets of our long TOA data set. However,a complication of Joshi & Rasio (1997) method arises when interpreting thesign and value of ν˙s, since it will contain a time-varying component due to theorbital Doppler shift along with the nominal components that are approxi-mately constant across typical data spans. We elaborate on this complicationin more detail in Appendix B.3 and also discuss potential components of ν˙sand ν¨s due to globular-cluster dynamics in Section 4.4.4.3 Timing Update for PSR B1620−26During the construction of an updated timing model for PSR B1620−26,it became clear that a global, two-orbit solution that accurately models allavailable TOAs – where ∆R = (∆R)i + (∆R)o – could not be obtained at thistime. An example of the complex behavior is shown in Figure 4.1, where TOA122Figure 4.1: TOA residuals for all data collected from PSR B1620-26, usinga two-orbit solution derived from data collected between 1987 and 2010 (seeSection 4.3.1 for a discussion of this model). TOAs are collected at 1400 MHz(unless otherwise noted) and are colored in the following manner: 140-ft radiotelescope at Green Bank (red circles - 430 MHz, blue circles - 1400 MHz);Lovell Telescope at Jodrell Bank (green circles); VLA (magenta circles); GBTusing the Spectral Processor (cyan circles), BCPM (yellow circles), GASP(black circles), and GUPPI (red crosses); and Effelsberg Telescope (bluecrosses). The two-orbit solution dramatically fails to model TOAs collectedafter 2010.residuals were computed using a two-orbit model with (Pb)o ≈ 38 years thatwe derive from all data taken between late 1987 and mid-2009. (We discussthis model in more detail in Section 4.3.1 below.) It is clear from Figure 4.1that the two-orbit solution, while adequately predicting TOAs between 1987and 2009, drastically fails to model TOA data taken after a ∼ 2-year gap in2009-2010.We nevertheless found that all data taken before the 2-year gap couldbe reasonably fit with a two-orbit solution, with small non-random structure123that may likely indicate significant three-body interactions. Moreover, usingthe Joshi & Rasio (1997) method, we successfully modeled the outer orbitwith a large number of spin-frequency derivatives when examining smallerportions of the whole data set, and several measures of the outer-orbitalparameters derived from these spin-frequency measurements were consistentbetween these subsets.In this section, we present separate timing analyses of three TOA subsets:one subset consists of TOAs collected before the late-2009 gap in GASP data;the second subset consists of GASP TOAs collected after the 2009 gap, aswell as all GUPPI TOAs; and the third analysis is performed on the entireTOA data set for PSR B1620−26. We demonstrate that both “pre-gap”and “post-gap” subsets yield a large number of spin-frequency derivativesthat are consistent with an unmodeled outer orbit with a period of ∼ 40years when using the Joshi & Rasio (1997) method, described in AppendixB, for estimating orbital elements. For the data subset collected prior tothe 2009 gap, we are also able to successfully fit a two-orbit model with anouter-orbital period of ≈ 38 years, which is consistent with the value derivedfrom the spin-frequency-derivative model of the outer orbit for the samedata subset. We also present our current analysis of the global data set usinga single timing solution with a large number of spin-frequency derivativesas well as significant first-order changes in the inner-orbital elements, andpresent results obtained from the Joshi & Rasio (1997) method.4.3.1 Analysis of “Pre-Gap” DataWe first considered the first portion of the whole data set, where this subsetconsists of TOAs collected between November 1987 and May 2009. This sub-set includes all data first published and analyzed by Thorsett et al. (1999),as well as a large portion of data collected with the Effelsberg, Green Bankand Jodrell Bank telescopes discussed in Section 4.1 above. While all avail-able BCPM and Effelsberg data are included in this subset, only half of the124GASP data (prior to the 2009 gap) are included here. In total, this subsetconsists of 2,876 TOAs that span nearly 22 years of observation.We first developed a timing solution that models the outer orbit using 8spin-frequency derivatives, while directly fitting for the inner orbit and vari-ations in x across the data set. The best-fit residuals for this timing modelare shown in the top panel of Figure 4.2 and the best-fit parameters are pre-sented in Table 4.1. The timing model we derive from this pre-gap subsetis in good agreement with the model first derived by Thorsett et al. (1999),with the largest differences in parameter values occurring in the higher-orderspin-frequency derivatives. This slight discrepancy is likely due to the largedegree of covariance between the higher-order derivative terms as model pa-rameters. However, the higher-order spin-frequency derivatives we measurefrom the 22-yr, pre-gap subset possess the same order of magnitude and signas those published by Thorsett et al.When we use the first five spin-frequency derivatives and the procedureoutlined in Appendix B, we estimate that the five outer-orbital Keplerianelements areao ≈ 14 AU(Pb)o ≈ 41 yearseo ≈ 0.178ωo ≈ 175 degrees(T0)o ≈ MJD 44420, or July 1980(4.1)The full list of outer-orbital parameters that can be derived from the Joshi &Rasio (1997) method are presented in Table 4.2. Using the above estimatesand Equation B.23, we computed the outer-orbital companion mass to be(mc sin i)o ∼ 1×10−3 M. The outer-orbital period and planetary-companion125Parameter Pre-Gap Post-GapStart date (MJD) 47104 55008Finish date (MJD) 54953 57231Reference Epoch (MJD) 51029 56290AstrometryRight Ascension, αJ2000 16h23m38.21610(6)s 16h23m38.20230(2)sDeclination, δJ2000 −26◦31′53.880(4)” −26◦31′54.1841(15)”µα (arcsec yr−1) -13.03(9) -12.2(2)µδ (arcsec yr−1) -21.7(5) −17(1)$ (arcsec) 0.59 0.59Spinνs (s−1) 90.287331294248(2) 90.2873323075030(18)ν˙s (10−15 s−2) -1.74145(6) 6.02434(18)ν¨s (10−23 s−3) 1.76871(10) 1.5478(7)ν(3)s (10−33 s−4) -7.40(3) 4.3(8)ν(4)s (10−41 s−5) 1.217(5) 2.91(17)ν(5)s (10−49 s−6) 7.04(17) -20(20)ν(6)s (10−56 s−7) -2.42(3) . . .ν(7)s (10−65 s−8) -5.7(5) . . .ν(8)s (10−72 s−9) 1.83(12) . . .DispersionDM (pc cm−3) 62.862983 62.862983˙DM (pc cm−3 yr−1) -0.0006997 -0.0006997Inner Binaryxi (lt-sec) 64.8093323(9) 64.8091202(6)x˙i -0.580(5) −0.52(2)x¨i (10−22 s−1) 1.6(3) 0(3)ei 0.025315412(9) 0.025315442(8)(T0)i (MJD) 51025.575809(10) 55428.760606(13)(nb)i (10−8 s−1) 6.0457079725(8) 6.0457080391(15)ωi (deg) 117.128143(19) 117.12786(2)Fit Statisticsχ2red 1.268 1.135Number of TOAs 2,876 639Weighted RMS residual (µs) 9.982 4.987Table 4.1: Best-fit parameters from TEMPO analysis of TOAs collected forPSR B1620−26 when separately analyzing two subsets collected before andafter a gap in data around the year 2009. Values in parentheses denote the 1σuncertainty in the preceding digit(s) as determined from TEMPO. Quantitieswith no reported uncertainties were held fixed at the listed values; see Section4.1 for a discussion.126Figure 4.2: TOA residuals of PSR B1620-26 using data collected up to mid2009. The top panel consists of residuals computed from a timing modelwhere the inner orbit is directly fitted for, and the outer orbit is approx-imately fitted for with 8 time-derivatives in νs. The bottom panel showsresiduals computed from a model where both Keplerian orbits are directlyfitted for, along with one time-derivative in νs. The residual colors denotethe same data sets presented in Figure 4.1.mass we derive from the spin-frequency derivatives are consistent with theinitial estimates made by Thorsett et al. (1999).The results shown in Equation 4.1 above and Table 4.2 are significantlydifferent from those that were derived by Ford et al. (2000a), who also usedthe method developed by Joshi & Rasio (1997) to derive the outer-orbitalelements from spin-frequency derivatives. Ford et al. used the timing solutionconstructed Thorsett et al. (1999) and derived a larger eccentricity (≈ 0.45)and orbital period (∼ 300 years) than we derive from the pre-gap subset.However, the Thorsett et al. solution only measured four significant spin-frequency derivatives, along with a fifth derivative that was consistent with127Parameter FJRZ Pre-Gap Post-Gap All DataJoshi & Rasio (1997) Parameterseo 0.448 0.178 0.306 0.131ωo (deg) 135 175 227 178u (deg) 18.9 165 185.3 175u˙ (rad yr−1) 0.055 0.111 0.094 0.210Derived Orbital Elements(m2 sin i)o (10−3 M) 6.48 0.80 1.27 0.18ao (AU) 56.5 14.0 13.2 9.62(Pb)o (years) 330 40.9 37.8 23.2(T0)o (MJD) . . . 44420 49088 48062Table 4.2: Unique solutions of the Joshi & Rasio (1997) method and severalderived elements for the outer orbit. In keeping with notation used in Chapter1, u is the true anomaly and u˙ is its time derivative. The parameters directlymeasured from the Joshi & Rasio (1997) method are then used to derivethe outer-orbital elements. The label “FRJZ” refers to the study conductedby Ford et al. (2000a) that used the Thorsett et al. (1999) solution and itsconstraint on ν(5)s .zero at the 68.3% confidence level. We therefore consider this difference tonot be problematic, as our estimate of ν(5)s is statistically significant and morerobust as a timing-model parameter.In conjunction with the TEMPO results discussed above, we used aniterative Markov Chain Monte Carlo (MCMC) procedure in order to explorethe phase space spanned by the fit parameters and sample the region of bestfit to obtain their posterior probability distributions for all model parameters.The MCMC method consists of a random walk in the parameter space thatis generally biased towards regions of more probable parameter values where,after each step in the walk, a likelihood probability is evaluated and comparedto a random number drawn from a uniform distribution between 0 and 1; theproposed step in the phase space is accepted if the likelihood is less than thanthe randomly drawn number, and is rejected otherwise. This process leads tothe construction of a Markov chain, where the acceptance or rejection of each128element in the chain only depends on the immediately preceding element.3While the algorithm generally forces the walk to probe the best-fit region ofthe parameter phase space, the probabilistic nature of the MCMC methodallows for the random walk to reach regions of the parameter space thatdo not fit the data well. This method of model determination is thereforeadvantageous for exploring complex local features in the χ2 phase space, aswell as for determining more robust confidence intervals in the case of nonzeroand/or nonlinear correlation.We implemented a MCMC method using a Metropolis-Hastings algorithmthat uses TEMPO to compute the χ2 for a set of fixed parameters at each stepin the Markov chain. The ratio of likelihood probability between adjacentelements in the chain is evaluated by computing the quantity exp(−0.5∆χ2),where ∆χ2 is the change in χ2 between the current and proposed coordinatein the parameter phase space. We used the covariance matrix (Σ) of thebest-fit timing solution discussed above, computed by TEMPO, in order toaccount for covariance between model parameters when sampling the joint-prior normal probability distribution of model parameters,f(x) =1√(2pi)kdetΣexp(− 12(x− µ)TΣ(x− µ))(4.2)where x is the vector of model parameters, µ is the vector of the mean pa-rameter values, “T” refers to the transpose of the vector (x − µ), and k isthe number of parameters (i.e. the dimension of the phase space). We per-formed 100,000 iterations of the MCMC method, and the step-acceptancerate for our simulation was ∼ 25%. The median values and 68.3% credibleintervals for all posterior distributions were consistent with the best-fit valuesand 1−σ uncertainties determined by TEMPO. Figure 4.3 displays the pos-terior distributions and two-dimensional phase spaces of the spin-frequencyderivatives used to model the pre-gap TOA data set. The MCMC method3An “element” of a Markov chain consists of a set of all model parameters.129Figure 4.3: MCMC results for spin-frequency derivatives when analyzing datacollected prior to the 2009 gap in GASP data. The scatter plots illustratecorrelation between posterior distributions, while the histograms are poste-rior distributions for the parameter denoted at the bottom of their respectivecolumns.confirms the high degree of linear correlation between the higher-order fre-quency derivatives used as timing-model parameters.One of the advantages of using an MCMC method is that, in principle,the posterior distributions obtained for the spin-frequency derivatives can beconverted to distributions of the outer-orbital elements through the use ofthe Joshi & Rasio (1997) method. We computed the posterior distributionsof outer-orbital elements by applying the Joshi & Rasio (1997) method usingthe first five spin-frequency derivatives within each element of the Markovchain. The results of this translation are shown in Figure 4.4. All 100,000sets of spin-frequency derivatives were successfully converted to estimates ofthe outer-orbital elements. The distributions are consistent with the best-fit130Figure 4.4: Posterior distributions of the outer-orbital Keplerian elements,computed from the MCMC posterior distributions of the first five spin-frequency derivatives measured from the pre-gap data set, using the Joshi &Rasio (1997) method discussed in Appendix B.estimates shown in Equation 4.1 above.We also derived a timing solution that explicitly models for the outerorbit, based on an extension of an earlier timing solution by Thorsett et al.(1999) that was developed over the past decade as more data was beingcollected. We display the best-fit residuals in the bottom panel of Figure4.2 and present the best-fit parameters in Table 4.3. The astrometric andinner-orbital parameters agree with those that were derived from the timingsolution that fitted the outer orbit with multiple spin-frequency derivatives.In the two-orbit model, the estimate of ν˙s is significantly different from theestimate obtained from the spin-frequency solution due to the fact that theouter-orbital Ro¨mer delay essentially accounts for the component of ν˙ dueto orbital motion. The outer-orbital Keplerian parameters are generally con-131Parameter Value (Uncertainty)Start date (MJD) 47104Finish date (MJD) 57231Reference Epoch (MJD) 52167AstrometryRight Ascension, αJ2000 16h23m38.21595(6)sDeclination, δJ2000 −26◦31′53.880(4)”µα (arcsec yr−1) -12.96(9)µδ (arcsec yr−1) -21.7(5)$ (arcsec) 0.59Spinνs (s−1) 90.2873322732(10)ν˙s (10−15 s−2) -1.4415(14)DispersionDM (pc cm−3) 62.862983˙DM (pc cm−3 yr−1) -0.0006997Inner Binaryxi (lt-sec) 64.8093261(8)x˙i (10−12) -0.600(5)x¨i (10−22 s−1) 4.7(3)ei 0.025315326(8)(T0)i (MJD) 51025.57574(10)(nb)i (10−8 s−1) 6.0457080216(8)ωi (deg) 117.128020(19)Outer Binaryxo (lt-sec) 2.422(4)eo 0.14836(2)(T0)o (MJD) 44441(4)(Pb)o (years) 38.215(19)ωo (deg) 183.737(3)Fit Statisticsχ2red 1.710Number of TOAs 2,829Weighted RMS residual (µs) 11.479Table 4.3: Best-fit parameters from TEMPO analysis of TOAs collected priorto the 2009 gap for PSR B1620−26, using a two-Keplerian-orbit model.132sistent between the two-orbit and frequency-derivative solutions. However,several outer-orbital timing parameters possess considerably larger uncer-tainties than those which are normally attainable with pulsar timing. Thislack of precision reflects the degeneracy obtained when directly fitting for theouter-orbital Ro¨mer timing delay using data that does not yet span a full(outer) orbit.4.3.2 Analysis of “Post-Gap” GASP and GUPPI DataWe also separately analyzed all data collected after the gap in GASP TOAsthat occurred around the year 2009. This data subset consists of post-gapGASP data and all TOAs collected with the GUPPI processor. In total,this subset consists of 639 TOAs that collectively spans over five years ofobservation with the Green Bank Telescope.From the post-gap GASP and GUPPI TOAs, we derived a timing solu-tion that used five spin-frequency derivatives to model the outer orbit; theTOA residuals for this timing solution are shown in Figure 4.5. The best-fitparameters of the post-gap timing solution are shown in Table 4.1, alongsidethe pre-gap solution discussed in Section 4.3.1, and indicate that only four ofthe five spin-frequency derivatives are significant. The frequency-derivativevalues are different from those quoted in the pre-gap solution because thereference epoch is different between the pre-gap and post-gap subsets. More-over, the best-fit estimate of ν˙s is positive, which indicates that the compo-nent of ν˙s due to outer-orbital motion is positive and larger than the sumof the other components due to intrinsic spin-down and acceleration in theGalactic potential.As discussed in Appendix B, we used the first four spin-frequency deriva-tives to find a family of solutions to the Joshi & Rasio (1997) method fordifferent values of eo. The family of solutions for the directly-measurableparameters of the Joshi & Rasio (1997) method are shown in Figure 4.6.While only four significant spin-frequency derivatives are measured from the133post-gap TOA subset, we can use the (insignificant) best-fit estimate of ν(5)sto obtain a unique solution of the outer orbit using the Joshi & Rasio (1997)method. We find that the five spin-frequency derivatives yield the followingouter-orbital elements:ao ≈ 13 AU(Pb)o ≈ 37 yearseo ≈ 0.306ωo ≈ 227 degrees(T0)o ≈ MJD 49088, or April 1993.