Bachelor ThesisTemporal pulse-shape variations in therelativistic pulsar - white dwarf binary

J1141−6545

Lorenz Matthias Haase

November 19, 2014

First Referee: Joris P. W. VerbiestSecond Referee: Stefan Os lowski

Contents

1 Introduction 11.1 Pulsars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Pulsar binaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1 Keplerian orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.2 Relativistic orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2.3 Geodetic precession . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Pulsar timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4 An outline of J1141−6545 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5 Thesis outline and justification . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Observations and preprocessing 112.1 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Removing RFI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Updating the timing model . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Scrunching and folding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 Aligning profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Analysis of pulse-shape variations 143.1 Fitting components using paas . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1.1 Component centres . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.1.2 Component amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . 163.1.3 Component compactnesses . . . . . . . . . . . . . . . . . . . . . . . 163.1.4 Refine alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.1.5 Fitting ambiguities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Profile shape evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3 Fitting components with bfit . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3.1 Fitting procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Conclusions and further research 26

In this thesis, the pulse-shape variations are investigated in order to obtaininformation on these variations. These modelled variations can be used inthe timing of the pulsar, thereby improving the precision and making thelimits on GR more stringent. After the comparison of different approachesand it was found that a constrained fit of components is essential, and thatpeak-normalisation of the uncalibrated archives also provided more reasonableresults. The application to timing has not been done because the results arestill in large parts dominated by noise. Therefore, further analysis is requiredto allow a clear model to be derived. At the end, several suggestions are givenof how this analysis can be taken further.

1 Introduction

According to the standard model of stellar evolution, every star has a finite lifetime.Just before ejecting its outer layers into space, the core of the star collapses due to thegravitational force, leading to a compact remnant. Among the many different kinds ofobjects in the Universe there are three main types of stellar remnants. Depending on themass of the progenitor star, the remaining object can either be a white dwarf (WD), aneutron star (NS) or a stellar black hole.Main sequence stars with masses up to approximately ten solar masses end up producing aWD at the end of their lives. A typical WD is thought to be a sphere slightly smaller thanEarth in diameter and having a mass MWD ≤ 1.4M, where the solar mass is denoted byM. If the progenitor star’s mass exceeds ∼ 20M (Fryer, 1999), the gravitational corecollapse leads to a stellar black hole. A NS forms when the mass of the main sequencestar lies between this upper and lower boundary.Although WDs already have a tremendous density beyond any matter we know on Earth(ρWD ≈ 109 kg

m3 ), the mean density of a neutron star is even greater than that of nuclear

matter, accounting for about ρNS ≈ 5 · 1017 kgm3 . All known NSs have masses between one

and two solar masses and are roughly 20 km in diameter. When observed from Earth,some NSs exhibit a pulsating radio signal, therefore objects belonging to this class aregiven the name pulsar (PSR). (The above is based on the review by Lorimer & Kramer(2005) as is most of the information in this chapter)

1.1 Pulsars

When the thermonuclear fusion in the core of a giant star stops because after the pro-duction of iron nuclei the fusion process does not continue exothermally, the thermal-gravitational equilibrium cannot be maintained and the core collapses due to gravity. Asthe radius r decreases, the initially moderately rotating core begins to speed up, mainlycaused by the conservation of angular momentum:

L = Iω =2

5r2ω. (1)

In this equation, the magnitude of the angular momentum is denoted by L. I = 25Mr2

is the moment of inertia of a uniform-density sphere of radius r and ω is the angularfrequency, which scales with spin frequency f as ω = f

2π . The mass M of the core canbe assumed constant, although some mass loss occurs during the supernova following thecollapse. Although this formula holds for a sphere-shaped, homogeneous and rigid body,it is a useful first approximation to describe the increase of the spin frequency during thecollapse.

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Since the magnetic field of the star is also conserved to first order, the magnetic fieldstrength is also expected to increase with the star’s compactness. Given such a strongmagnetic field, a consideration of the magnetic dipole radiation generated by a rotatingmagnet is sensible:

E =2

3cµ2Ω4sin2α. (2)

The emitted power E is given as a function of the velocity of light c, the magnetic momentµ, the angular frequency of rotation Ω and the magnetic inclination angle α (Figure 1).This occurs at the expense of rotational energy, so the pulsar is expected to slow downgradually. The magnetic field strength on the surface of the pulsar BS is given by

BS = k√PP . (3)

The constant k includes the NS’s radius, the total moment of inertia and the magneticinclination angle α. Both equations (2) and (3) are taken from Lorimer & Kramer (2005).Once measurements of these variables including the rotational period P and its timederivative P are made, assuming a typical value for the moment of inertia, one can inferestimates of the magnetic field strengthBS on the star’s surface from Equation (3), yieldingfield strengths of 108 to 1014 G.

Although the exact emission processes are not fully understood yet, a beam of radioemission is generated in the areas around the magnetic poles with the emission takingplace in a constrained region centred on the magnetic pole. The radio emission is all con-tained within a so-defined cone of emission. In general, the magnetic poles of the pulsardo not coincide with the geographic poles, thus the inclination angle between the two axesis α 6= 0.

