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The detectability of millisecond pulsars in eccentric binary systems Madsen, Erik 2013

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The detectability of millisecond pulsars in eccentric binarysystemsbyErik MadsenBSc, University of British Columbia, 2011A 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)August 2013c? Erik Madsen, 2013AbstractWith new, highly sensitive telescopes, increased computational power, and im-proved search algorithms, the present century has seen a great increase in the dis-covery of pulsars in globular clusters. These are typically fast-spinning ?millisec-ond? pulsars, and more often than not they are members of binary systems. Unlikein the Galactic field, millisecond pulsars in globular clusters are often found in ec-centric systems because of disruptions and exchanges due to the high stellar densityof the cluster environment.A long-standing problem is that of characterizing our sensitivity to pulsars inbinary systems, particularly those with non-zero eccentricity. A pulsar?s orbitalmotion modulates its observed pulse period, making its detection through stan-dard Fourier analysis difficult or impossible. A common technique to mitigate thisproblem is the ?acceleration search?, which corrects for uniform line-of-sight ac-celeration, but not higher-order variations. This is often a valid approximation,and many pulsars have been found this way. However, it is not clear where such asearch breaks down.This is a problem with a many-dimensional phase space that includes all ofthe binary parameters, the pulsar parameters, and the various search inputs. Paststudies have approached the problem analytically, and have made valuable insights;however, until recently they have been restricted to circular orbits, and have notaccounted for pulsar brightness or signal digitization.Here I approach the problem empirically. I simulate 1.8 million pulsars ina variety of orbital configurations and explore the frequency of pulsar recoveryacross various dimensions of the phase space. I find in particular that, at very shortorbital periods, high eccentricities make binary systems easier to detect.iiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 What is a pulsar? . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Millisecond pulsars . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Globular clusters and their MSPs . . . . . . . . . . . . . . . . . . 51.4 Why are unusual MSP binaries interesting? . . . . . . . . . . . . 71.5 Pulsar surveys of globular clusters . . . . . . . . . . . . . . . . . 91.6 Investigating search biases . . . . . . . . . . . . . . . . . . . . . 102 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1 Pulsar observations . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.1 Pulse profiles and TOAs . . . . . . . . . . . . . . . . . . 122.1.2 Dispersion measure . . . . . . . . . . . . . . . . . . . . . 132.1.3 Pulsar timing . . . . . . . . . . . . . . . . . . . . . . . . 132.2 PRESTO and acceleration searching . . . . . . . . . . . . . . . . . 152.3 Detectability studies . . . . . . . . . . . . . . . . . . . . . . . . . 16iii3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.1 Parameters of the artificial pulsars . . . . . . . . . . . . . . . . . 203.2 The pulse profile . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3 The time series . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.4 Creating and searching a large quantity of data . . . . . . . . . . . 273.5 Matching candidates . . . . . . . . . . . . . . . . . . . . . . . . 294 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 314.1 Detections across parameter space . . . . . . . . . . . . . . . . . 314.2 Comparisons with recent analytical work . . . . . . . . . . . . . . 374.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43ivList of TablesTable 3.1 Randomly-selected pulsar parameters . . . . . . . . . . . . . . 20vList of FiguresFigure 1.1 A simple pulsar model . . . . . . . . . . . . . . . . . . . . . 3Figure 1.2 The P-P? diagram . . . . . . . . . . . . . . . . . . . . . . . . 6Figure 3.1 Non-uniformly selected pulsar parameters . . . . . . . . . . . 21Figure 3.2 Sample profiles of fake pulsars . . . . . . . . . . . . . . . . . 23Figure 3.3 Sample PRESTO candidate . . . . . . . . . . . . . . . . . . . 28Figure 4.1 Detections across eccentricity and orbital period . . . . . . . 32Figure 4.2 Phase dependence of detectability . . . . . . . . . . . . . . . 32Figure 4.3 Another look at the phase dependence . . . . . . . . . . . . . 33Figure 4.4 Detections as a function of flux density . . . . . . . . . . . . 34Figure 4.5 Detections across longitude of periastron and eccentricity . . . 35Figure 4.6 Detections across mass function and orbital period . . . . . . 36Figure 4.7 Detections across spin and orbital period . . . . . . . . . . . 37Figure 4.8 Spin-orbit histogram with increasing eccentricity . . . . . . . 38Figure 4.9 Averaged line-of-sight acceleration and jerk . . . . . . . . . 39Figure 4.10 Globular cluster pulsars discovered with Arecibo and GreenBank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40viAcknowledgmentsI would like to thank Dr. Ingrid Stairs for her patience and advice, and for alwaysbeing an impressive source of knowledge with regards to all things pulsar. By nowit comes as no surprise to me that even in a year as busy and exciting as this onehas been for her, she finds a way to make time for her students.Thanks to Dr. Jeremy Heyl and Dr. Scott Ransom for taking the time to readthrough this thesis even though I withheld it until the last possible moment.I would also like to acknowledge the support of CANARIE for providing fund-ing for the computing system used in this work.Finally, thanks to Chelsea for keeping me sane and being patient with me de-spite my less-than-ideal organizational skills. I never would have made it half thisfar without her love and support.viiChapter 1Introduction1.1 What is a pulsar?The first pulsar was discovered serendipitously in 1967 (Hewish et al., 1968). Avery fast, steady series of radio-frequency pulses was observed from a particularpoint on the sky, evidently originating in deep space. Following this discovery,many more pulsars were found in quick succession, and before long it was agreedupon that they were highly magnetized, rapidly rotating neutron stars (Gold, 1969;Pacini, 1968); radio emission beamed from the stars? magnetic poles was sweepingacross our line of sight like a lighthouse beam, and this was the reason it wasobserved as regular radio pulses. This remains our basic description of pulsarsnearly half a century later.Neutron stars were proposed by Baade & Zwicky (1934) as theoretical compactproducts of supernovae, and the first direct evidence for their existence was in thediscovery of pulsars. Pulsars are a class of neutron star, though not every neutronstar can be observed as a pulsar. It is very likely that there are many neutron starswith beamed radio emission that simply never points towards Earth, any manywithout such emission at all.A very good description of the modern pulsar model is laid out in Lorimer &Kramer (2005), and much of the information in this section can be found there andin references within?where necessary, I provide additional references. Neutronstars are formed in core-collapse supernovae. When an evolved massive star is no1longer able to create energy through nuclear fusion in its core (i.e., the core nucleiare iron), the hydrostatic equilibrium between radiative expansion and gravitationalcollapse breaks down, and an enormous amount of mass rapidly falls inward. Be-cause multiple electrons cannot exist in identical states, electrons are forced tooccupy higher and higher energy states as the volume is compressed. This createsan outward electron degeneracy pressure, which is the pressure that withholds awhite dwarf star from collapse. If there is sufficient stellar mass to compress thecore beyond this state, then electrons and protons form neutrons via inverse betadecay, and an analagous neutron degeneracy pressure withholds further collapse.If the compression can proceed no further, we are left with a neutron star, with aradius of about 10 km. Further collapse would produce a black hole or perhapsanother exotic object such as a quark star, though the latter remains speculative.While the physics of neutron star interiors remains a subject of debate, muchhas been observationally established about their masses and exteriors. The Chan-drasekhar limit of 1.4 M has often been used as a canonical neutron star mass,though it has become clear that one ?typical mass? is not sufficient to describethe current distribution of measurements, with the mass distribution increasinglyappearing multi-modal and dependant upon formation scenario (e.g., O?zel et al.,2012; Schwab, Podsiadlowski & Rappaport, 2010). Their masses do tend to bescattered in a small range about this value, nonetheless; thus far they have beenmeasured between 1.25 and 2 M (Demorest et al., 2010; Kramer et al., 2006).The Chandrasekhar limit is the mass beyond which a white dwarf star should theo-retically no longer be able to withstand gravitational pressure, so it is not surprisingthat neutron stars are not found well below this mass (though a neutron star?s mea-sured mass will be less than that of a white dwarf that collapses to form it, due toits higher gravitational binding energy, as discussed by, e.g., Podsiadlowski et al.