(4.3)which are also presented in Table 4.2 for comparison with other portionsof the TOA data set for PSR B1620−26. The post-gap estimates of theouter-orbital Keplerian elements generally differ from those determined fromthe pre-gap data analyzed in Section 4.3.1, though the derived estimate ofao and (Pb)o are similar between the pre-gap and post-gap subsets. Thediscrepancies in eo, ωo (T0)o between data subsets are likely due to the factthat ν(5)s is not measured with statistical significance; its constraint on theouter-orbit parameters is considerably weak and will change as its significanceimproves over time with the collection of additional GUPPI TOAs.We used the same MCMC algorithm described in Section 4.3.1 aboveto obtain posterior distributions of all fit parameters; we then convertedthe spin-derivative distributions into ones for the outer-orbital Keplerian ele-ments using the Joshi & Rasio (1997) method, which are also shown in Figure4.7. In this case, only ∼ 72% of all Markov-chain elements produced solu-tions of the Joshi & Rasio (1997) method. The small fraction of non-solutionsto the Joshi & Rasio (1997) method is likely related to the fact that ν(5)s isconsistent with zero at the 68.3% confidence level, and so a small fraction of134Figure 4.5: TOA residuals of PSR B1620-26 using data collected after the2009 gap in GASP data, using a frequency-derivative model to account forthe outer orbit. Points in red are GASP data, while points in green areGUPPI TOA residuals.posterior samples for ν(5)s will have orders of magnitudes and signs that areinconsistent with the component of ν(5)s due to orbital motion. The resultantdistributions of the outer-orbital Keplerian elements are wider than thosederived from the pre-gap analysis in Section 4.3.1 above, which also indicatea weak constraint from the observed ν(5)s . Moreover, the distributions for(T0)o do not overlap, and likely indicates an inability of the Joshi & Rasio(1997) to accurately determine all outer-orbital elements based on only ∼4years of TOA data.135Figure 4.6: Numerically stable solutions of the Joshi & Rasio (1997) methodusing time-derivatives in νs measured for PSR B1620−26 for the post-gapdata set. The red-solid lines are solutions found for a fixed value of eo,while the vertical blue-dashed lines are unique solutions obtained from a fullNewton-Raphson method that allows eo to be determined.4.3.3 Changes of Inner-Orbital Parameters over TimeGiven the greatly-improved TOA precision and the uncertainties in parame-ter measurements4, the inner-orbital Keplerian elements presented in Table4.1 are significantly different when measured relative to the pre-gap and post-gap epochs of periastron, where the difference is ∆(T0)i = 4,403 days ≈ 12years in time. For example, the minimum difference in the ei measurementsbetween the pre-gap and post-gap estimates of (T0)i is nearly 4σ when usingthe larger of the two uncertainties in ei. The ∆ei measured between the twodata subsets implies a rate of change e˙i ≈ ∆ei/∆(T0)i ≈ 10−9 yr−1. Similarly,4The reduced χ2 for the pre-gap and post-gap fits are slightly greater than 1.0, indi-cating that uncertainties for the TOAs and best-fit model parameters are underestimated.136the best-fit values of ωi change between the two data subsets such that theminimum difference is 12σ; this difference implies a significant rate of changeω˙i ≈ ∆(ω)i/∆(T0)i ≈ −2× 10−5 deg yr−1. The largest change occurs in themeasured (nb)i, where the minimum difference in inner-orbital frequency isnearly 46σ and the inferred (n˙b)i ≈ ∆(nb)i/∆(T0)i ≈ 5× 10−17 s−1 yr−1.We consider these apparent changes in the inner-orbital elements to re-flect real variations due to one or more physical mechanisms affecting theinner binary system. As we discuss in Section 4.4.3, the variation in (Pb)o isdominated by the component due to mean-field acceleration in the globular-cluster potential; this measurement allows for an evaluation of the componentof ν˙s do to the same mechanism, which we show to be a significant bias inthe observed first-order change in spin frequency. In Section 4.4.5, we showthat the estimates of e˙i and ω˙i computed above possess orders of magni-tude consistent with those expected from prolonged interaction with a thirdcompanion in a wide orbit about the inner binary.4.3.4 Global Analysis of All TOA DataThe current, complete TOA data set for PSR B1620−26 triple system spans27+ years and is well fit by a timing solution that models the outer orbit using15 spin-frequency derivatives. The best-fit timing solution, which directly fitsfor the inner orbit and first-order variations in all inner-orbital elements, issummarized by the parameters listed in Table 4.4 and shown in Figure 4.8.As with the separate pre-gap and post-gap analyses presented above, wecurrently do not fit for variations in DM over the time span since nearly alldata collected was observed using a single receiver centered at 1400 MHz.Instead, we used the DM parameters determined by Thorsett et al. (1999)and held them fixed in our current solution. Since our primary goal is torobustly fit for (∆R)o, we do not consider unaccounted variations in DM tobe a significant source of systematic error that prevents or heavily impacts adirect measurement of the time delay due to the outer orbit.137Parameter Value (Uncertainty)Start date (MJD) 47104Finish date (MJD) 57231Reference Epoch (MJD) 52167AstrometryRight Ascension, αJ2000 16h23m38.21276(3)sDeclination, δJ2000 −26◦31′53.964(2)”µα (arcsec yr−1) -12.39(5)µδ (arcsec yr−1) -19.2(3)$ (arcsec) 0.59Spinνs (s−1) 90.287331207875(4)ν˙s (10−17 s−2) 1.435(6)ν¨s (10−23 s−3) 1.7505(3)ν(3)s (10−33 s−4) 1.43(7)ν(4)s (10−40 s−5) 1.46(3)ν(5)s (10−48 s−6) 1.58(8)ν(6)s (10−55 s−7) -1.03(4)ν(7)s (10−63 s−8) -3.75(9)ν(8)s (10−71 s−9) 7.4(4)ν(9)s (10−78 s−10) 3.99(10)ν(10)s (10−86 s−11) -4.6(3)ν(11)s (10−93 s−12) -3.15(9)ν(12)s (10−101 s−13) 2.07(18)ν(13)s (10−108 s−14) 1.76(6)ν(14)s (10−117 s−15) -5.0(6)ν(15)s (10−124 s−16) -5.3(2)DispersionDM (pc cm−3) 62.862983˙DM (pc cm−3 yr−1) -0.0006997Inner Binaryxi (lt-sec) 64.8092852(5)x˙i -0.564(3)x¨i (10−22 s−1) 1.14(14)ei 0.025315465(13)e˙i (10−8 yr−1) -1.01(15)(T0)i (MJD) 51982.790123(18)(nb)i (10−8 s−1) 6.04570849(4)(n˙b)i (10−24 s−2) 1.60(4)ωi (deg) 117.12850(3)ω˙i (10−5 deg yr−1) -6.0(4)Fit Statisticsχ2red 1.801Number of TOAs 3,515Weighted RMS residual (µs) 10.287Table 4.4: Best-fit parameters from TEMPO analysis of TOAs collected forPSR B1620−26 when using all data and modeling the outer orbit using spin-frequency derivatives. Values in parentheses denote the 1σ uncertainty in thepreceding digit(s) as determined from TEMPO. Quantities with no reporteduncertainties were held fixed at the listed values, and taken from Thorsettet al. (1999).138Figure 4.7: Posterior distributions of the outer-orbital Keplerian elements,computed from the MCMC posterior distributions of the first five spin-frequency derivatives measured from the post-gap data set, using the Joshi& Rasio (1997) method discussed in Appendix B.The observed behavior in the pulsar’s spin period across the 27-year dataspan can be directly computed using the measured time-derivatives in spinfrequency and Equation 1.12, and is shown in Figure 4.9. The inclusion ofhigher-order terms in the Taylor expansion, which are due to the outer-orbitalmotion of the planetary companion about the inner-binary’s center of mass,produces a wave-like variation in Ps over time. It is clear from Figure 4.9that the outer orbit possesses a large orbital period, as the variation has notbeen fully covered and observed to repeat. This feature is consistent withinitial estimate made by Thorsett et al. (1999) that (Pb)o is much longerthan their initial data set, and that the outer-orbital period is ∼ 100 years.Moreover, Figure 4.9 indicates that the current value of Ps is approachingthe value observed in 1987, when PSR B1620−26 was discovered by (Lyne,139Figure 4.8: TOA residuals of PSR B1620-26 using the complete data set. Thetiming solution used to compute these TOAs directly fits the inner orbit andusing 15 spin-frequency derivatives to model the outer orbit. The residualcolors denote the same data sets presented in Figure 4.1.1988). Assuming that the observed variations are purely due a third body ina bound orbit, this feature then most likely indicates that the outer orbit willsoon reach its point of inflection within the next few years. The observationof another inflection of Ps may make a numerically-stable fit for eo, andtherefore (∆R)o, more robust and allow for a unique determination of bothsets of orbital elements from direct modeling of the orbit.We attempted to implement the Joshi & Rasio (1997) method using thefirst five spin-frequency derivatives measured in the global timing model, butno convergent solution of the method could be obtained. The fitted value of(ν˙s)obs = −1.722(4)× 10−17 s−2 is two orders of magnitude smaller than thevalue published by Thorsett et al. (1999), which corresponds to the time inFigure 4.9 where the orbit-induced curve is turning over and the first deriva-140tive is approximately zero. Therefore, the assumption that (ν˙s)obs ≈ (ν˙s)orbdoes not hold at the quoted reference epoch of the timing solution. More-over, as discussed in Appendix B.3 below, the various mechanisms that cancontribute to an observable first-derivative in νs complicates a robust deter-mination of outer-orbital elements. However, the large number of requiredspin-frequency derivatives are likely dominated by the outer-orbital motion,and so an analysis based of higher-order derivatives could lead to more “un-biased” estimates of the Keplerian parameters.As described in Appendix B.3, we extended the Joshi & Rasio (1997)method by re-writing the expected derivatives due to orbital motion in termsof ν¨s, instead of ν˙s, and thus avoiding the contributions from pulsar spin-downand various terms from kinematic, non-orbital acceleration. This extensionof the Joshi & Rasio (1997) method requires a measurement of ν(6)s in order touniquely solve for the approximate orbital elements, which we readily madeusing our full, 27-yr data set. We applied the same Newton-Raphson proce-dure for determining the “best-fit” values of the outer-orbital elements, andderived the following orbital elements based on the higher-order derivativesup to ν(6)s :ao ≈ 10 AU(Pb)o ≈ 23 yearseo ≈ 0.131ωo ≈ 178 degree(T0)o ≈ MJD 48062, or June 1990.(4.4)The outer-orbital eccentricity and semi-major axis are relatively consistentwith the estimates made when analyzing the pre-gap TOA subset discussedin Section 4.3.1 above. However, the 23-year outer-orbital period inferred141Figure 4.9: Ps versus time for the complete TOA data set for PSR B1620−26,derived from the timing model shown in Figure 4.8.from the higher-order frequency derivatives is more than a decade smallerthan the 40-year period inferred from both the pre-gap and post-gap anal-yses described above. More strikingly, the outer-orbital period determinedfrom the entire data set is slightly smaller than the time span of the set itself,meaning that a robust fit for (∆R)o can theoretically be made. However, asdiscussed at the beginning of Section 4.3, a two-orbit solution based on theentire TOA data set is not currently attainable. Furthermore, our globalestimate of (Pb)o from the extended Joshi & Rasio (1997) method is con-siderably smaller than the period suggested from the long-term behavior ofthe spin period illustrated in Figure 4.9. We discuss a variety of plausiblemechanisms that could affect our estimation of the outer-orbital elementsusing the Joshi & Rasio (1997) method in Section 4.4 below.For the first time, we measure first-order variations in all inner-orbitalelements, as well as a significant second derivative in xi, when deriving ourtiming model on the full TOA data set for PSR B1620−26. The measurement142of x˙i was first made by Arzoumanian et al. (1996), and our estimate of x˙i inTable 4.4 is consistent with the last measurement made by Thorsett et al.(1999) when computing the value at their timing-solution reference epoch.The signs and orders of magnitude of the best-fit variations are consistentwith the values derived from computing changes in the elements over timebetween the pre-gap and post-gap estimates as discussed at the end of Section4.3.2. As we discuss in Section 4.4.5 below, the variations in ei and ωi arelikely due to third-body perturbations from the planetary companion.4.4 DiscussionWe demonstrated in Section 4.3 that PSR B1620−26 exhibits complex timingbehavior due to a long-period outer orbit. While the analysis of data subsetsyields consistent estimates of the outer-orbital elements, and a two-orbitmodel can be applied to the majority of our complete data set, a globalanalysis of the entire TOA set does not yet yield a two-orbit solution. In thissection, we discuss additional sources of complication in our analyses andperform calculations to determine the likelihood of additional biases fromglobular-cluster dynamics, as well as to check the veracity of the observedinner-orbital secular variations.4.4.1 Is PSR B1620−26 a Triple System?In principle, our use of higher-order derivatives other than ν˙s when imple-menting the Joshi & Rasio (1997) method generally allows for an “unbiased”estimate of the outer-orbital parameters, since it avoids any consideration ofthe various non-binary processes that contribute to the observed first deriva-tive in spin frequency. However, when using the extended Joshi & Rasio(1997) method on the complete TOA data, the derived outer-orbital ele-ments illustrate an eccentric orbit with a period of ∼ 23 years. While theinferred eccentricity is generally in agreement with estimates made by analyz-143ing subsets of all available TOAs, the derived outer-orbital period is shorterthan the time span of the whole data set and implies that a direct fit of (∆R)ocan be obtained. However, this is inconsistent with the observed behaviorof the pulsar’s spin period shown in Figure 4.9, where the measurement ofhigher-order frequency derivatives suggest an outer-orbital period of ∼ 100years. Moreover, we are currently not able to obtain a two-orbit solution forthe entire TOA data set collected for PSR B1620−26.At this time, it is not clear why the extended Joshi & Rasio (1997) methodgives conflicting answers when applied to the entire TOA data set for PSRB1620−26. It is also unclear why the two-orbit solution discussed in Section4.3.1, where ∆R = (∆R)i + (∆R)o, currently fails to reasonably model datacollected after the late-2009 TOA gap.Given its globular-cluster association and the complications discussedabove, we are confronted with the possibility that PSR B1620−26 is notactually a bound hierarchical triple system. The observed variations in TOAresiduals and inner-orbital elements nonetheless indicate that a sustainedgravitational interaction is occurring between the pulsar-WD binary and atleast one other massive object. One possible scenario involves a hyperbolicencounter of the pulsar-WD binary with a nearby, low-mass star, likely an-other globular-cluster white dwarf. As with bound orbits, the hyperbolic fly-by will induce a number of time-derivatives in spin frequency in the same waythat we observe a large number of time-derivatives when extending the TOAdata set for PSR B1620−26. Moreover, the spin-frequency analysis of thepost-gap data subset for PSR B1620−26 (Section 4.3.2) yields outer-orbitalelements that are consistent with the results obtained from the pre-gap sub-set, which could be caused by both bound and hyperbolic orbits.While such hyperbolic encounters are possible in globular clusters, thefact that PSR B1620−26 is outside of the central regions of the cluster core(where the stellar-number density is comparatively smaller than at the cen-ter) makes this scenario unlikely. However, high-resolution, multi-wavelength144optical imaging and spectroscopy