The fact that we observe an apparently pulsating radio signal from Earth can be ex-plained by the lighthouse effect: if the geometric setting allows the line of sight, i.e. theimaginary line connecting the Earth with the pulsar, to traverse through the emissioncone as the pulsar rotates, the Earth periodically receives a radio signal from this object.This is comparable to the rotating light signal of a lighthouse; the geometry is shown inFigure 1.Considering this model, the pulse period P equals the rotational period of the pulsar. Theperiods of all currently known pulsars lie between about a millisecond and a few secondscorresponding to a frequency range of 0.08 Hz . f . 700 Hz1 (Manchester et al., 2005),spanning four orders of magnitude. Despite pulsar periods being very regular over time, agradual spin slow-down P because of E (see Equation (2)) ranging from ∼ 10−21ss−1 to10−12ss−1(Manchester et al., 2005)1, thus covering nine orders of magnitude, is observedin all pulsars.

At the time of writing (2014), about 2300 pulsars are known and their distribution inthe sky mostly coincides with the plane of the Milky Way disk. Most currently knownpulsars are part of our own Galaxy, with a few exceptions found in the Magellanic clouds.

Aside from the pulse period, the mean pulse shape (i.e. the flux variation during a pulseperiod) itself can also vary considerably depending on the respective pulsar, the observa-tion frequency and - in some cases - time (see Figure 2).

Depending on the pulse period, a distinction between ‘ordinary’ (P ∼ 0.5 s) and ‘mil-lisecond’ (P . 30ms) pulsars has been made. As the name suggests, millisecond pulsars

1ATNF Pulsar Catalogue, Catalogue Version: 1.50, URL: http://www.atnf.csiro.au/people/pulsar/psrcat/

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Figure 1: This figure shows the geometry of the emission cone relative to the line of sight.α is the misalignment angle of the magnetic and rotational axes (also knownas magnetic inclination angle), the angles ρ and β are the opening angle of thecone and the impact parameter, which is the minimal angle between the line ofsight and the magnetic axis in the course of a pulse period. (Lorimer & Kramer,2005)

Figure 2: Pulse shapes of different pulsars (Lorimer & Kramer, 2005). For PSRs B1237+25and B1937+21, the pulse profile is shown at two different frequencies, demon-strating the frequency dependence of the pulse shapes. PSR B1913+16 exhibitsgeodetic precession (see Section 1.2.3) so its pulse shape varies slightly in time.

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Figure 3: This P − P diagram shows radio pulsars and their different classes. Most ofthe millisecond pulsars have a significantly smaller spin-down rate P and areconstituents of a binary system. (Lorimer & Kramer, 2005)

have much shorter pulse periods than their slower spinning counterparts. As shown inFigure 3, the normal and the millisecond pulsars can be found in clearly identifiable clus-ters in P − P space.The key to the high rotational frequency of millisecond pulsars probably lies in their bi-nary nature: when the more massive star of a main-sequence binary first evolves into a NSand the companion subsequently develops to a red giant, the NS can accrete matter fromhis companion, because its outer layers are only loosely gravitationally bound to the star.This process of mass transfer also transfers angular momentum to the NS, increasing itsspin frequency.This model explains why most millisecond pulsars have a companion.

Pulsars can be used for a variety of investigations in physics and astronomy. Below,we list a few of these, but for a more detailed treatment, the reader is referred to Lorimer& Kramer (2005).

• Exploring the interstellar medium (ISM): the ISM can be examined by measuring thearrival times of pulses as a function of observing frequency. Different densities of the

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ISM along the line of sight cause the delays of the pulse arrival times with frequencyto vary. Since most pulsars have a large proper motion, the structure of the ISM canbe measured in this way (Keith et al., 2013). Moreover, measurements of scintillation(the brightness of the pulse as a function of time and observing frequency) providesinsight into the structure of the ISM on smaller scales (Brisken et al., 2010). Anotherway the ISM affects the pulsar signal is via scattering. This effect is also largest atlow radio frequencies and causes the radio emission to be delayed by changing itspath length (Bhat et al., 2004).

• Measuring polarisation properties reveals properties of the Galactic magnetic field.When polarised electromagnetic radiation passes regions with a magnetic field, theparallel component of the field with respect to the propagation direction of theradiation causes the polarisation vector to rotate (Faraday rotation). Since pulsarradiation is often highly linearly polarised, effects of large-scale magnetic fields aremeasurable by the time variations of the position angle across the bandwidth, as thepulsar moves through the Galaxy.

• Pulsars are also outstanding objects for astrometry: fhe position on the celestialsphere can be determined with high precision by using a radio interferometer orthrough pulsar timing. Timing positions are determined by measuring the shiftsof the arrival times of pulses with high precision. Yearly variations of these pulsearrival time are caused by the geometric orientation of our planet’s orbit relative tothe pulsar, depending on the epoch.

• In some cases pulsars allow the detection of extrasolar planets in an orbit aroundthem. Indeed, the first planet discovered outside our solar system was already foundin 1992 accompanying a NS (Wolszczan & Frail, 1992)! There is also a planet ofLunar mass in this planetary system, making it one of the lightest extrasolar planetsknown so far. But the abundance of planets around pulsars is strongly limited dueto the violent formation process of these stellar remnants.