(2005)). The analagous upper-limit mass for neutron stars is not well-determined,but it might be as high as about 3 M, depending upon the correct neutron starequation of state.Neutron stars spin fast and have strong magnetic fields. Angular momentumconservation and magnetic field compression during core collapse can basicallyaccount for these properties. Because of the magnetic field, there is a corotatingmagnetosphere, extending no further than the light cylinder (Figure 1.1), the dis-2Figure 1.1: The basic pulsar model. Magnetic field lines that extend beyond the lightcylinder cannot close on the neutron star?s opposite pole, as corotation wouldrequire them to move faster than the speed of light. Particles streaming alongthese ?open? field lines are thought to play a part in the formation of the radiobeams.3tance beyond which anything corotating with the star would exceed the speed oflight; a faster-spinning star has a tighter light cylinder. Field lines that originatenearer the star?s equator turn back and close before reaching this distance. Fieldlines that originate closer to the poles, however, extend beyond the light cylinderand do not close on the star?s opposite pole. It is in these polar cap regions that aprocess involving the streaming of charged particles is thought to be responsiblefor the beamed radio emission observed in pulsars.We are able to observe a pulsar?s beamed emission as a series of short pulsesbecause of a misalignment between the magnetic and spin axes; this causes thebeams aligned with the magnetic axis to be swept about, and if at least one of thebeams points towards the Earth once per sweep, we see a series of pulses, muchlike the brief flashes of a lighthouse beam. Probably some neutron stars have theirspin and magnetic axes aligned, but even if radio beams are emitted, we would notsee these as pulsars.1.2 Millisecond pulsarsA pulsar?s emission is powered by its rotational energy, and this leads to a gradualslowing of the pulsar?s spin rate. The youngest pulsars have lower spin periods (P)and higher period derivatives (P?), while older pulsars spin more slowly (periodsfrom hundreds of milliseconds up to a few seconds) and their rotational speeds de-cay more slowly. This is shown on the P-P? diagram of pulsars shown in Figure 1.2,where diagonal lines show characteristic ages, defined as ?c = P/2P?. The maincluster of points on this plot represents these ?typical? pulsars, but there is anotherdistinct cluster at much lower spin periods and period derivatives. These are themillisecond pulsars (MSPs), whose positions on the diagram do not correspond totheir ages, as they are old pulsars that have had their spin rates increased throughmass-accretion; looking at the plot, it is immediately striking how many of the pul-sars in this region are members of binary systems. It has been widely thought forsome time that these binary systems were at one time much like the accreting sys-tems known as low-mass X-ray binaries (LMXBs) (e.g., Kiziltan & Thorsett, 2010;Phinney & Kulkarni, 1994). Recently, direct links between MSPs and LMXBshave even been established, with the 3.93-ms globular cluster pulsar J1824?2452I4(Papitto et al., 2013, in press) apparently switching between rotation-powered andaccretion-powered emission, and the 1.69-ms field pulsar J1023+0038 (Archibaldet al., 2009) showing evidence for a recent accretion disk. The angular acceler-ation that occurs during this accretion is often called spinning up, and MSPs aresometimes referred to as recycled pulsars because of this process. The spin-up line,drawn as a thick black line in Figure 1.2, represents the minimum period allowedby Eddington-limited accretion.Tidal forces in an accreting binary system also tend to circularize the orbit,so that binary systems with MSPs typically have very low eccentricities (e). Anexception in the Galactic field is PSR J1903+0327, an MSP in a binary system witha main-sequence star that has a high eccentricity of 0.44 (Champion et al., 2008).Initial possible explanations included the system having been formed in and ejectedfrom a globular cluster; alternatively, Freire et al. (2011) and Portegies Zwart et al.(2011) both suggest that the system previously had a third body, that which spun upthe pulsar, which was later ejected due to chaotic three-body interactions. Such ascenario could have resulted in the ejection of the MSP from the system instead ofthe donor star, and may thus explain the presence of isolated MSPs in the Galaxy.Whatever the explanation for its own properties, it is clear that J1903+0327 is nota typical system.It is in globular clusters, however, that a population of eccentric MSP systemsexists; in these clusters, very high stellar densities greatly increase the probabilityfor close encounters that disturb binaries.1.3 Globular clusters and their MSPsGlobular clusters are gravitationally bound systems of hundreds of thousands ofstars. They orbit in the Galaxy?s halo, formed early in the history of the Universe,and contain some of the oldest stars ever observed. It has long been thought that allof the stars in a globular cluster formed at the same time, and while star-formationclearly ceased early in the clusters? formation, the discovery of multiple distinctmain sequences with different helium abundances in some clusters has shown thatthis idea is not quite true (e.g., D?Antona et al., 2005; Milone et al., 2012). It ap-pears as though there can be a few generations of star-formation, with the chemical510-3 10-2 10-1 100 101Period (s)10-2110-2010-1910-1810-1710-1610-1510-1410-1310-1210-1110-1010-9Period derivative (s/s)0.1 Myr1 Myr10 Myr100 Myr10 10 G10 11 G10 12 G10 13 G10 14 Gisolated pulsarspulsars in binariesmagnetarsFigure 1.2: The pulsar P-P? diagram, featuring all known pulsars except those inglobular clusters, because the Doppler effect due to their intra-cluster accel-eration makes it difficult or impossible to accurately establish intrinsic P andP? values. A point is placed based on the pulsar?s spin period and its deriva-tive. Dashed lines show constant characteristic age and constant magnetic fieldstrength. Pulsars known to be members of binary systems are shown as red opencircles and magnetars are shown as green stars. Note that the millisecond pul-sars, in the lower-left region of this diagram, are typically members of binarysystems. Pulsar radio beams are thought to ?shut off? with slow rotation speedand low magnetic field strength, hence the lack of points in the lower right por-tion of the diagram. The thick black line is the spin-up line, below which weexpect all of the recycled pulsars. The points in this diagram are not correctedfor Shklovskii or Galactic-acceleration contributions to P?. Based on Figure1.13 from Lorimer & Kramer (2005); data from the ATNF pulsar database at (Manchester et al., 2005).6content of each generation slightly different from the last due to contaminationfrom stellar winds and supernovae. Richer et al. (2013) have even measured dy-namical differences between stellar generations in 47 Tucanae. Even with thisinteresting complication, the nearly uniform age of a globular cluster?s populationprovides valuable insights into stellar evolutionary models. The most massive starshave long since evolved beyond the main sequence, becoming white dwarfs, neu-tron stars, or black holes. Of more than 150 known globular clusters in orbit aboutthe Milky Way (Harris, 1996), 144 pulsars are currently known to reside in 28 ofthem.1Most of the pulsars found in globular clusters are MSPs, which is perhaps notvery surprising; if a cluster?s neutron stars were formed billions of years ago whenthe early-type stars were running out of fuel for nuclear fusion, enough time haspassed that we would expect ?normal? pulsars to have spun down and ceased radioemission, and those in binary systems have had time to accrete matter and be spunup to MSP speeds. Adding to this, the high stellar densities in globular clusterscan lead to binary systems losing or exchanging companions?such exchanges arethought to be responsible for a number of the pulsar systems that are observed inclusters. It is also the likely reason that MSP systems in globular clusters, unlikethose in the Galactic disk, frequently have high eccentricities, with 8 currentlyknown that have P < 20 ms and e > 0.2.Exchange scenarios are also thought to be responsible for the high density (perunit mass) of LMXBs in globular clusters compared to the Galactic disk, withprimordial neutron stars capturing main sequence companions that subsequentlyevolve off of the main sequence and accrete onto the neutron stars (Clark, 1975).Thus, despite the very old age of cluster populations, pulsars are still being activelyrecycled.1.4 Why are unusual MSP binaries interesting?There is much scientific interest in finding neutron stars in globular clusters, partic-ularly in eccentric systems. Eccentric systems are scientifically valuable becausethe orbit?s shape introduces a reference direction into the system: the longitude of1 With this reference direction present and easily measured through pul-sar timing, we also gain the ability to measure whether it changes with time, and atwhat rate. While the orbital trajectories of an ideal, classical two-point-mass sys-tem are static, various physical phenomena, including general relativistic effects,can cause the longitude of periastron to precess. General-relativistic precession is afunction of the total mass of the two bodies; it is one of five major ?post-Keplerian?