may be able to constrain the likelihood ofan interacting white dwarf nearby the PSR B1620−26 system.For the remainder of the discussion presented below, we assume that thetriple-system status of PSR B1620−26 remains valid.4.4.2 Complications from Moons and Pulsar GlitchesAssuming that the PSR B1620−26 system is indeed a hierarchical triplesystem, one possibility that is not considered in the above analyses is thatthe outer planetary object possess a moon. Lewis et al. (2008) consideredthe idealized scenario where a moon is in a circular orbit about a planetthat is in a wider orbit with PSR B1620−26, and found that the mooncould be detected through TOA perturbations if its mass is > 5% of theplanet’s mass and its distance from the planet’s center of mass is ∼ 2%of the planet-pulsar separation. Thorsett et al. (1999) estimated that thedistance between the planet and the inner-binary’s center of mass is ∼ 35AU, meaning that the hypothetical moon could be ∼ 1 AU away from theplanet. However, the TOA perturbations predicted by Lewis et al. areexpected to be comparatively smaller (∼1 µs) than those seen in Figure 4.1,and so it is unlikely that a moon could be the dominant cause of the observedTOA variations.Another possible source of complication arises if PSR B1620−26 exhibitsa “glitch” – a sudden, abrupt change in its rotational period that is sometimesfollowed by a period of relaxation towards the pre-glitch period – sometimeduring or immediately after the 2009 gap. While still an active area of re-search, a glitch is believed to indicate a sudden transfer of angular momentumdue to either a truncation of an oblate crust towards a more spherical shape(Baym et al., 1969), or the disconnection of vortices between the neutron-star superfluid interior and the remaining solid component of the star (e.g.Glampedakis & Andersson, 2009). The Crab and Vela pulsars are known toexhibit a large number of glitches during the past few decades of observation145(e.g. Espinoza et al., 2011; Lyne et al., 2015). While not commonly observedin MSPs, glitches have been seen in PSR B1821−24 (Cognard & Backer,2004) and more recently in PSR J0613−02005 (McKee et al., 2016). It iscurrently difficult to assess whether an intrinsic glitch event has occurredaround the time of the 2009 gap, since a glitch can be approximately mod-eled using a large number of spin-frequency derivatives and therefore servesas another biasing effect in our modeling of the outer orbit.4.4.3 Bias in ν˙s from Cluster AccelerationPhinney (1992) pointed out that globular-cluster dynamics due to the col-lective “mean-field” potential can also contribute to a significant first-orderrate of change in Doppler shifts, which affect both spin and orbital periods(as discussed in Section 1.5 above). The globular-cluster association of PSRB1620−26 therefore introduces a possible third component of ν˙s, along withthe components from spin-down and kinematic acceleration. For a sphericalstar cluster, the component of acceleration due to mean-field cluster dynam-ics corresponds to a change in Doppler shift, such that(P˙sPs)GC=(P˙bPb)GC= −1cGM(< r)r2hr(4.5)where r is the radial distance of the source to the cluster center, h is a pro-jected distance of the source to the plane of the sky that intersects the clustercenter, and M(< r) is the total mass contained within a sphere or radius rcentered on the cluster core. The value of h can be positive or negative,corresponding to the pulsar being in the front half or back half of the clusterrelative to a distant observer. For the first-derivative in spin frequency, wecannot uniquely separate the cluster-dynamics term from the other promi-nent components due to spin-down, kinematic bias and outer-orbital motion.5We analyze TOAs collected from PSR J0613−0200 as part of the NANOGrav programin Section 3.4.1 of this dissertation.146However, we can place limits on the significance of the cluster-dynamics com-ponent (Equation 4.5) by analyzing the likely mechanism that produced theobserved change in (Pb)i.We used the best-fit astrometric parameters for the global timing solutionand a distance d = 1.72 kpc (Peterson et al., 1995) in order to compute thecomponents of (P˙b)i due to Galactic acceleration, differential rotation andsignificant proper motion, using Equation 1.45,(P˙bPb)D=D˙D=azc− cos b(Θ20cR0)(cos l +βsin2 l + β2)+µ2dc.Using the acceleration model developed by Kuijken & Gilmore (1989) for az,we find that these three non-globular terms produce an expected value of(P˙b)i,D = 3.78× 10−11 s−1. This estimate of the kinematic, non-globular biasis positive and an order or magnitude smaller than the observed negativevalue, (P˙b)i,obs = −4.36(2)×10−10 s−1. If the residual amount of P˙b is due tomean-field cluster acceleration, (P˙b)i,GC = (P˙b)i,obs − (P˙b)i,D ≈ −5 × 10−10,then it is clear from Equation 4.5 that h > 0, and that the PSR B1620−26triple system is being accelerated away from observers on Earth.Furthermore, assuming that (P˙b)i,GC ≈ −5×1010 and using the left-handside of Equation 4.5, we find that the spin-frequency derivative due to mean-field cluster acceleration is (ν˙s)GC ≈ 2× 10−15 Hz−2, which is comparable inorder of magnitude to the value of ν˙s published by Thorsett et al. (1999) andcould therefore likely be a significant component in our measurement of thefirst derivative in spin frequency. While (ν˙s)GC > 0, the corrected value of ourfirst-derivative measurement for the two-orbit model presented in Table 4.3is (ν˙s)corr = (ν˙s)obs−(ν˙s)GC < 0, and is compatible with the expected changesin rotation from pulsar spin-down. Given the computation of (P˙b)i,D above,the component of ν˙s due to non-cluster kinematic acceleration is smaller inorder of magnitude than the spin-down component.1474.4.4 Bias in ν¨s from Cluster JerksThe discrepancy in the Joshi & Rasio (1997) estimate of (Pb)o shown inTable 4.2 could possibly be due to further bias in several spin-frequencyderivatives associated with accelerations and jerks from nearby stars in theM4 globular cluster. Blandford et al. (1987) first pointed out that globular-cluster pulsars would experience observable time-varying perturbations fromgravitational fields of nearby cluster members, especially those that residein the denser environments of cluster cores. The net change in gravitationalaccelerations over time will produce an observable second derivative in spinfrequency, (ν¨s)GC, that results from a time-averaged jerk. The second-orderchange in spin period varies with an order of magnitude of(P¨sPs)GC∼ 10−29(σ10 km s−1)3(rc1 pc)−2s−2 (4.6)where σ is the mean velocity dispersion of the globular cluster and rc is char-acteristic radius normally taken to be the core or tidal radius. For Messier4, σ = 3.5(3) km s−1 and rc ∼ 40′ ∼ 10 pc assuming a distance of 1.72 kpcto the cluster (Peterson et al., 1995), which suggests that (P¨s)GC ∼ 10−33s−1. This is significantly smaller than the observed second-derivative in spinperiod, where |(P¨s)obs| ∼ 10−27 s−1. We therefore do not believe that biasesin ν¨s due to jerks from nearby cluster star are present in our timing model.4.4.5 Variations of the Inner-Orbital ElementsThe observed variations of xi, ei and ωi are likely due to three-body inter-actions between the inner components and the outer planet. Arzoumanianet al. (1996) argued that the highly significant x˙i is mostly dominated bythe three-body component (Equation 4.7), since the maximum componentdue to kinematic bias from proper motion (Equation 1.43) only constitutes∼ 10% of the observed value. To first order, the secular perturbations ofhierarchical inner orbits due to outer companions are expected to exhibit the148following rates of change in several elements (Rasio, 1994):x˙i =3pi(ap cos i)iw2c(Pb)isin 2θo cos(ωi + φo) (4.7)ω˙i =3piw(Pb)i[sin2 θo(5 cos2 φo − 1)− 1](4.8)e˙i = −15pieiw2(Pb)isin2 θo sin 2φo (4.9)where the various terms are summarized as follos: w = (m2/m1)[ai/ro]3;m2 = (mc)o; m1 = mp + (mc)i; (ap cos i)i is the product of the inner-binarycos i times the semi-major axis of the pulsar relative to the inner-binary centerof mass; ai is the total semi-major axis of the inner binary; and (ro, θo, φo)are the spherical coordinates of the outer planet in a fixed coordinate systemcentered on the inner-binary center of mass, with φo measured from theinner-orbital periastron argument in the inner-orbital plane and θo measuredrelative to the inner-orbital angular momentum vector (with θo = 0 pointingin the same direction as the angular momentum vector).As discussed in Section 4.1 above, we have not yet fully characterized andadjusted the uncertainties in our TOA measurements for the global analysisdiscussed in Section 4.3.4. Moreover, Figure 4.10 shows non-random struc-ture in TOA residuals at ∼ 10− µs level that is likely due to an indirect fitof the outer orbit by using spin-frequency derivatives. We therefore do notyet utilize our measurements of x˙i, e˙i and ω˙i to constrain the geometry of theouter orbit. We can nevertheless perform an order-of-magnitude comparisonof the observed perturbations to their expected values, given a set of assump-tions motivated by our study of PSR B1620−26. For instance, we determinedthat m2 ∼ 10−3 M in the above analyses of spin-frequency data from ourTOA set, and Sigurdsson et al. (2003) determined that (mc)i = 0.34(5) Mfrom a WD cooling-sequence analysis of optical colors and magnitudes, whichlikely means that m1 ∼ 2 M. For the purposes of calculation, we assume an149Figure 4.10: A zoomed-in view of the TOA residuals shown in Figure 4.8.The residual colors denote the same data sets presented in Figure 4.1.inclination of the inner binary i = 45◦, so that ap = xi/ sin i ≈ 0.2 AU andso ai ∼ 1 AU. If we assume that ro = 30 AU and neglect the (θo, φo) trigono-metric terms in Equations 4.7-4.9, we find that x˙i ∼ 10−13, ω˙i ∼ 10−5 yr−1and e˙i ∼ 10−9 yr−1. These order-of-magnitude estimates of the inner-orbitalsecular variations are consistent with the observed changes between pre-gapand post-gap analysis presented in Section 4.3.3, as well as the variationsmeasured from the global solution derived from all data discussed in Section4.3.4.4.5 Further WorkAs discussed above, we could not successfully apply a two-orbit model to thefull, 27-year TOA data set we have collected for PSR B1620−26 at this time.We nonetheless derived comparable estimates of the outer-orbital parameters150when examining subsets of the whole TOA data set. However, we derivedestimates of the outer-orbital period using the Joshi & Rasio (1997) methodthat are shorter than suggested from the model of the spin period as a func-tion of time shown in Figure 4.9. We considered the possibility of significantcomponents in ν˙s and ν¨s due to globular-cluster dynamics, and found thatthere is a likely bias in the first time-derivative. Future work will assess thepossibility that a pulsar glitch occurred sometime during or after the late-2009 gap, which could add further, significant bias into the spin-frequencymodel of the outer orbit for the global analysis if left unaccounted.An explicit two-orbit model will likely make the measurements of inner-orbital variations more robust, since the use of many spin-frequency deriva-tives to model the outer orbit is an incomplete parametrization of the orbitand introduces large degrees of statistical correlation between derivatives.Once a two-orbit model is obtained, we will implement an analysis of theinner-orbit perturbations similar to the method used by Joshi & Rasio (1997)and Thorsett et al. (1999) to uniquely constrain the geometry of the outerbinary. The increasing significance of x¨i over time will yield an additionalconstraint on the analysis of inner-orbital perturbations and will allow fordirect constraints on the inner-binary mass components, which have so farnot been possible.Finally, we will eventually use the three-body integrator developed byA. Archibald for the analysis of the PSR J0337+1715 stellar triple sys-tem performed by Ransom et al. (2014), which has been shown yield ad-ditional interaction parameters and allow for unique determination of thethree hierarchical-component masses.151Chapter 5Long-Term Observations of theRelativistic PSR B1534+12Binary SystemThe population of “double-neutron-star” (DNS) systems – binary orbits thatconsist of two neutron stars – is expected to be comparatively small due tothe need for both components to undergo a supernova event; if the binarysystem ultimately survives and both components remain bound, then thepost-formation system eccentricity is expected be e ∼ 0.1 or greater due tothe injected energy from both supernovae. Indeed, DNS systems are obser-vationally rare as only 11 such systems are currently known of among the∼ 2, 500 pulsars observed in the Galaxy (see Table 1 in Martinez et al., 2015).The identification of a binary system as a DNS type is less straightforwardthan for a pulsar-WD system, where WD companions in relatively nearby sys-tems can be observed with sufficiently sensitive optical telescopes. However,DNS systems typically have comparable eccentricities and spin parameters,with Ps ∼ 50 ms, that reflect a common evolution; their evolutionary historyis generally thought to consist of minimal mass transfer between componentssince their massive progenitor stars are expected to have evolved on short152and comparable timescales (e.g. Stairs, 2003).Despite their rarity, DNS systems currently offer the most precise meansfor testing the predictions of Einstein’s relativity theory in the “strong-field”regime. The most relativistic pulsar-binary systems can exhibit a large num-ber of PK secular variations of orbital elements that are measurable ondecadal timescales. This is especially true for DNS systems with orbital peri-ods on the order of hours, where relativistic secular variations are expected tobe comparatively large in magnitude. A classic example of a relativistic DNSsystem is PSR B1913+16, famously known as the “Hulse-Taylor” system andthe first pulsar-binary system to be discovered, for which the first PK mea-surements in a pulsar-binary system were made. The eventual measurementof orbital decay in the PSR B1913+16, consistent with the prediction fromGR, provided the first evidence for the existence of gravitational radiation(Taylor & Weisberg, 1989). Long-term timing of PSR B1913+16 produceda set of pulse profiles that were observed to be secularly changing in time,consistent with the notion of geodetic precession of the pulsar’s spin axisabout the orbital angular momentum vector occurring in the relativistic sys-tem and ultimately leading to an evolving view of the two-dimensional radiobeam (Weisberg et al., 1989). As with the PK timing parameters discussedin Section 1.5, the rate of geodetic precession (Ωspin1 ) can be computed underthe assumption of a gravitational theory. In GR, this rate is given as (e.g.Barker & O’Connell, 1975)Ωspin1 =12T2/3 n−5/3bmc(4mp + 3mc)(1− e2)(mp +mc)4/3 . (5.1)However, the measurement of geodetic precession in PSR B1913+16 requiresa model of the two-dimensional beam shape (Kramer, 1998), which is notimmediately known. This DNS system nevertheless remains a valuable high-precision laboratory that for many years yielded only one test of GR usingthe ω˙-P˙b-γ combination, as the Shapiro timing delay was not significant formany decades due to low inclination (Weisberg et al., 2010). However, with153the aid of substantial periastron advance over the years, the Shapiro delayhas recently become significantly measurable for the first time (Weisberg &Huang, 2016).PSR B1534+12 is a 37.9-ms radio pulsar in a 10.1-hour orbit with anotherneutron star that currently exhibits up to six PK deviations in the system’sorbital parameters and orientation due to strong-field gravity. In their firstlong-term analysis of PSR B1534+12, Stairs et al. (1998) provided the first-ever measurements of at least five PK timing parameters from one pulsar-binary system; this also corresponded to the first time that multiple tests ofGR could be performed using a single gravitationally-bound system immersedin the strong-field regime.In 2014, we published a study that built on work done by Stairs et al.