• Matter in conditions not reproducible on Earth can be analysed: by the observationsof small sudden spin-ups (glitches) and the subsequent decay towards the pre-glitchrotation period, one can deduce some basic properties of the pulsar’s highly com-pressed interior, which is likely to be superfluid and superconducting. The pulsar’splasma atmosphere and its interaction with the extraordinarily strong magnetic fieldis an important subject of research, since there is still little known about the un-derlying physics (Melrose & Gedalin, 1999). Yet, it is possible to probe the radioemission region in close double neutron star (DNS) binaries where the strong rel-ativistic wind emanating from one constituent affects and measurably distorts theemission region of the other. Moreover, if we observe the binary nearly edge on,the radio beam of one pulsar may periodically pass through the magnetosphere ofthe other, offering a unique opportunity to investigate the plasma envelope and theradio emission region.

• Gravitation and Cosmology: although GR is one of the most successful theoriesto date, it is imperative to find ways of further testing Einstein’s theory. Binarypulsars provide the chance of testing GR and alternative theories of gravity in thestrong-field regime. Some additional orbital parameters compared to the classicalKeplerian orbit can be measured and matched against the theoretical predictions.Furthermore, in certain circumstances it is possible to measure the relativistic time

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dilation of the pulsar signal, adding powerful tools to test GR outside the limitingscope of our solar system with its weak gravitational interactions. The tests of GRare described below in more detail.

1.2 Pulsar binaries

Pulsars are not necessarily solitary objects: There are many pulsars with a companion,which can be either another NS, constituting the above-mentioned DNS systems, a WD, amain sequence star, a red giant or even a black hole (BH). Especially DNS and pulsar-WDbinaries have been proven to be valuable systems to test not only GR but also restrictalternative theories of gravity. The tests would be taken a lot further if and when a NS-BHbinary would be found (Liu et al., 2014).

1.2.1 Keplerian orbits

The properties of a classical elliptical orbit of a celestial body around the barycentre2 of atwo-body system was first proposed by Johannes Kepler more than 350 years ago. Thereare six Keplerian parameters uniquely describing such an orbit (see Figure 4):

• The inclination angle i relative to a reference plane, that in astronomy is usuallydefined to be the plane perpendicular to the line of sight at the distance of thebinary’s center of mass.

• The ascending node marks the point where the orbiting object passes the referenceplane while receding from the observer. The angular separation between the ascend-ing node and a reference direction (typically North) is denoted by Ω, the longitudeof the ascending node.

• The semi-major axis ap is a measure of the orbital size.

• The orientation of the orbit (or more precise - of the line of nodes within the orbitalplane) is given by the argument of periastron ω, the angle between the ascendingnode and the periastron, the nearest approach of the orbit to the barycentre.

• The eccentricity e determines the shape of the orbit itself. The higher its value, themore elongated is the corresponding ellipse.

• There are three quantities fixing the position of the body as a function of time: Theeasiest understandable from a geometrical point of view is the true anomaly ν as theangle between the current position of the body and the periastron. Besides that, themean anomaly M and the eccentric anomaly E are frequently used, but all threeare interchangeable and mutually related.

Figure 4 gives an overview of some orbital parameters. Although relativistic parametersplay a key role in the determination of the pulsar and companion mass, the mass functionalready permits to impose constraints on the mass of the companion Mc, providing thatPb, i and ap are known. The mass function is given by

f(Mp,Mc) =(Mc sin i)3

(Mp +Mc)2 =

4π2

G

(ap sin i)3

P 2b

. (4)

2barycentre = centre of mass

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Figure 4: Visualisation of some key orbital parameters of a classical Keplerian orbit. Theperiastron is also more generally referred to as periapsis.

In many cases, the inclination angle i is unknown, such that it is only possible to placea lower limit on the companion mass assuming the orbit to be seen edge on. The pulsarmass and gravitation constant are denoted by Mp and G, respectively.

1.2.2 Relativistic orbits

In the context of GR, more parameters have to be added to properly describe the orbitsof compact objects, namely NSs and WDs in close orbits. The strong gravitational fieldsinvolved make the description according to GR necessary and the so-called post-Keplerianparameters quantifiable:

• The argument of periastron slowly rotates around the center of mass leading to ashift of the periastron with the angular velocity ω also known as periastron advance.

• Oribital shrinking: Due to the loss of energy by the emission of gravitational waves(GW), GR predicts the semi-major axis of the orbit to decrease. As a result, theorbital period is expected to shorten. The rate of this gravitational wave dampingis denoted by Pb.

• The Shapiro delay parameters r and s are derived from the passage of a pulsars radiobeam through the gravitational field of the companion. When light is deflected bya massive object, the travelled distance and therefore time increases before reaching

2“Orbit1” by Lasunncty. Licensed under Creative Commons Attribution-Share Alike 3.0 via WikimediaCommons

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a distant observer3. The delay of the radio pulses can only be detected if i is closeto 90.

• The Einstein delay parameter γ is a combination of the gravitational redshift of thesignal caused by the companion and time dilation due to varying gravitational fieldstrengths of the companion as a function of distance.