(PK) parameters that can in principle be measured in a relativistic system (e.g.,Damour & Deruelle, 1985, 1986), a measurement of any two of which allows bothmasses to be determined. Each further PK parameter then provides an indepen-dent measurement of the two masses and acts as a test of general relativity or othertheories of gravity.A famous example of the use of PK parameters is the double-pulsar systemPSR J0737?3039A/B, the only case so far in which both neutron stars could beobserved as pulsars: an MSP with period 22.7 ms and a slow pulsar with period2.77 s. In this system, all five PK parameters (and a sixth contraint in the massratio) were used to constrain the two masses, and all contraints proved mutuallyconsistent (Kramer et al., 2006). The double-pulsar system has a short orbital pe-riod (2.5 hours) and an eccentricity of 0.088, which is a significant eccentricity fora recycled system, and is probably due to the supernova explosion that created thesecond pulsar (Burgay et al., 2003). Because of the possibility of binary exchangescenarios, there are more ways in which a double-pulsar system can form in aglobular cluster than in the field, and an exchange is probably the only way to forma double-MSP binary. An eccentric MSP-MSP binary would provide an excellentastrophysical laboratory for high-precision measurements of orbital precession andother general relativistic effects.Another scientifically valuable system would be a pulsar in orbit with a blackhole. A close orbit with such a companion could provide very precise tests ofgravity (thanks to the high mass) and black hole physics. Such systems have yet tobe discovered, but globular clusters may be the best places to find them, and theymight have very high (& 0.9?0.99) eccentricities (Sigurdsson, 2003).81.5 Pulsar surveys of globular clustersSignificant efforts have been made to find globular cluster pulsars, with increas-ing success in the last decade thanks to improvements in computing and searchalgorithms, upgrades to the 305-metre Arecibo telescope in Puerto Rico, and theconstruction of new high-sensitivity instruments, especially the 100-metre NRAOGreen Bank Telescope (GBT) in West Virginia. While a typical survey for fieldpulsars consists of many adjacent telescope pointings that are each only a few min-utes in duration, globular cluster pulsar surveys operate quite differently; becauseof its small angular size on the sky, a globular cluster typically fits entirely withinthe beam of a radio telescope. This is convenient, as it is necessary to point a tele-scope at a single cluster for several hours in order to get sufficient signal-to-noisefrom such distant sources. Here I give a brief overview of some of the most recentsearches and their discoveries.In 2001 and 2002, Hessels et al. (2007) surveyed the 22 globular clusters ob-servable with Arecibo and within a distance of 70 kpc from the Sun. They tookdata using the Wideband Arecibo Pulsar Processor (WAPP) backend in the L-bandfrequency range, generally with a sampling time of 64 ?s. 11 new MSPs werefound in M3, M5, M13, M71, and NGC 6749, all but one of which are in binarysystems.Nearly half of the pulsars currently known in globular clusters have been foundsince 2005 with the GBT. Many of these are in the cluster Terzan 5, with 21 MSPdiscoveries reported by Ransom et al. (2005), and the discovery of the fastest-spinning pulsar yet known, PSR J1748?2446ad, with a spin period of only 1.40 ms(Hessels et al., 2006). These were found in data taken with the Pulsar Spigot back-end (Kaplan et al., 2005). This system acquired data with 3-level sampling. Thedata were then correlated and accumulated and each cumulative sample was savedwith 16-bit precision with a time resolution of 81.92 ?s. To date, 34 pulsars havebeen found in Terzan 5 (e.g., Lynch et al., 2013), and 18 of these are in binarysystems, 7 of which have high eccentricities.Be?gin (2006) and collaborators searched for pulsars in the globular clustersM28, NGC 6440 and NGC 6441, and one observation each in NGC 6522 andNGC 6624, using data taken on the GBT, again mostly with the Spigot backend.923 new MSPs were discovered in these searches: 11 in M28 (7 in binaries, 2 withhigh eccentricities), 5 in NGC 6440 (3 in binaries, 1 with high eccentricity), 3 inNGC 6441 (1 in a binary), 2 in NGC 6522, and 3 in NGC 6624. The eccentric bina-ries that were found have periods from 8?30 days, while many of the non-eccentricbinaries have periods less than 1 day. The eight new pulsars in NGC 6440 andNGC 6441 were followed up in Freire et al. (2008), those in NGC 6624 are dis-cussed in Lynch et al. (2012), those in M28 will be reported in Be?gin et al (in prep.),and the best information currently available for the new pulsars in NGC 6522 is inthe online catalogue of globular-cluster pulsars referred to in Section 1.3.All of the above surveys used the PRESTO2 software package to reduce andsearch the telescope data, using a method known as acceleration searching. Thisallows the detection of a periodic signal whose origin is accelerating due to or-bital motion, and will be described in more detail in Chapter 2. Without such atechnique, many of these pulsars? fast orbits would keep them from being detected.1.6 Investigating search biasesA major challenge of these surveys is in characterizing our sensitivity to variousconfigurations of binary systems. In short, pulsars are detected as periodic signalsin time-series data, and the varying Doppler shift caused by a pulsar?s orbital mo-tion dilutes the signal?s power across a range of frequencies so that it is not stronglypeaked at the pulsar?s rotation frequency. This effect gets worse as the fraction ofthe pulsar?s orbit contained in the data becomes larger, and so it?s a problem thatparticularly affects our ability to find short-period binaries.Various attempts have been made to characterize our sensitivity to pulsars inbinary systems (e.g., Johnston & Kulkarni, 1991; Jouteux et al., 2002), all takingan analytical approach. Until very recently (Bagchi, Lorimer & Wolfe, 2013), allof these assumed circular orbits in order to make the problem tractable. They havealso needed to come up with a metric by which to judge whether or not a system is?detected.?I take an empirical ?brute force? approach to this problem; I have written codeto generate large quantities of simulated survey data based on the GBT Spigot2, and I run this data through standard search algorithms. Upon determiningwhich systems are found and which are not, the results can be histogrammed tosee which parameters or combinations of parameters lead to a significant drop inpulsar-recovery.In Chapter 2, I briefly discuss pulsar-observing. I then explain the accelerationsearch method and the software package PRESTO, which implements it. At the endof the chapter, I discuss previous efforts to characterize our sensitivity to pulsarsin binary systems, including a brief explanation of their methods and their short-comings. In Chapter 3, I describe the software I wrote to simulate large quantitiesof time series data and how this was run. In Chapter 4, I show the results obtainedwith this software and discuss them in the context of previous results as well as the-oretical ideas concerning cluster pulsars. I end with a brief discussion of potentialfuture directions for this work in Chapter 5.11Chapter 2Background2.1 Pulsar observations2.1.1 Pulse profiles and TOAsWhen one observes a known pulsar, data are acquired at a telescope for some du-ration, and the signal is dedispersed and folded at the pulsar?s rotation period. Thismeans cutting the time series up into segments that each contain one rotation ofthe neutron star and stacking these to obtain a high signal-to-noise pulse profile:an ?image? of one pass of the beam across our line of sight. While individualpulses vary dramatically in profile shape (in cases where they are bright enoughto be seen above the noise level), a pulsar?s average profile is very steady in mostcases; clearly there are short-timescale variations that cancel out when the signal isaveraged. Thus, a pulsar?s ?standard profile? may be constructed by stacking manypulses to reduce the noise to a low level.A well-constructed standard profile can then be matched to future observationsthat have been folded in order to measure the precise time at which the pulsarpassed through a particular point in its rotation?the peak of its pulse profile, forinstance. This surprisingly powerful piece of data is known as a time of arrival(TOA), and I will discuss its application shortly.122.1.2 Dispersion measureThe vast space between stars is populated by free electrons that act as a dispersingmedium. This is the ionized component of the interstellar medium (ISM) and itseffect is a frequency-dependent delay in the TOAs of the pulses as seen at Earth.Since any observation has some finite bandwidth, it is always necessary to correctfor this dispersion, otherwise the broad-band profiles are smeared out. The con-ceptually simple way of doing this is to split the full band into narrow subbands,each of which has a relatively negligible level of dispersion smearing. These bandscan then be rotated into alignment before being summed to obtain the full signal.This is known as incoherent dedispersion. A more sophisticated method, coherentdedispersion (Hankins & Rickett, 1975), removes