(1998, 2002) and analyzed TOAs from PSR B1534+12 that collectively span22 years; this data was obtained exclusively with the 300-m Arecibo tele-scope using three generations of signal processors (Fonseca et al., 2014). Weimproved the quality of the best test obtained from this system using theγ − s − ω˙ combination and showed that GR is correct to within 0.17% ofits predictions, nearly a factor of 8 smaller than the previous timing studyperformed by Stairs et al. (2002). Stairs et al. (2004) made the first quanti-tative (but low-precision) beam-model-independent measurement of the rel-ativistic spin-precession rate by comparing aberration effects and precession-induced changes in the total-intensity profile of PSR B1534+12. In our 2014study, we confirmed this detection and achieved a precision measurement ofΩspin1 = 0.59+0.12−0.08 degrees per year, which is in excellent agreement of the GRprediction of 0.51 degrees per year. We also used evolving polarization prop-erties to uniquely determine the full orientation of the system, confirm therelativistic precession of the system, and constrain the misalignment anglebetween spin and orbital angular momenta to be 27± 3 degrees.The majority of the timing analysis presented by Fonseca et al. (2014), aswell as an initial investigation of the total-intensity pulse profiles, constituted154the bulk of the M .Sc. thesis written by E. Fonseca. However, work was doneafter the formal granting of the M. Sc. degree that was incorporated into theFonseca et al. (2014) publication, including the determination of the preciserelativistic spin precession rate. In this chapter, we summarize the work donefor the analysis of TOAs and pulse profiles from PSR B1534+12 during thePh. D. program. In Section 5.1, we briefly discuss the logistics of data usedby Fonseca et al. (2014) since most of the results presented in this chapterare derived from these data. In Section 5.2, we present an analysis of DMmeasurements and ISM turbulence made using the 22-yr data set. In Section5.3, we present the tools and analyses used to quantify relativistic precessionin pulse-structure data. In Section 5.4, we present initial results obtainedfrom ongoing observations of PSR B1534+12 using the PUPPI coherent de-dispersion backend. In Section 5.5, we summarize the results presented byFonseca et al. (2014) and discussed in this chapter, as well as discuss futureprospects in PUPPI observations of PSR B1534+12.5.1 Observations & ReductionData were obtained exclusively with the 305-m Arecibo Observatory in PuertoRico, using two observing frequencies and three generations of pulsar signalprocessors. Basic information regarding the data and backends used in thisanalysis are presented in Table 1 of Fonseca et al. (2014) while a more detailedaccount of observing information can be found in Fonseca (2012).Part of this set of pulse profiles and times-of-arrival (TOAs) were recordedwith the Mark III (Stinebring et al., 1992) and Mark IV (Stairs et al., 2000)pulsar backends. The Mark III system employed a brute-force pulse de-dispersion algorithm by separating each receiver’s bandpass into distinctspectral channels with a filterbank, detecting the signal within each channel,and shifting the pulse profile by the predicted amount of dispersive delay foralignment and coherent averaging. A small amount of Mark III data was ob-155tained using the coherent-dedispersion “reticon” subsystem; these data wereused only in the polarization analysis. The Mark IV machine was an in-strumental upgrade which employed the now-standard coherent de-dispersiontechnique (Hankins & Rickett, 1975) that samples and filters the data streamprior to pulse detection. A series of digital filters applied in the frequencydomain completely remove the predicted dispersion signatures while retain-ing even greater precision than the Mark III counterpart. See Stairs et al.(1998, 2002) for more details on these observing systems and reduction ofPSR B1534+12 data obtained with these two backends.Recent data were collected with the Arecibo Signal Processor (ASP; De-morest, 2007), a further upgrade from the Mark III/IV systems that retainsthe coherent de-dispersion technique, but first decomposes the signals acrossa bandwidth of 64 MHz into a number of 4-MHz spectral channels that de-pends on the observing frequency. We used data collected with the four inner-most spectral channels centered on 430 MHz, and typically sixteen channelscentered on 1400 MHz with some variability, due to limits in computer pro-cessing and available receiver bandpass. While the Mark IV machine used4-bit data sampling in 5-MHz-bandpass observing mode and 2-bit samplingin 10-MHz-bandpass observing mode, ASP always used 8-bit sampling. Thecoherent de-dispersion filter is applied to the raw, channelized data, whichare then folded modulo the topocentric pulse period within each channel andrecorded to disk, preserving polarimetric information.Observations were generally conducted at semi-regular intervals, with typ-ical scan lengths of an hour for each frequency. Several extensive “campaign”observations were also conducted at 430 MHz, which consisted of several-hourobserving sessions performed over 12 consecutive days, in order to obtainhigh-precision snapshots of the pulse profile at different times. Campaignsessions occurred during the summers of 1998, 1999, 2000, 2001, 2003, 2005,and 2008. We used all available data for the timing analysis, and only usedmost of the campaign profiles and several strong bi-monthly scans for the pro-156file analysis. We excluded the 2008 ASP data from the profile-shape analysisdue to weak, heavily scintillated signals recorded during this epoch, but usedseveral stronger observations during this campaign for the RVM analysis (seeSection 5.3 below).We used the template cross-correlation algorithm developed by Taylor(1992) for determining pulse phases, their TOAs and uncertainties using astandard-template profile. A standard template was derived for the MarkIII and Mark IV backends at each frequency by averaging several hours ofconsecutive pulse profiles; ASP TOAs were derived using the Mark IV tem-plates. We added small amounts of error in quadrature or as factors to theoriginal TOA uncertainties, in order to compensate for apparent systematicerrors in TOAs. We also ignored TOAs with uncertainties greater than 10 µs;only 10% of all available TOAs – including points affected by radio frequencyinterference – were excised when using this cut.It is important to note that there is a overlap in pulse TOAs collectedwith the Mark IV and ASP data sets between MJD 53358 and 53601. Weincorporated TOAs acquired from both machines during this era, despite theoverlap, due to the substantially larger ASP bandwidth; we believe that thisdifference in bandwidth does not produce many redundant data points. Theimprovement in bit sampling between backends has a measurable effect onthe pulse profile shape, as discussed in Section 5.3 below.5.2 DM Variations in PSR B1534+12The DM of any pulsar traces the amount of free electrons in the ISM alongthe observer’s line of site. Three-dimensional motion of the pulsar will changethe instantaneous line of sight and ultimately lead to a changing number ofelectrons in between the observatory and pulsar, which alters the amount ofdispersion experienced by a broadband electromagnetic signal. This effectmanifests itself as an inherently unpredictable change in DM over time as157the Galactic electron distribution is not known.1 If left unaccounted, thestochastic signal from DM variations will bias other timing parameters dueto a suboptimal model fit. Moreover, if one seeks to test the predictionsof GR or any other viable theory of gravitation, then all physical processes(whether deterministic or inherently random) must be modeled in order toobtain unbiased estimates of PK variations and the Shapiro timing delay.In the case of PSR B1534+12, DM variations are easily observed and havedramatic, piecewise structure over the course of the 22-yr data set analyzedby Fonseca et al. (2014). These variations are shown in Figure 5.2. As firstnoted in the M. Sc. thesis for E. Fonseca, we chose to probe DM evolutionacross our data set in two different ways. For one method, shown as theblack data points in Figure 5.2, we fit for ∆DM in 80-day bins using theMark IV and ASP TOAs, where dual-receiver TOAs are available. For thesecond method, shown as solid, sloped lines in Figure 5.2, we instead fitfor four distinct DM bins that account for any temporal evolution as timederivatives of DM. Both methods adequately model the DM variations inPSR B1534+12, though we ultimately chose the few-bins/gradients methodas it reduced the number of free parameters in our timing model.These long-term DM measurements are useful for a statistical analysisof turbulence within the interstellar medium (e.g. Kaspi, Taylor, & Ryba,1994), which usually assumes that the power spectrum of spatial variationsin electron density is a power law within a range of length scales (Rickett,1990),P (q) ∝ q−β, qo < q < qi (5.2)where q = 2pi/l is a spatial frequency and l is a scattering length. Thefrequency range in Equation 5.2 corresponds to a range between an “inner”1In fact, the combination of distance and DM measurements from pulsar TOA analyseshave allowed for rough models of the Galactic ne distribution to be inferred (e.g. Cordes& Lazio, 2001).158Figure 5.1: DM variations from TOAs collected from PSR B1534+12.Horizontal-dashed lines represent the timespan where the denoted pulsarbackend was used. Points with uncertainties represent fits of DM to seg-ments of data contained within 80-days bins. Black-solid lines represent fitsof DM and rate of change in DM across larger segments of time. This figurewas first published by Fonseca et al. (2014).(li) and “outer” (lo) length scale where the power-law form is valid. Theobserved spatial fluctuations due to a relative transverse velocity v are relatedto a time lag τ by l = vτ . The power spectrum P (q) can therefore beestimated by computing a pulse-phase structure function Dφ(τ) = 〈[φ(t +τ) − φ(t)]2〉, where the angle brackets represent an ensemble average overobserving epoch t. The pulse’s electromagnetic phase φ is linearly related toDM, which therefore relates Dφ(τ) to a DM structure function DDM(τ) =〈[DM(t+ τ)−DM(t)]2〉,Dφ(τ) =(2piCf)2DDM(τ) (5.3)159where C = 4.148 × 103 MHz2 pc−1 cm3 s, and f is the observing frequencyin MHz. Moreover, Dφ(τ) is a power law in τ within the inner length scalesdefined in Equation 5.2, which finally requires thatDφ(τ) =(ττ0)β−2(5.4)where τ0 is a normalization constant with units of time. Scintillation theoryrequires that τ0 = τd, where τd is the timescale of diffractive scintillation ofthe signal (e.g. Cordes et al., 1985), if the inner length-scale li ≤ vτd.We computed values of Dφ(τ) at f = 430 MHz using the Mark IV andASP small-bin measurements of DM shown in Figure 5.2. The Mark III DMpoint was measured using all Mark III TOAs collected over several years,which were generated with a different standard profile than the one used forthe Mark IV and ASP data; we therefore chose to ignore this measurement inorder to avoid incorporating bias in the structure function. Uncertainties inDφ(τ) were determined by propagating errors from our DM(t) measurements.Our estimate of Dφ(τ) is shown in Figure 5.2, and illustrates a power-lawevolution between time lags of roughly 70 and 900 days. We fitted Equation5.4 to this segment of data, and found thatβ = 3.70± 0.04τ0 = 3.0± 0.8 minutes (5.5)which is shown as a solid black in in Figure 5.2.The measured spectral index β is consistent with the value for a “Kol-mogorov” medium, βKol = 11/3. Furthermore, β and τ0 in Equation 5.5are consistent with the structure-function estimates reported by Scheiner &Wolszczan (2012). Our estimate of τ0 is also consistent with the value of τdmeasured from the autocorrelation function of a dynamic spectrum of PSRB1534+12 (Bogdanov et al., 2002).160Figure 5.2: Phase structure function Dφ as a function of time lag τ . Thesolid line is a best-fit model of Equation 5.4 for data with lags between 70and 900 days. This figure was first published by Fonseca et al. (2014).At large timescales, the structure function departs from the fitted modelat a lag τo ≈ 900 days, which suggests thatlo ≈ 52(v100 km/s)AU (5.6)Bogdanov et al. (2002) derived an interstellar scintillation (ISS) velocity of192 km/s. They noted in their study that ISS velocities of pulsars are typ-ically dominated by the systemic transverse component, which means thatv ≈ 192 km/s for PSR B1534+12, and lo ∼ 100 AU ∼ 1015 cm from Equa-tion 5.6. This estimate is consistent with the upper limit of lo observed forseveral pulsars by Phillips & Wolszczan (1991). By contrast, there is no ev-idence for a significant inner scale from our data set, since bins with mean161values less than 70 days contain only one or two pairs of DM(t) and weretherefore ignored in the analysis. We did not apply any correction for thesolar-wind contribution of our DM(t) measurements, due to a covariance be-tween the TEMPO solar-wind DM model and a fitted timing parameter thatis discussed in Fonseca et al. (2014).5.3 Geodetic Precession and Secular Evolu-tion in Pulse StructureAn observed pulse produces a set of Stokes-vector pulse profiles, from whicha TOA can be derived when cross-correlating the total-intensity profile Iwith a standard template as described in Chapter 1. As with pulsar tim-ing, gravitational strong-field effects can give rise to observable changes inparameters that describe the electromagnetic structure of pulses, such as thetotal-intensity shape and polarization properties, over a variety of timescales.Such variations will occur if there is a misalignment between the spin axisof the pulsar and the orbital angular momentum vector of the binary system(de Sitter, 1916). This spin-orbit coupling, generally referred to as geodeticprecession2, ultimately leads to an evolving view of the radio beam emittedfrom the pulsar and different slices of the radio cone over time. The effectsof relativistic spin precession on pulse structure have also been observed inPSR B1913+16 (Weisberg et al., 1989), the double-pulsar system (Bretonet al., 2008), and most dramatically in PSR J1141-6545 (Manchester et al.,2010).In order to detect such changes in PSR B1534+12, we shifted our 430-MHz profiles to a common phase using the derived DD-binary timing modeldescribed by Fonseca et al. (2014). Each set of “campaign” data, where PSRB1534+12 was observed for the entirety of its observable track in the sky for2Pulsar literature also refers to geodetic precession as relativistic spin precession, as wedo at the beginning of this chapter.16214 consecutive days, was then binned into twelve orbital-phase cumulativeprofiles. Several particularly strong, non-campaign scans that were takenduring the observing year were integrated into single profiles recorded at theirrespective epochs and included in this analysis. We subsequently performedtwo distinct types of analyses on these total-intensity and polarization datain order to extract gravitational information from independent techniques,as described below.5.3.1 MethodologiesFor the first analysis, we employed the general model developed by Stairset al. (2004) that establishes pulse-structure data as functions of time andlocation within the relativistic orbit. Estimates of the total-intensity profileshape (F ) at a given epoch were derived by first applying a principal com-ponent analysis (PCA) on a set of total-intensity profiles collected over time.The first and second principal components correspond to an average (P0)and “difference” (P1) profile, respectively, and an observed profile within thetimespan of the PCA input can be approximately represented by a linearcombination of the two PCA components: P = c0P0 + c1P1. The coeffi-cients c0, c1 were estimated using a cross-correlation algorithm between theobserved profiles and principal components in the Fourier domain, and theshape F of each profile was then estimated by calculating the ratio F = c1/c0in order to negate epoch-dependent scintillation effects.The shape F of a profile recorded at time t and eccentric anomaly E canalso be determined using the relationFmod =dFdtt+ δAF (E) + F0 (5.7)where F0 is an intercept parameter and dF/dt and δAF are the terms thatdescribe secular and aberrational changes in the pulse shape, respectively.These two important terms are functions of the pulsar’s precession rate Ωspin1163Figure 5.3: Difference between cumulative 2005 campaign profiles for theMark IV and ASP backends. This figure was first published by Fonsecaet al. (2014).and the angle  between the line of nodes and the projection of the spin axison the plane of the sky:dFdt= F ′Ωspin1 cos  sin i (5.8)δAF = F′ β1sin i[− cos S(E) + cos i sin C(E)] (5.9)The parameter F ′ = dF/dζ characterizes the unknown beam structure asa function of the auxilary “viewing” angle ζ, β1 = 2pix/(Pb√1− e2) is themean orbital velocity of the pulsar, and164C(E) = cos[ω + u(E)] + e cosωS(E) = sin[ω + u(E)] + e sinωare time-dependent orbital terms that depend on the true anomaly u(E).The