Stability of black-widow

Philipp Krause

Bielefeld University Faculty of physics October 7, 2019

Supervisors: JProf. Joris P. W. Verbiest Dr. Ann-Sofie Bak Nielsen

Contents

1. Introduction2 1.1. formation ...... 2 1.2. Single pulsars and millisecond pulsars ...... 4 1.3. Binarypulsars...... 5 1.4. Observing pulsars ...... 5

2. Pulsar timing and observations of PSR J0636+51287 2.1. Observations ...... 8 2.2. Pre-processing ...... 11 2.3. Processing the data ...... 13 2.4. Pulsartiming ...... 15

3. Results for PSR J0636+5128 16 3.1. Orbital variations of PSR J0636+5128 ...... 17 3.1.1. Analysis ...... 19

4. Pulsar timing and observations of PSR J2051−0827 22 4.1. Observations ...... 22 4.2. Processing the data and pulsar timing ...... 22

5. Results for PSR J2051−0827 22 5.1. Orbital variations of PSR J2051−0827 ...... 24 5.1.1. Analysis ...... 25 ˙ 6. Calculated vs. measured Pb for different companion types 28 ˙ 6.1. Formulae for calculations of Pb contributions ...... 28 6.2. Dataused ...... 29 6.3. Doubleneutronstars ...... 29 6.4. Ultra-light companions ...... 31 6.5. Helium white dwarf companions ...... 32 6.6. Main-sequence ...... 35 6.7. Carbon-Oxygen white dwarf companions ...... 36 6.8. Allbinaries ...... 38 6.9. Cluster 47 Tucanae ...... 39

7. Conclusions 40

Statutory declaration 44 ˙ A. Pb-contribution script 46 Abstract This thesis is about pulsar timing of PSR J0636+5128 and investigates the common assumption that black-widow systems are unstable. After investigation of some orbital parameters, derived from the timing solution, the pulsar seems to be stable or at least not as unstable as is often assumed for black widows. With the help of that timing solution, a deviation of the calculated orbital-period derivative from the measured orbital-period derivative was found, which is followed by a more general investigation of orbital-period derivatives in binary pulsar sys- tems. Specifically the measured and calculated P˙b values for different companion types were checked and it was found that the orbital-period derivative for dou- ble neutron- binaries conform to the theoretical expectation. This is also the case for most of the binaries with carbon-oxygen white-dwarf companions. But for ultra-light and main-sequence stellar companions the orbital-period derivative deviates several orders of magnitude from the predictions. It is interesting that the orbital-period derivative for helium white-dwarfs, that are not in a globular cluster, is conform to theory, but for those who are in a globular cluster the orbital-period derivative is inconsistent with theory. This behaviour could be explained by the chaotic state that prevails in a globular cluster.

1. Introduction

The following sections are about basic theory of pulsars. First the formation of pulsars and the different types of pulsars are described. Afterwards the basic idea of observing pulsars and some observational effects that can occur at observations are explained.

1.1. Pulsar formation During their lifetime, stars pass several evolutionary stages and use nuclear fusion to gain energy. They start with burning hydrogen until their hydrogen reservoirs in the core are used up. After that, they continue with fusion of the next heavier element, helium. The heavier the star the heavier the elements it can burn. The last element a star burns is silicon into iron because the binding energy per nucleon has a maximum in iron (Demtr¨oder, 2017, p. 341). If that process ends, the life of the star ends too. The end of a star’s life depends on its . If the star is heavy enough, above 25 M 30 (1 M ≈ 2 × 10 kg), it ends in a black hole. If the mass is low, less than ∼ 8 M , it ends in a white dwarf (WD), like 95% of all stars do (Kepler, 2014). If the mass is between 8-25 M , it ends in a (Demtr¨oder, 2017, pp. 346-365). After silicon burning, the Fermi-energy of the electrons is high enough that protons, colliding with electrons, convert into neutrons in a process, called inverse beta decay − (p + e → n + νe). So the electron density decreases and the Fermi-pressure of the electrons is getting smaller than the gravitational pressure. When the gravitational pressure is too high, the core collapses further. The star repels a huge part of its mass in a supernovae type II explosion (Demtr¨oder, 2017, pp. 346-365). What is left is a small star, called a neutron star, with a diameter of about 10 km.

2 The mass of the neutron star is between 1.4 M and 3 M (Demtr¨oder, 2017, pp. 346- 365), although this range is not defined very precisely. Because of its small diameter and conservation of angular momentum, the neutron star rotates very rapidly. With contraction of the electrons and protons in the core during the collapse, the magnetic field 4 increases from B . 10 G (Spruit, 2008), which is the maximum known magnetic field of main sequence stars (MS), to the order of B ≈ 1012 G (Lorimer and Kramer, 2004, p. 27). The magentic field axis often deviates from the axis of rotation. Due to the strength of the magnetic field and the fact that it co-rotates with the pulsar, it can accelerate electrons nearly to the speed of light, so the electrons emit synchrotron radiation along a radiation cone (Demtr¨oder, 2017, pp. 346-365). This ”lighthouse principle” is illustrated in Fig. 1. As a consequence, the radio waves arrive at the observer as a sequence of highly precise, regularly spaced short pulses. The individual pulse shapes can differ, but the average pulse shape is very stable. That makes the pulsar a very precise clock.

Fig. 1: A schematic overview of the pulsar’s magentic field. One can see the closed and open magnetic field lines and the radiation beam of the pulsar, which emits the radiation one can detect on . Also, the axis of rotation deviates from the magnetic axis, which causes the lighthouse effect, as shown. (NRAO, 2010).

The first pulsar was discovered in 1967 by Jocelyn Bell Burnell and Anthony Hewish, after observing the angular structure of compact radio sources where they investigated the scintillation caused by the interplanetary medium (Hewish et al., 1968). That dis- covery led to the Nobel prize in physics for Hewish in 1974.

3 1.2. Single pulsars and millisecond pulsars

A normal pulsar has a rotational period between 0.1 and 10 s (Hewish, 1970, p. 267) and slows down with P˙ ≈ 10−15 s s−1 (Lorimer and Kramer, 2004, p. 27). But there is a specific category of pulsars which is called ”millisecond pulsars” (MSPs). These MSPs have rotational periods between 1.5 and 30 ms and slow down with P˙ ≈ 10−20 s s−1 (Lorimer and Kramer, 2004, p. 27). This distribution of pulsars is shown in Fig. 2, where the MSPs are in the bottom left corner and the normal pulsars in the top right corner. Most of the MSPs are in binary systems (≈ 80%) wheras normal pulsars are mostly without an orbiting companion (less than 1% in a binary system) (Becker, 2009). Also the MSPs have much weaker magnetic fields of the order of B ≈ 108 G, than normal pulsars have (B ≈ 1012 G, Lorimer and Kramer (2004, p. 27)). The MSPs accrete ma- terial and angular momentum from their companion, so the rotational period decreases to milliseconds (Bhattacharya and van den Heuvel, 1991).

Fig. 2: The P -P˙ diagram shows the distribution of normal pulsars on the upper right-hand side and MSPs on the lower left-hand side. Binary systems are shown as circles and are mostly found with MSPs and rarely with normal pulsars. The dot-dashed lines indicate the magentic field strength labeled on the right-hand side, which shows that normal pulsars have stronger magnetic fields than MSPs. (Lorimer and Kramer, 2004).