dispersion effects completelyby convolving the signal (whose amplitude and phase are both recorded) with theinverse of the ISM transfer function.The amount by which the signal needs to be corrected can be characterized by asingle number, the dispersion measure (DM). The DM is a measure of the columndensity of electrons along the line of sight to the pulsar. This turns out to be auseful property: using a model of the electron distribution in the Galaxy (Cordes& Lazio, 2002), a pulsar?s DM can be used to estimate its distance. This is notextremely accurate in practice, but it is often the only distance estimate available.Conversely, if we have independent measurements of the distances to a numberof pulsars, their DM values can be used to help construct a model of the Galacticelectron distribution.2.1.3 Pulsar timingWhile this work does not use the technique of pulsar timing itself, a brief discussionon the subject is warranted since the technique will be mentioned and pulsar-timingsoftware is used for some important calculations.Although pulsar timing involves very complicated many-parameter models, theprinciples behind it are quite simple. A pulsar spins (and spins down) with pre-dictable regularity, and yet there are apparent variations in its spin-rate as seen byan Earthbound observer. There are many distinct sources for these variations bothlocal and distant, mostly due to the Doppler effect, and we see a superposition of13all of them. The goal of pulsar timing, then, is to account for all of these variationsin a model whose parameters (when fit to data) contain a wealth of informationabout the behaviour of the observed system.The data that are needed for this are pulse TOAs. These are precise times atwhich the pulsar is known to be at some chosen fiducial point in its rotation, forinstance the peak of the pulse profile, and are obtained by taking several minutesof data and folding to get a high signal-to-noise pulse profile. There is no need(nor is it possible) to have such a measurement for every single rotation; we justneed to ?check in? occasionally to see if the rotation phase is where the modelpredicts that it should be, and to make minor adjustments if necessary. As longas the time between two TOAs is short enough that the current model is wrongby only a small fraction of the rotation period, phase connection is maintainedand we can be confident that every rotation that took place between TOAs has beencorrectly accounted for. If simply adjusting parameters cannot account for the data,additional parameters may be needed. Pulsar timing becomes a game of lookingfor patterns in fit residuals and recognizing which parameters need to be added toremove those patterns.Distant sources of pulse-period variations include a periodically varying Dopplereffect from motion within a binary system and delays due to the signal traversingstrong gravitational fields, while local causes come from the Earth?s rotation and itsmotion within the Solar System, which need to be corrected using accurate SolarSystem ephemerides and a good model of the Earth?s rotation.A commonly-used software package for pulsar timing is TEMPO1 (and its suc-cessor, TEMPO22). It takes TOAs and parameter estimates as inputs and performsa fit using those parameters. TEMPO also has a ?prediction mode? that uses a setof parameters (including binary system elements) to predict the observed spin fre-quency and the phase at a particular point in time and at a particular telescope orthe Solar System barycentre. This is done using a polynomial expansion, and theoutput is a set of polynomial coefficients, or polycos. This is the TEMPO functionthat is used as part of the data-simulation routine in this work.1http://tempo.sourceforge.net2http://tempo2.sourceforge.net142.2 PRESTO and acceleration searchingPRESTO is a software package designed primarily to detect pulsar signals in timeseries data. It performs many functions, both closely and peripherally related to thisgoal, including detecting and removing radio-frequency interference (RFI); dedis-persing and barycentring data; acceleration, single-pulse, and sideband pulsar-searching; pulsar candidate sifting and folding; and, through various utilities, visu-alizing, manipulating, and analyzing data. Ransom (2001) provides an overview ofPRESTO, while Ransom, Eikenberry & Middleditch (2002) describe the accelera-tion search in particular, which is the primary method employed in globular clusterpulsar surveys with the GBT, and is therefore the method that is tested in this work.A typical pulsar search using PRESTO follows a series of steps. Raw telescopedata contain information over time but also over frequency, the range of the lat-ter depending on the observing bandwidth. Ideally any signals present above thenoise level are cosmic in origin, but of course there is always radio-frequency in-terference (RFI). The PRESTO script rfifind masks out the most obvious RFI,which it identifies as a signal that is not persistent either in time or across the fre-quency band?either of these indicates that the signal does not come from a pulsar.PRESTO currently handles raw data from various machines at the GBT, Arecibo,Parkes, and Jodrell Bank, as well as the pulsar standard SIGPROC format and sim-ple floating-point time series.Before searching for pulsar signals, the data are barycentred?the arrival timesof the data corrected to the Solar System barycentre?and the frequency dimensionis collapsed by de-dispersing over a range of DM values one wishes to search,generating a one-dimensional time series for each step in DM. It is beneficial tofirst create a topocentric (non-barycentred) time series at DM = 0, however, asanother check for RFI. Even if a signal is found in this time series that appearsvery pulsar-like, it must be terrestrial in origin or else its signal would be dispersedand undetectable in a DM = 0 time series.Once a set of de-dispersed, barycentred time series is made, a Fourier transformis carried out on each of them. One of the strengths of PRESTO is its ability toefficiently calculate the fast Fourier transform (FFT) of very long high-resolutiontime series that may be much too large to work with in their entirety in computer15memory.Having thus prepared the data, the accelsearch script carries out the actualsearch, known as an acceleration search. Because a pulsar signal is typically a re-peated narrow spike, much of the power in Fourier space is in harmonics rather thanthe fundamental frequency. Harmonic-summing is performed in order to recoverthis power, and the user can specify whether to sum 1, 2, 4, 8, or 16 harmonics. Asignal from a source that is accelerated due to motion in a binary system will haveharmonics that drift across Fourier bins (linearly if the acceleration is constant),with higher harmonics drifting proportionally further than lower harmonics. For aparticular search, one specifies the maximum number of Fourier bins the highestharmonic is allowed to drift; if this is set to zero, a standard Fourier-domain searchis conducted rather than an acceleration search. When this value is set higher, pul-sars undergoing stronger linear accelerations may be detected. However, the time ittakes to perform a search increases linearly with the number of harmonics summedand with the number of bins we allow the highest harmonic to drift.Once the acceleration search is complete, we are left with a list of pulsar candi-dates and their properties. Additional functions in PRESTO are used to sift throughthese and to match candidates at neighbouring DM values that may be the samesource.2.3 Detectability studiesIt has been evident for many years that we are not equally sensitive to all regionsof parameter space when it comes to searching for pulsars in binary systems, butcharacterizing our sensitivity has presented an ongoing challenge. Here I brieflycover a few significant efforts that have been made.An early approach to the problem can be found in Johnston & Kulkarni (1991).Motivated by early globular cluster searches and the expectation for many clusterpulsars to be in short-period binary systems, they approach the problem analyti-cally, defining a stationary pulsed signal as a sum of Fourier components whosepower spectrum is straightforward to calculate. They consider a time-varying dis-tance to the source that can be expanded in a Taylor series, providing a velocityterm, an acceleration term, and higher derivatives. They show that the acceleration16and higher-order terms produce time-dependent phase errors that cause a reductionin the amplitude of the signal?s harmonics in the power spectrum. The extent towhich these amplitudes are reduced is taken as a measure of how difficult thesesystems are to detect.To quantify the reduction in detectability, they define the function?(?1,?2,T ) =1T????? T0exp[im?pc?(v0t +a0t22!+j0t33!