Keplerian binary parameters in Equations 5.8 and 5.9 were determinedthrough pulsar-timing techniques described in Chapter 1, and are presentedby Fonseca et al. (2014).We fitted Equation 5.7 to our 430-MHz data using an MCMC implemen-tation with a Metropolis algorithm in order to obtain posterior distributionsof Ωspin1 , , F′, and F0 from uniform priors. We assumed that the joint like-lihoo probability density of the model, J(Ωspin1 , , F′, F0|F, t, E), is a normaldistribution in the χ2 goodness-of-fit statistic for the profile-shape model,J ∝ exp[− 12∑i(Fi − Fmod(ti, Ei)σi)2](5.10)where σi is the uncertainty in Fi determined from the cross correlation be-tween the ith profile and the two principal components. The results of thisfitting procedure are summarized in Table 5.1 and discussed in Section 5.3.2below.As a second analysis, we used Equation 1.2 to fit an RVM to availablepolarization position-angle data on each full-sum campaign profile. As dis-cussed in Section 1.1, an RVM fit to significant measurements of Ψ as afunction of the pulse-rotation phase φ yields α and β simultaneously.3 Whileno evolution is expected in α, geodetic precession will cause β to evolve withtime such that dβ/dt = Ωspin1 cos  sin i (Damour & Taylor, 1992). While the3As first noted in Chapter 1, we use the angle-measurement convention adopted byDamour & Taylor (1992) when measuring and reporting values of α and β; this conventionis not consistent with the IAU standard. However, we are ultimately interested in the ratesof change in the RVM orientation angles and their association with geodetic precession.The analysis of their time derivatives does not depend on choice of convention.165Figure 5.4: Full-sum profile data for PSR B1534+12 collected during theJune 2003 campaign. (Top.) Ψ as a function of φ, are shown as blue points.The thick red line is the best-fit RVM for this profile. (Bottom.) Stokes I, Land V as a function of φ. (For clarity, uncertainties in Ψ are not shown inthis figures, but were used in all RVM fits.)166sign convention used in Equation 1.2 is inconsistent with the IAU standard,we are only concerned with secular variation of β over time and its connec-tion to Ωspin1 ; the choice in sign convention will ultimately make no differencein the measured estimate of the precession rate, and we therefore choosethe convention employed by Damour & Taylor (1992). The combination ofMCMC and RVM analyses therefore yields a test on observed profile evolu-tion due to relativistic spin precession from two independent measurements,since the profile-shape analysis yields an estimate of .The differences in data quality between Mark IV and ASP profiles can beseen as slight differences in the profile shape across pulse phase, as shown inFigure 5.3. This introduced slight discrepancies in the results obtained fromthe PCA analysis described above when performed using all available data,which subsequently affected the derived profile shapes and MCMC results.Two separate studies between backends were not possible as the ASP era con-sisted of fewer profiles and a smaller timespan, with campaign data collectedduring the 2008 observing year being excluded from the MCMC analysis dueto having many low signal-to-noise profiles. We therefore decided to performa PCA on all Mark IV profiles only, and then use the derived principal com-ponents to estimate the shapes for all high signal-to-noise Mark IV and ASPprofiles. This approach does not account for observed scintillation or pro-file evolution across observing frequency in the ASP data. We therefore onlyused ASP data collected with the two innermost frequency channels centeredon 430 MHz for both analyses in order to minimize such effects.5.3.2 ResultsResults from the MCMC fit on several data sets can be found in Table 5.1,and the posterior distribution for Ωspin1 derived from our Mark IV and ASPdata sets is shown in Figure 5.5. We generated 3 × 105 samples for eachapplication of the algorithm, after burning the first 5000 samples in order toremove non-convergent iterations. We provided the original results obtained167Parameter STA04 STA04-MCMC Mark IV AllΩspin1 (◦/yr) 0.44+0.48−0.16 0.51+0.10−0.08 0.48+0.09−0.07 0.59+0.12−0.08 (◦) . . . . . . ±103+10−10 ±99+2−2 ±118+10−15 ±139+16−25F ′ . . . . . . . . . n/a −5.9+0.9−1.0 −2.2+0.6−0.7 −1.3+0.3−0.5F0 (10−3) . −1.5+0.3−0.3 −1.90+0.08−0.09 −1.21+0.08−0.08 6.67+0.07−0.07Table 5.1: MCMC results for measurement of Ωspin1 and the orientation angle for different data subsets. Uncertainties reflect 68.3% confidence intervalsof posterior distributions. “STA04” denotes the results obtained by Stairset al. (2004), whereas “STA04-MCMC” denotes the results obtained whenimplementing our MCMC analysis on the same data set used by Stairs et al.by STA04, as well as a reproduced set of results from the STA04 data setusing the MCMC algorithm, for comparison with our extended Mark IV andASP profiles. We assumed that values of F ′ must be negative while usingthe MCMC algorithm, since the simultaneous-linear-fit technique used anddescribed by STA04, which avoids any consideration of F ′, estimates thatcos  < 0. These results agree well with predictions from general relativity,where Ωspin1 = 0.51◦/yr using the derived masses from PK timing parameters,and previous measurements made by STA04. General improvements in pre-cision come from the new fitting procedure, which permitted direct samplingof the precession rate and other free parameters, as well as the addition ofthe ASP 2005 campaign and several strong bi-monthly observations.The RVM analysis yielded values of α and β at different times using theMark III (reticon), Mark IV and ASP campaign profiles. The values of βmeasured for each campaign are shown in Figure 5.6. Measurements of α,with average values of α = 103.5(3)◦, are consistent with no evolution intime, while the values of β are found to change significantly, where dβ/dt =-0.23 ± 0.02 ◦/yr. This is consistent with the STA04 result of -0.21 ± 0.03◦/yr. The assumption that general relativity is correct requires that dβ/dt =Ωspin1 sin i cos , and therefore yields  = ±117 ± 3◦ (68% confidence), whichagrees with the value determined from the MCMC analysis described above.168Figure 5.5: Top: MCMC posterior distribution of Ωspin1 obtained from theprofile-shape analysis of Mark IV and ASP data discussed in Section 5.3.2.Bottom: Markov chain for Ωspin1 determined from the MCMC algorithm. Thisfigure was first published by Fonseca et al. (2014).With these values, the misalignment angle δ between the spin and orbitalangular momentum axes can be derived through spherical trigonometry bycos δ = − sin i sinλ sin +cosλ cos i. The sign ambiguity in  and i, as well asthe requirement that cos i tan  > 0 pointed out by STA04, gives an expectedvalue of δ = 27.0 ± 3.0◦ or δ = 153.0 ± 3.0◦. Physical arguments basedon alignment of angular momenta prior to the second supernova suggestthat the smaller angle is correct (Bailes, 1988), and therefore requires that = −117± 3◦ and i = 77.7± 0.9◦.The consistency between the MCMC and RVM analyses serves as animproved, independent check of precession within this relativistic binary sys-tem. These results also confirm the geometric picture of this pulsar-binarysystem derived in STA04.169Figure 5.6: Impact angle β between the magnetic axis and line of sight as afunction of time. The black line is a best-fit slope of -0.23 ± 0.02 ◦/yr. Thisfigure was first published by Fonseca et al. (2014).5.4 Timing Observations with PUPPIThe PUPPI instrument began formal operation during the early months ofthe 2012 observing year at the Arecibo Observatory. We submitted telescopeproposals every year since 2012 in order to collect TOAs using the PUPPIbackend, which can process and record data collected with the 1400-MHz re-ceiver that samples the full 800 MHz in bandwidth using 512 thin frequencychannels. The technical specifications of PUPPI are a dramatic improvementover the ASP processors for the case of PSR B1534+12, since the ASP ma-chine could only process up to 64 MHz in bandwidth due to limitations inhardware functionality (see Fonseca, 2012, for details). All submitted propos-170Parameters 430 MHz 1400 MHzFull frequency range (MHz) 421−445 1147−1765Effective Bandwidth (MHz) 20 600Number of Channels 4 4Integration Time (s) 180 180Number of TOAs 984 668RMS residual (µs) 8.47 6.32χ2red 1.013 1.004Table 5.2: Parameters of TOAs from PSR B1534+12 obtained with PUPPI.Note that “effective bandwidth” means the total width of broadband datathat is usable after excision of narrow-band radio frequency interference.als for continued timing of PSR B1534+12 were accepted4 and all requestedtime was granted; this includes two additional dense campaigns occurringduring the month of August in 2013 and 2015.Our initial analysis of the PUPPI data for PSR B1534+12 is shown graph-ically in Figure 5.7, and statistics of the current set of PUPPI TOAs are pre-sented in 5.4. We employed the same general methods for excision of radiofrequency interference (RFI) used by Fonseca et al. (2014) when generatingPUPPI TOAs and pulse profiles, in order to compare the data quality be-tween PUPPI TOAs and those published by Fonseca et al. (2014). Moreover,we averaged sections of data collected across the bandwidth of each receiverin order to obtain 4 channelized profiles per 3-minute integration for each re-ceiver. We used the PSRCHIVE5 software suite in order to reduce raw TOAdata into calibrated pulse profiles, using the same methodology described in3.1 of Chapter 3.From the logistics shown in Table 5.4, it is immediately clear that theprocessing capabilities of PUPPI are superior for data collected with the1400 MHz receiver at Arecibo; this improvement is entirely due to the greater4These projects were given the following designations: P2719; P2820; P2906; andP2990.5http://psrchive.sourceforge.net/index.shtml171Figure 5.7: Current state of the timing data on PSR B1534+12. Black pointsare data published by Fonseca et al. (2014). Red points are TOAs measuredat 430 MHz collected with PUPPI, and blue points are TOAs measured at1400 MHz collected with PUPPI.access of the 1400-MHz receiver bandwidth. The ASP machine was able toprocess up to 128 MHz in real time, though only the innermost 64 MHzwas generally usable. In stark contrast, PUPPI can process and record dataacross the full bandwidth at 1400 MHz, though in practice only ∼600 MHzof bandwidth was usable after removing channels that were affected by RFI.An example of a PUPPI observation at 1400 MHz is shown in Figure 5.8.Furthermore, the expanded bandwidth and additional pulsar signal allows usto average TOA data into a smaller number of channels across the 800-MHzbandwidth, which can help mitigate the effects of complex profile evolutionacross the band while folding profiles for improving S/N. For our currentdata set shown in Figure 5.7, we were able to derive 668 usable TOAs withthe 1400-MHz receiver using PUPPI over the course of two years, which is172Figure 5.8: Heat map of TOA data for PSR B1534+12 collected with the1400-MHz receiver, using PUPPI, on MJD 56326. Several frequency channelswere excluded due to the presence of narrow-band RFI.nearly three times as many TOAs recorded at 1400 MHz during the entireASP era using the same receiver.The improved consistency in dual-receiver measurements with PUPPIalso allows for estimates of DM using widely separated observing frequencieson a nearly per-epoch basis. Figure 5.9 shows measurements of DM versustime for our current PUPPI data set, using 430-MHz and 1400-MHz TOAdata on nearly all observing epochs, illustrating a gradual change over timesimilar to the pre-PUPPI DM measurements shown in Figure 5.2. The twoDM points in Figure 5.9 with comparatively large uncertainties, derived from173Figure 5.9: DM versus time for PSR B1534+12, using the PUPPI backend.PUPPI data collected at the beginning of the 2014 and 2015 observing years,were derived from the channelized 1400-MHz data set for these two epochs;no 430-MHz TOAs collected during these two observations were found tobe usable after excision of RFI. It is clear that, when using the broadbandPUPPI processor, the DM offset is measurable from channelized 1400-MHzreceiver data alone, which was not previously possible due to a lack of signalacross the bandwidth recorded by ASP. However, the lack of precision in thesetwo PUPPI points is mostly due to the intrinsically weaker pulsar signal at1400 MHz when compared to the profile observed at 430 MHz.5.5 ConclusionsThe timing and pulse-profile results presented by Fonseca et al. (2014) covera wide range of astrophysical scope and application, including improved tests174of GR from extended timing observations of PSR B1534+12. In Sections 5.2and 5.3 of this chapter, we discussed additional results obtained by furtheranalysis of TOA and profile data that was ultimately presented by Fonsecaet al. (2014). We found that the DM variations seen in PSR B1534+12were consistent with local ISM fluctuations due to a turbulent, Kolmogorovmedium. We also carried out an extensive analysis of the pulse structureand observed secular variations over time, modeling the long-term changes intotal-intensity pulse shape and polarization properties to determine a preciseestimate of the Ωspin1 .Future long-term timing observations of PSR B1534+12 are crucial forthe improvement of its formidable constraints on relativity theory using high-quality Arecibo data. We have demonstrated that full use of Arecibo data,both in the timing of pulse profiles and in the electromagnetic propertiesof the profiles themselves, contain a wealth of information that has so farbeen used to obtain unique and precise measures of relativistic phenomena.We continue to use the PUPPI signal processor in an effort to improve thesignal-to-noise ratio and profile quality at both proposed observing frequen-cies (particularly at 1400 MHz, where the source is intrinsically weaker). Thefactor of ∼ 5 increase in bandwidth at 1400 MHz has so far provided a largeboost in S/N, as shown in Figure 5.8, such that we have so far collected moreuseable PUPPI 1400-MHz TOAs than we have during the entire ASP era.Moreover, we have reduced and included PUPPI data collected up to March2016 (and as early as March 2012) into our current data set and will continueto process ASP/PUPPI data collected before/after then.The full, current timing model for PSR B1534+12 uses TOAs collectedwith four signal processors (including PUPPI) that collectively span 25 yearsof observation. We use the same timing methodology for our current model,which fits for variations in DM during the PUPPI era with a single gradient.A full evaluation of the current measurements of relativistic parameters inthe PSR B1534+12 system, using the updated PK timing parameters, are175Figure 5.10: PK parameters measured in the PSR B1534+12 system, shownas colored bands in the (mp, mc) plane with black labels that denote the par-ticular effect. The width of each colored band represents the 68.3% credibleinterval measured for the denoted parameter. The values and uncertain-ties of the PK timing parameters were determined from an updated timingmodel that used the data analyzed by Fonseca et al. (2014) and the PUPPIobservations presented in Section 5.4 of this dissertation.shown in Figure 5.10. The five PK timing parameters remain consistentwith those estimated by Fonseca et al. (2014) and in the previous studiesundertaken by Stairs et al. (2002, 1998); the TEMPO uncertainties in thesecular PK variations (P˙b and ω˙) have decreased by nearly a factor of 2 sincethe publication of the Fonseca et al. (2014) study, which is expected sincewe’ve extended our timespan by nearly three years.As first noted in Stairs et al. (1998), the use of the DM-based estimateof d = 0.7(2) kpc (Cordes & Lazio, 2001) in the Doppler correction for176(P˙b)obs produces an estimate of (P˙b)int = (P˙b)obs− (P˙b)D that does not agreewith the predictions from the other PK parameters at the 68.3% uncertaintylevel. Moreover, the uncertainty in (P˙b)int is dominated by the uncertaintyin the DM-based estimate of d. This slight bias in the P˙b correction is shownin Figure 5.10 as the blue-shaded region that narrowly misses the commonregion of intersection for all of the other PK parameters. Fonseca et al. (2014)used this discrepancy in the observed orbital decay to derive a distance tothe system when assuming the validity of GR, dGR = 1.051(5) kpc, thatproduces the corresponding (P˙b)D. PSR B1534+12 was recently included intothe MSPSRpi radio-interferometry program6, which will eventually producean independent measure of d – and, ideally, a robust correction of (P˙b)obs–within the