4 1.3. Binary pulsars The orbital companion of a pulsar can be another neutron star, a WD, a main sequence star or a black hole. If the mass of the companion mc is below 0.1 M the pulsar is called a black-widow pulsar (BWP), named after the black-widow spider. It is called that way because the pulsar was originally thought to be ablating its companion. If the mass is between 0.1 M and 0.4 M the MSP is called a redback (RBP). The Pb of both these MSP types is usually less than a day. The formation of BWPs and RBPs is not well understood. A possible scenario, proposed by King et al. (2003, 2005) is the formation of a typical MSP binary in a globular cluster where it is spun up by accretion from its binary companion. In the next phase of the formation of the BWP, the helium WD gets replaced by a main sequence star. That leads to ablation of the companion, because of the energetic irradiation of the MSP. After that formation the binary system can be ejected into the Galactic field. Even though most BWPs and RBPs were observed in globular clusters in the past, in the last years more and more BWPs and RBPs have been found in the Galactic field. (Chen et al., 2013).

1.4. Observing pulsars As the dish of the radio telescope points towards the sky, it receives radio waves. A big part of them are man-made signals from for example FM stations or mobile phones. These signals have to be removed from the data, because the radio emission of a pulsar detected with Earth-based telescopes is much weaker than the man-made noise. To get more signal than noise, the collecting area can be increased. The Effelsberg telescope in Bad Munstereiffel¨ for example has a dish diameter of 100m (MPIFR, 2019). The collected signal gets focussed on the receiver which passes the signal to the computer after passing through several amplifiers and filters. The next steps are to de-disperse and fold the data. The whole dataflow is shown in Fig. 4. The signal has to be de-dispersed because pulses with higher frequencies arrive earlier than pulses with lower frequencies, as seen in Fig. 5. This frequency-dependent delay is caused by the interstellar medium (ISM). The ISM is a highly turbulent medium which causes phases modulations on the propagating pulse signal. The result is a fluctuation of the observed intensity on different frequencies and timescales. This effect is called scintillation. That effect can be seen in Fig. 3. After de-dispersing, the signal has to be folded to a mean pulse profile, because the individual pulse shapes differ, but the shape of the mean pulse profile is stable. So the signal-to-noise ratio (S/N) increases as well. After getting a mean pulse profile, the time of arrival (TOA) of the pulse can be measured. The data are now cross- correlated with a template and the difference between the theoretically predicted TOA and the observed TOA is called the ”timing residual”. The aim of pulsar timing is to minimize the RMS of the timing residuals and deriving orbital and other pulsar-related observable parameters from fitting the parameters of the timing model to the measured TOAs. Observable parameters are for example the orbital period Pb which is the period

5 in which the companions orbit each other. Another parameter is the derivative of the ˙ orbital period Pb which gives the change of the orbital period. The of ascending node TASC is the time when the observed object passes through the plane of reference. Furthermore, the eccentricity e determines the amount by which the orbit deviates from a circle. It is given by EPS1 (e×sin(ω)) and EPS2 (e×cos(ω)), where ω is the Longitude of periastron. Periastron is the nearest point of an orbiter to its host.

Fig. 3: Effect of Scintillation on the pulse signal of PSR B1133+16. The intensity fluctuates noticeably at 626 MHz and 1412 MHz. The effect also occurs at the other frequencies, but it is much weaker. (Lorimer and Kramer, 2004)

6 Fig. 4: Dataflow of pulsar timing (Lorimer and Kramer, 2004)

Fig. 5: Pulsar dispersion. (Lorimer and Kramer, 2004)

2. Pulsar timing and observations of PSR J0636+5128

In this chapter the pulsar timing of PSR J0636+5128 is described. First the telescopes used for the observations and their different properties are described. Also, the pre- processing and processing of the data are specified. At the end the actual workflow of pulsar timing is explained.

7 2.1. Observations The observations for this thesis were used from the radio telescopes in Nan¸cay, West- erbork, Effelsberg and Jodrell Bank. The specific observation properties are shown in Tab. 1 and the observed timespan for each telescope is illustrated in Fig. 6.

Telescope (Backend) fc [MHz] BW [MHz] # TOAs Westerbork (PuMa2) 346 70 23 Nan¸cay (NUPPI) 1484 512 177 Jodrell Bank (Roach) 1520 384 83 Effelsberg (Asterix) 1347 200 16 1397 400 4

Tab. 1: The telescope recording system, centre frequency fc, bandwidth BW and the number of TOAs used in our analyis.

1 6 0 0

1 4 0 0 ] z

H 1 2 0 0

M W e s t e r b o r k ( 3 4 6 M H z ) [ E f f e l s b e r g ( 1 3 4 7 M H z ) y

c 1 0 0 0 E f f e l s b e r g ( 1 3 9 7 M H z ) n

e N a n c a y ( 1 4 8 4 M H z ) u

q J o d r e l l B a n k ( 1 5 2 0 M H z ) e

r 8 0 0 f

e r t n

e 6 0 0 C

4 0 0

2 0 0 2 0 1 3 2 0 1 5 2 0 1 7 2 0 1 9 Y e a r s

Fig. 6: Observation timespan of each observing system

The radio telescope in Nan¸cay in France consists of two parts. The main dish is an adjustable plane mirror of 200 m × 40 m which reflects the radio waves towards the smaller spherical mirror of 300 m × 35 m, which concentrates the waves in the focal

8 point from where the data are send to several receivers. The backend used for the data in this thesis is called NUPPI. The telescope was build in 1965 and is the fourth largest telescope of the world. It receives radio waves from 1.06 to 3.5 GHz (NRT, 2019). In Fig. 8 one can see the movable receiver in front of the primary mirror. The radio telescope in Effelsberg in Germany is a fully steerable 100 m dish, build in 1972. It is the second largest fully steerable telescope after Green Bank in the US. The frequency range in which the telescope operates is between 0.395 to 95.5 GHz. The backend called ASTERIX is based on the ROACH receiver from Jodrell Bank (EPTA, 2014a). The telescope can be seen in Fig. 9. The observatory of Jodrell Bank in the UK contains the fully steerable 76.2 m Lovell Telescope, which was build in 1957. The backend used for the data of this thesis is called ROACH which processes the observations with a bandwidth of 384 MHz and mostly with a frequency of 1.4 GHz (EPTA, 2014b). The Westerbork Synthesis Radio Telescope in the Netherlands is an array, which consists of 14 fully movable parabolic dishes with a diameter of 25 m (ASTRON, 2019). It receives radio waves from 0.12 to 8.3 GHz. The backend used for the data of this thesis is called PuMa2 (EPTA, 2014c). One can see a part of the array in Fig. 10. It should be mentioned that the time of the observations of the Lovell Telescope are much shorter than observations of the other telescopes. The mean time of observations of the Lovell Telescope is between 500 and 1800 s, while the observations of the other telescopes could be up to 8000 s long. The data of that four telescopes were taken as part of the European Pulsar Timing Array (EPTA), which is a collaboration of European research institutes, established in 2006. Another member is the Sardinia Radio Telescope in Italy, which is currently being commissioned so there were no data of this telescope used in this thesis (Desvignes et al., 2016). The provided data of these instruments were coherent dedispersed, so the disturbing effect of interstellar dispersion has already been removed. That effect can be seen as small gaps between the frequency channels in Fig. 7.