These effects are quantified as follows (Stairs, 2003):

ω = f1(M, Pb, e)(Mp +Mc)23 (5)

Pb = f2(M, Pb, e)MpMc

(Mp +Mc)13

(6)

r = f3(M)Mc (7)

s = f4(M, Pb, ap, i)(Mp +Mc)

23

Mc(8)

γ = f5(M, Pb, e)Mc (Mp + 2Mc)

(Mp +Mc)43

(9)

The functions fi with i = 1, 2, ..., 5 depend on the respective values in parentheses, whereM is the solar mass, Pb the orbital period, Mp the mass of the pulsar and Mc the compan-ion mass. The denotation of the remaining quantities on the right-hand side of equations(5) to (9) is consistent with those of Section 1.2.1. As well as in the previous section, allmasses should be expressed in solar masses.In some situations, all parameters Pb, e, ap sin i and i are accessible via pulsar timing,allowing determination of the masses of the binary constituents in multiple ways, sincethe system of equations (5) to (9) is threefold overdetermined. This provides a powerfultesting method for GR by performing a consistency check, which was conducted for theDNS J0737-3039, as shown in Figure 5. All constraints on the masses Mp and Mc of thetwo NSs in J0737−3039 are consistent within the uncertainties.So far, measurements of the orbital decay Pb by the shift of the periastron passage as afunction of time for the first discovered DNS, B1913+16, are also consistent with GR (seeFigure 6). This system was the first to demonstrate this GW emission, resulting in theNobel prize for Taylor and Hulse in 19934.

1.2.3 Geodetic precession

Another predicted effect of GR is the so-called geodetic precession. Unlike the precessionof the rotation axis of the Earth, explained by the deviations from a purely spherical shapeas well as inhomogeneities and tidal forces of Sun and Moon, geodetic precession occurs asa result of space-time curvature near a compact body and even affects perfectly sphericalbodies of uniform density. This precession Ωgeod about the total angular momentum ofthe binary is given by

Ωgeod =

(2π

Pb

) 52 Mc (4Mp + 3Mc)

2 (Mp +Mc)43

1

1− e2T

23 (10)

3http://www.astro.ucla.edu/ wright/deflection-delay.html4http://www.nobelprize.org/nobel prizes/physics/laureates/1993/press.html

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Figure 5: Parameters of GR in the M1-M2 diagram of J0737-3039. The large shaded areaof the M1-M2 plane is forbidden by the mass function in Equation (4) (Krameret al., 2006).

Figure 6: Measured periastron passage (dots) in comparison with the GR prediction (line)for PSR B1913+16 (Weisberg et al., 2010).

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where T = GMc3

. Geodetic precession leads to relativistic coupling of the spin andorbital angular momentum, comparable to the spin-orbit coupling in atomic physics. Thechanging orientation of the spin axis as a function of time is indirectly detectable fromEarth by time-variable polarisation properties and long-term pulse-shape changes as theline of sight cuts through different parts of the emission beam over time.

1.3 Pulsar timing

Pulsar timing is a powerful technique that allows the determination of the above-describedparameters with high precision. Small deviations in the arrival times of the highly periodicpulsar signals can be measured, because small changes of the pulse period accumulatequickly, reaching values comparable to the pulse period. Apparent changes of the pulseperiod in pulsar binaries for example can be attributed to the Doppler effect of orbitalmotion. Since this motion is related to the orbital parameters (see above), it is possibleto deduce the orbital properties.The key to all this is to measure the time of arrival (TOA) of pulses. This can be achievedusing a template profile generated by the addition of many previous observations. Thetime shift τ between the template and the profile, calculated using the cross-correlationmethod, yields the desired TOA. Timing models, such as the one by Damour & Deruelle(1986), allow all Keplerian and relativistic orbital parameters to be fitted to the TOAs,thereby improving the timing model. For a complete treatise on pulsar timing, the readeris again referred to, Lorimer & Kramer (2005).

1.4 An outline of J1141−6545

In this thesis we investigate the pulse-shape variations of the binary pulsar J1141−6545lying far south in the constellation Musca. Due to its locus, it cannot be observed by mostradio telescopes. It consists of a normal pulsar with average mass in a relatively closeorbit around its solar-mass WD companion, with an orbital period of less than five hours.Some relevant parameters are listed in Table 1.4.This pulsar was already the subject of recent research including GR tests (Bhat et al.,2008) and mapping of the emission region (Manchester et al., 2010). Measurements ofthe post-Keplerian parameters, performing consistency checks of the inferred masses Mp

and Mc as described in Section 1.2.2 were positively supporting GR. More specifically,the orbital decay rate was found to be 1.04±0.06 times the GR prediction and the orbitalinclination angle i derived from the Shapiro delay was consistent with other independentmeasurements performed by Ord et al. (2002).Later, Manchester et al. (2010) presented results of mapping a large fraction of the emissionregion, showing that it has a rather patchy structure with no symmetry with respect tothe magnetic pole. This was only feasible because of geodetic precession changing thepath of the line of sight through the emission region of the pulsar.

1.5 Thesis outline and justification

The main goal of this thesis is to get information about the variations of the pulse-shapeas a function of epoch. Different approaches to modelling these variations are discussedand compared, trying to find the best approach to be used for precise timing of the pulsar,allowing stringent tests of GR in the strong field regime.

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Discovery year 1999Right ascension, α (J2000) 11h41m07s.01Declination, δ (J2000) −6545′19′′.11Pulse period, P 0.393899 s

Pulse period time derivative, P 4.307 · 10−15

Orbital period, Pb 0.197651 daysEccentricity, e 0.1719Longitude of periastron, ω 5.310yr−1

Einstein delay, γ 0.00077 ms

Orbital period derivative, Pb −0.4 · 10−12

Pulsar mass, Mp 1.27(1)MCompanion mass, Mc 1.02(1)M

Table 1: An overview of J1141−6545 with some selected parameters (Bhat et al., 2008).