+ . . .??1t2??2t)]dt???? , (2.1)where T is the observation length, m is the harmonic number of the power spectrumpeak under consideration, and ?p is the angular frequency of the pulsar. c and iare the speed of light and??1 respectively, and v0, a0, and j0 are the line-of-sightvelocity, acceleration, and jerk (first derivative of acceleration) respectively at timet = 0. ? takes values between 0 and 1.?2 is the ratio of the peak height of the mth harmonic in the power spectrumwhen the signal is accelerated compared to when it has a constant velocity. ?1 and?2 are free parameters. The authors test the effectiveness of a standard Fourieranalysis (no acceleration search) by setting ?1 = 0, and ?2 is chosen to maximize? . ?2 can be thought of as a mean velocity over the integration time T . For testingthe effectiveness of an acceleration search, both ?1 and ?2 are varied to maximize? . The authors wrote code that maximizes ? for circular orbits with chosen bi-nary parameters such as inclination angle, orbital period, and companion mass, aswell as integration time T , and they run this over a grid of orbital and spin periodvalues. They take ? = 0.5 as their detection threshold, acknowledging that this isarbitrary, as a strong signal reduced to a fraction of its strength may still be mucheasier to detect than an undiminished weak signal. Regardless, they are able todemonstrate that a pulsar signal is reduced most severely for fast-spinning pulsarsin tight binaries.Jouteux et al. (2002) define their own set of ? functions for various search meth-ods, including the standard Fourier analysis and the acceleration search. Thesefunctions carry a meaning similar to the ? of Equation 2.1, in that they repre-sent the loss of signal strength due to source acceleration, with a larger value of17? (closer to 1 than 0) implying a better-recovered signal. Their main comparisonis between the acceleration search method and their proposed ?partial coherencerecovery technique? (PCRT). A method implemented in PRESTO known as a ?side-band search? is based on the same principles, which involve taking advantage ofthe binary periodicity. So, unlike the acceleration search, which breaks down if thetime series contains whole orbits or even an appreciable fraction of an orbit, thePCRT method works best if the time series covers several orbits.While interesting insights are made in both of these studies, there are notablelimitations owing both to computing power and the analytical expressions used.Neither can test for the effect of pulse amplitude compared to the noise level of thetime series, and both are restricted to circular orbits for the purpose of tractabil-ity. In Johnston & Kulkarni (1991), the amount of additional calculation requiredto handle eccentric orbits would have been unfeasible with the computing poweravailable. Jouteux et al. (2002) describe the Fourier response of their signal usingsimple Bessel functions, which would not be possible had they used non-circularorbits. Recently, the non-circular-orbit problem was addressed by Bagchi, Lorimer& Wolfe (2013), who extend the methods of Johnston & Kulkarni (1991) to includeorbital eccentricities.Bagchi, Lorimer & Wolfe (2013) define ?1m and ?2m, identical to Equation 2.1with and without ?1 = 0, respectively, and ?3m, which includes an additional t3term, testing the idea of a ?jerk search?, which corrects not just a linear accelera-tion of the source, but also the first derivative of that acceleration (this is rarely usedin practice as it greatly increases the processing time). The subscript m is simply anotational clarification, denoting the harmonic under examination. The extensionto eccentric systems significantly increases the calculation required to determinethe line-of-sight terms in these expressions; in particular, Kepler?s equation mustbe solved iteratively to find the true anomaly at each point in time. Furthermore, aweighted-average ?nm is calculated over the integration time in each case, account-ing for the varying amounts of time spent at different points in an eccentric orbit.This extension to eccentric systems means that more parameters need to be set,namely the eccentricity and argument of periastron, though tools are provided bythe authors to test the signal degradation for any given set of parameters.The results of the new study are consistent with those of Johnston & Kulkarni18(1991), but the inclusion of arbitrary eccentricties makes them more robust. In eachcase, the importance of a higher-order search, such as an acceleration search com-pared to a straightforward Fourier analysis, is greater when searching for faster-spinning pulsars.I have approached the problem of characterizing our sensitivity over the param-eter space empirically, rather than analytically. In keeping with the recent surveypractices, particularly those of the GBT, I use PRESTO?s accelsearch functionto search large quantities of artificial survey data made up of noise and randomly-generated binary-system pulsar signals. While past studies use ?2 as their bestanalytical stand-in for the fraction of pulsars recovered, here I am able to use therecovered fraction directly, and the amplitude of individual pulses is a variable pa-rameter. In the next chapter, I discuss my methods for generating the artificial dataand searching it with PRESTO.19Chapter 3MethodsIn this chapter I outline the basic ingredients of my simulated survey data, howthey are generated, and how they are searched using the PRESTO software.3.1 Parameters of the artificial pulsarsI generate pulsars in binary systems by drawing parameters from a set of chosendistributions. These distributions are listed in Table 3.1 with those that are non-uniform illustrated in Figure 3.1. The inclination angle i is actually drawn from adistribution that is uniform in cos i, as this is the actual distribution of orbits that areoriented completely at random. For the companion mass, most of the distribution isuniform between 0.05 and 3 M, since pulsar companions tend to be white dwarfsTable 3.1: The distributions from which pulsar parameters are selected. Those thatare not uniform are illustrated and explained in Figure 3.1.Parameter Range Uniform?Period (P) 1?20 ms yesEccentricity (e) 0?1 yesBinary period (Pb) 10?1440 minutes yesLongitude of periastron (?) 0?360 degrees yesInclination angle (i) 0?90 degrees noCompanion mass (m2) 0.5?10 M noPulse amplitude 0.01?0.1 no200 30 60 90Inclination angle (degrees)0 2 4 6 8 10Companion mass (M?)0.00 0.02 0.04 0.06 0.08 0.10Pulse amplitude (fraction of noise)Figure 3.1: Distributions for three of the parameters randomly chosen when gener-ating pulsars. These are the only three that are not uniformly distributed. Theinclination angle i is actually uniform in cos i, as is the case for any completelyrandom distribution of orbit orientations. The companion mass is split betweentwo uniform distributions with 80% between 0.05 and 3 M and 20% between 3and 10 M. The pulse amplitude distribution decreases from 0.01 to 0.1 becausethe detection of lower-amplitude pulsars is more interesting.21or neutron stars, but 20% of the distribution is uniform between 3 and 10 M,for companions that are non-degenerate stars or black holes. This distribution wassimply chosen out of a desire for better statistics in the lower-mass range.The orbital periods are distributed between 10 minutes and 1 day. The shortestpulsar orbital period currently known is that of PSR J1311?3430, at 94 minutes(Pletsch et al., 2012). Because short-period compact objects spiral inwards dueto the emission of gravitational radiation, neutron stars should exist in extremelyshort-period binaries, although it is not clear to what point they would continue tobehave as pulsars. A lower limit of 10 minutes is taken as plausible while suffi-ciently below the current observed limit.The fastest-spinning pulsar is PSR J1748?2446ad with a spin period of 1.40 ms(Hessels et al., 2006), so 1 ms is taken as a suitable lower limit for the spin perioddistribution. The upper limit of 20 ms is a period sometimes taken as that be-low which pulsars may be considered millisecond pulsars (though recycled pulsarsslower than this are often still referred to as MSPs).Once the parameters for a particular pulsar are selected, they are saved in aTEMPO-style parameter file, or parfile. Use of these parfiles allows me to takeadvantage of TEMPO?s prediction mode for performing orbital calculations.3.2 The pulse profileA simulated pulse profile is generated as a closely-spaced superposition of 1 to5 narrow Gaussian curves, in order to mimic the structure observed in real pulseprofiles; the peaks are scaled so that the maximum point of the superposition is atunit height. The profile is then a function with values between 0 (the baseline) and1 (the peak) whose height can be returned as a function of phase, which is also avalue between 0 and 1. Each profile is associated with a TEMPO parfile, containingthe pulsar?s rotational and binary parameters.The placement of the Gaussian curves proceeds as follows. An overall ?widthscale? W is chosen from a truncated normal distribution with mean 1.0 and standarddeviation 0.2, constrained between 0.1 and 1.9. The width (Gaussian ? ) of eachcurve is chosen from a normal distribution with mean 0.01W and standard devia-tion 0.001W . The position in phase of a particular curve is chosen from a normal22Figure 3.2: A set of twelve randomly-generated pulse profiles constructed as de-scribed in the text. Profiles such as these are made for every pulsar generated.The horizontal axis in each frame is rotation phase and runs from 0 to 1. Thevertical axis is dimensionless pulse amplitude, with the baseline at 0 and thepeak of the pulse at 1.distribution with mean 0.5 and standard deviation 0.02W . The first curve placedis given a height of 1. The height of each subsequent curve is set by multiplyingthe previously placed curve?s height by the square of a uniform random numberbetween 0 and 1. Once all of the Gaussian curves are set, they are scaled so thatthe maximum point of