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Wideband timing of the double pulsar (PSR J0737-3039A) Grandy, Victoria Rebecca 2016

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Wideband Timing of the Double Pulsar (J0737-3039A)byVictoria Rebecca GrandyB.Sc. (Hons), Memorial University of Newfoundland, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Astronomy)The University of British Columbia(Vancouver)December 2016c© Victoria Rebecca Grandy, 2016AbstractPulsars are neutron stars (NS) that produce beamed radio-frequency emission. Dueto their rapid, steady rotation rate, this signal is detected as a series of pulses whoseintegrated profile is unusually stable over time. Pulsars in double neutron star(DNS) binary systems are a rare, but extremely useful, astronomical tool and havebeen used in tests of gravity theories in the strong-gravitational field limit. Rarerstill are DNS systems in which both objects have been detected as pulsars; onlyone such system has been found thus far – PSR J0737-3039A/B. Discovered overa decade ago (Burgay et al. 2003; Lyne et al. 2004), this system consists of onerecycled pulsar, PSR J0737-3039A, and its companion, PSR J0737-3039B, whichhas since become undetectable.In any pulsar-related research, precise timing is necessary to produce meaning-ful results. The pulse time of arrivals (TOAs) are greatly affected by the mediumthrough which the electromagnetic (EM) signal travels in both frequency-dependentand -independent ways. Even after accounting for such effects, many pulse pro-files still exhibit frequency-dependent shape changes, which can greatly affect theprecision of the timing results. Traditionally, corrections are applied to the TOAsafter calculation in an ad hoc manner. In contrast to this, we explored the wide-band timing algorithm developed by Pennucci et al. (2014) which accounts forfrequency-dependent profile changes through a two-dimensional Gaussian pulseportrait model implemented in the TOA calculations. It was found that the por-trait model is well-representative of the pulse profile shape over a wide frequencyrange. This method is also able to produce a robust set of wideband TOAs. Thesubsequent timing model, determined with TEMPO timing software, was foundto be comparable to those produced from subbanded TOAs derived though moreiitraditional methods. Some inconsistencies between the timing model astrometricand spin parameters of the wideband and subbanded data of this well-studied pul-sar imply potential difficulties in achieving precise timing results not only for thispulsar, but for others, such as those used in pulsar timing arrays aiming to detectgravitational waves.iiiPrefaceSome data and the main algorithm employed in analyses within this work havebeen utilized in other studies. As well, the acquisition of PSR J0737-3039A datahas been a collaborative effort, as noted below:• The wideband timing algorithm used in the work of this thesis was devel-oped by Pennucci et al. (2014) as the basis of his doctoral studies (Pennucci,2015). This algorithm, which utilizes python packages and PSRCHIVE pul-sar analysis software, is freely available online1.• Data used in the timing analysis of PSR J0737-3039A were acquired at theRobert C. Byrd Green Bank Telescope (GBT) using the Green Bank UltimatePulsar Processing Instrument (GUPPI), with central observing frequenciesof 820 MHz and 1500 MHz. These data were collected by M. Kramer, I.Stairs, M. McLaughlin, A. Possenti, P. Freire, R. Ferdman, and V. Grandy.• Data files employed in radio frequency interference (RFI) mitigation of theobservational data were provided through private communication with R.Ferdman.• The “subbanded” time of arrival (TOA) values were provided by I. Stairsand R. Ferdman and span from June 2009 to June 2015. These will be in-cluded in analysis presented in a forthcoming paper by Kramer et al. TOAswere derived from data taken at the Robert C. Byrd GBT using GUPPI, withcentral observing frequencies of 820 MHz and 1500 MHz. The pulse profile1https://github.com/pennucci/PulsePortraitureivtemplate was produced using the psrsmooth routine developed by Demorestet al. (2013).• Data from the Parkes 64-m telescope was provided by R. Manchester. Thesefilterbank observations were taken between May 2003 and August 2004,with centre frequencies of 680 MHz, 1390 MHz, and 3030 MHz. Thesedata were the subject of analysis by Manchester et al. (2005).Everything other than the above is an original, unpublished, work by the author, V.Grandy, with support from many others, including Tim T. Pennucci, and invaluableguidance from I. H. Stairs.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction to Pulsars . . . . . . . . . . . . . . . . . . . . . . . 21.1.1 Discovery . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Properties of Pulsars . . . . . . . . . . . . . . . . . . . . 41.1.3 Importance of Pulsar Research . . . . . . . . . . . . . . . 71.2 The Double Pulsar . . . . . . . . . . . . . . . . . . . . . . . . . 82 Pulsar Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1 Pulse Profiles and Arrival Times . . . . . . . . . . . . . . . . . . 112.2 Effects of the Interstellar Medium . . . . . . . . . . . . . . . . . 132.2.1 Dispersion Measure . . . . . . . . . . . . . . . . . . . . 132.2.2 Scintillation . . . . . . . . . . . . . . . . . . . . . . . . . 14vi2.2.3 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 De-dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.1 Incoherent De-dispersion . . . . . . . . . . . . . . . . . . 162.3.2 Coherent De-dispersion . . . . . . . . . . . . . . . . . . 162.4 Profile Evolution with Frequency . . . . . . . . . . . . . . . . . . 172.5 Timing Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.1 Data Acquisition and Handling . . . . . . . . . . . . . . . . . . . 203.1.1 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . 203.1.2 Data Handling . . . . . . . . . . . . . . . . . . . . . . . 223.2 Traditional TOA Fitting Method . . . . . . . . . . . . . . . . . . 233.3 Wideband Timing Algorithm . . . . . . . . . . . . . . . . . . . . 243.3.1 Gaussian Modelling . . . . . . . . . . . . . . . . . . . . 253.3.2 Time of Arrival Calculations . . . . . . . . . . . . . . . . 334 Results & Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.1 Gaussian Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.1.1 Profile Evolution . . . . . . . . . . . . . . . . . . . . . . 344.1.2 Comparison of Model to Observational Data . . . . . . . 404.1.3 Qualitative Comparison with Wider Range of Bands . . . 504.2 Timing Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.2.1 DMX Fitting and Dispersion Measure . . . . . . . . . . . 544.2.2 Timing Model and TOA Residuals . . . . . . . . . . . . . 585 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68A Technical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73A.1 Zapping Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73A.2 Wideband Timing Algorithm . . . . . . . . . . . . . . . . . . . . 77A.2.1 PPALIGN . . . . . . . . . . . . . . . . . . . . . . . . . . 78A.2.2 PPGAUSS . . . . . . . . . . . . . . . . . . . . . . . . . 78viiA.2.3 PPTOAS . . . . . . . . . . . . . . . . . . . . . . . . . . 78viiiList of TablesTable 1.1 Properties of PSR J0737-3039 . . . . . . . . . . . . . . . . . . 8Table 3.1 Observational date range . . . . . . . . . . . . . . . . . . . . 21Table 3.2 Epochs included in Gaussian modelling from GUPPI coherentde-distpersion mode 820 MHz data set . . . . . . . . . . . . . 26Table 3.3 Epochs included in Gaussian modelling from GUPPI coherentde-distpersion mode L-band data set . . . . . . . . . . . . . . 26Table 3.4 Join file offset parameters for 19 Gaussian model . . . . . . . . 30Table 3.5 Gaussian Model Parameters . . . . . . . . . . . . . . . . . . . 32Table 4.1 EFAC parameter values used in final timing model . . . . . . . 54Table 4.2 Timing parameter uncertainties . . . . . . . . . . . . . . . . . 60Table 4.3 Astrometric and spin parameters . . . . . . . . . . . . . . . . 65ixList of FiguresFigure 1.1 Pulsar magnetosphere . . . . . . . . . . . . . . . . . . . . . 6Figure 1.2 Earth’s line of sight to PSR J0737-3039A . . . . . . . . . . . 9Figure 2.1 Integrated pulse profile of PSR J0737-3039A . . . . . . . . . 12Figure 3.1 Example initial Gaussian fit of PSR J0737-3039A from wide-band timing algorithm . . . . . . . . . . . . . . . . . . . . . 28Figure 4.1 Profile evolution of 820 MHz coherent de-dispersion mode data 36Figure 4.2 Profile evolution of 820 MHz incoherent de-dispersion modedata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Figure 4.3 Profile evolution of L-band coherent de-dispersion mode data 38Figure 4.4 Profile evolution of L-band incoherent de-dispersion mode data 39Figure 4.5 PSR J0737-3039A Normalized Pulse Portrait . . . . . . . . . 41Figure 4.6 19 Gaussian Model Normalized Pulse Portrait and Residuals . 42Figure 4.7 Normalized ‘19 Gaussian model’-predicted pulse profile around820 MHz . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Figure 4.8 Normalized ‘19 Gaussian model’-predicted pulse profile around1500 MHz (L-band) . . . . . . . . . . . . . . . . . . . . . . 44Figure 4.9 Comparison of 19 Gaussian model with 820 MHz coherent de-dispersion mode data . . . . . . . . . . . . . . . . . . . . . . 46Figure 4.10 Comparison of 19 Gaussian model with 820 MHz incoherentde-dispersion mode data . . . . . . . . . . . . . . . . . . . . 47Figure 4.11 Comparison of 19 Gaussian model with L-band coherent de-dispersion mode data . . . . . . . . . . . . . . . . . . . . . . 48xFigure 4.12 Comparison of 19 Gaussian model with L-band incoherent de-dispersion mode data . . . . . . . . . . . . . . . . . . . . . . 49Figure 4.13 Qualitative comparison of 19 Gaussian model and Parkes 64-mTelescope data . . . . . . . . . . . . . . . . . . . . . . . . . 52Figure 4.14 DM values from wideband timing algorithm . . . . . . . . . . 56Figure 4.15 DMX and bin-averaged wideband DM values . . . . . . . . . 57Figure 4.16 DMX value correlation . . . . . . . . . . . . . . . . . . . . . 58Figure 4.17 Daily average timing residuals . . . . . . . . . . . . . . . . . 62Figure 4.18 Orbital phase bin-averaged timing residuals . . . . . . . . . . 63xiGlossaryDM Dispersion MeasureDNS Double Neutron StarEM ElectromagenticFWHM Full-Width Half-MaximumGBT Green Bank TelescopeGR General RelativityGUPPI Green Bank Ultimate Pulsar Processing InstrumentISM Interstellar MediumISS Interstellar ScintillationMJD Modified Julian DateMSP Millisecond PulsarNANOGrav North American Nanohertz Observatory for Gravitational WavesNS Neutron StarRFI Radio Frequency InterferenceS/N Signal-to-NoiseTOA Time of ArrivalxiiAcknowledgmentsForemost, I would like to express sincere gratitude to my supervisor, Dr. Ingrid H.Stairs, for her support, patience, and guidance throughout this project.Besides my supervisor, I would like to thank all my professors and collabora-tors, specifically Dr. Timothy T. Pennucci, without whom this project would nothave been possible.I must also thank my fellow lab-mates, past and present: Emmanuel Fonseca,Kathryn Crowter, Cherry Ng, and Wei Wei Zhu. I will be forever grateful for theirsupport and friendship throughout the last three years.Finally, I thank my friends and family: Justin, Kelsey, Javiera, Nick, April,Kathleen, Kevin, Sheri, Noa, sister Renee, and my parents, all of whom have con-vinced me, at one point or another, not to quit and high-tail it back to Newfound-land. I have found those who support me through thick and thin, and will neverforget their endless encouragement.xiiiChapter 1IntroductionWe’re still pioneers. And we’ve barely begun. Our greatestaccomplishments cannot be behind us, because our destiny lies aboveus. — Interstellar (2014)Radio pulsars are a fascinating class of astronomical object. It is estimatedthat there are between 105 and 106 active pulsars within the Milky Way Galaxyalone, mostly concentrated within a centralized disk of the galactic plane (Lyne& Graham-Smith, 2006). These rapidly rotating, highly magnetized neutron stars(NS) weigh more than the Sun, but have an estimated diameter of only 20 km, mak-ing them extraordinarily dense. As implied by the name, a radio pulsar producesradio emission; like a deep-space lighthouse, its beams sweep across the night skyas the pulsar rotates about its axis. If we’re lucky enough, a pulsar will be orientedsuch that we on Earth can detect its presence (Lorimer & Kramer, 2005).This thesis explores a particularly unique pulsar system: PSR J0737-3039A/B.This system, often referred to as the ‘double pulsar,’ is a binary consisting of twoneutron stars, both of which have been detected as pulsars (Burgay et al. 2003; Lyneet al. 2004). Double neutron star (DNS) systems themselves are rarely detected,and a double pulsar rarer still. PSR J0737-3039A/B is still the only known systemof its type.Systems such as this are invaluable tools in many areas of astronomical re-search. For instance, as discussed further in this chapter, they have been used inthe study of general relativity, as well as NS mass measurements. Precise pulsar1timing is necessary to produce meaningful results in any related study. This thesisexplores the timing analysis of PSR J0737-3039A, a millisecond pulsar (see Sec-tion 1.1.2), using a new method developed by Pennucci et al. (2014), in hopes ofimproving upon this pulsar’s timing solution.We begin with a brief, general introduction to pulsars and their discovery, fol-lowed by a more detailed description of PSR J0737-3039A/B.1.1 Introduction to Pulsars1.1.1 DiscoveryIn the decades following the Second World War, radio astronomy flourished. Dur-ing this time, the introduction of numerous observational methods lead to manynew fields of astronomical research. The identification of quasars can be attributedto new radio telescope techniques designed to investigate radio galaxies (Schmidt1963; Oke 1963; Greenstein 1963). The first discrete radio source, Cygnus A, wasfound in a study of the radio background (Hey et al., 1946). Initially, each newtechnique was designed for a specific problem, but observers were often guidedin unexpected directions by the vast richness of the observable radio sky. Similarcircumstances paved the way for the discovery of radio pulsars in 1967.In the 1960s, Professor Antony Hewish, with graduate student Jocelyn Bell(now Professor Bell Burnell) worked on a sky survey which focused on interplan-etary radio scintillation. For this, they constructed a large receiving antenna for arelatively long radio wavelength, 3.7 m (a frequency of 81.5 MHz). These speci-fications made the telescope sensitive to weak, discrete radio sources, now knownto be a requirement for detecting pulsars.In July 1967, within a month of starting regular survey recordings, Jocelyn Bellhad made a crucial discovery. She observed large fluctuations