2 Observations and preprocessing

In the following section the available data and the processing methods are described.

2.1 Observations

The data of PSR J1141−6545 come from the 64-metre single-dish radio telescope at theCSIRO Parkes Observatory in New South Wales, Australia5. They consist of 914 pulsar-archive files taken between July 25, 2003 and October 18, 2009 (MJD range 52845-55122),covering observations on 71 days. Each file consists of many roughly one-minute sub-integrations, which are the result of folding using a pulsar-specific timing model. Allfiles have four polarisation channels and are recorded at two different centre frequenciesat 1341 and 1405 MHz with a bandwidth of 64 MHz covered by 256 frequency channels.Every sub-integration spans over one pulse period and has a time resolution of 1024 bins.

2.2 Removing RFI

The program pazi allows the removal of sub-integrations and frequency channels whichcontain man-made radio signals or other disturbing signals distinguishable from noise andpulsar emission, also referred to as Radio Frequency Interference (RFI). It is possible tointeractively remove unwanted RFI from frequency vs. pulse phase or time vs. pulse phaseplots of a given data file. One example of RFI removal is shown in Figure 7.

2.3 Updating the timing model

Since most pulsars are very weak radio sources, timing models of a pulsar are crucial tofold (or “phase-average”) their profiles to reduce the noise and therefore obtain a moreprominent profile. This is already done during the preprocessing of the data to make sub-integrations and align the sub-integrations as depicted in Figure 7. Nevertheless timingmodels are never perfect. In the later stage of the zapping process it became apparentthat the applied timing-model is not correct for some observations in the last third ofthe observation time span. When plotting the sub-integrations it became visible that theprofiles are not located at the same pulse phase but instead were shifted by a constant

5http://www.parkes.atnf.csiro.au/

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Figure 7: Example of RFI zapping with pazi : On the left-hand side one can see the plotof an archive file with all the sub-integrations. The timing model applied herealigned all the profiles on the left. Sub-integrations with RFI are clearly visiblewhere the detected off-pulse intensity is considerably higher than noise (darkgray). The right-hand side figure shows the same data, after the RFI-affectedtime channels were manually removed.

value each sub-integration or had a curved or seemingly random distribution across thepulse phase (see Figure 8).

Some of the deviations of the pulsar’s behaviour from the timing model can be attributedto a glitch occurring at epoch ∼54280 (June/July 2007, Manchester et al., 2010). Thenew situation requires the old timing model (ephemeris) to be updated to a new versionfor the affected files. For this purpose a shell script was used to check each file for the rightephemeris version writing the results in a text file that is subsequently used by anotherscript that among others invokes tempo2 to transform the old ephemeris version to a newone.

2.4 Scrunching and folding

Afterwards all files with observation epochs less than 10000 seconds (∼1.16 days) apartwere combined, that is, added up using psradd to make the profile more prominent. Duringthis process, all sub-integrations of the respective files were scrunched (i.e. averaged) to asingle integration before adding up individual files with the epochal proximities describedabove. In this process 81 folded files are generated to undergo further processing. Untilnow, every folded file still consists of 256 frequency channels and four polarisation chan-nels. After this procedure, some folded files exhibited more than one distinct profile peak.A likely explanation for this discrepancy lies in the timing model. Due to differences inthe parameters of the timing model for some pre-scrunched files and the resulting differentlocations of the profile across the pulse phase in different pre-scrunched files, the processof adding up these files created two or more peaks across the pulse phase.

This issue could finally be resolved by refitting the pulse frequency F0 and its time deriva-tive F1 of the timing model on the TOAs of sub-integrations with tempo2. The fitting isdone individually for every folded multiple-peaked profile, as including more pre-scrunchedfiles in a larger time span would unnecessarily complicate the analysis, since at this stageonly accurate alignment of pulse profiles within an observation were required (and not areliable long-term timing model).

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Figure 8: Non-aligned profiles as a result of wrong parameters in a timing model. It isclearly visible that the profile drifts in time and changes its position in pulsephase.

After that, the sub-integrations are scrunched again and the obtained files now show aclear single-peaked pattern, indicating that indeed wrong parameters in the timing modelwere the culprit. In a next step the remaining frequency and polarisation channels arescrunched to one channel of total intensity applying the program pam.

2.5 Aligning profiles

The software pas permits to align profiles of two files by hand. At this stage of the analysisit was obvious that the profile shows a remarkable stability at its wings and that the moststriking variations occurred in the center region, which is why the profiles were aligned atthe lowest possible edges not biased by noise, corresponding to ∼5-10% of the peak height.

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3 Analysis of pulse-shape variations

3.1 Fitting components using paas

After separating the files based on their centre frequencies, two equally-sized groups offiles with centre frequencies of 1341 MHz and 1405 MHz are obtained.Now it was attempted to fit Gaussian components to the profile with the help of thepaas program which can be used to set initial values of the curves interactively with themouse and to then fit those components to the profile. The initial parameters of thefirst profile were chosen from scratch. After they were fitted to the first observation, theresulting fitted parameters were chosen as initial values for the second file and so on.This technique which I henceforth refer to as ‘consecutive fitting’, was applied to eachcentre frequency separately. It should ensure that the parameters of the components varysmoothly as a function of epoch, unless there are rapid changes of the pulse profile withinshort periods of time.A total number of three components were allowed to be fitted to the pulse profile, eachcomprising of three parameters. These parameters are:

• Centre/Centroid: This is the abscissa of the peak of the Gauss curve.