9 Fig. 7: Dispersed archive of the added data, which was split into four parts.

Fig. 8: The movable receiver in front of the primary mirror of the Nan¸cay radio telescope. (Wikimedia, 2019)

10 Fig. 9: The fully steerable 100 m single-dish-telescope in Effelsberg. (EPTA, 2014a)

Fig. 10: The view along the dishes of the Westerbork Radio Synthesis Telescope. (EPTA, 2014c)

2.2. Pre-processing The data were pre-processed using the PSRCHIVE software package (van Straten, 2019). First the data were scrunched in polarisation using the command pam. After that the radio frequency interference (RFI) was removed using pazi. RFI is a man-made product, for example emitted by FM radio stations. This noise has to be cut out to get a clearer signal. The data were cleaned by hand, so every single archive was checked by eye. If there was RFI present, the relevant frequency channel or sub- integration was cut out by weighting it with 0, so it was not deleted but it does not appear in the used data. An example of this pre-processing step is illustrated in Fig. 11, where one can see the plots before the RFI removal and afterwards.

11 Fig. 11: Comparison of an archive before and after removing RFI. The integrated pulse profile is seen as a function of pulsephase in the top row. The middle row shows intensity as a function of time and phase, where two sub-integrations were cut out. The bottom row shows intensity as a function of observing frequency and pulse phase, where one can see that the top frequency band was cut out.

The data were then scrunched in time, polarisation and frequency to get smaller files to work with. Except the data from Nan¸cay, which were scrunched to blocks of 300 s to get multiple TOAs per observation. It was scrunched in polarisation, but only partially scrunched in frequency because of Scintillation, which will be explained in the following section 2.3. This step only have to be considered with the data from Nan¸cay because it

12 is the most sensitive data. After getting initial pulsar timing results, an updated parameter file was installed in the Effelsberg data to update the ephemeris and improve the TOAs, because the par-file in the Effelsberg data were too bad and the signal got smeared.

2.3. Processing the data To create a standard template for creating TOAs, the cleaned data were sorted by S/N to only use data that contribute significant signal. The cut-off values were S/Nmin > 26 for Effelsberg, S/Nmin > 18 for Nan¸cay and S/Nmin > 11 for Westerbork. The data were split up in groups of the same frequency and bandwidth so the data could be added together using the command psradd. The data were neglected if there were not enough data points left of the same frequency. To add up the data, the archives were aligned in phase during the adding. This results in only one file containing all the information of all archives and a high S/N. The provided observations of Jodrell Bank already have been processed and only the tim-file, which will be introduced in the next section 2.4, has been used for this thesis. For Effelsberg and Westerbork the observations have to be pre-processed as described in the previous section 2.2, but they were already coherent dedispersed as mentioned in section 2.1. The Nan¸cay data provide a large bandwidth and high sensitivity, which is susceptible to Scintillation. Therefore the data were split up into four parts with 32 channels for each part, using the command psrsplit, to time the data parts separately. For each frequency band an individual template was created. The pulse profiles of two of these frequency bands were over-plotted to verify, if there are differences. As seen in Fig. 12 the pulse shapes of these two parts have slight differences on the left edge of the peak, which become clearer in Fig. 13 which shows the difference of the two pulse profiles.

13 1 2 9 5 M H z ( C e n t r e F r e q u e n c y ) 1 6 8 2 M H z ( C e n t r e F r e q u e n c y )

1 . 0 y t i s n e t n I 0 . 5

0 . 0 8 5 0 9 0 0 9 5 0 1 0 0 0 1 0 5 0 1 1 0 0 b i n s

Fig. 12: The pulse profiles of two frequency bands over-plotted, with intensity in arbitrary units. Furthermore, in the leading edge of the profile a clear evolution in shape can be recognized.

0 . 1 y t i 1 σ s n

e 0 . 0 t n I x

- 0 . 1 5 0 0 1 0 0 0 1 5 0 0 b i n s

Fig. 13: Difference between the pulse profiles of the two frequency bands shown in Fig 12, with intensity expressed as a multiple of the peak pulse intensity. The green line represents the mean value and the blue line represents the standard deviation (1σ).

14 After preparing the data one has to create the actual standard template which will be cross-correlated with the data. The command paas creates that standard template by fitting von Mises functions to the added data.

2.4. Pulsar timing The basic idea of pulsar timing was described in Section 1.4, whereas this section fo- cusses more on the exact description of how the software was used. By cross-correlating the data with the standard template one creates the so-called tim- file which contains the TOAs. This file was created with the command pat. The observations from Westerbork had a time offset which was fixed with the help of a time jump of +10 s that was added to the affected TOAs between MJD 56808 and MJD 56903. TEMPO2 uses this tim-file and an initial parameter-file to fit and creates a param- eter-file which contains all fit parameters. The fit parameters for PSR J0636+5128 are shown in Tab. 3. After fitting for the orbital parameters to minimize the χ2 its value is set to 1 with T2EFAC. This rescales the TOA uncertainties for each backend multiplying all TOA uncertainties with the square root of the χ2, calculated for the individual backend (see Tab. 2).

Telescope red. χ2 Westerbork 4.57 Nan¸cay 1.39 Jodrell Bank 1.26 Effelsberg 8.28

Tab. 2: Reduced χ2 values for each telescope used to rescale the TOA uncertainties with T2EFAC.

15 3. Results for PSR J0636+5128

Since PSR J0636+5128 was discovered in 2014 (Stovall et al., 2014), no complete timing solution has been published. Also there are two papers about PSR J0636+5128 which used the same orbital period of Pb = 96 minutes (Draghis and Romani, 2018; Kaplan et al., 2018) and also the same DM = 11.1 cm−3pc (Draghis and Romani, 2018). The other parameters, derived by pulsar timing with Tempo2, shown in Tab. 3 have not been determined by somebody else before. The timing residuals are shown in Fig. 14. They were taken from TEMPO2 1. The different colours show the timing residuals of the respective telescopes.

E f f e l s b e r g N a n c a y

- 5 W e s t e r b o r k 3 x 1 0 J o d r e l l B a n k

2 x 1 0 - 5 )

s - 5 (

1 x 1 0 s l a u d i 0 s e R

- 1 x 1 0 - 5

- 2 x 1 0 - 5

5 6 5 0 0 5 7 0 0 0 5 7 5 0 0 5 8 0 0 0 5 8 5 0 0 M J D

Fig. 14: Timing residuals plotted against MJD. The different colours show the residuals of the respective telescopes. The errorbars are rescaled with T2EFAC. The residuals were taken from TEMPO2.