in the recorded sig-nal, which repeated around the same time on consecutive days. These fluctuationswere unlike scintillation (see Section 2.2.2) and much more like terrestrial interfer-ence. Because of this, Hewish initially dismissed them. However, after a brief lull,the signal reappeared and continued to appear intermittently. Eventually it becameclear that it appeared four minutes earlier with each passing day, indicative of a2celestial origin.At first, researchers were stumped. They only knew that this was somethingnew. Subsequent observations showed that the signal appeared with a regular pe-riod of about 1.337 s, and speculation brought up the possibility of an intelligentextraterrestrial origin. Not wanting to cause public hysteria from this disturbinghypothesis, communication on the matter was quelled until the true source couldbe determined (Lyne & Graham-Smith, 2006).A now classic paper appeared in Nature in February 1968, less than a year afterJocelyn Bell’s initial discovery. The paper indicated that the source must be outsideof the Solar System, and be very small due to the rapid rate of pulsation (Hewishet al., 1968). The researchers speculated that it was a white dwarf, or possibly aneutron star (which was still only a hypothetical entity at the time). The paper alsomentioned the detection of three other similar sources, for which investigationswere underway. A slew of research into these still unclassified objects followed inthe wake of this publication (Lyne & Graham-Smith, 2006), and within the spanof months, significant contributions were being made by at least eight other radioobservatories.It took nearly two years of research to conclude the nature of the source. Thiswas in spite of several papers published around the sane time which showed linksbetween pulsars and neutron stars. A paper published by Franco Pacini (Pacini,1967) demonstrated that a rapidly rotating neutron star could act as a generator,assuming a strong, dipolar magnetic field, providing an energy source for radiationfrom a surrounding nebula. Similar, but separate, work presented by T. Gold (Gold,1968) clearly outlined the case for identifying pulsars as neutron stars. Betweenthe two, both the basic theory and observational connections were established.As the work of Pacini was unknown to Hewish, he supported the theory thatthe source was a white dwarf. However, the short-period pulsars known now asthe Vela and Crab pulsars were used to prove otherwise. Theoretical calculationsshowed that white dwarfs might have a period as small as 1 s (from textbook). TheVela pulsar was found to have a period of 89 ms (Large et al., 1968), and the Crabpulsar a period of just 33 ms (Staelin & Reifenstein III, 1968). These small periodswere theoretically impossible for a white dwarf, but a neutron star would be quitecapable of rotating at these rates. In 1968, the final piece of evidence required to3classify these new objects as neutron stars was provided by Richards & Comella(1969). A slowdown was discovered in the period of the Crab pulsar. This was atell-tale sign of a neutron star, as rotational energy is lost through magnetic dipoleradiation (Gold, 1968). At this point, the debate was effectively ended, and a newepoch in radio astronomy research was launched.1.1.2 Properties of PulsarsFirst proposed as a theoretical entity by Walter Baade and Fritz Zwicky (Baade& Zwicky, 1934), a neutron star (NS) is formed from the supernova collapse of astar with mass of approximately 8-15 M. The neutron star is born, along witha supernova remnant, as the end product of this violent death. Although thosestudying pulsars and neutron stars have made much progress since their discoverynearly 50 years ago, there is still a great deal left unknown. Many questions, suchas, ‘what is the composition of a neutron star?’ and, ‘what is the structure of themagnetic field?’ are left without clear-cut answers as no definitive ‘pulsar model’exists to explain many of their observed properties (Lorimer & Kramer, 2005).Because of this, pulsar researchers are constantly in need of better, more preciseobservational data and processing methods.In early investigations, it appeared as though neutron star masses lay within anarrow bracket of 1.35 M (Thorsett & Chakrabarty, 1999). However, all currenttechniques used to measure NS mass rely on binary systems, in which orbital mo-tions can be precisely tracked through pulse time of arrivals (TOAs) (O¨zel & Freire,2016); few systems of this type had been discovered at the time of these prematureestimates. Recent analysis has suggested that NS masses are actually split into two(or possibly more) distributions (Schwab et al. 2010; O¨zel et al. 2012; Antoniadiset al. 2016). The most recent of these works, conducted by Antoniadis et al. (2016)focusing on millisecond pulsars (described below), suggests a bimodal mass distri-bution with the lower mass component centering around ∼ 1.4 M, and the higheraround ∼ 1.8 M. A comprehensive review of pulsar mass and radii calculationscan be found in O¨zel & Freire (2016).Neutron stars are very complex objects, and extreme by nature. Due to theirrelatively small sizes and high masses, their makeup is unlike anything observable4on Earth; therefore, it is impossible to definitively determine their internal struc-tures. However, several theories have been developed. Most suggest that the bulkof a neutron star is composed of a neutron fluid, which is in equilibrium with about5% protons and electrons. This core, which may have an average density of up toeight times the nuclear density (O¨zel & Freire, 2016), is enclosed in one or moreouter layers, or crusts. The crust, thought to consist mainly of iron and degenerateelectrons (Lorimer & Kramer, 2005), accounts for only a few percentage of star’stotal mass, but has significant effect on the NS’s properties, such as magnetic fieldshape and strength.Pulsars themselves fall into two different age categories. Those that follow atypical evolutionary path are called ‘normal’ pulsars. They are born in a supernova,with periods on the order of tens of milliseconds. Over the pulsar’s lifetime, its pe-riod gradually slows until it reaches death. At this point, any radiation becomesundetectable, as the dying pulsar lacks the minimum energy required to acceleratecharged particles to the necessary speeds to produce electromagnetic (EM) radia-tion strong enough to be observed on Earth. Millisecond pulsars (MSPs) start outlike their normal counterparts, but, due to interactions with binary partners, theytake a different evolutionary path. Through mass transfer from either its companionor a common envelope, a pulsar gains angular momentum. This ‘spin-up’ processcan increase a NS’s spin period to hundreds of rotations per second, even in caseswhere the pulsar had already reached ‘death’ (Lyne & Graham-Smith, 2006). It isbecause of this that some MSPs are often referred to as recycled pulsars instead;PSR J0737-3039A is thought to have undergone a short-lived mass transfer from acommon envelope, and is therefore considered to be a recycled pulsar.Pulsars are differentiated from other neutron stars by the detection of beamed,time-dependent radio emission; they are thought to have deviated in some wayfrom the typical NS evolutionary path. One possible theory suggests that a pulsarwill result from the collapse of a normal star with a polar field on the order of100 gauss, whereby the flux is conserved in the collapsing stellar material. Thesuperfluid in the star’s interior allows for a decay time that is long in comparisonto the life of the pulsar. This gives rise to some extraordinary field strengths, foundeven in older pulsars. Polar field strengths range from as high as 1015 G in youngpulsars (Parent et al., 2011) to 108 G in some older, evolved millisecond pulsars5(Lyne & Graham-Smith, 2006). Although this field is only thought to minimallyaffect the structure of most pulsars, the external magnetic field dominates manyphysical processes.Figure 1.1: Pulsar magnetosphere schematic (not shown to scale). A pulsar sits in aregion called the velocity of light cylinder, with radius rc = c/Ω. The magneticfield lines touching the cylinder’s edge define the polar caps. Charged parti-cles restrained by the polar caps are bound within the velocity of light cylinder.Those on open field lines, through some yet-unknown mechanism, produce theobservable radio emission the characterizes pulsars. Figure based on Lorimer &Kramer (2005), Figure 3.1.Many of a pulsar’s observable characteristics are the direct or indirect resultof this magnetic field. As shown in Figure 1.1, the rotation axis and magnetic6dipole are misaligned. Through this offset, the pulsar’s rotational energy is carriedaway by an electromagnetic wave at the rotational frequency, and consequently thepulsar’s rotation rate will slow over time.Most importantly (from an observational perspective), it is theorized that apulsar’s radio emission is caused by its magnetic field. As shown in Figure 1.1, apulsar sits in a region called the velocity of light cylinder. This region extends toradius rc = c/Ω, where c is the speed of light and Ω is the pulsar’s angular veloc-ity. Inside the cylinder exists an ionized magnetosphere of co-rotating high energyplasma, with polar caps defined by the magnetic field lines touching the cylinder’sedge. Since the charged particles can only move along magnetic field lines, thoserestrained by the polar caps are bound within the velocity of light cylinder, whereasthose on the open field lines, which extend beyond the region, can escape. This,through some yet-unknown mechanism, produces the observable radio emissionthat characterizes pulsars (Lorimer & Kramer, 2005).1.1.3 Importance of Pulsar ResearchThe versatility of pulsars as a laboratory tool is astounding. In addition to high pre-cision timekeeping, the ability to accurately determine their positions is extremelyuseful for celestial mechanics and astrometry. The most celebrated use of pulsarsis with gravitational physics. DNS binary systems present a near ideal pair of pointsources for testing theories of gravity in the strong-gravitational field limit. Theseparation between the two objects is much larger than the objects themselves,minimizing mass transfer and tidal effects. PSR B1913+16, also known as thefamous Hulse-Taylor pulsar after its discoverers, Russell A. Hulse and Joseph H.Taylor (Hulse & Taylor, 1975), was the first DNS system found to contain a pulsar.For this system, the orbital period was observed to decrease at the rate predictedby general relativity (GR), a consequence of wave damping (Taylor & Weisberg1989; Weisberg & Huang 2016), providing the first evidence for the existence ofGR gravitational radiation.71.2 The Double PulsarThe discovery of PSR J0737-3039 (Burgay et al. 2003; Lyne et al. 2004) markedthe first of its kind — a DNS system in which both neutron stars have been detectedas radio pulsars; it was found as part of a high-latitude multibeam survey of thesouthern sky using the 64-m Parkes Radio Telescope. Initially, only one of the twoneutron stars (now known as J0737-3039A) was determined to be a pulsar, but italready promised to be an excellent system for the study of relativistic astrophysics.The pair was found to have a short orbital period (Pb) of 2.4 hours and an extremelyclose orbit 3 light-second orbit, which would enhance many relativistic effects.These qualities, combined with the detailed pulses of the known pulsar, made it anexcellent candidate for these studies.Amazingly, within a year, PSR J0737-3039’s binary companion (henceforthknown as PSR J0737-3039B) was also identified as a pulsar through the analysisof subsequent data acquired for detailed study of the system. A comprehensiveoutline of this discovery can be found in Lyne et al. (2004). PSR J0737-3039A, amildly recycled pulsar, and the younger, normal pulsar, PSR J0737-3039B, remainthe only known pair of this type. The individual properties of each pulsar are listedin Table 1.1. With PSR J0737-3039B’s discovery, it was also determined that thesystem has a very high inclination angle (i), such that it is viewed nearly edge-onfrom Earth’s point of view. Recent studies show this angle is approximately 88.69◦(Kramer et al., 2006b). This line of sight is indicated in Figure 1.2, with referenceto PSR J0737-3039A and the pair’s orbital plane.Property PSR J0737-3039A PSR J0737-3039BMass (m) 1.3381 M 1.2489 MPeriod (P) 22 ms 2.7 sTable 1.1: Properties of individual pulsars in PSR J0737-3039 (Kramer et al., 2006b)This distinctive pair has provided a fortuitous opportunity for astronomers andphysicists to delve into many unexplored branches of study. It is still the most rel-ativistic double neutron star system discovered to date; the stars have much highermean orbital velocities and accelerations than other DNS systems. This has al-lowed for measurement of post-Keplarian parameters with unparalleled accuracy8Figure 1.2: Earth’s line of sight to PSR J0737-3039A in reference to the orbital planeof the DNS system. The plane of the observer’s sky is defined by −→I and −→J ,with −→K pointing away from the observer. The orbital plane of the pulsar isdefined by −→i and −→j . In this case, −→k points in the direction of both the orbitalangular momentum and spin axis. As PSR J0737-3039A rotates about its axis,its radio beam is detected as a series of pulses on Earth (located at a near verticalinclination i of 88.69◦) with period 22 ms. Figure based on Ferdman et al.(2013) and Damour & Taylor (1992).