• Amplitude/Height: This is the ordinate value of the peak and the maximum of thecomponent.

• Compactness: This value is high for narrow components and becomes smaller whenthe curve’s width increases; there is an inverse relation between the compactnessand the Full Width at Half Maximum (FWHM) of the component.

Gaussian curves are very important in statistics for describing the normal distribution

G(x, x0, σ) =1√2πσ

exp

(−(x− x0)2

2σ2

). (11)

The multiplicative factor arises from the fact that the normal distribution is normalizedby the integral over the range of the real numbers. The standard deviation σ and theFWHM of the curve are proportional. Since the fitted Gaussian components are notnormalized, the parameters Centre, Amplitude and Compactness, denoted by x0, A, andC, respectively, yield the following function

G(x) = A exp

(−(x− x0)2C2

k2

), (12)

where C is given by C = kσ with some constant k. (The function used by paas is in

fact a von Mises function which is periodic and smooth when adjacent pulse periods areregarded, unlike Gaussian distributions which show a kink at the boundary between twoperiods, but the above discussion can help to visualise the process even so.)It is striking that one fitted component at a centre frequency of 1341 MHz could not beprevented from becoming very low and wide, corresponding to small values in both am-plitude and compactness, suggesting that two components are sufficient to fit the profileproperly at later epochs; the third component could thus be neglected (see Figure 9). Theresiduals, that is, the difference between the actual profile and the sum of all fitted compo-nents, could not be distinguished from noise in all the files before the first low-amplitudeand non-compact curve showed up, indicating that three components are sufficient to de-scribe the profile shape in this range. From the point where one curve drops down, the

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Figure 9: This figure shows two examples of profiles fitted with Gaussian functions by paas.The left-hand side shows a profile at early epochs where three components arerequired to obtain a sensible fit. At later epochs, two components often sufficedfor this purpose (right-hand side of the figure).

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next few profiles could be described quite well (displaying noise-like residuals) with onlytwo components until at some later epoch a third curve was needed again to reduce theresiduals to noise. Two examples of two and three fitted curves are shown in Figure 9.The fact that the third component disappears and reappears at later epochs a couple oftimes at different phases and the occurrence of a dramatic centroid drift of the majorcomponents when the third vanishes is unsatisfying.

However, the consecutive fits to the files at 1405 MHz centre frequency resulted in a moresatisfying outcome with no components disappearing. But, as is visible in figures 10, 11and 12, some parameters still exhibit rapid temporal changes.A discussion of the observed variations in the 1405 MHz data, follows below.

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Figure 11: Fitted component heights depending on epoch (1405 MHz).

3.1.1 Component centres

The component centres all seem to follow an overall trend, therefore showing, that thealignment from scratch is relatively robust, but with an unexplainable, small overall shiftto later phases. This was subsequently corrected, as described in Section 3.1.4.

3.1.2 Component amplitudes

In contrast to this, the amplitudes seem to be scattered throughout the epoch range.Despite two outliers at the beginning, the height of component one shows a fluctuation butsignificantly increases across the observation span, whereas the heights of the second andthird component show a decreasing trend at early MJDs and an overall constant behaviourat later epochs. The initially unsteady behaviour of all amplitudes can be attributed todifferent observation lengths of the scrunched files, whereas the random scatter at theend could be caused by a lower signal-to-noise ratio due to decreasing flux density. Thesame reason might hold for the behaviour of the compactness in later epochs, althoughthe relative stability of component one remains unexplained.

3.1.3 Component compactnesses

The compactness of the second and third components have a visibly smoother behaviouruntil MJD ∼54200, when large variations start to occur. Especially the third componentshows a high, distinct and steep-sloped peak at the end of the MJD range. Since thispeak consists of not just one outlier point, it is unlikely to be a random fluctuation causedby noise. paas does not provide uncertainties for the components, so they have to beguessed from the fluctuations of neighbouring points, allowing the implication that theuncertainties significantly grew towards the end of the observation epoch span.

3.1.4 Refine alignment

In order to remove the phase drifts of the centroids the profiles were again aligned bymeans of a shell script at the 5 % level of the profile’s peak intensity. First, the channelnumber of the peak flux density was determined and from that point, going down the

16

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Figure 12: Time evolution of component compactnesses at a centre frequency of 1405 MHz.

right and left-hand slopes of the profile, the first channels that undercut the 5 % level ofthe peak on either side of the profile were averaged. The difference of the averaged valueswith respect to that of a chosen reference file determined the number of bins the profileshould be shifted. This job was eventually accomplished by the pam software.

3.1.5 Fitting ambiguities

Due to ambiguities of fitting three components to a specific profile (see Fig. 13), four fitswith different component parameters were made from scratch on the first profile with acentre frequency of 1341 MHz.