1https://www.atnf.csiro.au/research/pulsar/tempo2/; see Hobbs et al. (2006)

16 Fit and data-set Pulsar name ...... J0636+5128 MJD range ...... 56655.2—58506.1 Data span (yr)...... 5.07 Number of TOAs...... 302 Rms timing residual (µs)...... 1.0 Weighted fit...... Y Reduced χ2 value ...... 1.0 Measured Quantities Right ascension, α (hh:mm:ss) ...... 06:36:04.846185(15) Declination, δ (dd:mm:ss) ...... +51:28:59.9655(4) Pulse frequency, ν (s−1) ...... 348.559226341734(4) First derivative of pulse frequency,ν ˙ (s−2)...... −4.1896(3)×10−16 Dispersion measure, DM (cm−3pc) ...... 11.107(5) −1 Proper motion in right ascension, µα cos δ (mas yr ) ...... 3.21(4) −1 Proper motion in declination, µδ (mas yr )...... −1.76(8) Orbital period, Pb (d) ...... 0.06655133902(9) Projected semi-major axis of orbit, x (lt-s) ...... 0.00898612(10) ˙ −12 First derivative of orbital period, Pb ...... 2.21(7)×10 TASC (MJD) ...... 56027.2485423(9) EPS1...... 1.1(23)×10−5 EPS2...... −4.1(21)×10−5 Set Quantities Reference epoch of frequency, position and DM determination (MJD) 56000 Derived Quantities

log10(Characteristic age, yr) ...... 10.12 log10(Surface magnetic field strength, G) ...... 8.00 log10(Edot, ergs/s) ...... 33.76 Assumptions Clock correction procedure ...... TT(BIPM2011) Solar system ephemeris model...... DE421 Binary model...... T2 Model version number ...... 5.00

Tab. 3: Timing-Model parameters for PSR J0636+5128

3.1. Orbital variations of PSR J0636+5128

This section is about the observed secular variations during the timespan of nearly five years, based on the analysis of Shaifullah et al. (2016). To study these, the timespan was divided into periods of 50 days. For each period the data were fitted for either TASC or x1 while all other values in the timing model were kept constant. For the change of TASC and x1 the difference between the fitted value and the reference value of

17 the parameter-file, seen in Tab. 3, were calculated. To analyse the change of Pb the following equation, taken from Shaifullah et al. (2016), was used:

TASC,1 − TASC,0 ∆Pb = × Pb,ref (1) t1 − t0

where TASC,0 is the time of ascending node at time t0, TASC,1 is the time of ascending node at time t1 and Pb,ref a reference value of the orbital period used from the timing solution. During that fit process, periods were left out if there were not enough data in these periods. Furthermore, single data points were deleted if their residual differed too much from the other data points. An example of that is shown in Fig. 15, where the deleted point is marked with a red circle. The fits were made with TEMPO2.

6 . 0 x 1 0 - 2

4 . 0 x 1 0 - 2 ) s µ

( - 2

l 2 . 0 x 1 0 a u d i s e

R 0 . 0

- 2 . 0 x 1 0 - 2

5 8 1 0 0 5 8 1 1 0 5 8 1 2 0 5 8 1 3 0 5 8 1 4 0 M J D

Fig. 15: Example of outlier rejection: The data point circled in red is a statistical anomaly that cannot be explained by any other effect in the timing model. Instrumental failure or leftover RFI are suspected. Data points like this one were omitted from the analysis.

18 The aim of this analysis is to check the stability of PSR J0636+5128 over time, because BWPs are known as unstable binaries. That means that it is not possible to create a timing solution over several years, instead the solution is only valid for short time periods (see Shaifullah et al., 2016; and Lazaridis et al., 2011). The observed parameter Pb is the orbital period of the binary system. The parameter x1 is the projected semi-major axis of the orbit, for which the following expression applies: x1 = a × sin(i), where a is the semi-major axis and i is the inclination angle. The last parameter TASC is the epoch of the ascending node.

3.1.1. Analysis

The resulting fit for the variations of Pb is shown in Fig. 16. The values fluctuate symmetrically at the beginning and at the end. Between MJD ∼ 57750 and ∼ 58250 the plot looks much more flat. There were three big outliers with big uncertainties which mostly contain data from Effelsberg with incomplete orbital coverage. These outliers were removed to get a clearer plot, but they were consistent with the rest of the data. Therefore, the periods were manipulated a bit. To obtain a higher resolution, the timespan of the period was chosen smaller around each of the removed outliers.

2 x 1 0 - 8

1 x 1 0 - 8

0 b P ∆ - 1 x 1 0 - 8

- 2 x 1 0 - 8

- 3 x 1 0 - 8

5 6 5 0 0 5 7 0 0 0 5 7 5 0 0 5 8 0 0 0 5 8 5 0 0 M J D

Fig. 16: Pb as a function of time. There are symmetric fluctuations at the beginning and at the end. The plot looks more flat between MJD ∼ 57750 and ∼ 58250.

19 Unlike the change of Pb, the variations of TASC , see Fig. 17, show more fluctuations. There were two outliers with big error bars which mostly contain data from Effelsberg, too. These outliers were left out in the plot to get a clearer plot, but they were consistent with the rest of the data. For this plot, the periods were manipulated a bit. Around each of the removed outliers the timespan of the period was chosen smaller to obtain a higher resolution. As seen in Fig. 17 the plot looks flat, but there are two slight peaks at MJD ≈ 57000 and one at MJD ≈ 58250 which do not have high uncertainties.

2 . 5 x 1 0 - 5

2 . 0 x 1 0 - 5

1 . 5 x 1 0 - 5

1 . 0 x 1 0 - 5 c s A 5 . 0 x 1 0 - 6 T ∆ 0 . 0

- 5 . 0 x 1 0 - 6

- 1 . 0 x 1 0 - 5

- 1 . 5 x 1 0 - 5

5 6 5 0 0 5 7 0 0 0 5 7 5 0 0 5 8 0 0 0 5 8 5 0 0 M J D

Fig. 17: TASC as a function of time. One can see that the plot looks mostly flat, but there are two slight peaks, one at MJD ≈ 57000 and one at MJD ≈ 58250. The distribution is noticeably asymmetric. All of the data points are positive, even the outliers.

20 The values of x1, see Fig. 18, have more fluctuations, too. Just like for TASC , most of the data points with big uncertainties contain mostly data from Effelsberg. Three small peaks can be recognized at MJD ≈ 56750, between MJD ≈ 57500 − 57750 and at the end at MJD ≈ 58250.

6 x 1 0 - 5

5 x 1 0 - 5

4 x 1 0 - 5

3 x 1 0 - 5 1 x - 5 ∆ 2 x 1 0

1 x 1 0 - 5

0

- 1 x 1 0 - 5

5 6 5 0 0 5 7 0 0 0 5 7 5 0 0 5 8 0 0 0 5 8 5 0 0 M J D

Fig. 18: x1 as a function of time. One can see that there are some significant fluctuations. At the beginning of the observations at MJD ≈ 56750, between MJD ≈ 57500 − 57750 and at the end at MJD ≈ 58250 one can see small peaks. It should be mentioned that the distribution is asymmetric. All data points are positive, even the outliers.

With respect to the uncertainties, the values of x1 and TASC look relatively stable. Although, there are small fluctuations of x1 with small error bars and similar fluctuations for TASC . The origin of these fluctuations cannot be explained. The change of Pb fluctuates symmetrically at the beginning and at the end, but it seems much more flat in the middle of the plot. This BWP seems to be stable with respect to other BWPs, for example PSR J2051−0827. Future observations will show if the fluctuations will continue.

21 4. Pulsar timing and observations of PSR J2051−0827

This section is about the observed secular variations of PSR J2051−0827, an extension to the work of Shaifullah et al. (2016). To extend his analysis we used the latest data of Shaifullah et al. (2016) and added new data from Nan¸cay.