(Kramer et al., 2006b).The system continues to be of great use in the study of neutron star and, moregenerally, compact object merger rates. To predict these merger rates, the massesof both NSs must be known. Most DNS systems have well defined masses, butonly a handful are known; therefore, the double pulsar plays a key role in theseestimations. Subsequently, this work been used in constraining the gravitationalwave detection rates from NS-NS in-spirals (Vangioni et al. 2015; Kim et al. 2015;Belczynski et al. 2016).It is known that the coherent radio emission produced by pulsars is connected totheir magnetospheric properties, such as structure and particle density. However,these properties have not been well constrained by either theory or observation.9Due to the nearly edge-on orbital plane of the double pulsar, the short, 30 s eclipseof PSR A by PSR B can be observed from the absorption of PSR A’s signal as itpasses behind PSR B and its magnetosphere. Flux modulations in PSR A’s signalat both one- and two-times the rotation frequency of PSR B have been found to beintrinsically linked to the eclipse mechanism. This has allowed for the investiga-tion of PSR B’s magnetospheric structure, as well as the properties of its plasma(Lyutikov & Thompson 2005; Breton et al. 2012). Studies such as these are criticalin understanding the fundamentals of pulsar emission behaviour.Due to the curvature of space-time near massive objects, the double pulsar un-dergoes a process known as geodetic precession – the orientations of the pulsars,with respect to Earth, change over time as their spin axes precess about the totalangular momentum vector. The geodetic spin precession of PSR B has been pre-cisely measured using the observed eclipses (Breton et al., 2008). In combinationwith observational information of the two pulsar’s orbits, this measurement hasallowed researchers to quantitatively constrain relativistic spin precession in thestrong-field regime within several gravitational theories, including GR.In March 2008, the weakening radio signal from PSR J0737-3039B completelyvanished; the beam had shifted out of Earth’s line-of-sight due to its geodetic pre-cession. Intriguingly, changes in PSR B’s pulse profile and flux density had beenobserved prior to its disappearance. By studying these changes, researchers haveproduced several possible models of the pulsar’s beam shape which have been usedto predict when it may become detectable again. One such model, now disproved,expected the signal to return in 2014, while another still plausible model forecastsa reappearance around 2035. A comprehensive description of these changes andpredictions can be found in Perera et al. (2010).10Chapter 2Pulsar TimingOne of the characteristic features of a pulsar is the clock-like stability of its rota-tions. Through the study of pulse arrival times and pulse profiles, researchers workto obtain precise timing of the source whilst exploring, amongst other things, var-ious phenomena affecting the propagation of these pulses through the interstellarmedium (Lorimer & Kramer, 2005). This chapter aims to highlight the main pointsof pulsar timing, and problems faced in doing so.2.1 Pulse Profiles and Arrival TimesPulsars are very weak radio sources, and as such, they must be observed with spe-cialized equipment to obtain a significant detection. The signal is collected by alarge area receiver, and further intensified by low-noise amplifiers. The individualpulses from a particular pulsar will show variation in their shapes, but a pulsar’sintegrated pulse profile, the classic ‘picture’ of a pulsar’s signal, is most often verystable for a set frequency. Once a pulsar’s period is determined, the individualpulses can be added together in a process known as ‘folding’. In the most basicfolding process, the signal is split in time into segments with length P, the pul-sar’s period. Each segment is subsequently split into a set number of bins, X . Theweighted intensity associated with each bin, IX , is summed over the total numberof segments, giving an added profile when plotting IX vs X . This allows the sig-nal to be discernible above the background noise, with the signal-to-noise (S/N)11ratio growing in proportion to the square root of time. The number of pulse pe-riods required to achieve a stable integrated profile – one whose shape remainsunchanged over time – is pulsar-dependent, but is typically on the order of a fewhundred to a few thousand (Lorimer & Kramer, 2005). The integrated pulse profileof PSR J0737-3039A, centered at 820 MHz with bandwidth 200 MHz, is shown inFigure 2.1.Figure 2.1: Integrated pulse profile of PSR J0737-3039A centered at 820.781 MHz,bandwidth of 200 MHz, total integration time of 25144 s (419.1 minutes). Sig-nal was recorded with the Green Bank Ultimate Pulsar Processing Instrument(GUPPI) in coherent de-dispersion mode at the Robert C. Byrd Green BankTelescope (GBT). Data were recorded on Modified Julian Date 56616 (Novem-ber 20th, 2013) and processed with the PSRCHIVE software suite.Most often, in order for pulsar observations to be meaningful, the pulse TOAsmust be accurately known. This quantity is normally defined as the arrival timeof the fiducial point of the pulse closest to that of the mid-point of the observation(Lorimer & Kramer, 2005). This fiducial point is usually set to coincide to a planewithin the pulsar characterized by its rotational and magnetic axes, and the ob-server’s line of sight. However, frequency-dependent effects, such those caused bythe interstellar medium (Section 2.2) and profile evolution (Section 2.4) influencea pulsar’s TOAs and make it more difficult to define the fiducial point. Techniques,12such as those described in Section 2.2.1, have been developed to combat thesecomplications and produce more precise TOAs.2.2 Effects of the Interstellar Medium2.2.1 Dispersion MeasureThe interstellar medium (ISM) refers to the gas (in ionized, atomic, and molecularform), dust, and cosmic rays that exist between star systems in a galaxy. As EMradiation from pulsars (and other sources) travels through the ISM, it is subject toa frequency-dependent index of refraction. This results in a frequency-dependentdelay of pulse TOAs. In pulsar astronomy, the dispersion measure (DM) is usedto correct for this. DM is the integrated column density of free electrons along theline of sight, defined as:DM =∫ d0nedl (2.1)Here, d is the path length between the pulsar and the observer, and ne is the electronnumber density of the ISM (Lorimer & Kramer, 2005). DM is typically expressedin pc cm−3. From this, the time delay of each frequency’s signal relative to infinitefrequency can be determined, as per Equation 2.2.t ≡ D x DMf 2(2.2)D is the dispersion constant (Equation 2.3; Lorimer & Kramer (2005)):D≡ e22pimec= (4.148808±0.000003) x103 MHz2 pc−1 cm3 s (2.3)This value was first introduced in its reciprocal form by Manchester & Taylor(1972) as 1/D ≡ 2.41 x10−4 MHz−2 pc cm−3 s−1. As such, it can be inferredthat higher frequency signals arrive at later times than their lower frequency coun-terparts. The time delay, ∆t, between frequency signals can be subsequently de-termined using the Equation 2.4, wherein fre f is the reference frequency in MHz,fchan is the frequency of the channel for which the time delay is to be measured,in13MHz, and DM is the dispersion measure in pc cm−3.∆t ≈ 4.15 x 106 ms x ( f−2re f − f−2chan) x DM (2.4)2.2.2 ScintillationThe ISM, by its complex nature, is both turbulent and inhomogeneous. As thepulsar signal experiences considerable environmental variability along its path, itundergoes phase modulations referred to as interstellar scintillation, a phenomenonwherein the observed signal intensity fluctuates over differing bandwidths andtimescales (Lorimer & Kramer, 2005). This behaviour was first recognized byLyne & Rickett (1968) as strong-intensity modulations in pulsar observations, af-ter which Scheuer (1968) developed the theory of scintillation. Scheuer modeledthe turbulent ISM as a thin screen of irregularities midway between Earth and thepulsar, and considered the phase perturbations such a screen could cause. Fromthis he was able to demonstrate that there is a characteristic bandwidth, denotedas the scintillation bandwidth, over which the intensity fluctuations should be cor-related. This bandwidth, ∆ f , was found to be proportional to the fourth power ofthe observing frequency, f , such that ∆ f ∝ f 4, which was in agreement with earlyobservations by Rickett (1969).Recent work by Rickett et al. (2014) focuses on the fluctuating interstellar scin-tillation (ISS) of the double pulsar. Due to the changing transverse velocities of thepulsars, the ISM, and the Earth, variation is seen in the ISS. Assuming the thinscreen model, they analyzed the ISS over both frequency and time, incorporatingthe location, anisotropy, turbulence level, and transverse phase gradient of the ion-ized ISM. From this, they determined that the ionized ISM can be well-modelledover the course of a few orbital periods (the length of each observation), but foundsignificant variation in its turbulence level and mean velocity over the course of the18 month study. Their analyses supported the value of sin(i) reported by Krameret al. (2006b), and were furthermore able to resolve the ambiguity of the sign ofcos(i).142.2.3 ScatteringPulse scattering is closely related to scintillation. The profiles of many distantpulsars exhibit ‘exponential tails’. Based on the thin screen model described inSection 2.2.2, these tails are a direct effect of the variable path lengths, and subse-quent arrival times, of the rays as they are scattered by the ISM’s irregularities. Thescattered rays are delayed in their arrival, resulting in the asymmetric broadeningof an otherwise intrinsically sharp pulse. This broadening can be estimated as theconvolution of the true pulse shape with a one-sided exponential function with a1/e time constant known as the scattering time, τs (Lorimer & Kramer, 2005).It has been found that τs is strongly correlated with DM (Bhat et al., 2004), andthat in general, more distant pulsars with larger DMs will exhibit more scatteringeffects in their profiles. The frequency dependence of these effects is also quite ap-parent. From the thin screen model, τs ∝ 1/∆ f ∝ f−4, and as such, for a particularpulsar, the scattering time will be smaller for larger frequencies, resulting in lessdeviation from the true profile shape.More recent studies suggest that the frequency scaling index can deviate fromthis simple model, in which this value is set to -4. For example, work by Lohmeret al. (2001) indicates that this value can be as low as -2.8 for highly dispersedpulsars. Follow-up studies focusing on a wider range of pulsars, such as thoseconducted by Krishnakumar et al. (2015) and Lewandowski et al. (2015), havefound some comparable values for individual pulsars, but conclude that the averagevalue of the frequency scaling index is close to 4.The effects of scattering are generally not problematic in observations of PSRJ0737-3039A, hence, they will not be considered in the profile modelling discussedin this thesis.2.3 De-dispersionA pulsar’s dispersion measure can significantly affect the signal seen by a filterbank– signal data that has been channelized either by an analog device or software. Pul-sars with higher DMs will show more smearing in their observed profiles, such thatan intrinsically sharp pulse peak will appear stretched in phase and have reducedintensity; this effect is even more prominent in wideband observations. However, to15properly perform pulsar timing, the timing must have very high resolution. Whileobservational sensitivity is maximized by covering a wide bandwidth of these weakradio sources, this is countered by the DM effects (Lyne & Graham-Smith, 2006).Fortunately, techniques have been developed to handle and remove many of theeffects of the ISM, including the dispersive effects, such that wideband timing canbe accomplished with meaningful results. To deal with dispersion, the most prob-lematic ISM effect seen in observations of PSR J0737-3039A, there are two mainmethods: incoherent de-dispersion and coherent de-dispersion.2.3.1 Incoherent De-dispersionThe simpler of the two, incoherent de-dispersion, utilizes a spectrometer, either inanalog hardware or with a digitized signal that is processed with software, to com-pensate for dispersion effects. The incoming frequencies are split by the spectrom-eter into a large number of independent frequency channels. Software employingEquation 2.4 is used to add the correct time delays to each channel, allowing dif-ferent frequencies in the signal to appear to arrive at the same time.2.3.2 Coherent De-dispersionCoherent de-dispersion is a more advanced technique, pioneered by Hankins &Rickett (1975). Using this method, the full bandwidth is processed before detec-tion, completely removing all dispersion effects and restoring the original pulseshape. This is achieved by passing the signal through a filter which delays thephase by a frequency-dependent amount. High speed sampling is required, as thesignal must be digitized before detection. A sequence of samples is first Fouriertransformed, preserving the phase of the individual components, after which a com-puted phase delay is applied to each component. Finally, a reverse Fourier transfor-mation is used to recover the signal without the delay caused by dispersion effects(Stairs et al., 2000). Using this method, pulse profiles appear sharper and morerefined than with incoherent de-dispersion, allowing for measurement of higherprecision TOAs.162.4 Profile Evolution with FrequencyIntegrated pulse profile shapes have been found to intrinsically evolve with fre-quency (Lyne & Manchester, 1988). Depending on the pulsar, this can be more orless pronounced, but is oftentimes significant. Normal pulsars will generally showan increase in pulse width and separation of profile components with decreasingfrequency, but this effect is much less pronounced, in general, for MSPs (Krameret al., 1999). In addition to this, both the number of profile components and theirrelative intensities can change with frequency. These phenomena, if not properlyaccounted for, can have considerable consequences on the accuracy of pulsar tim-ing. As such, these effects must be taken into consideration to produce precisetiming results.Once the dispersion effects, which cause TOA delays proportional to 1/ f 2(see Equation 2.4), have been removed, small frequency-dependent residuals arestill apparent in wideband observations using different instruments and telescopes.Lower frequency signals (e.g. 820 MHz data) have been found to lead higher fre-quency signals (e.g. L-band data) by microseconds (Zhu et al., 2015). The precisecause of these delays is unknown, but researchers have developed methods to cor-rect for some of the resultant TOA biases. Demorest et al. (2013) implemented“jumps” – free parameters in the timing fit that set a constant offset in time fora particular time period or channel for their analysis of 17 MSPs. In their anal-ysis a jump was set for each 4 MHz frequency channel. This reduced TOA bias,but, as the constant-in-time per-channel offsets are completely covariant with theconstant-in-time dispersion measure, both parameters could not be determined in-dividually.Zhu et al. (2015) used a frequency-dependent (FD) model, discussed further byArzoumanian et al. (2014), to analyze PSR J1713+0747 – one of the most precisepulsars. This incorporates an additional timing delay, ∆tFD, into all timing models.This parameter is defined below, wherein the coefficients, ci, are fit parameters inthe timing models.