The job of consecutive fitting based on each of these four initial files separately was doneby a script, creating four strands of fitting (see Figure 14). It was hoped that there mightnot be a component drop-out in one of the fitting strains or that the overall evolution of thecomponents would be smoother, but this was not the case. Three of the strands convergequite quickly at epoch ∼52900, with the fourth joining the convergence at epoch 53204.Subsequently one component almost vanishes as discussed at the beginning of this sec-tion. The parameters of this curve (the second one in Figure 14) show a peculiar behaviour:

• The centroid rapidly decreases and even becomes negative at epoch 53932, lying farout of the pulse phase range of the uppermost figure. This phase shift is so dramatic,that it crosses the phase boundary and reappears at the other side of the phase dueto the periodic property of the profile. Since the period is p = 1, the relation φmod 1 can be used to shift the phase without changing any parameters.

• Along with the centre shift comes the decline of the flux density making this com-ponent negligible in comparison to the other peak flux densities.

• The drop of the compactness of the initially narrowest component is not as suddenas in the parameters mentioned before, it is indicated since MJD ∼53500 but thensteepening until reaching zero. Later, negative values are also adopted, remaininginexplicable in this context.

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Figure 13: The fitting of Gaussian components with all parameters allowed to vary doesnot have a unique solution: for this profile obtained at the earliest epoch 52845there are at least four distinctly different possibilities to keep the residuals atnoise level.

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Figure 14: This visualisation of the strands of fitting shows the convergence of all strandsafter approximately one third of the observation time span. (Note the left-handside of the figure is stretched for clarity)

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3.2 Profile shape evolution

After omitting profiles with a noise level higher than 5% of the peak, the aligned profileswere plotted as a function of epoch to visualize the evolution of the pulse shape variationsover time at both centre frequencies (see Figure 15).

One can clearly see a significant pulse-shape variation in the observing time spanningapproximately six years. At early epochs some kind of ramp preceding the peak canbe seen which gradually disappears towards later epochs, growing to become part of theprofile. The profile becomes more symmetric and adopts a more rounded shape over thecourse of time. An increase of the pulse width in terms of the peak is also evident. Hotanet al. (2005) pointed out that the origin of these long-term variations may lie in geodeticprecession of the pulsar’s spin axis as the two bodies of this system orbit each other.The pulse-shape evolution is similar at both centre frequencies and is consistent with thepulse-shape variations Manchester et al. (2010) showed in their paper, during the windowof overlap.

3.3 Fitting components with bfit

Since the fitting with paas did not produce a really satisfactory result, the software bfitby Michael Kramer was used, which permits to keep the centres of the components fixed.As in the fitting procedure with paas, the components of one profile (this time chosen tobe approximately at the centre of the observing time span), were chosen from scratch andfrom this point, the files were fitted consecutively to the newest epoch and then backwardsin time to the earliest epoch, keeping the centres fixed. Unlike paas, this software usesreal Gaussian curves for fitting and not von Mises functions.A further advantage of bfit is the ability to obtain uncertainties for the fitted param-eters, which paas lacks, making it possible to match the scatter of the values to theiruncertainties.

3.3.1 Fitting procedure

A total number of between three and seven components are needed to obtain a reasonablefit with residuals comparable to noise. If a fit did not yield noise-like residuals with thisnumber of components, one or more components were added by hand and subsequentlyfitted until reasonable outcomes were achieved. In the case one component drops to almostzero and/or becomes very low and wide, this component was deleted before a final fit wasdone.The variation of the parameters can be seen in Figure 16. When loading the parametersof the previous fit into bfit alongside the profile to be fitted, a tiny shift of the centrescould be noticed, requiring corrections when the accumulated shift reached values of onequarter of a bin, which took place roughly every tenth fit.

3.3.2 Results

Of all components, three are continuously present across all epochs, centred at bins 832,850 and 860, respectively. Since these main components contribute the most to the fittingin the left, middle and right fraction of the profile, respectively, they are henceforth referredto as the leading, central, and trailing components. A fourth component spanning the firstthird of the time span and centred on the 855th channel, is denoted simply by the fourthcomponent, if necessary.

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Figure 15: Pulse variations as a function of epoch observed at centre frequency 1341 MHz(left) and 1405 MHz (right), aligned at the 5 % level.

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Figure 16: Temporal evolution of the component amplitude and width of the manuallyfitted profiles with bfit without normalisation in terms of the profile peaks.

22

Amplitudes After an initial scatter (see Figure 16), the component amplitudes begin tofollow a general trend leading to a sharp height increase of all three components followedby a rapid decrease and a region of moderate scatter with a visible upward trend. At laterMJDs all curves vary in the same manner without intersecting each other, exhibiting aclear trend. Some of these rapid variations may be smoothed by the normalisation withrespect to the profile peak value, although this affects only changes that concern an overalltrend. The seemingly random scatter at early epochs is not expected to be removed bythis method.In the next steps the amplitudes were normalised by the peak flux density of the profilesas well as the Root Mean Square (RMS) of the baseline, that is, the pulse phase rangewhere no profile is visible. If the baseline contains N channels and the intensity of the nthchannel is denoted by In, the RMS is defined as

IRMS =

√√√√ 1

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I2n. (13)

Figure 17 shows the results of the two normalisations. The normalisation with respectto RMS did not yield an improved outcome compared to unnormalised heights: there aretwo peaks at early epochs and in the last third of the epoch range. Almost all componentsshow the same noisy peaks, which is unfavourable.