their superposition (i.e., the pulse profile) has a height of 1.All of this simply comes from tuning numbers to obtain what look to be reasonablyconsistent and realistic pulse profiles with some random variation. A set of sampleprofiles is shown in Figure The time seriesArtificial time series are created with a time resolution of 81.92 ?s, the same asthe resolution in the GBT globular cluster surveys. They are 1000 s (12.2 milliondata points) long. Typically, in real surveys, long (several-hour) globular clusterobservations are made, but the time series are split up into overlapping segmentsof about this length in order to find pulsars in tight binaries. By generating theseshorter time series directly, I am able to spend my computation time on a muchlarger set of artificial pulsars than if I generated long time series and split themup. There is certainly an orbital-phase dependence to the difficulty of detectinga pulsar in an eccentric orbit, as will be seen in Chapter 4, and the reader shouldbear in mind that in a real survey with long observations, a pulsar in a short orbitthat is highly accelerated along the line of sight in one ?1000 s time series may beaccelerated much less in another from the same long observation. To ensure thatorbital phase is realistically sampled, at the start of a time series each pulsar is ata uniformly random mean anomaly between 0 and 360?. This means that a pulsarin an eccentric orbit is more likely to be near apastron than periastron, due to therelative durations spent at each point in the orbit.The effects of DM are ignored in these time series, or to put it another way,all of the pulsars are assumed to have the same single DM to which the telescopedata would have been de-dispersed. This is not far from the real situation whenobserving globular clusters, since all of the pulsars in the beam are at almost thesame distance from us, and therefore have almost the same DM. Thus, I can addnoiseless pulsar signals directly to an array of white noise.If I wished to construct a noisy time series containing a set of signals belong-ing to pulsars that are inertial relative to the reference frame (the Solar Systembarycentre), I could simply add the appropriately-scaled repeated pulse profile to atime series of noise. At each time step, the code would check precisely how muchtime had passed, find the rotational phase of the pulsar at that moment (trivial foran inertial pulsar), and get the value of the pulse signal at that phase. I might wantto take a typical pulsar spin-down into account, but over a single observation itseffect on the spin period would be negligible.However, matters become much more complicated when the pulsar orbits an-24other body. The apparent spin period is modulated about the intrinsic period ac-cording to a varying Doppler shift, which can be determined by the pulsar?s veloc-ity along the line of sight to the Solar System at any given time. The tricky stepis calculating the instantaneous line-of-sight velocity, whose periodic behaviourdepends upon the orbit?s shape and orientation, and at a far more precise level,on general relativistic effects too, which cause the orbit to deviate from a simpleKeplerian ellipse. All of these factors can be accounted for by using TEMPO?sprediction mode.The particular binary system model that is used is the DD (Damour & Deruelle,1986) theory-independent relativistic model, although I calculate and include all ofthe post-Keplerian parameters with the appropriate general relativistic equations,effectively making it equivalent to the DDGR variation of the DD model (Taylor& Weisberg, 1989), in which general relativity is assumed to be correct. I didthis because of an error (subsequently fixed) in the DDGR code that sometimescaused the program to crash when making predictions for systems with very higheccentricities.Given a set of parameters for a pulsar system (including a reference epochfor the parameters), a start time for the range of validity of the expansion, and thelength of the range of validity, a set of polynomial coefficients, or polycos, is outputthat can be used to calculate the rotation phase ? as? = ?re f +60 f0?T + c1 + c2?T + c3?T 2 + . . . (3.1)at a particular time T , where ?T = (T ? Tmid)? 1440. Here f0 (the referencerotation frequency) is in units of hertz and T and Tmid (the midpoint of the chosenrange of validity) are in days, so the factors of 60 and 1440 scale everything tominutes. This is an arbitrary choice of units, of course, as ? itself is dimensionless.The coefficients cn are the polycos themselves and are determined by the samekind of many-parameter model TEMPO uses in pulsar timing?in this case, theparameters are simply input, not fit.I use 12 polycos per segment, where a segment is the range of validity of thepolycos, which is taken to be the whole time series if the pulsar has a long enoughorbit, but which I never allow to be greater than one-fifth the length of the orbit.25When necessary, multiple sets of polycos are generated throughout a time series.To keep memory-usage at a manageable level, time series are made in chunks ofsize 222 (4.2 million) points, each chunk appended to a file on disk before generat-ing the next.Pulsar signals are superposed onto white noise, with the pulse amplitude de-fined relative to the noise level. This amplitude is proportional to the signal-to-noise ratio (SNR) that appears in the pulsar radiometer equation (Dewey et al.,1985; Hessels et al., 2007),Smin =(SNR)?TsysG?n??tobs(wP0?w)1/2, (3.2)which is used to estimate the flux density detection threshold for a survey. Typicallythe SNR here is the cutoff below which a candidate is not considered. The ? termis a correction factor to account for the signal digitization, and is taken to be 1.2for the 3-level sampling of the Arecibo and GBT surveys (Hessels et al., 2007).Tsys is the system temperature and G is the telescope gain which converts betweentemperature and flux density. These are 20 K and 2.0 K/Jy respectively for theGBT survey, and 40 K and 10.5 K/Jy for Arecibo. The number of orthogonalpolarizations summed is n, which is 2 for both surveys. The observing bandwidth?? is 600 MHz for the GBT (Ransom et al., 2005) and 100 MHz for Arecibo,and the observation length is tobs. The pulse period and pulse width are P0 and w.Equation 3.2 produces a value with dimensions of flux density.To convert the amplitudes of my simulated pulses into flux density, I begin withEquation 7.1 from Lorimer & Kramer (2005),SNR =1?p?weqnbins?i=1(pi? p?), (3.3)where ?p is the noise level in the folded pulse profile, and thus ?p = ?r?tobs/P0,where ?r is the noise level in the raw time series. The equivalent width weq isthe width of a top hat function with the same integrated area and peak height asthe pulse profile, in bins. Thus, if wp = w/P0 is the width as a fraction of thefull rotation period (wp = 0.03 is the average value in my simulation), weq = wp?26P0/tsamp, where tsamp is the sampling rate (81.92 ?s). The last term is simply asum of the amplitudes of the bins across the profile. If we imagine again the tophat function with the pulse?s peak height h and width wp, we can replace this termwith wpP0tsamp h. Combining all of these we getSNR =?tobswptsamph?r, (3.4)noticing that the last term h/?r is equivalent to the peak pulse amplitude as a frac-tion of the time series noise level, as I have defined my amplitudes.Pulse-to-pulse fluctuations are not modeled; individual pulses are equivalent inshape to the average summed pulse profile. The rotation phase at each time step isefficiently calculated using the Numerical Python1 polynomial evaluation. ?T inEquation 3.1 is input as a precise array of times, and ? is returned as an array ofphases, which is then used as input to obtain an array of signal strengths between0 and 1. This array is scaled as needed and added to the time series array, and thisprocess is repeated for each pulsar. The final array is saved in the PRESTO timeseries format (a simple array of 4-byte numbers), and a corresponding time seriesinformation file is generated, containing the start time of the data, the number ofbins, the time resolution, and a flag indicating that the data are barycentred.3.4 Creating and searching a large quantity of dataI generate these data on a parallel-processing system, using 36 CPUs: there are 3nodes that each consist of 12 CPUs. For every run of the program, each CPU gen-erates 10 time series 1000 s in length, with each time series randomly containingup to 20 pulsars. A time series has a corresponding directory containing the TEMPOparfiles for all of its pulsars. These time series are then searched using PRESTO?saccelsearch function. As explained in Section 2.2, the number of harmonicsto be summed and the number of Fourier bins the highest harmonic can drift mustboth be input. Here the first 8 harmonics are summed and the maximum drift is setat 300 bins. Frequencies between 1 and 10,000 Hz are searched.1 3.3: An example of a PRESTO candidate (upper plots and text) and the profileof the artificial pulsar that is detected here (lower plot). The uppermost PRESTOplot is the summed profile and the plot below that shows the persistence ofthe signal over time, both plotted over two rotations in phase. The plot to theright is the significance of the signal in period-vs-period-derivative space. Theamplitude of this pulsar is 8.5% of the noise level, which is quite high in theamplitude distribution.28The product of