∆tFD =n∑i=1ci(log f )i (2.5)For a given pulsar, the number of terms, i, can be determined by an F-test with17significance value 0.0027. For example, PSR J1713+0747 investigated by Zhuet al. (2015) only required n= 4 FD parameters. This algorithm is an improvementon the simpler method executed by Demorest et al. (2013), significantly reducingthe number of free parameters, but it is still an ad hoc technique. It is only appliedafter the TOAs have been calculated, and does not incorporate the profile data,which contains information about the pulse shape.In recent years, alternatives to these techniques, which deal with frequency-dependent effects before the TOAs are calculated, have been increasingly underinvestigation. Liu et al. (2014) investigated a channelized Discrete Fourier Trans-form method that was found to greatly mitigate the influence of these effects whenmeasuring TOAs from broadband data. The timing algorithm developed by Pen-nucci et al. (2014) (see Section 3.3), explored in this thesis, utilizes a broadbandprofile-fitting approach that directly incorporates the profile’s evolution. The pro-file model (see Section 3.2) is derived accounting for changes in the pulse profileshape with frequency. Hence, TOAs calculated with such techniques should notrequire any subsequent ad hoc per-channel jumps or additional timing delays.As discussed in detail in Section 4.1.1, the pulse profile of PSR J0737-3039Adisplays several frequency-dependent trends. The relative intensities of the peakschange as frequency increases, and several key features are found to evolve, modi-fying the overall profile shape. To improve the timing model of this pulsar, frequency-dependent profile evolution must be taken into account. As presented above, thewideband timing algorithm accounts for the evolution in a more physically mean-ingful way than other current methods by incorporating the pulse shape informa-tion in the profile model. Because of this, the wideband timing algorithm was thechosen method for the profile model development carried out in this thesis.2.5 Timing ModelsThe ultimate goal of pulsar timing is to produce the most accurate representationof the object through a ‘timing model’. To do so, one must take into account amultitude of parameters, ensuring that all processes and system properties affectingthe TOAs are considered. These parameters include, for example, period, periodderivative, and astrometric parameters, as well as the motion of the Earth and Solar18System in relation to the pulsar. In the case of pulsars in binary systems like PSRJ0737-3039A, properties of the orbit must also be included in the timing model.There are several programs available which can be used to determine timingmodels from pulsar TOAs, such as TEMPO1, TEMPO22, and PINT3, but TEMPO,developed by J.H. Taylor and collaborators, is the industry standard. This programwas specifically designed for the analysis of pulsar timing data, reading in pulseTOAs, pulsar model parameters, and user-specified instructions. The TOAs arethen fit to a pulse timing model, incorporating a transformation to the Solar-Systembarycenter (the centre of mass of the Solar System; for example, the JPL DE430planetary ephemeris4 was used in the timing analysis reported in this thesis) andtaking into account the pulsar’s rotation and spin down rates. If the pulsar is ina binary, this program has several binary models available for use. TEMPO iscapable of determining the values of various parameters and their uncertainties,residual pulse arrival times, and chi-squared statistics.1http://tempo.sourceforge.net2http://tempo2.sourceforge.net3https://github.com/nanograv/PINT4http://naif.jpl.nasa.gov/naif/19Chapter 3MethodologyThis chapter outlines both the traditional method of pulsar TOA calculation and anew algorithm designed to improve wideband timing.3.1 Data Acquisition and Handling3.1.1 Data AcquisitionAll data used in the following analyses were obtained with the Robert C. ByrdGreen Bank Telescope (GBT) in Green Back, WV, United States using the GreenBank Ultimate Pulsar Processing Instrument (GUPPI) in timing mode. PSR J0737-3039A was observed with both the 820 MHz and L-band (1.5 GHz) receivers,using bandwidths of 200 MHz and 800 MHz, respectively. Data were recordedin both coherent and incoherent mode, as GUPPI transitioned from incoherent tocoherent mode observations over the time-scale of data acquisition. The date rangefor each mode are listed in Table 3.1 with both Gregorian dates and Modified JulianDates (MJD; an abbreviated version of the old Julian Date system where MJD =JD−2400000.5).GUPPI is a flexible digital signal processor, designed for pulsar observations.This instrument allows for an 8-bit sampling rate, large bandwidths, full polariza-tion capabilities, high resolution in time and frequency, and better immunity fromradio frequency interference (RFI) than its predecessor, the GBT Spigot. Addi-20Band 820 MHz L-BandBandwidth 200 MHz 800 MHzIncoherent Mode Dates MJD 55002 – 55430 MJD 55121 – 56314June 2009 – Aug. 2010 Oct. 2009 – Jan. 2013Coherent Mode Dates MJD 55502 – 57436 MJD 56367 – 57524Nov. 2010 – Feb 2016 March 2013 – May 2016Table 3.1: Observational date range for acquisition of 820 MHz and L-band datationally, GUPPI is flexible in its parameter values, such as number of channels andintegration times, and can be used in both search and timing modes with the op-tion for real-time folding, rather than exclusively folding the data post-observation(DuPlain et al., 2008). This is an advantage over many previous pulsar signal pro-cessing instruments, as the folded data takes up much less storage space than itsun-folded counterpart. However, it may be necessary to fold the data after the ob-servation, such as for the study of PSR J0737-3039A/B’s eclipse. Observationaldata analyzed in this thesis were taken in time series and folded after observationwith integration time of 30 s, so that the data may be used in other analyses.In its initial phase, starting in April 2008, GUPPI processed signals using in-coherent de-dispersion (DuPlain et al., 2008). Usable observations of PSR J0737-3039A began in June 2009; prior to this, there were non-deterministic offsets inthe data. In this mode, observations centered at 820 MHz were recorded with 2048frequency channels (channel bandwidth of ∼0.098 MHz), while those in L-bandwere recorded with 1024 frequency channels (channel bandwidth of ∼0.78 MHz).As GUPPI moved from its initial phase into its secondary phase of operation,coherent de-dispersion processing was introduced, alongside other improvements.As per Table 3.1, this took place in steps. Coherent mode observations began withthe 820 MHz receiver in late 2010, while it was early 2013 before the switch wasmade with the L-band observations. The transition for the L-band was delayedbecause the DM smearing is much less pronounced in an L-band channel, and itwas thought that the switch from incoherent to coherent de-dispersion would makelittle to no difference. However, it was later decided that it may have an effecton the residual smearing and the switch was made. Data for the 820 MHz band21and L-band were recorded into 128 channels and 512 channels, respectively, withindividual channel bandwidths of 1.5625 MHz in both cases.3.1.2 Data HandlingPSRCHIVE software1, developed by A.W. Hotan and collaborators for analysis ofpulsar astronomical data (Hotan et al. 2004; van Straten et al. 2012a; van Stratenet al. 2012b), was used to process the data acquired from observations at the GBT.Data from both bands and de-dispersion modes (other than noted exceptions) werehandled with the following steps.RFI was mitigated using PSRCHIVE’s psrsh program in a process known as‘zapping’. Through this procedure, full channels and subintegrations exhibitingRFI were removed (‘zapped’), along with individual subintegration / channel pairs.Each observational data file had a corresponding ‘zap’ file containing this informa-tion, which was subsequently read by a short python script (see Section A.1 forscript and further details) and converted to a format that could be interpreted bypsrsh. These files, initially derived for use with another data reduction package,were provided by Robert Ferdman (private communication). This alone was suffi-cient to remove the RFI in the 820 MHz data, but the automated zapping processwas used as a starting point for the L-band data. This data required some additionalzapping, completed manually with PSRCHIVE’s pazi software.Following RFI removal, the zapped pulsar observation files were flux cali-brated. When a pulsar is observed by a telescope, its signal is not recorded inphysically meaningful units, as the output of pulsar data acquisition devices is de-signed to be in a compact form. Simple ‘machine units’ or ‘counts’ are used toindicate relative intensities of the signal, rather than Janskys (Jy), the standard fluxdensity units. Hence, the signal intensities must be converted from Counts to Jan-skys.Data calibration occurs through a series of steps, utilizing several calibrationscans taken during the observation. A pair of scans are taken on and off a sourceof known flux density – in this case, the quasar B1442+101. During both scans, asemi-conductor diode is used to introduce a pulsed white noise square-wave signal.1www.psrchive.sourceforge.net/22In the off-source scan, this gives a signal strength ‘b’ (the height of the noise incounts), whereas the on-source scan will have a signal strength of ‘a + b’ (theintensity of the quasar plus the noise height, in counts) in the on-phase of thediode, and simply ’a’ in the off-phase. As the intensity of a in mJy, Iquasar, is astandardized value available in astronomical catalogs, the following ratio can beused to find the intensity measure of the calibrator value, calb, in mJy:b(counts)a(counts)=calb(mJy)Iquasar(mJy)(3.1)Subsequently, the value of calb is used to convert the intensity of the pulsar signalfrom counts to mJy. A third calibration scan is recorded on the pulsar itself, againwith the injected diode signal. The noise from the diode is treated as a pulsar signal,creating a square-wave profile. Similar to the above equation, ratios are used todetermine the pulsar signal intensities. This is accomplished using PSRCHIVE’s‘pac -x’ command, completing the flux calibration process.The final step was to update the ephemeris file, accomplished using the PSRCHIVEprogram pam. An ephemeris file contains the parameters of the pulsar in questionand, in the case of pulsars in DNS systems such as PSR J0737-3039A, those of thesystem as a whole. Peak alignment between different observational epochs can beimproved by installing the most up-to-date version of pulsar’s ephemeris. Here, thenew ephemeris was based on the 2006 version of the PSR J0737-3030A parameterfile, but with updates to encompass the modern TOAs derived from the standardfitting model. Once complete, these aligned pulse signal files, henceforth knownas ‘aligned’ files, were ready to be processed by the wideband timing algorithm.3.2 Traditional TOA Fitting MethodBefore diving into the new wideband timing method, it is important to discusstraditional TOA fitting methods so that comparisons can be made. Conventionally,multi-channel TOAs are calculated using the followings steps, described in detailin Lorimer & Kramer (2005). For a singular band (e.g. 820 MHz or L-band),the channels of each epoch’s calibrated integrated pulse profile are divided intoa set number of subbands (typically 4, 8, or 16). The profile of each subband isthen cross-correlated with a smoothed, high S/N template file (or ‘profile model’),23which is determined from the average of all data from a given receiver for thespecific pulsar and band. The template matching algorithm assumes that the profile,D(t), is a scaled and shifted version of the template profile, P(t), with added noise,N(t), as per Equation 3.2 (Lorimer & Kramer, 2005):D(t) = aP(t− τ)+b+N(t) (3.2)τ is the time shift between the profile and the template, a is a scaling factor, andb is an arbitrary offset. This yields the TOA relative to the fiducial point of thetemplate and the reference time of the observer (typically chosen from the middlesubintegrations which minimizes error from non-linear phase errors). The cross-correlation may be completed in either the time domain, or the frequency domainafter Fourier-transforming the template and profile as per Taylor (1992). In thetime domain, precision is limited to approximately 1/10 of the sampling interval.This implies that, with bin-by-bin cross-correlation and fitting for where the cross-correlation function peaks, the maximum precision is about 1/10 of a bin. For thefrequency domain, precision is dependent on the number of harmonics involved,but again, it is generally less than one bin, with 1/10 of a bin being the conventionalestimate of the best possible performance.Dispersion measure values are determined as either one value for the full dataset, or with per-epoch shifts, typically with bins containing around 60-70 days.These values are fit alongside all other parameters in the timing solution. DailyDM values are not calculated using this method.3.3 Wideband Timing AlgorithmPulse TOA and DM values were calculated with an algorithm developed by Timo-thy T. Pennucci and collaborators (Pennucci et al., 2014) for use with widebandtiming of radio pulsars. Further detail on the algorithm can be found in Pen-nucci (2015). The wideband timing algorithm was created using publicly-availablepython code, along with the pulsar data analysis package, PSRCHIVE (Hotan et al.2004; van Straten et al. 2012b). An up-to-date version of the code is available freeon-line2.2www.github.com/pennucci/PulsePortraiture24This code uses an extension of J. Taylor’s FFTFIT algorithm (Taylor, 1992)to simultaneously measure the TOAs and DMs from folded wideband pulsar data.Unlike traditional methods, the wideband timing algorithm utilizes a summationof Gaussian components to produce a frequency-dependent pulse profile template.Hence, this model can be used to account for frequency evolution within, and evenbetween, observational bands, and has been shown to improve upon the timingresults of many pulsars (Pennucci et al. 2014; Pennucci 2015).3.3.1 Gaussian ModellingThe wideband timing algorithm is comprised of three smaller algorithms: ppalign,ppgauss, and pptoas. A frequency-dependent Gaussian-model portrait template iscreated using a combination of ppalign and ppgauss, which can be followed byTOA and DM calculations using pptoas.To start, a high signal-to-noise averaged profile was produced for each of thecoherent mode L-band and 820 MHz data using ppalign. This algorithm first uti-lizes PSRCHIVE’s psradd program to create an average profile. To achieve thesmoothest averaged profile and minimize scintillation effects, epochs from eachband with the highest S/N ratios were chosen, listed in Tables 3.2 and 3.3. Unlikethe process described in Pennucci (2015), the phase alignment option in ppalignwas not selected, as profile alignment was dealt with when updating the ephemeris(as per Section 3.1.2). Initially, the individual profiles are ‘incoherently’ stackedand averaged, ignoring offsets due to changes in DM. The algorithm then uses theaverage profile to measure the