4.1. Observations In this chapter we present the timing of PSR J2051−0827, which contains new data from Nan¸cay (NUPPI) with 462 TOAs during a MJD range from 56698 to 58663. The centre frequency is fc = 1484 MHz and the bandwidth is 512 MHz. The system shows an eclipse, which has been removed from the dataset due to low or no signal in the eclipse region. The remaining dataset contains data in a MJD range from 56698 to 58542.

4.2. Processing the data and pulsar timing The pre-processing was similar to the procedure described in section 2.2. The data were scrunched in frequency and polarisation. The cut-off value for the S/N used for the standard profile is S/Nmin > 19. The data were grouped in 5 minute sub-integrations to have multiple TOAs per observation and then added together and phase aligned, using psradd. Afterwards, a standard template was created by fitting von Mises functions to the data with paas. Then the TIM-file was created by cross-correlating the data and the standard template with PAT. The timing solution was found with tempo2, where the χ2 got minimized. With T2EFAC the errorbars were rescaled and the χ2 set to 1 due multiplying the TOAs with χ. The reduced χ2 values of each backend can be seen in Tab. 4.

Telescope red. χ2 Effelsberg (Shaifullah et al., 2016) 4.44 Jodrell Bank (Shaifullah et al., 2016) 1.54 Nan¸cay (Shaifullah et al., 2016) 1.75 Nan¸cay (New data) 3.01

Tab. 4: Reduced χ2 values for each telescope used to rescale the errorbars with T2EFAC.

5. Results for PSR J2051−0827

As the origin of this analysis is from Shaifullah et al. (2016), the latest data from the paper is used to find a good timing solution. This timing solution contains data of Shai- fullah et al. (2016) during a MJD range from 56698 till 56729 and the new data during a MJD range from 56698 till 58663. As the new data are only data from Nan¸cay, the

22 data of Shaifullah et al. (2016) contain data from Nan¸cay, Jodrell Bank and Effelsberg. The fit results are shown in Tab. 5 and the timing residuals can be seen in Fig. 19.

Fit and data-set Pulsar name...... J2051−0827 MJD range ...... 56678.6—58747.3 Data span (yr)...... 5.66 Number of TOAs...... 601 Rms timing residual (µs)...... 5.1 Weighted fit...... Y Reduced χ2 value ...... 1.0 Measured Quantities Right ascension, α (hh:mm:ss) ...... 20:51:07.52060(6) Declination, δ (dd:mm:ss)...... −08:27:37.744(3) Pulse frequency, ν (s−1) ...... 221.796283594380(8) First derivative of pulse frequency,ν ˙ (s−2)...... −6.2560(11)×10−16 Dispersion measure, DM (cm−3pc)...... 20.710(10) −1 Proper motion in right ascension, µα cos δ (mas yr ) 4.7(4) −1 Proper motion in declination, µδ (mas yr )...... 1.6(12) Orbital period, Pb (d) ...... 0.0991102217(6) Projected semi-major axis of orbit, x (lt-s)...... 0.045076(3) ˙ −11 First derivative of orbital period, Pb ...... 1.123(16)×10 First derivative of x,x ˙ (10−12)...... −8.0(94)×10−15 TASC (MJD) ...... 54091.034792(9) EPS1...... 8.7(16)×10−5 EPS2...... −2.9(14)×10−5 Set Quantities Epoch of frequency determination (MJD) ...... 56738 Epoch of position determination and DM (MJD) . . . 56308 Derived Quantities

log10(Characteristic age, yr) ...... 9.75 log10(Surface magnetic field strength, G) ...... 8.38 log10(Edot, ergs/s) ...... 33.74 Assumptions Clock correction procedure ...... TT(TAI) Solar system ephemeris model ...... DE421 Binary model...... T2 Model version number ...... 5.00

Tab. 5: Parameters for PSR J2051−0827

23 N a n c a y ( S h a i f u l l a h , 2 0 1 6 ) J o d r e l l B a n k ( S h a i f u l l a h , 2 0 1 6 )

- 5 E f f e l s b e r g ( S h a i f u l l a h , 2 0 1 6 ) 6 . 0 x 1 0 N a n c a y ( N e w D a t a )

4 . 0 x 1 0 - 5

2 . 0 x 1 0 - 5 ) s (

s l 0 . 0 a u d i

s - 5 e - 2 . 0 x 1 0 R

- 4 . 0 x 1 0 - 5

- 6 . 0 x 1 0 - 5

- 8 . 0 x 1 0 - 5 5 6 5 0 0 5 7 0 0 0 5 7 5 0 0 5 8 0 0 0 5 8 5 0 0 M J D

Fig. 19: Timing residuals of data of Shaifullah et al. (2016) and the new data from Nan¸cay plotted against MJD. The different colours show the residuals of the respective tele- scopes. The data of Shaifullah et al. (2016) contain data from Nan¸cay, Jodrell Bank and Effelsberg, while the new data are only from Nan¸cay. The residuals were taken from TEMPO2.

5.1. Orbital variations of PSR J2051−0827 The analysis of the orbital variations of the BWP J2051−0827 are similar to the analy- sis described in section 3.1. To show the change of the orbital parameters over a longer timespan, the original data were taken from Shaifullah et al. (2016). It should be men- tioned that the data have not been exactly reproduced, there is a small offset in the data, but it is sufficient to recognize the evolution of the change of the orbital parame- ters. This original data were extended by new data from Nan¸cay. To compare the change of the orbital parameters of this thesis with the results of Shai- fullah et al. (2016), the same period of 365 days with an overlap of 30 days, which was also used by Shaifullah et al. (2016), was chosen here. Each data point represents the mean MJD of the period, while the x-errorbars represent the timespan over which the measurement was made. The periods are overlapping within 30 days. For each period the data were fitted for TASC and x1 while the other values were kept constant. The variations in Pb were calculated with equation 1. For the variations of TASC and x1 the difference between the fitted value and the reference value, taken from the timing solution of Shaifullah et al. (2016), were calculated.

24 5.1.1. Analysis

The resulting fit for the variations of TASC is shown in Fig. 20. One can see that at first the variations of TASC are high and are getting smaller with time. The variations of TASC for the new data seem to be small. Maybe the pulsar has entered a more quiet phase.

D a t a f r o m S h a i f u l l a h e t a l . ( 2 0 1 6 ) 8 . 0 x 1 0 - 5 N e w D a t a

6 . 0 x 1 0 - 5

4 . 0 x 1 0 - 5 C S A

T - 5

∆ 2 . 0 x 1 0

0 . 0

- 2 . 0 x 1 0 - 5

5 0 0 0 0 5 2 0 0 0 5 4 0 0 0 5 6 0 0 0 5 8 0 0 0 M J D

Fig. 20: TASC as a function of time. One can see that it looks quiet, compared with the plot in Fig. 5 in Shaifullah et al. (2016). The eras are 365 days long with an overlap of 30 days.

25 In comparison with the plot seen in Fig. 5 of Shaifullah et al. (2016) the variations in Pb look much smaller. There are only small fluctuations in the new observed timespan. The resulting fit is shown in Fig. 21.