In contrast to this, the normalisation in terms of peak flux density of the profile shownat the top of Figure 17, stresses the evolution of the individual component heights. Thereare no distinct peaks which are followed by most or all components.

At the beginning of the observation time until January 2006, two of the componentsare rising almost steadily, starting at relatively low values, whereas the two other con-tinuous components show a rather decreasing trend. At mid-epochs, all main amplitudesshow large short-term variations, often surpassing the errorbars of individual points by afactor of three or larger. Although scattered, it is visible, that the height of the centrecomponent in this area is significantly lower than the other heights. Since January 2008,all components remain approximately constant without showing rapid variations or inter-secting each other.In the highly scattered central MJD range, the leading and the central curve exhibit a lowcompared to the two edges of this region, forming a trough. The opposite is true for thethird curve, it shows an overall high with a hump-like structure.

Widths The widths of the Gaussian components correspond to the standard deviationσ, giving the width at 1√

eof the curve maximum.

The widths show a rather smooth behaviour at early epochs as well as at late epochs,staying approximately constant, except for the central component, falling slightly at initialepochs. As in the case of the normalised amplitudes, the scatter suddenly increases andbecomes significantly larger than the uncertainties particularly in the central componentof the profile.

Throughout the MJD range, the central component is the widest one, with a large offsetfrom the other curves particularly at early epochs. All curves show an overall increase inthe most scattered region at centre epochs; here the trailing component is much higher thanthe leading component, having comparable values elsewhere. Due to the large increaseof the trailing curve, the difference with the central curve diminishes and the two sigmasshow almost the same variations.

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Figure 17: This figure shows component heights depending on epoch, normalised by thepeak flux density of the profile (top) and by the Root Mean Square (RMS) ofthe baseline noise (bottom).

24

Figure 18: Scintillation in an early-epoch observation of PSR J1141−6545. The intensityfluctuations in the frequency-time space cannot be missed. It is caused byscattering of the pulsar beam in the ISM when scattered radio waves interfererather constructively or destructively.

Fourth component In figures 16 and 17 one can see a component centred at bin 837 inthe third quarter of the epoch range. This component disappears in one observation beforereappearing again. To improve the possible fitting of functions to the curve parameters(see Section 4), an attempt to ‘connect’ the two distinct parts was made by calculating themean values of all parameters taking the left and right edge of this gap. A new componentbased on these averaged parameters was added to the main components in the interruptedregion. After it became clear that this method resulted to bad residuals, this componentwas fitted to the profile, resulting in a very high width value incompatible with a smoothlyvarying curve. The amplitude remained very low, comparable to the two edges.

25

4 Conclusions and further research

• The origin of the rapid variations of the component parameters has not become clearduring the analysis. Maybe the large scatter is intrinsic to the pulsar or due to noisecaused by the digitisation of the signal over the whole bandwidth by the telescope.It might be possible to take smaller parts of the band to get a more time steadypulse.

• Even though scintillation (i.e. the intensity variations of the pulsar’s mean flux den-sity depending on frequency and time) occurs in PSR J1141−6545, it can be ruledout that it is related to these variations, because the profile shape is only a veryweak function of frequency, as shown in Figure 15. One example for scintillationcan be seen in Figure 18. Also the scintillation time scale is short enough that ina multi-hour observation we would always average scintles from across the wholebandwidth.

• Calibrations should be performed to avoid profile-shape corruptions caused by im-perfections in the receiver hardware.

• Possible further steps may include the removal of some outlier points with smallerrorbars and add profiles of close-lying points to lower noise.

• In addition to that, components with small amplitudes can be deleted due to theirsmall contribution to the profile.

• The two last steps facilitate the fitting of piecewise defined functions to the compo-nent parameters. These functions can then be used to obtain a template profile asa function of time for pulsar timing.

26

References

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Bhat, N. D. R., Cordes, J. M., Camilo, F., Nice, D. J., & Lorimer, D. R. 2004, ApJ, 605,759

Brisken, W. F., Macquart, J.-P., Gao, J. J., Rickett, B. J., Coles, W. A., Deller, A. T.,Tingay, S. J., & West, C. J. 2010, ApJ, 708, 232

Fryer, C. L. 1999, ApJ, 522, 413

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Keith, M. J., Coles, W., Shannon, R. M., Hobbs, G. B., Manchester, R. N., Bailes, M.,Bhat, N. D. R., Burke-Spolaor, S., Champion, D. J., Chaudhary, A., Hotan, A. W.,Khoo, J., Kocz, J., Os lowski, S., Ravi, V., Reynolds, J. E., Sarkissian, J., van Straten,W., & Yardley, D. R. B. 2013, MNRAS, 429, 2161

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Manchester, R. N., Hobbs, G. B., Teoh, A., & Hobbs, M. 2005, AJ, 129, 1993

Manchester, R. N., Kramer, M., Stairs, I. H., Burgay, M., Camilo, F., Hobbs, G. B.,Lorimer, D. R., Lyne, A. G., McLaughlin, M. A., McPhee, C. A., Possenti, A., Reynolds,J. E., & van Straten, W. 2010, ApJ, 710, 1694

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Final declaration

I confirm that I have written this thesis on my own and that I did not use any literaturethat is not stated in the bibliography.

(Lorenz Matthias Haase)

27