phase and DM of each individual profile includedin the average. Subsequently, the files are then stacked ‘coherently’, meaning thatphase shifts due to differing DM values are taken into account (Pennucci, 2015).Next, these two high S/N averaged profiles were used as input for ppgauss, uti-lizing the metafile option with allows for multiple profiles to be fit at once. Optionswere also chosen such that the initial model was centered at 1499.22 MHz (the cen-tral frequency of the L-band observations), with a bandwidth of 200 MHz, coveringthe range 1400-1600 MHz. The spectral features in the L-band profiles are moreclearly defined than those of the 820 MHz data, which can be seen when comparingFigures 4.1, 4.2, 4.3, and 4.4. Due to the frequency-dependent profile evolution,25Epoch (UTC) MJD (day) Length (min)2013-06-22 56465.72 2822013-11-20 56616.23 4192013-11-21 56617.30 2962014-11-20 56981.24 4002014-11-21 56982.25 382Table 3.2: Epochs included in Gaussian modelling from GUPPI coherent mode 820MHz data set; total length of 1779 minutes (29.7 hr)Epoch (UTC) MJD (day) Length (min)2013-10-30 56595.32 2492013-12-17 56643.21 3002014-07-27 56865.61 2532014-09-22 56922.41 2582015-03-25 57106.93 3062015-11-14 57340.31 3002016-01-16 57403.11 293Table 3.3: Epochs included in Gaussian modelling from GUPPI coherent mode L-band data set; total length of 1959 minutes (32.7 hr)and based upon the work of Pennucci et al. (2014), only a section of the L-banddata bandwidth was chosen. Each spectral feature in a pulse profile represents theaverage relative height of that feature over the selected bandwidth. When mod-elling with larger bandwidths, features that change significantly with frequency aremore difficult to account for. Subsequently, a shorter frequency range was foundto typically give sharper, more detailed profiles.Initial Gaussian components are chosen by hand with an interactive viewer.The user selects the location, width, and amplitude of each component via a ‘drag-and-click’ method, overlaying a plot of the pulse profile. After the addition of anynumber of components, the user can choose to have the program fit the Gaussiansto the pulse profile, whereby the ‘guess’ Gaussians are scaled in width and height,and shifted in phase to minimize the data fit residuals. This is demonstrated in Fig-ure 3.1, which shows the final fit for a preliminary model containing 21 Gaussiancomponents. For this, Gaussian components were added one at a time, with the26data fit assessed after each addition. This was repeated until the data fit residualsfor the pulse peaks were at the level of those for the baseline.It was found by Pennucci (2015) that beyond some number of Gaussian compo-nents, additional components had a negligible effect on the final TOA uncertainties.Erring on the side of caution, he used a maximum of 19 components in his analysisof 37 pulsars, with the next highest being 16. This decision was reaffirmed throughour initial models of PSR J0737-3039A; after about 20 Gaussian components, ad-ditional components had some zero-value parameters, preventing the model fromconverging. As discussed below, our final best-fit model contained 19 Gaussiancomponents.27Figure 3.1: Example of Gaussian fitting using coherent mode pulsar profiles of PSR J0737-3039A, centered at reference fre-quency fo = 1499.22/MHz with 200 MHz bandwidth. Top – Initial Gaussian components from wideband timing algorithm’sppgauss; Gaussian components shown in colour, overall model fit shown in black, profile shown in gray. Bottom – Datafit residuals. Gaussian components are added one at a time until the data fit residuals for the peaks are within the range ofthose for the baseline. This example includes 21 individual Gaussian components; additional components showed negligibleimprovement in the data fit residuals.28Once the user is satisfied with the model’s fit, the program fits the Gaussiansto each frequency channel in the initially selected bandwidth / frequency range. Itis then reiterated over the channels in the selected data files (i.e. the full range ofthe L-band and 820 MHz data), producing a set of parameters for the frequency-dependent model. Ppgauss also offers the option to reiterate a model once it hasbeen produced. This can be done using any average profile(s) created with ppalign,within range of the original profiles. The final profile model was derived from thisby reiterating the above preliminary 21 Gaussian model with the data from epochslisted in Tables 3.2 and 3.3, and including ppgauss’s ‘normalization’ option. Thisoption normalizes the average portrait to the profile maximum in each channel, re-ducing covariance between the amplitude and width parameters (Pennucci, 2015).The best-fit model that resulted in non-zero parameter values had 19 Gaussians,and will henceforth be known as the ‘19 Gaussian Model’. Although the num-ber of Gaussians in this model exceeds those of the pulsars studied by Pennucci(2015), the complexity of PSR J0737-3039A’s profile also surpasses most, if notall, of those as well.In cases where multiple frequency bands are used in the initial input, ppgausscreates a ‘join’ file. When fitting a model over multiple bands, there is an offsetin rotational phase and/or dispersion measure. This join file allows the user toremove these offsets when comparing the model in different bands, as well as tothe observational data. The join parameters were also used as one indicator ofcompleteness when re-iterating the model. When the offsets reached near-constantvalue between iterations, the model also appeared to exhibit minimal change in itsplotted profile. The join file parameters from the final iteration of the 19 Gaussianmodel are listed in Table 3.4. These values are used in profile model comparisonsonly, and are not required for subsequent TOA or DM calculations.29Band Phase Offset (rot) DM Offset (cm−3 pc)820 MHz 0.0000000000 ± 0.0000000000 0.029312 ± 0.000195L-band 0.0071397243 ± 0.0000360391 0.015795 ± 0.000373Table 3.4: Join file offset parameters for final iteration of 19 Gaussian model, pro-duced by the wideband timing algorithm’s ppgauss software. Offsets are givenas negatives of the true offset values.The Gaussian model itself can be described through the following equations,further explained in Pennucci et al. (2014). The profile intensity at a particular fre-quency ( f ) and pulse phase point (φ ) is the summation of all Gaussian componentsat that point:P( f ,φ) =∑igi( f ,φ) (3.3)Each individual Gaussian component, gi, is given by:gi( f ,φ |Ai,φi,σi) = Ai( f )exp(−4ln(2)(φ −φi( f ))2σi( f )2)(3.4)where Ai (amplitude), φi (position), and σi (FWHM) are modeled as power-lawfunctions of frequency, f :Xi( f |Xo,i,αX ,i, fo) = Xo,i(ffo)αX ,i(3.5)for the ith Gaussian component, Xi (Xi = Ai,φi,σi), and model reference frequencyfo.The model file produced by ppgauss lists six parameters for each independentGaussian: A, φ , and σ , with an evolutionary parameter, αX , for each. For the 19Gaussian model, the phase positions of all Gaussians were fixed (not dependenton frequency). Therefore, all αφ ,i parameters have been set to zero. The referencefrequency, fo, a DC offset, and scattering timescale are also saved to the modelfile. However, the scattering timescale was not fit for the 19 Gaussian model,as scattering is not considered for this timing analysis of PSR J0737-3039A. TheGaussian fit parameters for the model are listed in Table 3.5.30This final model was produced using only coherently de-dispersed data. Pre-liminary tests showed that the modelling with incoherently de-dispersed data pro-duced usable results. Therefore, the model derived from the coherently-dedisperseddata was adopted for both the coherent and incoherent data sets.31i φo,i [rot] δ (φo,i) αφ ,i δ (αφ ,i) σo,i [% rot] δ (σo,i) ασ ,i δ (ασ ,i) Ao,i δ (Ao,i) αA,i δ (αA,i)1 0.79745 5 × 10−5 0.0 0.0 0.0079 1.1 × 10−4 -0.14 2 × 10−2 0.147 3 × 10−3 -0.63 3 × 10−22 0.6901 3 × 10−4 0.0 0.0 0.0458 6 × 10−4 -0.11 2 × 10−2 0.0534 5 × 10−4 0.43 3 × 10−23 0.2392 4 × 10−4 0.0 0.0 0.0812 8 × 10−4 -0.009 1.5 × 10−2 0.248 5 × 10−3 -0.47 2 × 10−24 0.3110 7 × 10−4 0.0 0.0 0.030 1.2 × 10−3 0.23 4 × 10−2 0.072 2 × 10−3 0.20 3 × 10−25 0.33432 6 × 10−5 0.0 0.0 0.0069 2 × 10−4 -0.19 5 × 10−2 0.058 3 × 10−3 0.23 6 × 10−26 0.21778 6 × 10−5 0.0 0.0 0.0116 1.2 × 10−4 0.15 1.4 × 10−2 0.221 7 × 10−3 -0.47 3 × 10−27 0.17866 9 × 10−5 0.0 0.0 0.0106 2 × 10−4 -0.41 3 × 10−2 0.074 2 × 10−3 -1.64 4 × 10−28 0.78778 4 × 10−5 0.0 0.0 0.0121 1.1 × 10−4 0.074 1.1 × 10−2 0.320 6 × 10−3 -0.506 1.3 × 10−29 0.76443 9 × 10−5 0.0 0.0 0.0103 5 × 10−4 0.61 8 × 10−2 0.034 2 × 10−3 -0.10 9 × 10−210 0.8084 1.0 × 10−4 0.0 0.0 0.0259 1.0 × 10−4 0.389 8 × 10−3 0.356 3 × 10−3 -0.026 8 × 10−311 0.7541 5 × 10−4 0.0 0.0 0.0569 5 × 10−4 -0.49 2 × 10−2 0.216 4 × 10−3 0.72 2 × 10−212 0.3259 3 × 10−4 0.0 0.0 0.01574 9.7 × 10−4 -0.01 4 × 10−2 0.061 5 × 10−3 0.00(3) 3 × 10−213 0.20682 5 × 10−5 0.0 0.0 0.00568 1.2 × 10−4 -1.03 3 × 10−2 0.069 1.5 × 10−3 -2.03 3 × 10−214 0.22955 8 × 10−5 0.0 0.0 0.0176 3 × 10−4 -0.066 1.3 × 10−2 0.321 5 × 10−3 -0.690 1.3 × 10−215 0.7764 2 × 10−4 0.0 0.0 0.0338 3 × 10−4 -0.157 7 × 10−3 0.677 6 × 10−3 0.331 7 × 10−316 0.26115 6 × 10−5 0.0 0.0 0.0105 2 × 10−4 -0.18 3 × 10−2 0.114 3 × 10−3 0.10 5 × 10−217 0.19191 8 × 10−5 0.0 0.0 0.0349 2 × 10−4 0.8251 9.8 × 10−3 0.352 2 × 10−3 -0.437 5 × 10−318 0.2484 2 × 10−4 0.0 0.0 0.0219 4 × 10−4 0.30 2 × 10−2 0.248 4 × 10−3 -0.21 2 × 10−219 0.17265 6 × 10−5 0.0 0.0 0.01099 7 × 10−5 -0.06 2 × 10−2 0.224 3 × 10−3 1.78 4 × 10−2Table 3.5: Gaussian model parameters for the ‘19 Gaussian Model’; derived using the wideband timing algorithm’s ppgausssoftware.323.3.2 Time of Arrival CalculationsThe pptoas software is used in the final step of the wideband timing algorithm, inwhich TOA and DM values are determined. This algorithm developed by Pennucciet al. (2014) extends on the FFTFIT by Taylor (1992), and draws from the workof Demorest (2007). A detailed description of this process can be found in Pen-nucci (2015), but the main difference to note between this process and that used intraditional TOA determination methods (Section 3.2) is the frequency-dependentpulse profile, D( f ,φ), referred to as a ‘pulse portrait’ due to its two dimensionalnature (changes with frequency and phase), and the frequency-dependent ‘templateportrait’, P( f ,φ). This is given as:D( f ,φ) = a( f )P( f ,φ −Φ( f ))+B( f )+N( f ,φ) (3.6)Φ( f ) contains information regarding both time dependent and independent phaseshifts, and B( f ) effectively represents the bandpass shape of the receiver (analo-gous to the the DC offset, or ‘bias’ term, when considering only one frequencysuch as in FFTFIT). This term can be thought of as the frequency-dependent meanof the noise term N( f ).In practice, pptoas was used to calculate the DM value and TOAs for eachepoch’s pulse portrait (in both bands, and for each of the coherent and incoherentmodes). The option was selected such that a single DM was determined for eachepoch, rather than fitting one DM per subintegration. For the purposes of this anal-ysis, over the timescale of a single observation, there should be negligible changein the DM. The files output by pptoas contain all of the information required toproduce a timing model.33Chapter 4Results & Discussion4.1 Gaussian ModelHerein, we discuss results from the use of a frequency-dependent Gaussian basedtemplate portrait.4.1.1 Profile EvolutionTo investigate the pulse profile evolution of PSR J0737-3039A, subbanded pulseprofiles were produced for each of the 820 MHz and L-band data in both incoherent-and coherent-de-dispersion timing modes (four data sets in total). These profileswere constructed using the PSRCHIVE software from GBT GUPPI data (pro-cessed via the methodology described in Section 3.1.2). Profiles for the epochsshown in this chapter were chosen due to their relatively high S/N. Using the pamprogram from PSRCHIVE, the data for each profile was averaged into 8 frequencychannels. Channels in the 820 MHz region have bandwidths of 25 MHz, whereasthose for the L-band data have bandwidths of 100 MHz. The frequency channeldata were then extracted to readable text files using PSRCHIVE’s pdv (Pulsar DataViewer) program, and plotted accordingly. These subbanded profiles are shown inFigures 4.1, 4.2, 4.3, and 4.4. Each figure presents the data in both normalized andun-normalized form.The profiles in both bands and observational modes were found to show evo-lution with frequency. As expected, overall intensity of the profile (as per the34un-normalized profiles) decreases with increasing frequency. However, the nor-malized profiles indicate that the height of the left-most peak decreases relative tothe right-most peak with increasing frequency. A more pronounced difference inpeak shape and the key components can be seen in the L-band subbanded profile,with the most notable of these is apparent in both the coherent- and incoherent-mode data (Figures 4.3 and 4.4). A sharp peak emerges around phase point 0.18 inthe left-most peak, increasing in height, relative to overall intensity, as frequencyincreases. As this occurs, the relative intensity around phase point 0.23 decreases.As well, in the main body of the right-most peak, lower frequency bands show twocusps, while those of higher frequency show three. Several other minor changescan also be observed. These changes found even over small frequency scales indi-cate the need for a frequency-dependent profile template for PSR J0737-3039A.35(a) Unnormalized Profile (b) Normalized ProfileFigure 4.1: Profile evolution of 820 MHz coherent de-dispersion mode data. Data were taken in timing mode at the Robert C.Byrd GBT using GUPPI on April 15th, 2012 (MJD 56032) for a length of 17516 s (219.9 minutes). Using PSRCHIVEsoftware, data were split into 8 subbands, each with bandwidth 25 MHz. Overlaying these profiles revealed frequency-dependent evolution over the full band. The pulse profile can be seen to clearly change with frequency in both intensity andshape. Subplot (a) shows the un-normalized profile, within which the pulse intensity is observed to decrease with increasingfrequency. The pulsar data and calibration observations were taken in different modes (post-observation vs real-time datafolding), resulting in an intensity scaling of 1/20 of the expected value, but this does not affect the overall result. Subplot (b)shows the data normalized by its maximum intensity. Here, the relative peak heights can be seen to change with frequency.The left peak, relative to the right peak, decreases in intensity as frequency increases.36(a) Unnormalized Profile (b) Normalized ProfileFigure 4.2: Profile evolution of 820 MHz incoherent de-dispersion mode data. Data were taken in timing mode at the Robert C.Byrd GBT using GUPPI on December 17th, 2009 (MJD 55182) for a length of 16687 