an acceleration search is a list of pulsar candidates. This list isparsed and used to fold the time series using the frequency and frequency derivativeof each candidate. We are thus left with a set of folded PRESTO candidate files, orpfd files, which PRESTO can display graphically. The graphical candidate for oneof the artificial pulsars is shown in Figure 3.3 along with the pulsar?s generatedprofile. Slight differences between the input and candidate profiles arise from thenoise and the coarser binning of the candidate. While not indicated in the figure,the amplitude of this pulsar is 8.5% the noise level, which is quite high in thedistribution (Table 3.1), so that a pulse every ?17 ms in a 1000 s time series leadsto a folded signal ?20? the height of the noise in the upper-left plot. A similarpulsar with an amplitude of 1% the noise level (the low end of the distribution)would have a folded profile at 2?3? the noise level.3.5 Matching candidatesAt this stage, I have many time series, for each time series I have a set of parfilescontaining the information about the pulsars contained within, and I have a set ofcandidates found by PRESTO. The goal is to figure out which of the pulsars wereactually found?i.e., to determine which of the pulsars are among the candidates.In a real survey, a human expert would look through the candidate plots andmark those that look like pulsars for further investigation. It is possible in thisscenario for a candidate that represents a real pulsar to be disregarded as randomnoise or RFI. In this work, I consider a pulsar that has been recognized by PRESTOas a candidate as ?detected? and ignore the possibility of human failure beyondthis point, since it would be very difficult to properly account for this in any kindof realistic way. The enormous number of candidates produced also prohibits mylooking at each one individually.There are certain to be candidates that, while present due to an injected pulsarsignal, show up not at the actual pulse period but rather at some harmonic ratio ofthe period. In an effort to reduce false positives, I do not attempt to identify these;instead, I assume that these pulsars will also show up as candidates at their trueperiods (modulated by binary motion).The matching process proceeds as follows for each time series.29? A list of the parfiles associated with the time series is created. For each entry,polycos are generated to determine the apparent spin period at the start of thetime series, which in general is not the same as the spin period listed in theparfile because of the pulsar?s line-of-sight velocity in a binary system. Eachentry contains the pulsar label and this apparent period.? A list of the candidates associated with the time series is created. Each entrycontains the candidate label and the spin period of the candidate.? For each candidate, a list of pulsars for which |1? P?PPC| < 10?5 is created,where P?P and PC are the apparent pulsar period and the candidate period,respectively. This is usually either an empty list or a list containing 1 item.? Taking the best match from each candidate?s list (the match with the smallestvalue of |1? P?PPC|), make a list of the pulsars that were detected.This is done for each time series, and a database of all of the pulsars is created,along with many of the parameters contained in their parfiles and whether or notthey were detected. In total, I generated 1,782,261 pulsars, of which 1,039,258were detected. I will explain these results in detail in Chapter 4.30Chapter 4Results and Discussion4.1 Detections across parameter spaceOf 1.8 million pulsars generated, about 60% were detected in the accelerationsearches described in Chapter 3. These results are shown as histograms of the frac-tion of pulsars recovered, displaying the detection distribution over one or two pa-rameters while marginalizing over the remaining parameters. The colour-mappingin the 2-dimensional histograms is chosen so that regions with more than 50% ofpulsars detected are in blue and regions with less than 50% detected are in red. Forease of communication, I will often refer to blue and red regions as ?detected? and?undetected? respectively, even though the colours represent a continuous rangebetween 0 and 1.One of the motivating goals of this simulation is to observe the effects of non-zero eccentricities on the detectability of pulsars in short-period binary systems.As seen in Figure 4.1, the detected pulsars actually push down to shorter orbitalperiods at higher eccentricities. This is because, while an eccentric system is veryhighly accelerated near periastron when compared to the acceleration of a non-eccentric system with the same orbital period, it moves through this portion of itsorbit very quickly and spends most of its time undergoing relatively little acceler-ation. This phase-dependence of eccentric orbits is evident in Figure 4.2, whichshows the fraction of pulsars detected as a function of mean anomaly and eccen-tricity. The detectability of pulsars at periastron (mean anomaly 0) begins to drop31Figure 4.1: Pulsar detections across eccentricity and orbital period. Because of therelatively long time spent on the slow, distant portion of its orbit, a pulsar in ahigh-eccentricity system can be found at somewhat shorter orbital periods thanone in a circular orbit. In this and all subsequent 2-dimensional histograms, thered-blue colour gradient represents the fraction of pulsars recovered.Figure 4.2: Orbital phase dependence of detectability. Pulsars with higher-eccentricity orbits are not detected at periastron (mean anomaly 0) due to veryhigh accelerations there but are easier to detect at other mean anomalies thanless eccentric systems.32Figure 4.3: Another look at the orbital phase dependence. Each of the four panelsintegrates one quarter of the total eccentricity range in order to show a pro-gression from low to high eccentricities. While pulsars near periastron (meananomaly 0) become increasingly difficult to detect at higher eccentricities, thoseat other orbital phases are detected at ever-shorter orbital periods. An eccentricbinary system spends very little time at periastron, so this explains the slightimprovement in detections seen at high eccentricities in Figure 4.1.noticeably at e ? 0.3, reaching almost zero by e ? 0.7. To understand the high-eccentricity behaviour in Figure 4.1, we can look at the three parameters (e, Pb,and mean anomaly) of Figures 4.1 and 4.2 simultaneously.Figure 4.3 plots detections as a function of mean anomaly and orbital periodintegrated over four eccentricity ranges. The phase-eccentricity relationship seen inFigure 4.2 is apparent as the increasing spike of the undetected region at periastron.330.000 0.005 0.010 0.015 0.020flux density with GBT survey parameters (mJy) of pulsars foundFigure 4.4: Pulsar detections as a function of GBT survey flux density. The greyregion surrounding the black line represents the statistical uncertainty in thehistogram due to the sample size. The sample size decreases linearly towardshigher amplitudes.However, as the central spike increases, the wings of this region are pushed down tolower orbital periods, meaning that shorter-period binaries are detected there. Thisimprovement over most of the orbit leads to the overall increase in the detectabilityof short-orbit pulsars with increasing eccentricity.The amplitude of the pulses are scaled to units of flux density using Equations3.2 and 3.4 with the GBT survey parameters. This is plotted in Figure 4.4, whereit is seen that the lowest amplitude pulses are not found at all; there is a steepincrease in detectability at low flux densities, but by ?0.01 mJy, brightness makeslittle difference. As discussed in Section 3.1, brighter pulsars are generated lessfrequently in this simulation, so the high-amplitude region of Figure 4.4 has greateruncertainty.When eccentricity is introduced, a symmetry is broken: the orbit has a well-defined periastron, and we may ask how the position of periastron affects de-tectability. A histogram across longitude of periastron ? and eccentricity e is34Figure 4.5: Detections across longitude of periastron ? and eccentricity e. To in-crease the contrast, this is only for orbital periods < 0.2 days (above which theeffect of ? is weaker), and the greyscale is mapped only onto the range of valuespresent in the plot. Gaussian smoothing is also applied, with ? of 3 bins. Thechange in behaviour at e? 0.7 is discussed in the text.shown in Figure 4.5. The effect of ? is quite small, but it is more noticeable atshort orbital periods, so I only include pulsars with Pb < 0.2 days. To increase thecontrast, I also use a greyscale colour map, only spanning the values present inthe histogram, and apply Gaussian smoothing to reduce noise. If we look back atFigure 4.1, it is no surprise that pulsars at the bottom of the histogram in Figure4.5 are poorly detected in this Pb range. The interesting feature of Figure 4.5 is thechange in the pattern of ?-dependence at higher eccentricities.Up to e ? 