D a t a f r o m S h a i f u l l a h e t a l . ( 2 0 1 6 ) N e w D a t a

1 . 0 x 1 0 - 8

5 . 0 x 1 0 - 9 b

P 0 . 0 ∆

- 5 . 0 x 1 0 - 9

- 1 . 0 x 1 0 - 8

5 0 0 0 0 5 2 0 0 0 5 4 0 0 0 5 6 0 0 0 5 8 0 0 0 M J D

Fig. 21: Pb as a function of time. One can see that it looks more quiet than the plot in Fig. 5 of Shaifullah et al. (2016). Each era contains 365 days with an overlap of 30 days.

26 Like the variations of TASC , the values of x1, see Fig. 22, in the new observed timespan also look much more stable, but there is one outlier with large error bars.

D a t a f r o m S h a i f u l l a h e t a l . ( 2 0 1 6 ) 4 . 0 x 1 0 - 5 N e w D a t a

2 . 0 x 1 0 - 5

0 . 0 1 x ∆ - 2 . 0 x 1 0 - 5

- 4 . 0 x 1 0 - 5

- 6 . 0 x 1 0 - 5

5 0 0 0 0 5 2 0 0 0 5 4 0 0 0 5 6 0 0 0 5 8 0 0 0 M J D

Fig. 22: x1 as a function of time. One can see that it looks stable with one outlier which has large error bars. The eras are 365 days long with an overlap of 30 days.

Overall one can see that the orbital variations of PSR J2051−0827 are on a smaller scale. The values of TASC and x1 seem to get more stable. The variations of Pb are smaller during the current observation time but there are still small fluctuations. Maybe future observations will show that this BWP continues to be in a less unsteady phase.

27 6. Calculated vs. measured P˙b for different companion types

In Section 3 the fitted value of the orbital-period derivative was mentioned. We checked if the fitted value agrees with the theoretically predicted value. For further investiga- tion the following sections discuss the comparison between the calculated and measured values with different types of companion. The calculated value is the sum of four contri- butions. The contributions considered in this analysis are the emission of gravitational waves (GW), the proper motion, the differential Galactic rotation and the acceleration perpendicular to the Galactic plane. The different companion types that are studied are binary systems where the second star is a helium WD, a carbon-oxygen WD, a main- sequence star and an ultra-light companion. Also included are binary systems with double neutron stars. The BWPs and RBPs fall into the ultra-light and main sequence star companion categories.

6.1. Formulae for calculations of P˙b contributions The formula for the emission of GWs is given by Weisberg and Huang (2016):

−192 × π × T 5/3 × 1 + 73 e2 + 37 e4 × m × m ˙ GW 24 96 p c Pb = , (2) 5/3 q √ Pb  2 7 3 5 × 2π × (1 − e ) × mp + mc

−6 with T = 4.9257929 × 10 s, the pulsar mass mp and the companion mass mc, both given in kilograms. Also e is the eccentricity and Pb is the orbital period in seconds. ˙ For the Pb contribution due to proper motion, we use (Nice and Taylor, 1995):

2 µ µ d P˙ = P × (3) b b c where µ is the proper motion in rad/sec, d the distance in meters, Pb is the orbital period in seconds and c = 3 × 108 m/s the speed of light. The contribution of Galactic differential rotation is given by Nice and Taylor (1995):

  2 d DGR Θ cos(b) − cos(l) P˙ = −cos(b) 0 cos(l) + R0  × P (4) b R × c   2  b 0 sin2(l) + d cos(b) − cos(l) R0

21 where b is the Galactic latitude, l is the Galactic longitude, R0 = 2.6±0.19×10 m (Reid 3 et al., 2009) is the distance of the Sun to the Galactic centre, Θ0 = 254 ± 16 × 10 m/s (Reid et al., 2009) is the rotation velocity of the Galaxy and Pb is the orbital period in seconds.

28 The formula for the contribution of acceleration perpendicular to the Galactic plane is from Nice and Taylor (1995):

  ˙ aGP az −19 1.25z Pb = sin(b) = 1.08 × 10 √ + 0.58z sin(b) × Pb (5) c z2 + 0.0324

where b is the Galactic latitude, Pb the orbital period in seconds and z = d × sin(b) is the distance above or below the Galactic plane in meters.

6.2. Data used

All the data used in this thesis are from the ATNF Pulsar Catalogue 2 (Manchester et al., 2005). The calculations were made with a self-written script (see Appendix A). Those script uses the sources and values from the catalogue. The pulsar and ˙ the companion masses were chosen from the paper that is quoted at the Pb-value in the catalogue. If there were no entries for used values, the values were taken from the same ˙ paper which provides the Pb-value. If there were no uncertainties given in the catalogue or the referenced paper, the uncertainty was taken as 0. For the distances, the newest measurements and newest electron density model were used. For most of the pulsars in the following sections, the distances were from Yao et al. (2017). For the pulsars marked with an asterisk the distance was taken from Verbiest et al. (2012). The distance for the pulsars of globular cluster 47 Tuc was taken from Freire et al. (2017).

6.3. Double neutron stars

First we discuss the double neutron stars, which are fairly well understood. The cal- culated and measured values are shown in Tab. 6 and the plot of these data is shown in Fig. 23. As can be seen in Fig. 23, the calculated values and the measured val- ues are highly consistent for all of these pulsars, which might be due to the fact that the measurements for these pulsars are highly reliable. Only PSRs J1518+4904 and J1756−2251 have an imprecise companion mass, which results in a large uncertainty for the orbital-period derivative.

2http://www.atnf.csiro.au/research/pulsar/psrcat; (Catalog version: 1.60)

29 M e a s u r e d V a l u e D o u b l e n e u t r o n s t a r b i n a r i e s C a l c u l a t e d V a l u e 2 . 0 x 1 0 - 1 2

0 . 0

- 2 . 0 x 1 0 - 1 2 b P

- 4 . 0 x 1 0 - 1 2

- 6 . 0 x 1 0 - 1 2

- 8 . 0 x 1 0 - 1 2 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5

m c

Fig. 23: The P˙b calculations from this thesis compared with the measured values from litera- ture, plotted against companion mass for the double neutron-star binaries.

˙ ˙ ˙ Pulsar JName mc [M ] Calculated Pb Measured Pb ∆Pb J0737−3039A* 1.2489(7) −1.25(17) × 10−12 −1.25(2) × 10−12 4(173) × 10−15 J0737−3039B* 1.3381(7) −1.25(17) × 10−12 −1.25(2) × 10−12 4(173) × 10−15 +1.21 +0.011 −13 −13 −14 J1518+4904 1.05−0.14 2.120−0.007 × 10 2.4(22) × 10 3(22) × 10 J1537+1155* 1.35(5) −1.30(9) × 10−13 −1.366(3) × 10−13 7(9) × 10−15 J1756−2251 1.2(6) −2.1(16) × 10−13 −2.29(5) × 10−13 2(16) × 10−14 J1757−1854 1.3946(9) −5.3(4) × 10−12 −5.3(2) × 10−12 2(45) × 10−14 J1906+0746 1.322(11) −5.7(11) × 10−13 −5.6(3) × 10−13 9(110) × 10−15 J1915+1606 1.3886(2) −2.42(7) × 10−12 −2.42(1) × 10−12 7(7) × 10−15 J2129+1210C 1.35(1) −3.92(19) × 10−12 −3.96(50) × 10−12 4(20) × 10−14

Tab. 6: Calculated and measured values of P˙b, the difference of both values and the associated companion mass mc for the double neutron-star binaries. For the pulsars marked with an asterisk the distance were taken from Verbiest et al. (2012), for all other pulsars distances were derived from the Yao et al. (2017) electron density model. Numbers in brackets denote the 1 − σ uncertainty in the last digit quoted.