s (278.1 minutes). Using PSRCHIVEsoftware, data were split into 8 subbands, each with bandwidth 25 MHz. As with the 820 MHz coherent mode data inFigure 4.1, overlaying these profiles revealed frequency-dependent evolution over the full band. The pulse profile can be seento clearly change with frequency in both intensity and shape. Subplot (a) shows the un-normalized profile, within which thepulse intensity is observed to decrease with increasing frequency. Subplot (b) shows the data normalized by its maximumintensity. Here, the relative peak heights can be seen to change with frequency. The left peak, relative to the right peak,decreases in intensity as frequency increases.37(a) Unnormalized Profile (b) Normalized ProfileFigure 4.3: Profile evolution of L-band coherent mode de-dispersion data. Data were taken in timing mode at the Robert C.Byrd GBT using GUPPI on March 25th, 2015 (MJD 57106) for a length of 18380 s (306.3 minutes). Using PSRCHIVEsoftware, data were split into 8 subbands, each with bandwidth 100 MHz. Overlaying these profiles revealed frequency-dependent evolution over the full band. The pulse profile can be seen to clearly change with frequency in both intensity andshape. Subplot (a) shows the un-normalized profile, within which the pulse intensity is observed to decrease with increasingfrequency. In this plot, it can be seen that the right-most peak begins with two cusps at the lowest frequency band, butdevelops a third with increasing frequency. The pulsar data and calibration observations were taken in different modes (post-observation vs real-time data folding), resulting in an intensity scaling of 1/20 of the expected value, but this does not affectthe overall result. Subplot (b) shows the data normalized by its maximum intensity. Here, the relative peak heights areobserved to change with frequency. The left peak, relative to the right peak, decreases in intensity as frequency increases. Inaddition to this, the left-most peak develops a sharp addition near the 0.18 phase point. This increases in height, relative tothe overall peak intensity, as frequency increases. As this occurs, the relative intensity around phase point 0.23 decreases.38(a) Unnormalized Profile (b) Normalized ProfileFigure 4.4: Profile evolution of L-band incoherent mode de-dispersion data. Data were taken in timing mode at the Robert C.Byrd GBT using GUPPI on November 25th, 2012 (MJD 56256) for a length of 17516 s (291.9 minutes). Using PSRCHIVEsoftware, data were split into 8 subbands, each with bandwidth 100 MHz. Overlaying these profiles revealed frequency-dependent evolution over the full band. The pulse profile can be seen to clearly change with frequency in both intensity andshape. Subplot (a) shows the un-normalized profile, within which the pulse intensity is observed to decrease with increasingfrequency. In this plot, it can be seen that the right-most peak begins with two cusps at the lowest frequency band, but developsa third with increasing frequency. Subplot (b) shows the data normalized by its maximum intensity. Here, the relative peakheights are observed to change with frequency. The left peak, relative to the right peak, decreases in intensity as frequencyincreases. In addition to this, the left-most peak develops a sharp addition near the 0.18 phase point. This increases in height,relative to the overall peak intensity, as frequency increases. As this occurs, the relative intensity around phase point 0.23decreases. Significant channel zapping (RFI mitigation; see Section 3.1.2) in the region of the subband centered at 1205 MHz(pink) resulted in a very low S/N profile. Despite this, the line still follows the overall trends.394.1.2 Comparison of Model to Observational DataIn Section 3.3.1, we described the steps taken to produce a Gaussian model utiliz-ing the wideband timing algorithm developed by Pennucci et al. (2014). To sum-marize, this frequency-dependent model, referred to as the ‘19 Gaussian model’,employed several high S/N epochs of both 820 MHz and L-band data, listed inTables 3.2 and 3.3. This data was acquired from observations with the Robert C.Byrd GBT using GUPPI in coherent de-dispersion timing mode and was processedvia the method discussed in Section 3.1.2.Before proceeding to time of arrival calculations, the model was comparedwith the observational data to ensure the spectral features, and their changes withfrequency, were being correctly replicated. Figures 4.5 and 4.6 show both dataand model normalized pulse portraits, along with residuals from comparison of thetwo. Some small structure is still apparent in the residuals plot, with the ‘checker-board’ pattern in the 820 MHz region as the most prominent feature. This patternmay be the result of slight offsets between the data and model, such as those seenin Figures 4.9 and 4.10. Some aliasing is also apparent in the L-band region ofboth the data and residuals, but is not replicated in the model.40Figure 4.5: Normalized pulse portrait of coherent mode 820 MHz and L-band dataincluded in 19 Gaussian Model (Table 3.2 and 3.3). Left – Concatenated portraitdata; top figure represents the average profile and left figure represents relativescattering between bands. The spectral index of γ = −0.08 indicates that scat-tering effects are minimal for these observations of PSR J0737-3030A. Right– Average pulse portraits and pulse profiles of two bands (Bottom – 820 MHzpulse portrait; Middle – L-band pulse portrait; Top/Green – 820 MHz pulse pro-file; Top/Blue – L-band pulse portrait). Within the L-band region of the pulseportrait, some aliasing is observed.41Figure 4.6: 19 Gaussian Model normalized pulse portrait and residuals. Left – Pulseportrait predicted by the 19 Gaussian model, spanning 720-1900 MHz. Right– Normalized residuals from comparison of 19 Gaussian model to data usedin modelling (Figure 4.5). Some structure is still apparent in the residual plot;the most prominent feature is the ‘checker-board’ effect seen in the 820 MHzregion. This may be due to slight offsets between the model and observationaldata, such as those seen in Figures 4.9, and 4.10. The aliasing observed in theL-band region (Figure 4.5) is also apparent in the residuals.42Figures 4.7 and 4.8 show the normalized pulse profiles predicted using the 19Gaussian model. The plots shows eight profiles, each at a set frequency. Both showvery similar evolutionary trends to those of the observational data in Section 4.1.1,with the intensity of the left peak clearly decreasing in intensity, relative to theright peak, with increasing frequency. Of particular interest is the evolution of thesharp peak in the L-band profile at phase point 0.18. This peak changes from anindiscernible feature at 1150 MHz to a prominent component at higher frequen-cies, matching the observed pulse observations. Changes in features such as thisfurther the argument for a frequency-dependent template model, especially in thewideband regime.Figure 4.7: Normalized ‘19 Gaussian model’ predicted profile around 820 MHz.The model shows a decrease in the relative intensity of the left-most peak withincreasing frequency, as in its observational counterpart, Figure 4.1b. Minorchanges in pulse shape are also observed.A direct comparison between model and observational data pulse profiles wasaccomplished using the subbanded data files, produced in the analysis describedin Section 4.1.1. The Gaussian model parameters (Table 3.5) were used to cal-culate a template profile for the centre frequency of each subband in all four datasets (820 MHz coherent de-dispersion mode, 820 MHz incoherent de-dispersion43Figure 4.8: Normalized ‘19 Gaussian model’-predicted profile around 1500 MHz(L-band). The model shows an overall decrease in the relative intensity of theleft-most peak with increasing frequency, as in its observational counterpart,Figure 4.3. Of particular interest is the evolution of the minor peak at phasepoint 0.18, which is non-existent at 1150 MHz, but grows in relative intensity tobecome a prominent feature at higher frequencies in the modeled pulse profile.mode, L-band coherent de-dispersion mode, and L-band incoherent de-dispersionmode). These templates were then overlain with their corresponding subbands. Asmentioned in Section 3.3.1, when comparing the model to data at different obser-vational bands, the parameters in the join file must be taken into account. Theseoffsets, a ∆DM and a rotational shift, ∆rot, for each frequency band used in the 19Gaussian model are listed in Table 3.4. The total rotational offset was calculatedusing the following:∆phase=(D x ∆DM x f−2P)+ ∆rot (4.1)where D is the dispersion constant (Equation 2.3), f is the frequency of the desiredtemplate profile, and P is the rotational period of the pulsar (PSR J0737-3039A).This ‘∆phase’ is then subtracted from the phase at each point on the profile. Notethat the values listed in the join file table are the negative of the offset. Both model44and observational data were normalized using the maximum intensity of the profile,setting this value to one.Figures 4.9, 4.10, 4.11, and 4.12 show comparison of the normalized model andsubbanded data. Note that the subbanded data are average profiles, with bandwidthof 25 MHz for the 820 MHz data, and 100 MHz for L-band. Hence, they willexhibit some smearing of profile features which may affect the fit comparison.45Figure 4.9: Comparison of 19 Gaussian model with 820 MHz coherent de-dispersion mode data. Data were taken in timing modeat the Robert C. Byrd GBT using GUPPI on April 15th, 2012 (MJD 56032) for a length of 17516 s (219.9 minutes). UsingPSRCHIVE software, data were split into 8 subbands, each with bandwidth 25 MHz. The model profile was computed forthe center frequency of each subband. Both data and model were normalized by the maximum intensity in the profile. Asmall phase shift is observed between the model and data profiles, with the model appearing to move left relative to the dataas frequency increases. Although we have not been able to conclusively determine the cause of this shift, it may be, in part,due to slight differences in DM on the epoch plotted compared to those used in the portrait modelling.46Figure 4.10: Comparison of 19 Gaussian model with 820 MHz incoherent de-dispersion mode data. Data were taken in timingmode at the Robert C. Byrd GBT using GUPPI on December 17th, 2009 (MJD 55182) for a length of 16687 s (278.1minutes). Using PSRCHIVE software, data were split into 8 subbands, each with bandwidth 25 MHz. The model profilewas computed for the center frequency of each subband. Both data and model were normalized by the maximum intensityin the profile. A small phase shift is observed between the model and data profiles, with the model appearing to move leftrelative to the data as frequency increases. Although we have not been able to conclusively determine the cause of this shift,it may be, in part, due to slight differences in DM on the epoch plotted compared to those used in the portrait modelling.47Figure 4.11: Comparison of 19 Gaussian model with L-band coherent de-dispersion mode data. Data were taken in timing modeat the Robert C. Byrd GBT using GUPPI on March 25th, 2015 (MJD 57106) for a length of 18380 s (306.3 minutes). UsingPSRCHIVE software, data were split into 8 subbands, each with bandwidth 100 MHz. The model profile was computed forthe center frequency of each subband. Both data and model were normalized by the maximum intensity in the profile. Asmall phase shift is observed between the model and data profiles, with the model appearing to move left relative to the dataas frequency increases. Although we have not been able to conclusively determine the cause of this shift, it may be, in part,due to slight differences in DM on the epoch plotted compared to those used in the portrait modelling.48Figure 4.12: Comparison of 19 Gaussian model with L-band incoherent de-dispersion mode data. Data were taken in timingmode at the Robert C. Byrd GBT using GUPPI on November 25th, 2012 (MJD 56256) for a length of 17516 s (291.9minutes). Using PSRCHIVE software, data were split into 8 subbands, each with bandwidth 100 MHz. The model profilewas computed for the center frequency of each subband. Both data and model were normalized by the maximum intensityin the profile. The subband centered at 1205 MHz has a very low S/N due to significant zapping in that region of the pulseportrait. A small phase shift is observed between the model and data profiles, with the model appearing to move left relativeto the data as frequency increases. Although we have not been able to conclusively determine the cause of this shift, it maybe, in part, due to slight differences in DM on the epoch plotted compared to those used in the portrait modelling.49Overall, we have found good agreement between the model and data, despiteusing the averaged subbanded profiles. Comparisons of single frequency channelswith the corresponding model frequency were found to provide very little mean-ingful information, due to the amount of noise present. A small phase shift wasobserved between the model and data pulse profiles. This shift in the model, moreapparent in the 820 MHz region, appears to move left relative to the data with in-creasing frequency. This may also be related to the ‘checker-board’ phenomenonobserved in the 820 MHz region of the pulse portrait residuals (Figure 4.6). Al-though we have not been able to conclusively determine the cause of this shift, itmay be, in part, due to slight differences in DM of the epoch plotted compared tothose used in the portrait modelling.Attempts were made to create a Gaussian model utilizing the incoherent de-dispersion mode data. However, no model could be produced to properly matchthe observation data in either band. It is possible that this resulted from insufficientavailability of calibrated data in the L-band region. A large number of the fluxand polarization calibration files were found to have been ‘corrupted’, and thus thedata from many epochs remained in their zapped form. Since this data were a lessaccurate representation of the true pulse signal, incorporating these epochs into atemplate model would have produced unsuitable results. However, the 19 Gaussianmodel constructed with the coherent mode de-dispersion data was found, by eye,to fit both coherent and incoherent de-dispersion mode data on a comparable level.4.1.3 Qualitative Comparison with Wider Range of BandsFollowing comparisons of the frequency-dependent wideband timing template withboth coherent and incoherent de-dispersion mode data, the functionality of the tem-plate with other PSR J0737-3039A observational data was explored. Data from theParkes 64-m radio telescope was provided through private communication with R.Manchester (Manchester et al., 2005). Filterbank observations were taken betweenMay 2003 and