0.7, pulsars with ? at 90? or 270? are easiest to detect?these areorbits whose major axis is oriented along the line of sight. Above this eccentricity,however, the easiest detections shift symmetrically to values above and below 90?and 270? until, at the highest eccentricities, the situation is completely reversed:pulsars in systems with ? at 90? or 270? are the hardest to detect, while those with? at 0 or 180? are easiest. These latter systems have their minor axis orientedalong the line of sight, and at such high eccentricties, the minor axis is very smallrelative to the major axis, so that highly eccentric systems with ? at 0 or 180? move35Figure 4.6: Detections across binary mass function and orbital period. The upper-right region of the plot is beyond the range of the parameters used to generatethese pulsars.and accelerate almost entirely transverse to the line of sight, resulting in very littleradial Doppler-shifting behaviour. At lower eccentricities, the greatest accelerationchanges, near periastron, are largely along the line of sight.Detection levels also depend on the mass of the pulsar?s companion and theinclination angle of the orbit. There is a degeneracy in these parameters? effectson the pulsar?s Doppler shift, and so they are often combined into the binary massfunction f (m1,m2) = (m2 sin i)3(m1 +m2)?2, where m1 and m2 are the pulsar massand the companion mass, respectively. The pulsar mass m1 is taken to be 1.4 Mhere. A larger mass function (due to a more massive companion or a greater incli-nation angle) leads to greater difficulty detecting pulsars with short orbital periods,as seen in Figure 4.6. For a given orbital period, a pulsar with a more massive com-panion experiences greater acceleration throughout its orbit, and thus the variationsin acceleration with time are more severe. In addition to the acceleration variations,pulsars with the largest mass functions in this simulation would at times be under-going line-of-sight accelerations that shift their Fourier signals across more binsthan the 300 allowed for in the acceleration search.36Figure 4.7: Detections across spin and orbital period. This is a qualitatively simi-lar plot to those of Johnston & Kulkarni (1991) and Bagchi, Lorimer & Wolfe(2013), who plot their analytical ? detectability functions against these two pa-rameters.4.2 Comparisons with recent analytical workHere I make some qualitative comparisons with the recent work of Bagchi, Lorimer& Wolfe (2013).Plotting detections against spin and orbital period, as in the log-log histogramof Figure 4.7, we see results similar to those of Bagchi, Lorimer & Wolfe (andJohnston & Kulkarni 1991) when they assume the use of an acceleration search. Intheir plots of the detectability parameter ? across spin and orbital period, detectionsare reduced most severely for faster-spinning pulsars in shorter orbits, which is thebehaviour seen here. The poorly-detected region below orbital period 0.1 days iscut off in Figure 4.7 so that the axes begin at the same values as the spin-orbit ?plots of these previous works.In Figure 4.8 I divide this histogram into four eccentricity ranges. As Bagchi,Lorimer & Wolfe observe, eccentricity has little effect except at high values, whereit results in a larger region of phase space becoming easier to detect. This improve-ment in the orbital period parameter is also seen in Figure 4.1.37Figure 4.8: Improvement in detections on the spin-orbit histogram with increasingeccentricity. The integrated eccentricity ranges are shown at the top of eachpanel. The axes and colour scheme are the same as in Figure 4.7. As noted byBagchi, Lorimer & Wolfe, e has little effect except at high values.Bagchi, Lorimer & Wolfe find that increases in inclination angle or companionmass make a larger portion of phase space difficult to detect, and we see this herein Figure 4.6 with the binary mass function.While these observations are made primarily by examining the behaviour oftheir ?1m parameter, which assumes no acceleration search, the increase in com-panion mass can be seen in their paper to worsen the ?2m (acceleration search)phase space, and observationally, an increase in inclination angle is like an in-crease in the companion mass. Bagchi, Lorimer & Wolfe do not test how the ?2mphase space changes with variations in eccentricity. They do note that an increase(between 0 and 90?) in the longitude of periastron ? makes a larger portion of the?1m phase space difficult to detect with e = 0.5, which is opposite to the behaviourseen at e = 0.5 in Figure 4.5 in this work, where pulsar detection improves as ?increases from 0 to 90?. This is a qualitative difference between a standard Fouriersearch and an acceleration search. The former is negatively affected by line-of-sight acceleration, even if that acceleration is varying quite slowly, while the latteris, to a large degree, immune to the effects of a slowly-varying acceleration. It is arapid change in acceleration, i.e., a large line-of-sight jerk, that negatively affectsacceleration search results.Using expressions for the line-of-sight acceleration and jerk from Bagchi, Lorimer& Wolfe (their Equations 26 and 27, respectively), I perform a crude test to demon-strate this point. Over a grid of e and ? values, I find the mean acceleration andjerk across the complete orbit (in mean anomaly, so that it is a time average), withthe results shown in Figure 4.9. For each step in e, I divide the acceleration or jerk38Figure 4.9: Line-of-sight acceleration (left) and jerk (right) averaged over the time ofa complete orbit on a grid of e and ? values. Details on scaling are given in thetext. Whiter values are lower and considered better for detection, via standardFourier search in the left plot and via acceleration search in the right plot. Notethe similarity of the righthand plot to Figure its average value across ? at that step so that the much larger values at highe do not obscure the behaviour at low e. Thus, both plots in Figure 4.9 show therelative variations in acceleration or jerk over ? , but not the variations over e. Thegreyscale is chosen so that whiter regions have lower values and are thus deemedeasier to detect; the attempted detection is assumed to be via a standard Fouriersearch in the left plot, and via an acceleration search in the right plot. While thereis very little variation across the e = 0.5 region of the left plot, the maxima are at90? and 270?, so this agrees with the increased difficulty of detection at 90? seen byBagchi, Lorimer & Wolfe The plot on the right is similar in appearance to Figure4.5, supporting my observations.4.3 ConclusionsBinary pulsar detectability has long been a well-known problem with a dauntingrange of parameters. Past studies have taken an analytical approach to character-izing our coverage of the expansive phase space, and here I have supplementedthese studies with an empirical approach, one tailored in particular to recent glob-ular cluster surveys, and covering a broad range of parameters including orbital39Figure 4.10: Globular cluster pulsars discovered with the Arecibo and Green Banktelescopes, plotted over my detection levels for comparison.eccentricity. The results are consistent with those of Johnston & Kulkarni (1991)and Bagchi, Lorimer & Wolfe (2013), the latter being the first analytical study toaccount for orbital eccentricity. Like that study, I find that pulsars in short-periodbinary systems are actually easier to detect when they orbit at very high eccentric-ities.Comparing my spin-orbital-period detection plot with binary globular clusterpulsars that have been found in Green Bank and Arecibo telescope surveys (Figure4.10), it does not appear as though discoveries are pushing down to the orbital-period detection limits, at least for MSPs in eccentric orbits. While it is difficultto draw strong conclusions based on the relatively little information here, this mayreflect a lack of very eccentric orbits with orbital periods in this range. For instance,if the eccentricity is the result of a binary exchange scenario, it is expected that thesemimajor axis of the resulting binary system will scale as the ratio of the mass of40the neutron star to that of its replaced companion (Heggie, Hut & McMillan, 1996;Ivanova et al., 2008). Since neutron star companions tend to have masses lowerthan the neutron star, this will often lead to an increase in semimajor axis followingthe exchange, and thus an increase in orbital period unless the new companion ismuch more massive than the old one.Faucher-Gigue`re & Loeb (2011) expect that a high-density environment withexchanges resulting in black-hole-MSP binaries would lead to highly eccentric or-bits with periods as fast as ?5 minutes, slightly lower even than the minimumorbital periods I examine here. However, it is not known how many stellar-massblack holes are retained in globular clusters, if any.41Chapter 5Future WorkThe data generated in this simulation take quite a long time to create and search,even with a 36-CPU system. With sufficient time, the present work could be ex-panded in several ways. It would be interesting to run the acceleration search withsome variation in both the number of harmonics summed and the maximum driftin Fourier bins in order to see the particular effects these search parameters haveon the behaviour across various orbital parameters. Even testing a few values ofeach could increase the total computing time by a factor of 10 or more. It wouldbe possible to make improvements in speed elsewhere to partially compensate forthis, such as including more pulsars per time series. It is likely that some improve-ments in computational efficiency could also be made. Testing multiple values oftime series length and time resolution might also be of interest, but would again becomputationally time-consuming. 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