next two or three years.6https://safe.nrao.edu/vlba/psrpi/home.html177Chapter 6Concluding RemarksAs we have shown throughout this dissertation, an analysis of secular vari-ations in orbital elements can yield constraints or significant measurementsof the component masses and angles of orientation, which are otherwise notaccessible when analyzing purely Keplerian motion. In a practical sense, themeasurement of these variations is also important for robust determinationof the full, accurate pulsar-timing model applied to TOAs from a given bi-nary radio pulsar. Accurate timing solutions are essential for subsequentinterpretation of the model parameters, and are especially important for theNANOGrav, EPTA and PPTA efforts that search for correlated structurein TOA residuals due to nanohertz-frequency gravitational waves among alarge ensemble of MSPs.In Chapter 3, we measured the relativistic Shapiro timing delay in four-teen of twenty five NANOGrav binary radio pulsars, which yielded directmeasurements of mc and sin i for each system with varying degrees of preci-sion. Using the mass function computed from Keplerian elements, we derivedvalues of mp for each of these fifteen systems. The most significant measure-ments of mp that we made, which typically possess relative uncertaintiesequal to or less than 20%, are shown in Figure 6.1. Four of the fourteenmeasured signals – in PSRs J0613−0200, J2017+0603, J2302+4442, and178J2317+1439 – have been characterized by our analysis for the first time.It is important to note that no detections of ∆S were made for J2317+1439using the EPTA data set (Desvignes et al., 2016), or for J0613−0200 us-ing either the EPTA or PPTA data sets (Desvignes et al., 2016; Reardonet al., 2016). We attribute these differences to the superior timing preci-sion obtained with the Arecibo Observatory and GBT using the PUPPI andGUPPI processors, respectively, as well as the targeted Shapiro-delay cam-paign devised by Pennucci (2015). Moreover, we improved the measurementsof two previously known signals in the J1918−0642 and J2043+1711 systems(Sections 3.4.11 and 3.4.14, respectively) such that mp has been measuredwith relative uncertainties of ∼20% or less for the first time.The most significant measurements of ∆S generally corresponded to sys-tems with comparatively large orbital inclinations, where the Shapiro-delayparameters can be measured with low statistical correlation and with littleabsorption of the relativistic signal between other Keplerian timing param-eters. From the Shapiro delay alone, we made high-precision estimates ofpulsar masses as low as mp = 1.18+0.10−0.09 M for PSR J1918−0642 and as highas mp = 1.928+0.017−0.017 M for PSR J1614−2230 (Section 3.4.4). We used prob-ability density maps computed from grids of χ2 values obtained for differentcombinations of the Shapiro delay parameters to find accurate uncertaintiesof the component masses and inclination angles that reflect the statisticalcorrelation present in our measurements.For five pulsars studied here, we used the statistical significance of ∆S andone or more observed Keplerian variations to constrain our estimates of mp,mc and i. For example, we confirmed that the observed ω˙ in the eccentricPSR J1903+0327 system (Section 3.4.9) is due to relativistic precession atits current level of precision. We assumed the validity of GR in order to usethe statistical significance of ω˙ to further constrain the region of preferredsolutions based on the χ2 grid over the two Shapiro-delay parameters. Wefound that the constrained mass of PSR J1903+0327 is mp = 1.65+0.02−0.02 M,179Figure 6.1: Estimates of pulsar masses measured in this dissertation. Labelsin red indicate that one or more secular variations were used to constrainthe masses and/or geometry of the binary systems. Labels with black starsindicate the first significant measurements of the pulsar mass in the denotedbinary systems.which is consistent with the initial assessment and calculations made byFreire et al. (2011) using an independent data set obtained for the same MSP.In the case of PSR J1741+1351 (Section 3.4.7), we measured a significant ∆Sas well as a significant change in x over time that we determined to be due toan evolving orientation of a system with significant proper motion on the sky.We similarly used the statistical significance of the observed x˙ to constrain thevalue of Ω and the Shapiro-delay parameters while acknowledging ambiguitiesin the sign of i and Ω.The impact that these orbital variations can have in determining other in-trinsic quantities is most dramatically seen in our analysis of PSR B1534+12(Chapter 5), a DNS system with a compact, 10.1-hour orbit. We continue180to resolve five significant PK timing parameters with improved precision, aswell as a significant kinematic bias in our observed P˙b, when incorporatingnearly four years of data collected with the PUPPI backend after the studyconducted by Fonseca et al. (2014). We assumed the validity of GR anddetermined the component masses to bemp = 1.33302(17) M (6.1)mc = 1.34553(17) M (6.2)where the uncertainties reflect 68.3% confidence intervals as determined byTEMPO. At this time, the component masses for the PSR B1534+12 DNSsystem are the most precise neutron-star masses currently known. As asecondary study of relativistic gravity, we also conducted detailed analysesof electromagnetic pulse structure in PSR B1534+12 and quantified the ef-fect of geodetic precession by modeling secular changes in the profile shapeand polarization position angles at 430 MHz along with special-relativisticaberration of the signal due to orbital motion. We derived an estimate ofΩspin1 = 0.59+0.12−0.08 deg yr−1 that is consistent with the value predicted by GR,yielding a sixth PK parameter in this system for the first time. At thispoint in time, only PSRs B1534+12, J0737−3039A/B (Breton et al., 2008)and B1913+16 (Weisberg & Huang, 2016) yield at least six quantifiable PKparameters within one relativistic binary system.We also conducted a preliminary analysis of 27+ years of TOA data col-lected for PSR B1620−26 (Chapter 4), a radio pulsar that has long beeninterpreted as being embedded in a hierarchical triple system with a He WDand Jupiter-mass planet in the Messier 4 globular cluster. We measuredfifteen time-derivatives in νs that are due to the long-period motion of theouter planet with the inner pulsar-WD binary system, though we demon-strated that ν˙s contains significant components associated with globular-181cluster dynamics and intrinsic pulsar spin-down. We used the method pro-posed by Joshi & Rasio (1997) to relate the time-derivatives in νs to theouter-Keplerian elements. For the first time, we used only time-derivativesof higher order than ν˙s to compute the outer-orbital quantities when analyz-ing the entire TOA data set, which allowed for a unique determination ofthe orbital component in ν˙s. We derived an outer-orbital period of severaldecades, though analyses of data subsets or the entire TOA collection yieldconflicting estimates (see Section 4.3 and 4.4). The improving measurementsof inner-orbital secular variations and x¨i, which we believe are due to pro-longed three-body perturbations from the outer object, will eventually yielda direct measurement of the pulsar mass. Ongoing work will eventually leadto a simultaneous fit for both orbits, either using the two-orbit model withTEMPO or the novel three-body integrator first described by Ransom et al.(2014).6.1 Projections of Future PK MeasurementsThe primary means for future work with binary pulsars is the prolongedacquisition of high-precision TOAs over time, as it allows for the eventualresolution of secular variations and other relativistic terms that contain high-impact information. For new discoveries of binary pulsars, the presence ofa significant Shapiro delay can be quickly assessed using strategic observa-tions of specific orbital phases where the signal is expected to have maximumharmonic structure. We used data collected from targeted campaigns, firstdescribed by Pennucci (2015), to make the first measurement of ∆S in thePSR J2302+4442 system (Section 3.4.16) and the first significant measure-ment of mp in the J2043+1711 system. In both aforementioned systems, wemade a measurement of ∆S that passed the orthometric h3-significance test(Freire & Wex, 2010) using data sets that spanned less than two years ofobservation.182Continued timing of known radio pulsars will likely yield interesting rela-tivistic effects in the coming years. For example, the PSR J1600−3053 binarysystem (Section 3.4.3) currently exhibits a statistically marginal change in ωover time that is consistent with the predictions of GR. The precision in theω˙ measurement will improve over time such that the fractional uncertaintyin its measurement decreases as t−3/2 (Damour & Taylor, 1992). With thisscaling law, a ∼10% fractional uncertainty ω˙ will be achieved for J1600−3053by the year 2020. Similarly, the observed P˙b in the J1909−3744 system (Sec-tion 3.4.10) is dominated by the kinematic-acceleration component, thoughits uncertainty possesses the same order of magnitude as the expected com-ponent of intrinsic orbital decay due to GR. Since the relative uncertaintyin P˙b scales as t−5/2, the relative uncertainty in (P˙b)GR will reach ∼20% bythe year 2029. However, the measurability of (P˙b)GR depends on the correc-tion for the kinematic bias, and forthcoming studies will need to address theuncertainty associated with the distance-dependent correction.A recent study of the Hulse-Taylor DNS system has yielded the firstmeasurement of the measurable shape correction for eccentric orbits (δθ inEquation 1.26) in any binary system, as well as a marginal estimate of x˙that is likely due to classical spin-orbit coupling (Weisberg & Huang, 2016).While the δθ parameter may yield another unique test of gravitation, a mea-surement of x˙ due to classical spin-orbit coupling can be directly related tothe moment of inertia (Irot) for one or both neutron stars that can yield un-precedented constraints on neutron-star EOS models (e.g. Lattimer & Schutz,2005). However, the δθ measurement contains a bias due to aberration of thesignal from pulsar rotation (e.g. Damour & Taylor, 1992). Moreover, thecorrection for the intrinsic component of δθ due to GR, as well as the deter-mination of Irot from the spin-orbit-coupling component of x˙, both requirea measurement of several pulsar-orientation angles that have not yet beenmeasured in the Hulse-Taylor system. It is therefore currently not possibleto make a direct measurement of Irot in the PSR B1913+16 DNS system.183However, the required angles for correction of δθ and x˙ – namely  and λ =pi−α−β – have been estimated for PSR B1534+12, first by Stairs et al. (2004)and in Chapter 5 of this dissertation, by modeling the secular changes inpulse-profile shape and polarization position-angle data to ultimately derivea measure of Ωspin1 . In principle, then, our geometric measurements allowfor a full determination of the various components of δθ and x˙, as well as aunique determination of Irot. With ongoing observations of PSR B1534+12using the PUPPI backend at the Arecibo Observatory, we may likely be ableto eventually measure and correct for the component of δθ due to GR for thefirst time in the near future, as well as determine Irot from measurements ofspin-orbit coupling component of x˙. However, we do not currently measure δθor x˙ with statistical significance. Simulations of TOA data sets are thereforerequired in order to determine when in the future such measurements can bemade with sufficient precision.In order to perform robust simulations for “times of detection”, we firstcomputed the various components of δθ and x˙ that are possible for the PSRB1534+12 system. For δθ, we expect that only the GR and rotational-aberration components are significant:(δθ)GR = (Tnb)2/3( 72m2p +mpmc + 2m2c(mp +mc)4/3)≈ 5× 10−6 (6.3)(δθ)A =PsPbcsc i(1− e2)1/2sin ηsinλ≈ 1× 10−6 (6.4)so that (δθ)obs = (δθ)GR +(δθ)A ≈ 6×10−6 when using the values determinedby Fonseca et al. (2014) to compute the components of δθ. For the compo-nents of x˙, there are five possible terms that arise with varying degrees ofsignificance:184(x˙)GR = −645sin i(Tnb)2(1 + (73/24)e2 + (37/96)e4)(1− e2)7/2mpm2cmp +mc≈ −1.32× 10−17 (6.5)(x˙)D = x(D˙D)≈ 5.72× 10−18 (6.6)(x˙)A = −xPsPbΩspin1(1− e2)1/2(cotλ sin 2η + cot i cos η)sinλ≈ −2.58× 10−17 (6.7)(x˙)SO ≈ x cot ic2(2 +3mc2mp)Irot(2pi/Ps)(2pi/Pb)2(mp +mc)(1− e2)3/2 sinλ cos = −1.49× 10−15 (6.8)(x˙)µ = (−0.943 or − 1.25)× 10−15 (6.9)where the above equations are taken from Damour & Taylor (1992) andthe values in Equations 6.5-6.8 were computed using values estimated byFonseca et al. (2014). The possible values for (x˙)µ listed in Equation 6.9were first computed by Bogdanov et al. (2002) after determining that Ω =(70 or 290)± 20 degrees from orbital-dependent scintillation measurements.For the spin-orbit-coupling component – which we refer to in Equation 6.8as (x˙)SO – we used the assumption made by Weisberg & Huang (2016) thatthe contribution from the neutron-star companion (whose rotation rate isunknown) is negligible. With these expected components determined, it isclear that (x˙)obs ≈ (x˙)µ + (x˙)SO ≈ −3× 10−15. The component due to spin-orbit coupling is therefore expected to be among the most dominant effectsthat produce a non-zero x˙.We can also simulate the times of detection for several other possiblecomponents of ω˙, due to proper-motion bias (Equation 1.44) and spin-orbitcoupling. As first discussed for PSR J1640+2224 (Section 3.4.5), the spin-orbit component of ω˙ will generally be approximately equal to di/dt ≈(x˙/x)SO tan i. We therefore expect the non-GR components of ω˙ to be:185(ω˙)µ = µ csc i cos(Θµ − Ω) = (−6.61 or + 6.91)× 10−6 deg yr−1 (6.10)(ω˙)SO = (x˙/x)SO tan i ≈ −6.67× 10−6 deg yr−1 (6.11)where we used the estimates of Ω made by Bogdanov et al. (2002) to com-pute the expected changes from proper-motion bias. While both non-GRcomponents are comparable in order of magnitude, the allowed values for(ω˙)µ possess opposite signs and produce composite values of (ω˙)non−GR =(ω˙)µ + (ω˙)SO ≈ (−1.33 or 0.024) × 10−5 deg yr−1. For the purposes of sim-ulation, we chose an intermediate value of (ω˙)non−GR = −5 × 10−6 deg yr−1to estimate an approximate time of detection.We determined times of detection for δθ = 6× 10−6, x˙ = −3× 10−15 and(ω˙)non−GR = −5 × 10−6 deg yr−1 for PSR B1534+12 by using the currentbest-fit timing solution with the DD binary model presented at the end ofChapter 5, holding all parameters fixed, and producing fake TOAs that aremodeled by said timing solution. We assumed that observations began onMJD 48718 (i.e. the same start date of the real B1534+12 data set) and wereconducted once every 60 days, and that each observing epoch lasted 90 min-utes to produce 30 TOAs with an uncertainty of 8 µs for each TOA. We alsoneglected DM variations, computed barycenter-corrected TOAs, and injectedrandom-normal noise with a standard deviation of 1 µs. The results from oursimulations are shown in Figure 6.2, where each point with an uncertaintycorresponds to a best-fit estimate of the denoted parameter determined byTEMPO when simultaneously fitting for all model parameters, including theδθ, x˙ and (ω˙)non−GR.Our simulations suggest that significant, accurate estimates of x˙ and(ω˙)non−GR will roughly be made starting after the year 2040. Figure 6.2also suggests that δθ and x˙ are highly correlated when weakly constrained,as the best-fit estimates of both parameters track each other prior to the186Figure 6.2: Simulated times of detection for δθ, x˙ and (ω˙)non−GR in thePSR B1534+12 DNS system. For ease of comparison, all values are scaled topossess an order of magnitude of unity. The solid lines indicate the expected,simulated parameter values. The dots and error bars represent the best-fitvalues and uncertainties determined by TEMPO when simultaneously fittingfor all parameters, including δθ, x˙ and (ω˙)non−GR.year 2040. Once the x˙ measurement becomes robust, the δθ estimates gradu-ally becomes more accurate. However, our simulations show that δθ will notbe measured on reasonable timescales. Nonetheless, the eventual measure-ment of x˙ will allow for a direction computation of Irot using the angle andprecession-rate measurements presented in Chapter 5.1876.2 The Era of Neutron-Star Mass Measure-mentsThe first measurement of mp with the Shapiro timing delay in a pulsar-binary was made by Ryba & Taylor (1991) in their analysis of TOAs fromPSR B1855+09. Since then, a large and increasing number of pulsar-massmeasurements have been made such that meaningful