August 2004 at three frequency bands: 680 MHz, 1390 MHz, and3030 MHz.The mean pulse profiles at each of the three Parkes data bands were overlainwith the 19 Gaussian model template, calculated for the centre frequency in each50profile. This is shown in Figure 4.13. As these profiles were not aligned in thesame manner as the GBT GUPPI data (and subsequently the 19 Gaussian model),an arbitrary phase offset was introduced into the plotted Gaussian model such thatthe two templates aligned, by eye, at the right-most peak in each plot. All tem-plates were normalized to their maximum intensity, setting this value to one. Thereare several obvious differences between the templates, although some of these maybe accounted for the by fact that the Parkes profiles are based upon averaged data,which have not taken frequency-dependent evolution into account. In addition tothis, the Parkes data are all filterbank, and the 680 MHz and 3030 MHz profiles falloutside the range used in determining the model. The relative peak heights, partic-ularly within the 1390 MHz and the 3030 MHz data are well represented, as is theevolution of the sharp component in the left-most peak, especially evident in the3030 MHz subplot. The phase shift of the left-most peak in the 3030 MHz pulseprofile is of particular interest. Although the spectral shape and features are reason-ably represented, maxima are out of alignment. This suggests that some movementof the centroids of the Gaussians should be taken into account, particularly oververy wide bands.These results are very promising for the applicability of Gaussian models astemplate profiles (or ‘portraits’). The 19 Gaussian model was obtained by fittingdata within a narrow bandwidth in the L-band range. It is unsurprising that the1390 MHz data and model profiles are very similar, as this frequency falls withinthe modelling range. However, it predicts, within reason, the evolution from thispoint to the 3030 MHz range without using any observations in this band. This pro-vides some evidence that this method of modelling is not just an analytic fit to theobservational data, but instead has a connection to the physical regions of the neu-tron star producing the radio emission. Further investigation using the widebandtiming algorithm and a larger range of data sets could yield some very interestingresults in regards to the model’s ability to predict changes in the spectral features.51Figure 4.13: Qualitative comparison of 19 Gaussian model (red) and Parkes 64-mTelescope data (blue), provided in private communication with R. Manchester;further observational and analysis details of the filterbank Parkes data can befound in Manchester et al. (2005). Top – 680 MHz band; middle – 1390 MHzband; bottom – 3030 MHz band. Due to its processing, the central frequencyof the 3030 MHz band (bottom) actually sits at 2934 MHz. Profiles usingthe 19 Gaussian model were computed based on the central frequencies of theaveraged Parkes profile bands, and that both the 19 Gaussian model and theParkes templates have been normalized to their respective maximums. Despitesome obvious differences (possibly due to the Parkes data resulting from aver-aged, non-frequency-dependent profiles), the relative peaks heights of the dataare well represented by the model, particularly for the 1390 MHz and 3030MHz profiles. The development of the sharp feature in the left-most peak is ofparticular interest.4.2 Timing ResultsAs per Section 3.3.2, the wideband timing algorithm’s pptoas was used to calcu-late TOAs and a dispersion measure value for each epoch and band of the widebandGBT GUPPI data employing the 19 Gaussian model. Subsequently, TEMPO (Sec-tion 2.5) was used to analyze this timing data, fitting for all parameters listed in52Table 4.2, and utilizing the DDS binary model (Kramer et al. 2006b; Kramer et al.2006a) and JPL DE430 Solar System barycenter transformation. A modified ver-sion of the ephemeris file noted in Section 3.1.2 was used as a starting point forthe parameters. TEMPO’s ‘DMX’ option was implemented in the fit. This allowsthe program to fit per epoch DM shifts and further accounts for the changes in DMover time; epoch bins of 120 days were chosen for this analysis, ensuring that atleast one 820 MHz band and one L-band observation were included per bin. Theprogram also fit for constant offsets, or ‘jumps’, between TOAs of different datasubsets (both bands and de-dispersion modes).Two steps were required to complete the TEMPO timing analysis. In the firststep, the DMX bin ranges and initial shifts were determined. TEMPO was thenimplemented a second time, utilizing the output parameter file from the previousstep, now including the DMX parameters, as an initial guess. The per-epoch (per-day) DM values from the wideband timing algorithm were not used in this fit, butare compared to the DMX results in Section 4.2.1.For comparison, this TEMPO procedure was repeated on GBT GUPPI datawhich has been processed using more traditional methods. The wavelet standard-profile for this data set was constructed by I. H. Stairs and R. Ferdman using aprogram called psrsmooth, developed by Demorest et al. (2013). As per traditionaltiming methods, this data was divided into 8 subbands for TOA calculation, andwill henceforth be known as the subbanded data. Only the second TEMPO stepwas performed on this data. To remain consistent, the initial output parameter filefrom the wideband timing analysis (containing the DMX bins) was used as theinitial guess for the subbanded data.The range of the two data sets differed slightly. Data were available for thesubbanded data set from MJD 55002-57199, whereas the wideband data spannedMJD 55002-57468. Because of this, the TEMPO procedure was repeated twice foreach of the wideband and subbanded data sets – once for the full date range (fullset), and once again between MJD 55002-57199 (truncated set).When fitting the TOAs to a timing model, TEMPO calculates the chi-squarevalue for each data subset, as well as for the set as a whole. Values of χ2TOA = 1.31and χ2TOA = 1.23 were determined when fitting the full sets of wideband and sub-banded data, respectively. However, the TOAs may have inaccurate uncertain-53ties, which are nominally estimated from the template-matching process (Pennucci,2015). To account for this imperfect representation of the true pulse, an extra freeparameter, EFAC, is often introduced, which linearly scales the TOA uncertaintiesby the square root of the reduced χ2 value. EFACs found here are mostly close to1.0, as GUPPI, like most modern instruments, has good profile reproduction. In thefollowing analysis, EFAC parameters were incorporated for each band and timingmode, with values listed in Table 4.1.Band/Timing Mode WB Full WB Truncated SB Full SB Truncated820/Incoherent 1.2110 1.2110 1.0524 1.0524L-band/Incoherent 1.2171 1.2174 1.1305 1.1305820/Coherent 1.0895 1.0870 1.0967 1.0967L-band/Coherent 1.1771 1.1897 1.1989 1.1989Table 4.1: EFAC parameter values used in final timing model; WB – Wideband data,SB – Subbanded data4.2.1 DMX Fitting and Dispersion MeasureThe wideband timing algorithm allows for the determination of per-day DM values,a feature unavailable in traditional TOA and timing model calculations. Figure 4.14shows the DM values for each of the four analyzed data sets. A clear offset canbe seen between the 820 MHz and L-band observations. This aspect was alsoobserved in pulsars explored by Pennucci (2015) and was considered a residual ofthe wideband timing model. Aside from this, there appears to be some differencein structure between the two bands.As expected, the DM changes within the time-span covered by the observa-tions; the DMX shifts computed by TEMPO are used to account for these changeswhen calculating a timing solution. Figure 4.15 shows the DMX values for boththe wideband and subbanded data sets, as well as the weighted bin-averaged DMvalues from the wideband data set. Trends in the wideband DMX values track rea-sonably well with the wideband DMs and the subbanded DMX values, outside the820 MHz incoherent-mode era. This lends support to the wideband timing algo-rithm as a whole, and implies that it can produce a reliable set of full band TOAs. It54also indicates that the profile template is well-representative of the observed pulseprofile.It is interesting to note the shift in DMX values between the full and truncatedwideband data sets. Although they follow a similar track, the truncated data setDMX values fall approximately 0.0003 cm−3 pc less than their full set counter-parts. This is most likely due to covariance between the DMX values and jumpparameters, as both fit for variation in phase shift over time. Figure 4.16 showsthe correlation between DMX values for the wideband and subbanded data sets,with a black dashed line representing a slope of 1. Although both the full and trun-cated sets show obvious variations, some correlation is apparent. In both cases,four points near the subbanded value of 2 × 10−3 sit away from the others. Thesevalues fall within the 820 MHz incoherent de-dispersion mode era and indicate apossible problem with the data in this range, likely stemming from intra-channeldispersive smearing.55Figure 4.14: DM values from wideband timing algorithm. A clear offset can be seen between the 820 MHz and L-band observa-tions. This aspect was also observed in pulsars explored by Pennucci (2015) and was considered a residual of the widebandtiming model. Some structural differences can be seen between the two bands.56(a) Full data set(b) Truncated data setFigure 4.15: Comparison of DMX trends with time for wideband and subbandeddata. The two DMX bins on either side of MJD 55500 have limited 820 MHzdata (see Section 4.2.2), and as such may be unreliable.57Figure 4.16: Correlation of wideband and subbanded DMX values for full and trun-cated data sets; black dashed line, with slope of 1, is used for comparison, andis not a fit line.4.2.2 Timing Model and TOA ResidualsThe timing parameters for the subbanded data are the subject of the forthcomingpaper by Kramer et al. Therefore, we do not present the binary parameters here, butinstead show the spin and astrometric parameters and a comparison of uncertaintiesThe uncertainties associated with the final timing model parameters are listedin Table 4.2. For a particular parameter, the uncertainties are comparable betweenthe four data sets, with the subbanded full and truncated values being equal, as58expected. Although we do not present the binary parameters, the values have beenfound to agree within 2σ between all data sets, aside from the SHAPMAX term,which agrees within 3σ between the full wideband set and the subbanded set. Ingeneral, the wideband full data set uncertainties are less than the wideband trun-cated set uncertainties which are in turn less than the subbanded full/truncated sets’uncertainties. From this alone, we cannot determine which model is best, but thesesimilarities between the wideband and subbanded data sets lend additional supportfor the validity of the wideband TOAs.59Paramter Description Units ∆WB Full ∆WB Truncated ∆SB Full ∆SB TruncatedRAJ J2000 Right ascension hh:mm:ss.sss 4.4 × 10−6 4.7 × 10−6 5.4 × 10−6 5.4 × 10−6DECJ J2000 Declination hh:mm:ss.sss 1.1 × 10−4 1.1 × 10−4 1.3 × 10−4 1.3 × 10−4PMRA Proper motion in right ascension mas/yr 1.5 × 10−2 1.8 × 10−2 2.3 × 10−2 2.3 × 10−2PMDEC Proper motion in declination mas/yr 2.7 × 10−2 3.1 × 10−2 3.9 × 10−2 3.9 × 10−2PX Parallax mas 2.3 × 10−1 2.4 × 10−1 2.7 × 10−1 2.7 × 10−1F0 Rotational frequency s−1 1.1 × 10−12 1.2 × 10−12 1.0 × 10−12 1.0 × 10−12F1 1st Derivative of Rotational Frequency s−2 2.2 × 10−20 2.7 × 10−20 2.6 × 10−20 2.6 × 10−20F2 2nd Derivative of Rotational Frequency s−3 1.9 × 10−28 2.6 × 10−28 2.8 × 10−28 2.8 × 10−28A1 Projected semi-major axis of orbit lt-sec 1.2 × 10−7 1.3 × 10−7 1.5 × 10−7 1.5 × 10−7E Eccentricity of orbit 7.0 × 10−8 7.2 × 10−8 8.1 × 10−8 8.1 × 10−8T0 Epoch of periastron MJD 2.5 × 10−8 2.5 × 10−8 2.9 × 10−8 2.9 × 10−8PB Orbital period days 1.8 × 10−12 1.9 × 10−12 2.3 × 10−12 2.3 × 10−12OM Longitude of periastron deg 8.7 × 10−5 8.9 × 10−5 1.0 × 10−4 1.0 × 10−4OMDOT Rate of advance of periastron deg/yr 2.2 × 10−5 2.4 × 10−5 2.8 × 10−5 2.8 × 10−5GAMMA Post-Keplarian gamma term s 1.1 × 10−7 1.3 × 10−7 1.5 × 10−7 1.5 × 10−7PBDOT 1st Time derivative of binary period s/s * 1012 2.3 × 10−4 2.8 × 10−4 3.3 × 10−4 3.3 × 10−4SHAPMAX − ln(1− sin(i)); i = inclination angle 1.2 × 10−5 1.3 × 10−5 1.5 × 10−5 1.5 × 10−5M2 Companion Mass Solar Masses (M) 5.3 × 10−3 5.6 × 10−3 6.3 × 10−3 6.3 × 10−3Table 4.2: Timing parameter uncertainties60To properly compare timing residuals (differences between the observed TOAsand those predicted by the current model parameters) between the wideband andsubbanded data sets, the subbanded TOAs were evaluated using a fixed version ofthe wideband timing model, in which all parameters remained constant. The tim-ing residuals yielded in this calculation are compared to those of the wideband datain Figures 4.17 and 4.18. The residuals are a way to evaluate how well a model fitsthe data; Figure 4.17 shows the weighted daily average timing residuals of the twodata sets, divided by band and de-dispersion mode. The two sets (and their sub-sets) track reasonably well, and have a comparable range of residual values. Theorbital phase bin-averaged timing residuals can be seen in Figure 4.18, whereinboth data sets (and their subsets) can be seen to track similar structures. The simi-larities between data sets in the each residual plot further legitimize the widebandTOAs, as both the TOAs from the traditional method and wideband algorithm givereasonable residuals using the final wideband timing model.A daily average residual plot produced from an earlier version of the widebandtiming model lead to the discovery of some inconsistencies in the subbanded data– namely on MJDs 55337, 55502, 55517, and 55518 from the early 820 MHzcoherent timing mode era. These gave timing residuals on the order of 15 µs,whereas all other residuals fell within a similar range as those in the figure shown.TOAs from these epochs were removed from those included in the timing modelcalculations for both the wideband and subbanded data sets, as they were no longerconsidered reliable; the resulting small number of 820 MHz epochs in the binsnear MJD 55500 may be the cause of the outlying DM values in these bins. Thedata on these specific days were taken with different instrumental configurationsfrom the one that became standard, and although the timing offsets were nominallycalculated, it appears they may be incorrect.61(a) Full data set(b) Truncated data setFigure 4.17: Daily average timing residuals62(a) Full data set(b) Truncated data setFigure 4.18: Orbital phase bin-averaged timing residuals63Table 4.3 lists the astrometric and spin parameters