30 6.4. Ultra-light companions In this section we look at the binaries with ultra-light companions, which also contain some known BWPs (marked with †). Ultra-light companions are stars which have very low masses of mc . 0.08M . The plot of the orbital-period derivatives is shown in Fig. 24, which shows that the calculated values typically do not agree with the measured values. The deviations become even clearer if one looks at the values in Tab. 7, where the measured and calculated values, as well as the companion masses, are shown. Most of the deviations are two orders of magnitude and there are many sign errors. So the ˙ assumption that PSR J0636+5128 is the only pulsar with an unexplained Pb is not true, ˙ as all considered pulsars with ultra-light companions have unexplained Pb values.

M e a s u r e d V a l u e U l t r a - l i g h t c o m p a n i o n s C a l c u l a t e d V a l u e

1 . 2 x 1 0 - 1 0

8 . 0 x 1 0 - 1 1

4 . 0 x 1 0 - 1 1 b P

0 . 0

- 4 . 0 x 1 0 - 1 1

- 8 . 0 x 1 0 - 1 1 0 . 0 0 0 . 0 5

m c

Fig. 24: The P˙b calculations compared to the measured values plotted against companion mass for binaries with ultra-light companions.

31 ˙ ˙ ˙ Pulsar JName mc [M ] Calculated Pb Measured Pb ∆Pb J0023+0923† 0.05 −2.8(4) × 10−14 2.8(2) × 10−12 3(20) × 10−13 J0024−7204I 0.0153 −3.0(24) × 10−15 −8.0(2) × 10−13 8(2) × 10−13 J0024−7204R 0.0306 −7.4(19) × 10−14 1.9(4) × 10−13 2.6(4) × 10−13 J0636+5128† 0.05 −1.2(2) × 10−13 2.5(3) × 10−12 2.6(3) × 10−12 J1959+2048† 0.025 1.21(2) × 10−13 1.47(8) × 10−11 1.46(8) × 10−11 J2051−0827† 0.06 −7.4(9) × 10−14 −5.9(3) × 10−12 5.8(3) × 10−12 J2115+5448† 0.02 −1.8(95) × 10−14 6(2) × 10−11 6(2) × 10−11 J2322−2650 0.0007588(2) 8.5(7) × 10−15 6(6) × 10−11 6(6) × 10−11

Tab. 7: Calculated and measured values of P˙b, the difference of both values and the associated † companion mass mc for the ultra-light companions. Pulsars marked with a are known BWPs (Arzoumanian et al., 2018; Guillemot et al., 2019; Roberts, 2012). PSR J0636+5128 is not yet confirmed to be a BWP.

6.5. Helium white dwarf companions In this section we go through the binaries with helium WD companions which are divided into two groups; those that are in a globular cluster and those that are in the Galactic field. The orbital-period derivative comparison of those that are in the Galactic field is shown in Tab. 8 and the corresponding plot of it can be seen in Fig. 25. As one can see, the calculated values do not agree with the measured values, but the values are of the same order of magnitude. Some of the calculated values fit with the measured values, because of their big uncertainties and most of the calculated values do not deviate that much from the measured values. They are better explained than the ultra-light companion binaries but not as well as the double neutron stars. For the pulsars in globular clusters, on the other hand, the calculated orbital-period derivative deviates much more from the measured value. The calculated and measured values are shown in Tab. 9 and the plot can be seen in Fig. 26. These values deviate much more than those that are not in a globular cluster and many of them have sign errors. The deviations are between one and three orders of magnitude. These deviations could be explained by the chaotic conditions that prevail in globular clusters. The distances between the pulsars and their stars are very short and the orbital parameters of the binaries can be heavily affected. As explained in Section 1.3 for binaries in globular clusters, companions can be replaced by other stars. Another more important reason is the gravity of the cluster which causes an acceleration and ˙ therefore a change of Pb. That could be the reason why none of the orbital-period derivatives of the considered pulsars, with respect to the uncertainties, are explained.

32 M e a s u r e d V a l u e H e l i u m W D i n G a l a c t i c f i e l d C a l c u l a t e d V a l u e 1 x 1 0 - 1 1

0

- 1 x 1 0 - 1 1 b P

- 2 x 1 0 - 1 1

- 3 x 1 0 - 1 1

- 4 x 1 0 - 1 1 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4

m c

Fig. 25: The P˙b calculations compared with the measured values with respect to the compan- ion mass for the helium WD companions that are in the Galactic field.

˙ ˙ ˙ Pulsar JName mc [M ] Calculated Pb Measured Pb ∆Pb J0348+0432 0.172(3) −2.55(8) × 10−13 −2.73(5) × 10−12 2.47(5) × 10−12 J0437−4715* 0.224(7) 3.75(3) × 10−12 3.728(6) × 10−12 2(3) × 10−14 J0613−0200* 0.12 2.9(12) × 10−14 5.4(18) × 10−14 3(2) × 10−14 J0751+1807* 0.12 −2.2(4) × 10−14 −3.5(3) × 10−14 1.3(5) × 10−14 J1012+5307* 0.16(2) 6.3(20) × 10−14 6.1(4) × 10−14 2(21) × 10−15 +0.141 −14 −14 −15 J1738+0333 0.181(7) −1.523−0.122 × 10 −1.7(3) × 10 2(3) × 10 J1909−37443 0.2067(19) 5.22(5) × 10−13 5.03(6) × 10−13 1.9(8) × 10−14 J2019+2425 4 0.35 9.0(0) × 10−12 −3.0(6) × 10−11 3.9(6) × 10−11 J2129−5721 5 0.14 9.7(40) × 10−13 7.9(36) × 10−13 2(5) × 10−13 J2234+0611 0.30(5) 4.4(0) × 10−12 3.1(25) × 10−12 1(3) × 10−12

Tab. 8: Calculated and measured values of P˙b, the difference of both values and the associated companion mass mc for the binaries with helium WD companions that are in the Galactic field. For the pulsars marked with an asterisk the distances were taken from Verbiest et al. (2012) and for those without a symbol the distances were taken from Yao et al. (2017).

3Distance taken from Reardon et al. (2015)

33 With regards to PSR J2129−5721, the distance measurement for that pulsar is very ˙ imprecise, so the Pb calculation is very imprecise, too. The biggest contribution is the proper motion term which scales with ∼ d. This could explain the high uncertainties of the measured value and the calculated value.

M e a s u r e d V a l u e H e l i u m W D i n a c l u s t e r C a l c u l a t e d V a l u e

4 . 0 x 1 0 - 1 2

2 . 0 x 1 0 - 1 2

0 . 0 b P - 2 . 0 x 1 0 - 1 2

- 4 . 0 x 1 0 - 1 2

- 6 . 0 x 1 0 - 1 2

0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0

m c

Fig. 26: The P˙b calculations compared with the measured values plotted against the compan- ion mass for binaries with helium WD companions that are in a globular cluster.