analyses of the neutron-star mass distribution have recently begun. The distribution of neutron-starmasses can be directly inferred from available measurements of the Shapirotiming delay and its physical parameters, along with mass estimates thatwere derived from two or more PK variations. Recent work has shown thatan increasing number of these measurements can help delineate the roles ofdifferent supernovae processes in the formation of double-neutron-star binarysystems (e.g. Schwab et al., 2010) and assess the possible range of componentmasses for such systems (e.g. Martinez et al., 2015), as well as derive thestatistics for pulsar-binary populations that have evolved along different post-supernova evolutionary paths (O¨zel et al., 2012; Kiziltan et al., 2013).In this thesis, the significant estimates of mp span a range of 1.2−1.95 Min neutron star mass. PSRs J1614−2230 and J1918−0642 are at the high andlow ends of our overall mass distribution, respectively. The low mass of PSRJ1918−0642 is particularly interesting since this MSP possesses spin param-eters that are typical of an old neutron star that experienced significant masstransfer and a substantial spin-up phase. Nonetheless, our measurement ofmp = 1.18+0.10−0.09 M is comparatively low. The implication of a low “birthmass” for J1918−0642, believed to be an old neutron star, is inherently dif-ferent from recent studies of the binary evolution of PSRs J0737−3039A/B(Ferdman et al., 2013) and J1756−2251 (Ferdman et al., 2014), which areboth young pulsars believed to have been formed through electron-capturesupernovae that reduce the electron-degeneracy pressure within the progen-itor core and induces gravitational collapse at lower Chandrasekhar masses.188Furthermore, Antoniadis et al. (2016) recently used available measurementsof pulsar masses to argue that the MSP mass distribution is in fact bimodal.Given the wide range of spin and orbital periods in the MSP populationand a lack of clear correlation between the period/mass parameters, the twocomponents of the bimodal distribution derived by Antoniadis et al. likely re-flect inherently different MSP birth masses, as opposed to complex processesrelated to the mass-transfer period.As discussed in Section 3.5, significant measurements of ∆S allow forindependent tests of the theoretical correlation between Pb and mc fromtidal interactions during the phase of long-term mass transfer (e.g. Tauris& Savonije, 1999). Additional measurements of ∆S for long-period binarysystems will help constrain the Pb − mc relation over a wider range of Pbthan is currently seen in Figure 3.26.One of the frontier goals in high-energy astrophysics is the understand-ing of neutron-star structure and the microscopic processes that govern theinteriors of stellar-mass objects with mass densities that exceed nuclear sat-uration (ρsat ≈ 2.8 × 1014 g cm−3). Within the interiors of neutron stars,the relativistic pressure-density relation becomes theoretically uncertain andcurrently allows for a large number of proposed equations of state (EOSs)that differ in fractional composition of hadronic and pure-quark matter (e.g.Lattimer & Prakash, 2001; Lattimer & Prakash, 2004). Figure 6.3 illustratesa large number of proposed neutron-star EOSs1 , as well as the experimentalconstraints posed by radio-timing measurements of mp. In the case of PSRJ0348+0432 (see Antoniadis et al., 2013), optical radial-velocity estimatesof the companion mass and mass ratio were combined with the radio-timingmeasurement of orbital decay to yield a high-precision estimate of the pulsarmass.As shown in Figure 6.3, the low-mp measurement made for PSR J1918−06421The EOS data shown in Figure 6.3 can be downloaded at http://xtreme.as.arizona.edu/NeutronStars/.189Figure 6.3: Mass-radius relations for EOSs of neutron stars, shown as solidcurves, that reflect different underlying assumptions of internal composition.The EOS data used in this figure was compiled by O¨zel & Freire (2016)and first shown in Figure 7 of their study. Red bands represent mass mea-surements made in this dissertation, while the blue band represents the massestimate for PSR J0348+0432 made by Antoniadis et al. (2013). The lighter,gray lines represent EOSs that do not predict neutron-star masses larger thanor equal to the estimate made for PSR J1614−2230 in Chapter 3 of this dis-sertation.does not provide a meaningful constrain on the neutron-star EOSs since allcurves predict similar mass values for a range of radii. However, the high-mass estimate for PSR J1614−2230 exceeds the maximum-mass values forseveral EOSs, which are shown as gray lines in Figure 6.3. In principle, thePSR J1614−2230 mass measurement (first made by Demorest et al. (2010))invalidates the gray EOSs as physically plausible models of the neutron-starinterior. Estimates of larger pulsar masses in the future will therefore pro-vide even more stringent constraints on allowed compositions and EOSs of190neutron stars.It is clear that high-precision estimates of pulsar masses offer uniquelyfar-reaching impact on the areas of stellar-binary evolution, supernovae andneutron-star birth masses, tests of strong-field gravitation, and the ongoingefforts to constrain the equation of state for neutron stars. Ongoing studiesand future discoveries with premier radio facilities will surely aid in theseefforts. The forthcoming Square Kilometre Array (SKA) is projected todetect Shapiro-delay signals from ∼80% of all pulsar-binary systems withmc > 0.1 M and RMS TOA residuals of ∼50 µs, yielding unprecedentedaccess into the Galactic distribution of neutron-star masses and pulsar-binarysystem inclinations (Watts et al., 2015). 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A., Pennucci, T., Stovall, K., & Swiggum, J. 2015, ApJ,809, 41211Appendix AProbabilistic Analysis ofShapiro-Delay ParametersUsing an initial set of parameter values, the TEMPO pulsar-timing packagecreates a timing solution that best fits the supplied TOA data. The un-certainties reported along with the best-fit values are determined from thediagonal elements of the resultant covariance matrix, which is approximatedby the least-squares method for model determination as based on χ2 statis-tics. However, nonzero correlation between parameters can produce slightlylarger degrees of uncertainty, as well as nonlinear correlation between modelparameters that yield asymmetric uncertainties, that are not reflected in thereported TEMPO uncertainties.In Chapter 3, we employed a statistically rigorous approach to deter-mine more accurate uncertainties for the parameters of the Shapiro delayin fourteen NANOGrav binary pulsars. This approach, based on a methoddiscussed by Splaver et al. (2002), analyzes the accuracy and behavior ofthe timing model for different, fixed values of mc and sin i by generating agrid of χ2 values. The resulting distribution of χ2 values is then convertedto a two-dimensional probability density in the (mc, cos i) phase space usingEquation 3.4,212p(data|mc, cos i) ∝ e−(χ2−χ20)/2 (A.1)and a series of mathematical operations are used to convert this density toa posterior probability distribution in the (mp, cos i) phase space.In this Appendix, we outline the theory and procedure in more detail.We also provide the equations used to obtain the two-dimensional probabil-ity distributions and one-dimensional probability density functions (PDFs)shown in Chapter 3.A.1 Bayesian Interpretation of χ2-grid Anal-ysisIn the context of a Bayesian analysis of probabilities, the use of prior informa-tion for certain outcomes takes on an important role when determining thefinal, “posterior” evaluation of said outcomes. This interpretation of prob-ability allows for a powerful set of analysis techniques, such as the MarkovChain Monte Carlo (MCMC; Wall & Jenkins, 2003; Gregory, 2005b) methodused in Chapters 3, 4 and 5 to probe the parameter space, evaluate regionsof larger or lower likelihood for best fits based on relative changes in χ2 val-ues, and bias the random walk towards regions of maximum likelihood. Thefundamental concepts of Bayesian statistics are embodied in Bayes’ theorem,which states the following: the posterior probability that a set of parametersdescribe the supplied data – p(model|data) – is given asp(model|data) = p(data|model)p(model)p(data)(A.2)where p(data|model) is the likelihood function, p(model) is the prior proba-bility of the model parameters, and p(data) is the “Bayesian evidence.” Theprior probability is interpreted as one’s prior knowledge in the distributionof probable values for each parameter that forms the model of the data in213hand. In the case where it is unknown which value a given parameter is morelikely to have, a uniformly random distribution is chosen for that parameter:in the absence of prior knowledge, all values between a chosen interval areequally likely to be the parameter value.The Splaver et al. (2002) method for determining accurate confidenceintervals of the Shapiro-delay parameters from a χ2 grid uses Equation A.2in order to relate the likelihood density (Equation 3.4) to the joint-posteriorprobability density of mc and cos i. In their formulation, they compute agrid of values of mc and cos i, and assume prior probability distributionsfor both parameters. From this assumed form of p(mc, cos i), the posteriordistribution can be readily computed since, from Equation A.2,p(mc, cos i|data) ∝ p(data|mc, cos i) (A.3)and the Bayesian evidence is a constant, normalizing factor. The uniformprior-probability distribution in mc reflects our ignorance of the binary com-panion and its stellar type. For example, in the absence of prior information,a main-sequence companion star can have any mass, whereas a white-dwarfcompanion has an upper limit on physically allowed masses that correspondsto the Chandrasekhar limit. In such cases, prior information could be ob-tained from optical photometry and/or radial-velocity measurements fromspectroscopy, which yield a measurement of the component-mass ratio. Asdiscussed in Chapter 1, the upper limit for neutron-star companion massesis less certain but is expected from recent numerical studies to be ∼ 3 M.The uniform prior distribution in cos i is chosen to reflect random orientationof the orbit, which is discussed in more detail in Section A.2 below.214A.2 Derivation of Uniform Distribution forRandomly Oriented OrbitsLet us first consider an orbit with inclination i and longitude of ascendingnode Ω, such as the example orbit shown in Figure 1.2. Since the orbitalangular moment vector (~Sb) is perpendicular to the orbital plane, the orien-tation angles (i, Ω) also quantify the direction of ~Sb.Let us now consider a collection of orbits with different Keplerian param-eters, as well as different values of i and Ω. The set of ~Sb in this sample issaid to be randomly distributed if it is isotropic. It is clear that the Keplerianelements do not affect orientation, and so the magnitude of ~Sb does not con-tribute to random orientation. We can therefore consider the probability of~Sb pointing in some direction and require isotropy when determining whichfunctions of i and Ω should have uniformly random distributions.The probability that ~Sb points in a certain direction can be evaluated byfirst considering an infinitesimal solid angle (dφ) that contains the directionof ~Sb. If the solid angle spans a range of i and i+ di, as well as a range of Ωand Ω +dΩ, then the probability that the direction of ~Sb is contained withindφ isp =dφtotal solid angle=14pisin ididΩ (A.4)We can integrate Equation A.5 to consider the probability that ~Sb is con-tained within a finite patch of the sky, subtended by orientation angles (i0,Ω0) and (i, Ω):P (i,Ω) =14pi∫ ii0sin i′di′∫ ΩΩ0dΩ′=14pi(cos i0 − cos i)(Ω− Ω0). (A.5)215In order to determine the probability distribution functions (PDFs) whoseuniformity correspond to random orbital orientations, we must make use oftwo properties of probability theory:1. the probability of the occurrence of two independent events A and B(e.g. a specific outcome from two rolls of dice) is given as P (A and B) =P (A)P (B), and2. a random variable x is said to be uniformly distributed if its cumulativedistribution function (CDF) is proportional to the variable.Using property 1, we can view Equation A.5 as a joint probability distributionthat is made up of the product of two one-dimensional probability distribu-tions, i.e. P (i,Ω) = Q(i)R(Ω). We also see that Q = (cos i0 − cos i)/2 andR = (Ω−Ω0)/2pi are each normalized cumulative distribution functions of iand Ω, respectively, since each integral in Equation A.5 yields the probabilitythat their values lie between the bounds of integration. Using property 2,we can finally impose isotropy of ~Sb by requiring that the uniformly-randomvariables be cos i and Ω, since Q ∝ cos i and R ∝ Ω.A.3 Translations of Probability DensitiesAs discussed in Chapter 3 and Section A.1, the computation of p(data|mc, cos i)and use of a joint-uniform prior – p(mc, cos i) ∝ constant – allow for astraightforward determination of the posterior probability density p(mc, cos i|data)for both Shapiro-delay parameters from a map of χ2 values. We eventuallywant to obtain the marginalized, one-dimensional posterior PDFs from thecomputed maps:216p(mc|data) =∫ 10p(mc, cos i|data)d(cos i) (A.6)p(cos i|data) =∫ mc,max0p(mc, cos i|data)d(mc) (A.7)where mc,max is the maximum value of the companion mass defined on theχ2 grid. It is important to note that, while −1 < cos i < 1, the Shapiros = sin i is always defined to be positive since 0 < i < pi. We cannotuniquely determine the true value of i by analyzing only the Shapiro delay,but can instead compute two values of i that correspond to positive andnegative values of cos i. We therefore restrict the χ2-grid values for cos i torange from 0 to 1 when only analyzing the Shapiro-delay parameters, sinceno new information is obtained when compute a χ2 grid over the completerange of cos i.The binary mass function (Equation 1.37) can be solved for the valueof mp as a function of both mc and sin i. In order to properly obtain theposterior PDF for mp, we must project the computed p(mc, cos i|data) into adifferent phase space that is spanned by mp along one of the two dimensions.We arbitrarily choose the (mp, cos i) space since, in practice, the probabilitydensity in the (mp, mc) space is heavily truncated to a small slice and difficultto resolve using finite bin sizes. The probability density in the (mp, cos i) canbe determined by requiring that the marginalized posterior PDF of cos icomputed in both phase spaces be equal. In other words,p(cos i|data) =∫ ∞0p(mp, cos i|data)d(mp)=∫ ∞0p(mc, cos i|data)∣∣∣∣∂(mc)∂(mp)∣∣∣∣d(mp) (A.8)where the second line of Equation A.8 was obtained by writing Equation A.7217as an integral over mp. The absolute value of the derivative ensures that theprobability density (and marginalized PDFs) remain positive. The top andbottom forms of Equation A.8 are equal if the integrands are equal, so thatp(mp, cos i|data) = p(mc, cos i|data)∣∣∣∣∂(mc)∂(mp)∣∣∣∣ (A.9)where the notation for partial derivatives is used to emphasize that cos i isfixed in Equations A.8. The derivative must be evaluated using the binarymass function (Equation 1.37),fm =n2bx3T=(mc sin i)3(mp +mc)2,which is a constant quantity1 that relates mp, mc, and sin i. The derivativecan be computed through implicit differentiation of Equation 1.37 to yield∂(mc)∂(mp)=2fm(mp +mc)3m2c sin3 i− 2fm(mp +mc)(A.10)Equation A.9 implies that the probability distributions are continuous.In practice, however, the χ2 grids and derived probability densities consistof finite grid bins, and so the integrals in Equation A.7 turn into discretesums. Moreover, several grid bins in one phase space can be encompassedby a singe bin in another phase space. However, p(mc, cos i|data) can beapproximately translated to the (mp, cos i) space by averaging together theprobability in finite bins across mc that fall within a corresponding bin inmp. The approximation becomes more accurate when smaller grid bins areused.1For the most relativistic binary systems (e.g. PSR B1534+12), Pb intrinsically changesover time due to the emission of gravitational radiation from the system. However, therates of change in Pb are typically very small, and the secular change ultimately does notsignificantly change the value of fm over the time span of the data set.218A.3.1 Probability in Orthometric SpaceThe orthometric model for the Shapiro timing delay (Freire & Wex, 2010)parametrizes the effect to using two different PK parameters. The choice ofspecific parameters depends on the degree of inclination and orbital eccentric-ity, as the orthometric method was