derived from the TEMPOanalysis. It is important to note that comparison of these values does not let usjudge which fit is best, though, as previously stated, the subbanded data had asmaller initial χ2. This study incorporated data only from GUPPI; values in theupcoming Kramer et al timing paper, incorporating data from multiple telescopes,will be significantly more robust. Despite this, the change in spin parameters be-tween data sets is still surprising as PSR J0737-3039A is a well-studied pulsar.Wideband TOA fitting procedures, such as the one used here, are increasinglyunder investigation for the MSPs in timing arrays such as the North AmericanNanohertz Observatory for Gravitational Waves (NANOGrav). These pulsars typ-ically have sparser data, and precise DM corrections are necessary to achieve pre-cise timing. Our results point to potential difficulties in achieving robust timingresults.64Paramter Units WB Full WB Truncated SB Full SB TruncatedRAJ hh:mm:ss.sss 07:37:51.2481171 ± 4 × 10−6 07:37:51.2481272 ± 5 × 10−6 07:37:51.248114 ± 5 × 10−6 07:37:51.248114 ± 5 × 10−6DECJ hh:mm:ss.sss -30:39:40.7052 ± 1 × 10−4 -30:39:40.7051 ± 1 × 10−4 -30:39:40.7058 ± 1 × 10−4 -30:39:40.7058 ± 1 × 10−4PMRA mas/yr -2.65 ± 1 × 10−2 -2.71 ± 2 × 10−2 -2.62 ± 2 × 10−2 -2.62 ± 2 × 10−2PMDEC mas/yr 2.05 ± 3 × 10−2 2.00 ± 3 × 10−2 2.22 ± 4 × 10−2 2.22 ± 4 × 10−2PX mas 2.2 ± 2 × 10−1 1.9 ± 2 × 10−1 1.7 ± 3 × 10−1 1.7 ± 3 × 10−1F0 s−1 44.054068835269 ± 1 × 10−12 44.054068835265 ± 1 × 10−12 44.054068835260 ± 1 × 10−12 44.054068835260 ± 1 × 10−12F1 s−2 -3.41572 × 10−15 ± 2 × 10−20 -3.41562 × 10−15 ± 3 × 10−20 -3.41556 × 10−15 ± 3 × 10−20 -3.41556 × 10−15 ± 3 × 10−20F2 s−3 -1.7 × 10−27± 2 × 10−28 -2.8 × 10−27 ± 3 × 10−28 -3.4 × 10−27 ± 3 × 10−28 -3.4 × 10−27 ± 3 × 10−28Table 4.3: Astrometric and spin parameters65Chapter 5ConclusionsWe have used the wideband timing algorithm developed by Pennucci et al. (2014)to produce TOAs and a timing solution for PSR J0737-3039A, along with per-dayDM values. This was achieved with data spanning MJD 55002-57468, acquiredfrom the GBT utilizing GUPPI.The frequency-dependent pulse portrait template model developed with the al-gorithm was found to well-represent the observational data. Through comparisonwith additional data from the Parkes 64-m Telescope, it was found that the modelhas predictive properties exceeding the frequency range of the data from which itwas calculated. This implies that the model has some connection to the physicalregions of the neutron star producing the radio emission.TOAs derived using the wideband timing algorithm have been found to bevery reliable. Predicted dispersion measure shifts track reasonably well with thoseachieved through more traditional means. As well, the timing model calculatedwith these TOAs gives comparable timing residual values for the wideband andwavelet-standard subbanded data.The significant differences found between the spin and astrometric parametersof the different data sets and subsets is surprising, and possibly a cause for con-cern, given that PSR J0737-3039A is such a well studied pulsar. Other MSPs, suchas those in pulsar timing arrays aiming to detect gravitational waves, have sparserdata, limiting the precision of DM corrections required for precise timing results.As such, our findings point to potential difficulties in achieving precise timing re-66sults.Future studies incorporating multi-telescope data should allow us to producea more robust wideband timing model, and potentially determine the cause of thephase shifts observed between the portrait model and wideband data profiles. 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J., & Taylor, J. H. 2000,MNRAS, 314, 459 → pages 16Taylor, J. 1992, Philos. Trans. R. Soc. London, Ser. A, 341, 117 → pages 24, 25,3371Taylor, J., & Weisberg, J. 1989, ApJ, 345, 434 → pages 7Thorsett, S., & Chakrabarty, D. 1999, ApJ, 512, 288 → pages 4van Straten, W., Demorest, P., & Oslowski, S. 2012a, Astronomical Research andTechnology, 9, 237 → pages 22—. 2012b, Astronomical Research and Technology, 9, 237 → pages 22, 24Vangioni, E., Goriely, S., Daigne, F., & Francois, P. 2015, MNRAS, 455, 17 →pages 9Weisberg, J., & Huang, Y. 2016, ApJ → pages 7Zhu, W. W., Stairs, I. H., Demorest, P. B., Nice, D. J., Ellis, J. A., Ransom, S. M.,Arzoumanian, Z., Crowter, K., Dolch, T., Ferdman, R. D., Fonseca, E.,Gonzolez, M. E., Jones, G., Jones, M. L., Lam, M. T., Levin, L., McLaughlin,M. A., Pennucci, T., Stovall, K., & Swiggum, J. 2015, ApJ, 809, 41 → pages17, 1872Appendix ATechnical MethodsA.1 Zapping DataThe following python script was used in zapping the original fits data files. The filecontaining channels and subintegrations to be zapped must contain two columns,wherein the following format must be used:• Subintegration zap – ‘subint 1’• Frequency channel zap – ‘1 frequency’• Subintegration/channel point zap – ‘subint frequency’The zap data file name must also be of the format [fits file name base].zap.dat, asillustrated in the script’s ’Options/Features’ section.The script takes 4 command line arguments:1. Metafile containing all fits-format files to be zapped2. Location to save output files3. Location of fits-format files to be zapped4. Location of zap data files73# Zapping guppi f i l e s’ ’ ’DESCRIPTION : Zaps l i s t [ me ta f i l e ] o f f i t s−type f i l e s using ASPZap formatzap data f i l e s . REQUIRES PSRCHIVE SOFTWARE.OPTIONS/FEATURES:− F i t s f i l e s loca ted i n d i r e c t o r y [ f i l e l o c ]− Requires one zap data f i l e per f i t s f i l e w i th name type<f i lename base>.zap . dat , loca ted i n d i r e c t o r y [ z a p d a t f i l e l o c ]−− Example : F i l e ’ GUPPI tes t f i l e 0001 . f i t s ’ r equ i res zap dataf i l e ’ GUPPI tes t f i l e 0001 . zap . dat ’− Saves output zap f i l e s to d i r e c t o r y [ ou t ou t l o c ] w i th name<f i lename base>.zap− Saves psrsh zapping s c r i p t to <f i lename base>.psrsh i n workingd i r e c t o r yUSE: python newzap . py [ me ta f i l e ] [ ou t pu t l o c ] [ f i l e l o c ] [ z a p d a t f i l e l o c ]’ ’ ’## Packagesimport numpy as npimport osimport subprocessimport mathfrom datet ime import datet imeimport sysimport t ime## S ta r t t ime ( p r i n t ed wi th end t ime when a l l c a l c u l a t i o n s f i n i s h )FORMAT = ’%Y−%m−%d %H:%M:%s ’t ime1 = datet ime . now ( ) . s t r f t i m e (FORMAT)s t a r t t ime = ’%s ’ % ( t ime1 )## Command l i n e argumentsf i l e l i s t = sys . argv [ 1 ] # Me ta f i l ezap loca t ion = sys . argv [ 2 ] # Locat ion f o r output zapped f i l e sf i l e l o c = sys . argv [ 3 ] # Locat ion o f f i t s f i l e s i n me ta f i l ezapdat loc = sys . argv [ 4 ] # Locat ion o f zap data f i l e s## Load l i s t o f f i t s f i l e swi th open ( f i l e l i s t ) as f l :f i l e s = [ x . s t r i p ( ’\n ’ ) for x in ( y for y in f l . r ead l i nes ( ) i f not \y . s t a r t sw i t h ( ’ # ’ ) ) ]n f i l e s = len ( f i l e s ) # Number o f f i t s f i l e s i n me ta f i l e74## Zapping f i l e s’ ’ ’For each f i l e i n meta f i l e , python reads i n zap data f i l e . Through thefo l l ow ing , the zap data f i l e i s re fo rmat ted to produce a PSRCHIVE psrshcommand s c r i p t . Python ’ s subprocess package i s used to execute t h i scommand, producing a zapped f i l e .’ ’ ’for i in range ( n f i l e s ) :f i lename = ’%s/%s ’ % ( f i l e l o c , f i l e s [ i ] ) # Name of f i t s f i l e w/ locf i lename2 = f i l e s [ i ] # Name of f i t s f i l ef i l e b a se = st r ( f i lename2 . s p l i t ( ’ . ’ ) [ 0 ] ) # Name of f i t s f i l e w i thou t# ext ; <f i lename base>z f i l e = ’%s/%s . zap . dat ’ % ( zapdat loc , f i l e b a se ) # F i l e ’ s zap datap s r s h f i l e = ’%s . zap . psrsh ’ % ( f i l e b a se ) # Name of psrsh s c r i p t## Get number o f channels , center frequency , and bandwidth f o r f i t s f i l eva ls raw = subprocess . check output ( ’ p s r ed i t −q −c nchan ,bw −Q \%s ’ % ( f i lename ) , s h e l l =True )va ls = va ls raw . s p l i t ( )nchan = f l oa t ( va ls [ 0 ] ) # Number o f channelsbw = f l oa t ( va ls [ 1 ] ) # Bandwidth## Read in zap data’ ’ ’The ASPZap zap data f i l e i s fo rmat ted i n t o two columns :[ 0 ] Sub in teg ra t i on number[ 1 ] Frequency valueFor sub in teg ra t i ons t ha t are to be f u l l y zapped , the frequency value i s−1. A l t e r n a t i v e l y , f o r channels ( f requency bins ) t ha t are to be f u l l yzapped , the sub i n t eg ra t i on number i s −1.’ ’ ’zva ls = np . l o ad t x t ( z f i l e , unpack=True )sub in ts=zva ls [ 0 ] # Sub in teg ra t i on numberf reqs=zva ls [ 1 ] # Frequency valueswi th open ( z f i l e ) as f :da ta len=sum(1 for in f ) # Number o f l i n e s i n zap data f i l e## Def ine center frequency ( c f ) based on bandwidth ( assumes c f w i l l be## set a t a s p e c i f i c bw)i f abs (bw) == 200: # 820 MHz data has bw=−200c f = 820. # Works f o r coherent and incoheren t mode75i f abs (bw) == 800: # L−band data has bw=800c f = 1500. # Works f o r coherent and incoheren t mode## Create l i s t o f f requency b ins based on number o f channels , center## frequency , and bandwidthf r e q s l i s t = np . l i nspace ( cf −(0.5∗bw) , c f +(0.5∗bw) ,num=nchan , \endpoint=False )## Convert f requenc ies i n zap data f i l e to channel numberschans=np . ar ray ( [ ] )for j in f r eqs :i f j == −1.0: # Ind i ca tes f u l l sub i n t eg ra t i on to zapchans = np . append ( chans , j )else :chans = np . append ( chans , np . where ( f r e q s l i s t == j ) )## Sor t i n t o zapping type ( f u l l sub in teg ra t i on , f u l l channel , or## sub i n t eg ra t i on / channel i n t e r s e c t i o n pa i r )q1 = −1 in chans # Are there f u l l sub in ts to be zapped?# ( q1=True i f t r ue )q2 = −1 in sub in ts # Are there f u l l channels to be zapped?# ( q2=True i f t r ue )f u l l s u b i n t s =np . ar ray ( [ ] ) # Array f o r f u l l sub i n t eg ra t i ons to zapf u l l c hans=np . ar ray ( [ ] ) # Array f o r f u l l channels to zapi f q1 == True : # Fu l l sub i n t eg ra t i ons to zapfor j in range ( da ta len ) :i f chans [ j ]==−1.0:f u l l s u b i n t s =np . append ( f u l l s u b i n t s , sub in ts [ j ] )i f q2 == True : # Fu l l channels to zapfor j in range ( da ta len ) :i f sub in ts [ j ]==−1.0:f u l l c hans=np . append ( fu l l chans , chans [ j ] )s ub i n t s I =np . ar ray ( [ ] ) # Sub in teg ra t ions o f i n t e r s e c t i o n pa i r schansI=np . ar ray ( [ ] ) # Channels o f i n t e r s e c t i o n pa i r sfor j in range ( da ta len ) :i f sub in ts [ j ] != −1.0 and chans [ j ] != −1.0:s ub i n t s I =np . append ( sub in t s I , i n t ( sub in ts [ j ] ) )chansI=np . append ( chansI , i n t ( chans [ j ] ) )## Wri te i n fo rma t i on to psrsh−readable s c r i p t76s1= ’ zap sub in t ’ # Command to zap f u l l sub i n t eg ra t i ons2= ’ zap chan ’ # Command to zap f u l l channels3= ’ zap such ’ # Command to zap sub in t eg ra t i on / channel# i n t e r s e c t i o n po in tf l =open ( p s r s h f i l e , ’w ’ )i f len ( f u l l s u b i n t s ) > 0:for i tem in f u l l s u b i n t s :s1+= ’ %s ’ % ( i n t ( i tem ) )f l . w r i t e ( s1 + ’\n ’ )i f len ( f u l l c hans ) > 0:for i tem in f u l l c hans :s2+= ’ %s ’ % ( i n t ( i tem ) )f l . w r i t e ( s2 + ’\n ’ )i f len ( chansI ) > 0:for j in range ( len ( chansI ) ) :s3+= ’ %s,%s ’ % ( i n t ( s ub i n t s I [ j ] ) , i n t ( chansI [ j ] ) )f l . w r i t e ( s3 + ’\n ’ )f l . c lose ( )## Run psrsh command to read i n s c r i p t and zap f i t s f i l es= ’ psrsh −e zap %s %s ’ % ( ps r sh f i l e , f i lename2 )subprocess . c a l l ( s , s h e l l =True )## End t ime of c a l c u l a t i o nt ime2 = datet ime . now ( ) . s t r f t i m e (FORMAT)end t ime = ’%s ’ % ( t ime2 )## P r i n t s t a r t and end t ime of f u l l c a l c u l a t i o npr in t ’\n\nSta r t : %s\nEnd : %s\n\n ’ % ( s t a r t t ime , end t ime )A.2 Wideband Timing AlgorithmThe following outlines the basic steps in using the wideband timing algorithm,developed by Pennucci et al. (2014).77A.2.1 PPALIGNThe algorithm’s ppalign program is used to combine observational data taken inthe same timing mode and at the same frequency band to create an average profile.It takes a command line argument that works on a metafile containing the list offiles to be added, as shown below:ppa l ign . py −−n i t e r =[ x ] −T −M [ me ta f i l e ]The number of iterations is indicated by ‘x’, typically on the order of 1-5. Withthis command, the archives are time scrunched for iterations, indicated by the ‘-T’ flag; this is the recommended setting. In this thesis, the program was usedon the calibrated, aligned data files. If data files have not been aligned (thoughthe addition of an updated ephemeris, etc ...), the ‘-P’ option can be used in thecommand line.A.2.2 PPGAUSSTo create a Gaussian template model, ppgauss is used. The following command,as used in the analysis performed for this thesis, selects the reference frequency,nu ref, and bandwidth, bw to be used in the initial fitting.ppgauss . py −−n i t e r =[ x ] −−norm −−nu re f =[ nu re f ] −−bw=[bw ] −M [ me ta f i l e ]All subsequent iterations take into account the full bandwidth, as set by the aver-aged profiles provided in the metafile (these are the profiles produced with ppalign).Gaussians are chosen by hand in an interactive viewer, then subsequently fit bythe algorithm to produce the Gaussian model and uncertainty values.A.2.3 PPTOASThe pptoas program is used to calculate dispersion measure and TOAs of observa-tional epochs using the Gaussian model produced with ppgauss. In this analysis,options were chosen such that only one dispersion measure value was calculatedper epoch, and the output files, in ‘tempo2’ format, were saved with the extension‘.tim’. The following short script was used to accomplish this.’ ’ ’DESCRIPTION : Uses wideband t im ing a lgo r i t hm ( pptoas . py ) to ca l cu l a t e TOAsand DMs f o r a l l f i t s type f i l e s i n l i s t [ me ta f i l e ] using Gaussian model78f i l e [ mode l f i l e ] .OPTIONS/FEATURES:− Calcu la tes one DM value per f i t s f i l e− Output f i l e i n tempo2 format− Output f i l e s named wi th format <f i lename>. t imUSE: python ge t toas . py [ me ta f i l e ] [ mode l f i l e ]’ ’ ’## Packagesimport numpy as npimport subprocessimport sys## Command l i n e argumentsf1=sys . argv [ 1 ] # Me ta f i l e con ta in ing l i s t o f f i t s f i l e sca l cu la ted mode l f i l e =sys . argv [ 2 ] # Gaussian model f i l e## Load l i s t o f f i l e sdata=np . l o ad t x t ( f1 , dtype=str , unpack=True )## Ca lcu la te TOA & DM valuesfor i tem in data :s1= ’ pptoas . py −−one DM −f tempo2 −o %s . t im −m %s −d %s ’ % ( item , \mode l f i l e , i tem )subprocess . c a l l ( s1 , s he l l =True )79

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