developed by Freire & Wex to carefullyreduce correlation between the PK parameters:• if the binary model is ELL1 and i < 50 degrees, the third and fourthFourier harmonics of ∆S (h3 and h4, respectively) best characterize theShapiro delay;• if the binary model is ELLI and i > 50 degrees, or if the binary model isDD, then h3 and the orthometric ratio ς = h4/h3 are the ideal Shapiro-delay parameters.As with the traditional (mc, cos i) parameters, a χ2-grid analysis can be per-formed using the orthometric parameters in order to evaluate correlation andcompute more robust confidence intervals. The nonlinear relation betweenthe traditional and orthometric Shapiro-delay parameters (Equations 3.1-3.3) require additional, careful translation of the posterior density derivedfrom the χ2 grid over the orthometric parameters to the physical parametersof interest. Furthermore, the nonlinear relation amounts to a difference inchoice of prior probability densities between the traditional and orthometricparameters.For instance, let us consider a two-dimensional posterior probability den-sity in the (h3, ς) phase space. This map can be converted to one in the(mc, cos i) space by using the translation rule for multivariate probabilitydistributions,p(mc, cos i|data) = p(h3, ς|data)∣∣∣∣ ∂(h3, ς)∂(mc, cos i)∣∣∣∣ (A.11)219where ∂(h3, ς)/∂(mc, cos i) is the determinant of the Jacobian matrix thatmaps volume elements between phase spaces:∂(h3, ς)∂(mc, cos i)= det[∂(h3)∂(mc)∂(h3)∂(cos i)∂ς∂(mc)∂ς∂(cos i)]=∂(h3)∂(mc)∂ς∂(cos i),since ς is only a function of cos i. The posterior density in the orthomet-ric space can therefore be directly translated to the (mc, cos i) space usingEquation A.11. The same procedures can then be applied to obtain themarginalized PDFs for mp, mc and cos i.A.4 Confidence IntervalsThe primary goal of the above computations is to determine robust mea-sures of the best-fit parameter and its degree of uncertainty. With the one-dimensional posterior PDFs at hand, we take the “best-fit value” of, say, thepulsar mass as the median value of its posterior PDF. The median value ofmp (and, similarly, mc and cos i) corresponds to the point on the PDF where50% of the probability lies above and below it. In other words, we integratethe PDF up to a value of mp,med such that∫ mp,med0p(mp|data) = 0.5. (A.12)The median values of mc and cos i are computed in a similar manner usingtheir appropriate posterior PDFs.There are several ways to compute confidence limits from the one-dimensionalposterior PDFs of the Shapiro-delay parameters. For example, Splaver (2004)define the 68% confidence interval as the range that both encompasses 68%of the total probability and spans the shortest range of the parameter val-ues. Splaver et al. (2002) used this definition of confidence limits in theiranalysis of the PSR J0621+1002 binary system and its Shapiro-delay param-eters. While the “shortest interval” method is a common one for uncertainty220determination, we instead choose an “equal tail” method to compute con-fidence limits. For the 68.3% confidence interval, we find the lower boundof the range (mp,lo) by integrating the posterior PDF up to a value thatencapsulates 15.85% of all probability,∫ mp,lo0p(mp|data) = 0.1585 (A.13)and find the upper bound of the same range (mp,up) by integrating up to avalue that encapsulates 84.15% of all probability,∫ mp,up0p(mp|data) = 0.8415. (A.14)We refer to this set of confidence intervals as “equal tail” since mp,lo andmp,up serve as lower and upper bounds of the posterior tails that have equalprobability.221Appendix BSpin-Derivative Model ofLong-Period OrbitsThe last radio-timing study of PSR B1620−26 performed by Thorsett et al.(1999) showed that a direct fit of the binary Ro¨mer timing delay for the outerorbit could fully explain the observed spin derivatives, but could not uniquelydetermine all outer-orbital elements. Thorsett et al. instead fitted (∆R)o totheir 11-year data set for different values of the outer-orbital eccentricityeo, and found that all orbital elements indeed varied with increasing eo.In particular, the smallest outer-orbital period obtained using this method,corresponding to eo = 0 (a circular outer orbit), was found to be (Pb)o ≈62 years, much longer than the 11 years spanned by their data set. Thedegeneracy of timing parameters for (∆R)o is therefore due to significantlack of coverage across a full outer orbit.Joshi & Rasio (1997, JR97) noted that, in the case where a significantfraction of a full pulsar-binary orbit has not been spanned by the data set,the measured Doppler-induced time derivatives of νs could be used to inferthe orbital elements of the system. To first order in its derivation, the JR97framework can be applied to TOAs from PSR B1620−26 since Thorsett et al.(1999) showed that both orbits could be jointly represented as the sum of222two non-interacting Keplerian orbits: ∆R = (∆R)i + (∆R)o. In this sense, ahierarchical triple system can be viewed as a “binary” system where one of the“binary” components has a mass equal to the sum of the inner-componentmasses, and a center of mass that is approximately equal to the center ofmass of the inner-binary system.In this section, we describe the model developed by JR97 to estimatethe elements using the frequency-derivative method, and derive additionalequations for higher-order time derivatives in order to analyze an updatedtiming solution for our current TOA data set for PSR B1620−26 that wediscuss in Chapter 4.B.1 Derivation of Orbit-Induced Spin Deriva-tivesWe first consider a binary orbit, with eccentricity e, of two bound componentswith masses m1 and m2. In the context of PSR B1620−26, m1 = mp + (mc)iis the “primary” mass and m2 = (mc)o is the “companion” mass. The cor-responding semi-major axes of each component are a1 and a2, respectively.Due to the symmetry of the (non-relativistic) two-body problem, the eccen-tricities and true anomalies of both orbits about the center of mass are equal,though the periastron arguments are shifted such that ω1 = ω2 + pi.For radio pulsars, the Doppler shift in νs due to binary motion is generallygiven as νs = −νs,0(v · nˆ)/c, where v is the orbital velocity of the observedcomponent, nˆ is a unit vector pointing along the line of sight to the system,and νs,0 is the spin frequency in a reference frame that is co-moving withthe pulsar in its binary motion.1 In our current timing solution, as well asthe model published by Thorsett et al. (1999), the inner orbit is explicitly1In practice, the absolute radial motion of pulsars and pulsar-binary systems throughspace induces a constant Doppler shift in the true, intrinsic values of certain timing param-eters (e.g. spin/binary periods, projected semi-major axis, etc.), and so the “observed”values are different than the true values.223modeled by determining the inner Ro¨mer delay; therefore, all time-derivativesin spin frequency are due to intrinsic spin-down, biases from secular motionof the system, or orbital motion of the outer binary. We discuss the effectsof the non-binary components on our analysis in Section B.3 below.Binary motion induces a number of higher-order time derivative in νs dueto periodic Doppler shifts:ν(l)s =dldtlνs ≈ −νs,0 (a(l−1) · nˆ)c(B.1)where a = v˙ is the acceleration of the observed component. In the ap-proximation of Equation B.1 we ignored terms that were nonlinear in timederivatives of νs, which are typically small for pulsar-binary systems. Forlarge orbits with widely-separated components, the general-relativistic cor-rections of the orbital motion can be ignored for the purposes of first-ordercalculations; the acceleration of the pulsar is therefore given by Newtonianinverse-square law, a = |a| = kr−21 , wherek = Gm32(m1 +m2)2(B.2)h = a1(1− e2) (B.3)r−11 = h−1(1 + e cosu) ≡ h−1A (B.4)(B.5)and where A = 1+e cosu is a function of the true anomaly u. By computingthe dot product in Equation B.1 and taking derivatives, we find that thecomponent of the first derivative in νs due to orbital motion isν˙s = −νs,0KA2 sin(u+ ω1), (B.6)where K = k sin i/(h2c). While the constant K is not immediately known,the higher-order derivatives will also be proportional to K, and so we can224use Equation B.6 to write the higher-order derivatives in terms of ν˙s, whichis usually measured for radio pulsars:ν(2)s =ν˙sA2 sin(u+ ω1)Bu˙ (B.7)ν(3)s =ν˙sA2 sin(u+ ω1)Cu˙2 (B.8)ν(4)s =ν˙sA2 sin(u+ ω1)Du˙3 (B.9)ν(5)s =ν˙sA2 sin(u+ ω1)Eu˙4 (B.10)ν(6)s =ν˙sA2 sin(u+ ω1)Fu˙5 (B.11)and where the coefficients have the following form:B = 2AA′ sin(u+ ω) + A2 cos(u+ ω1) (B.12)C = B′ + 2BA′A(B.13)D = C ′ + 4CA′A(B.14)E = D′ + 6DA′A(B.15)F = F ′ + 8EA′A(B.16)where the apostrophe denotes a derivative with respect to u. EquationsB.7-B.11, along with the definition A = 1 + e cosu, therefore show thatfour quantities are directly measurable using the JR97 technique: {e, ω1, u,u˙}. At a minimum, five measured time derivatives are needed in order touniquely solve for the four orbital parameters, since four of the orbit-inducedderivatives (Equations B.7-B.10) are written in terms of ν˙s.225B.2 Determination of ParametersWe used a multi-dimensional Newton-Raphson method (Press et al., 1986)in order to solve a system of nonlinear equations fk(x) = 0, where x is a“state” vector with the unknown orbital parameters as components, and fk isa vector of functions with components (fk)i = ν(i+k)s −(ν(i+k)s )obs. The i indexdenotes the i-th component of fk, which has the same number of elementsas x, while the k index denotes the set of spin-frequency derivatives to beused for computation of the orbital elements. For example, if x = {ωo, u, u˙},then i runs from 1, 2, and 3, and the components of fk=1 are the threetime-derivatives given by Equations B.7-B.9, minus their observed values.If instead x = {eo, ωo, u, u˙}, the “full” state vector of orbital parametersthat are directly measurable, then the components of fk=1 are the four time-derivatives given by Equations B.7-B.10 minus their observed values. We usethis notation in order to simplify the discussion below when using differentsets of time-derivatives to derive the components of x, which is presented inthe following section of this Appendix.Using the iterative Newton-Raphson method for finding roots of a systemof equations, the best approximations of x can be determined by computingxn+1 = xn − J−1fk(xn), (B.17)where J is the Jacobian matrix of partial derivatives, with components J ij =∂(fk)i/∂xj. With an initial guess of x, Equation B.17 is computed repeatedlyusing the iteratively-updated state vector until a chosen criterion for the bestapproximation of x is satisfied.For a one-dimensional Newton-Raphson problem, one typically choosesthe “best fit” criterion that the function under consideration be approxi-mately equally to a small value close to zero, say f < 10−12. For the multi-dimensional case we consider here, a complication occurs from the fact thatthe components of fk each have different physical units and orders of mag-226nitude. Since all components of fk are theoretically equal to zero, we scaleeach component of fk by factors that yield values with orders of magnitude∼ 100. After this arbitrary scaling, we assumed the best-fit criterion to bethat the length of fk is less than 10−12.If only four time-derivatives are significantly measured, then x = {ωo, u, u˙}and a value of eo must be chosen and held fixed in order to use EquationB.17 to obtain a solution of x. In this way, one can obtain a “family” ofsolutions using the JR97 method, where a set of x is determined for a rangeof values of eo. JR97 used initial estimates of their significantly-measuredfrequency derivatives, from ν˙s up to ν(4)s , to find a family of solutions fordifferent, fixed values of ωo = ω1 − pi, while Thorsett et al. (1999) used theirupdated values of the spin-frequency derivatives to find a set of solutions interms of eo. For the purposes of comparison, we used the definitions adoptedby Thorsett et al. (1999), where x = {ωo, u, u˙} and k = 1, so that EquationsB.7-B.9 made up the components of fk=1. We then used Equation B.17 tofind the best approximations of x for different, fixed values of eo.If at least five time-derivatives are significant measured, then the fullstate vector x = {eo, ωo, u, u˙} can be uniquely approximated to find a sin-gle solution for fk=1(x) = 0. Ford et al. (2000a) provided the first uniquesolution for x using the time-derivatives reported by Thorsett et al. (1999),and determined the outer orbit to be highly eccentric, with eo ≈ 0.45 and(Pb)o ≈ 308 years. However, the value of ν(5)s was not measured with statis-tical significance.B.3 Complications from Spin-downOne of the major complications of the JR97 method is that the observedν˙s is not purely due to Doppler shifts from binary motion. As discussed inSection 1.1, radio pulsars generally exhibit an intrinsic spin-down in the formof a first derivative in νs due to magnetic dipole radiation. Moreover, the227significant kinematic effects from proper motion, differential Galactic rotationand gravitational acceleration in the Galactic potential discussed in Section1.5 will produce a change in Doppler shifts of Ps in the same manner observedfor the orbital periods of PSRs B1534+12 (Chapter 5), J1614−2230 (Section3.4.4) and J1909−3744 (Section 3.4.10). The observed first-derivative in spinfrequency is therefore a sum of three components:(ν˙s)obs = (ν˙s)int + (ν˙s)o + (ν˙s)D (B.18)where the “D” subscript denotes the component secular accelerations thatproduce changes in the Doppler shift that is discussed in Section 1.5. Ingeneral, the intrinsic and secular-acceleration terms in the first-derivativeare not separately measurable unless (P˙b)D is significant. While (ν˙s)int < 0from physical arguments, the sign of (ν˙s)o can be positive or negative andwill vary over time.JR97 and Thorsett et al. (1999) pointed out that assuming (ν˙s)o = (ν˙s)obsdid not produce a significantly large difference in their results when comparedto those obtained under the assumption that (ν˙s)o = 0.01(ν˙s)obs. This bias in(and the ad hoc adjustment of) the first derivative nonetheless complicatesthe unique determination of the outer-orbital elements.A solution to this problem can be obtained by noting that, in the absenceof timing noise and encounters with nearby stars, the second and higher-order time-derivatives in νs should be entirely due to binary motion. We cantherefore use Equation B.7 to put Equations B.8-B.11 in terms of ν(2)s ,228ν(3)s =ν(2)sBCu˙ (B.19)ν(4)s =ν(2)sBDu˙2 (B.20)ν(5)s =ν(2)sBEu˙3 (B.21)ν(6)s =ν(2)sBFu˙4 (B.22)and entirely avoid the use of ν˙s in the computations of Equation B.17 to ap-proximate x. The use of these “unbiased” derivatives requires a sufficientlylong data span in order to measure them and derive the elements. However,in the case of our ongoing analysis of PSR B1620−26 (see Chapter 4), wemeasure a large number of spin-frequency derivatives with statistical signif-icance. We refer to the sets of derivatives listed in Equation B.19 as fk=2in subsequent discussion, and use this approach in Section 4.3.4 in orderto compare the “biased” and “unbiased” results obtained using the JR97method.In Section 4.4, we discuss another potential component of ν˙s and ν¨s dueto accelerations and jerks, respectively, from stars in the Messier 4 globularcluster in which PSR B1620−26 resides. Such components are unique toglobular-cluster pulsars and can be used to constrain information on localmass densities in the cluster (Phinney, 1992; Blandford et al., 1987). How-ever, such analyses are not currently possible with PSR B1620−26 due tothe frequency-derivative model of the outer orbit, and we instead considertheir impacts on measurements we present and discuss in Chapter 4.229B.4 Derived Quantities of the OrbitThe directly-measurable quantities of the JR97 method, which collectivelyform the components of x, can be used to derive other orbital parametersof interest. JR97 derived the relations for a2 = m1a1/m2 and m2 sin i, usingEquations B.2 and B.3, in terms of the first time-derivative in spin frequency:m2 sin i ≈ − ν˙scνs sin(u+ ω)(m21A2Gu˙4)1/3(B.23)a2 =m1ν˙scA2νs(m2 sin i) sin(u+ ω)u˙2(1− e2) (B.24)where the approximation in B.23 uses the assumption that m2 << m1. Inthe case of PSR B1620−26, m1 = mp + (mc)i and m2 = (mc)o. We assumethat m1 ≈ 1.65 M, and Thorsett et al. determined m2 ∼ 10−3 M. Thiscorresponds to m2/m1 << 1, which satisfies the approximation made inEquation B.23. The outer-orbital period can therefore be computed usingKepler’s third law and carrying through the lowest-order approximation oflow companion mass:Pb ≈ 2pi√a32Gm1(B.25)230

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