4Distance taken from Lorimer et al. (1996) 5Distance taken from Reardon et al. (2015)

34 ˙ ˙ ˙ Pulsar JName mc [M ] Calculated Pb Measured Pb ∆Pb J0024−7204H 0.196 8.9(2) × 10−14 −7(6) × 10−13 8(6) × 10−13 J0024−7204Q 0.212 4.14(4) × 10−14 −1.0(2) × 10−12 1.0(2) × 10−12 J0024−7204S 0.105 3.40(5) × 10−14 −4.9(4) × 10−12 4.9(4) × 10−12 J0024−7204T 0.2 3.8(2) × 10−14 2.5(11) × 10−12 2(1) × 10−12 J0024−7204U 0.146 4.2(90) × 10−16 6.6(5) × 10−13 6.6(5) × 10−13 J0024−7204Y 0.164 5.8(7) × 10−15 −8.2(7) × 10−13 8.3(7) × 10−13 J1518+0204C 0.038 −4.2(6) × 10−14 −9.14(23) × 10−13 8.7(2) × 10−13 J1701−3006B 0.12 −6.7(90) × 10−14 −5.1(6) × 10−12 5.1(6) × 10−12

Tab. 9: Calculated and measured values of P˙b, the difference of both values and the associated companion mass mc for binaries with helium WD companions that are in a globular cluster.

6.6. Main-sequence stars We now move on to main-sequence star companions which also contain some known redbacks (marked with †). The calculated and measured orbital-period derivatives, as well as the companion masses, are given in Tab. 10 and the corresponding plot is shown in Fig. 27. It can be seen that, the deviation between the orbital-period derivatives is between three and six orders of magnitude. The highest deviation was obtained by PSR J0045−7319, for which the companion mass is given with a high uncertainty. In Tab. 10 it is furthermore evident that three out of six orbital-period derivatives have sign errors. This clearly shows that the orbital-period derivatives for the main-sequence star companions are not explained.

˙ ˙ ˙ Pulsar JName mc [M ] Calculated Pb Measured Pb ∆Pb J0045−7319 8.8(18) 9.4(62) × 10−13 −3.03(9) × 10−7 3.03(9) × 10−7 J1023+0038† 0.2 −7.3(43) × 10−14 −7.32(6) × 10−11 7.31(6) × 10−7 J1227−4853† 0.142 1.1(32) × 10−14 −8.7(1) × 10−10 8.7(1) × 10−10 J1302−6350 24(4) 1.3(0) × 10−11 1.4(7) × 10−8 1.4(7) × 10−8 J1723−2837† 0.43 4.6(88) × 10−14 −3.5(1) × 10−9 3.5(1) × 10−9 J1957+2516† 0.12 −3.96(288) × 10−14 1.20(2) × 10−11 1.20(2) × 10−11 J2215+5135† 0.23 −1.3(160) × 10−13 −4.4(0) × 10−10 4.39(2) × 10−10

Tab. 10: Calculated and measured values of P˙b, the difference of both values and the associated companion mass mc for binaries with main-sequence companions. Pulsars marked with † are known RBPs (Johnson et al., 2015; Roberts, 2012) and pulsars marked with a † are known BWPs (Stovall et al., 2016).

35 M e a s u r e d V a l u e M a i n - s e q u e n c e s t a r s C a l c u l a t e d V a l u e 1 x 1 0 - 7

0

0 . 0 - 1 x 1 0 - 7

- 1 . 0 x 1 0 - 9 b P b P - 2 x 1 0 - 7 - 2 . 0 x 1 0 - 9

- 3 . 0 x 1 0 - 9

- 7 - 3 x 1 0 - 4 . 0 x 1 0 - 9 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5

m c

- 4 x 1 0 - 7 0 1 0 2 0 3 0

m c

Fig. 27: The P˙b calculations compared with the measured values compared to the companion mass for the main-sequence stellar companions.

6.7. Carbon-Oxygen white dwarf companions The last type of possible companion stars are the carbon-oxygen WD companions, which we look at in this section. The calculated and measured orbital-period derivatives, as well as the companion masses, are shown in Tab. 11 and the associated plot is shown in Fig. 28. All of the calculated values are of the same order of magnitude and two out of four values agree with the measured values, with respect to the uncertainties. The distance measurements of the two pulsars for which the orbital-period derivatives are not explained, are very inaccurate. Maybe future distance measurements can provide higher accuracy and therefore better-explained orbital-period derivative values.

36 M e a s u r e d V a l u e C a r b o n - O x y g e n W D C a l c u l a t e d V a l u e

4 . 0 x 1 0 - 1 2

3 . 0 x 1 0 - 1 2

2 . 0 x 1 0 - 1 2 b P

1 . 0 x 1 0 - 1 2

0 . 0

- 1 . 0 x 1 0 - 1 2 0 . 0 0 . 5 1 . 0 1 . 5

m c

Fig. 28: The calculated and measured P˙b values with respect to the companion mass for carbon-oxygen WD companions.

˙ ˙ ˙ Pulsar JName mc [M ] Calculated Pb Measured Pb ∆Pb J1141−6545* 1.02(1) −2.5(15) × 10−13 −4.03(25) × 10−13 2(2) × 10−13 J1603−72026 0.346 1.0(9) × 10−13 3.1(2) × 10−13 2(2) × 10−13 J1614−2230 0.493(3) 2.9(11) × 10−12 1.7(2) × 10−12 1(1) × 10−12 J2222−0137 1.293(25) 2.9(12) × 10−13 2.0(9) × 10−13 9(15) × 10−14

Tab. 11: Calculated and measured values of P˙b, the difference of both values and the associated companion mass mc for the carbon-oxygen WD companions. For the pulsar marked with an asterisk the distance was taken from Verbiest et al. (2012), for the two pulsars without superscript the distances were taken from Yao et al. (2017).

6Distance taken from M. Tauris and Savonije (1999)

37 6.8. All binaries Here we briefly summarize the behaviour of the orbital-period derivative of the various types of binaries. A plot that contains the orbital-period derivatives of almost all con- sidered pulsars (the systems with main-sequence star companions are left out to get a clearer plot) is shown in Fig. 29, where the different companion types are marked with different colours and symbols. The PSRs J2115+5448, J2322−2650 and J2019+2425 were removed to get a clearer plot. A slight dependence on the companion mass can be seen, meaning that the orbital-period derivatives of pulsars with a higher compan- ion mass are better constrained than for the systems with a lower companion mass. It could be that the orbital parameters for double neutron stars are just much more accu- rate than for other companion types, because there was more research done on them. ˙ Another advantage of the double neutron stars is that the Pb amplitude is larger so a higher significance is more easily obtained. Future measurements may provide more accurate orbital parameters for the systems with high uncertainties, which would allow more precise calculations.

D o u b l e N S , U l t r a - l i g h t D o u b l e N S ( M e a s . ) - 1 1 2 . 0 x 1 0 H e l i u m W D a n d C O W D D o u b l e N S ( C a l c . ) U l t r a - l i g h t ( M e a s . ) U l t r a - l i g h t ( C a l c . ) 1 . 5 x 1 0 - 1 1 H e l i u m W D ( M e a s . ) H e l i u m W D ( C a l c . ) C O W D ( M e a s . ) - 1 1 1 . 0 x 1 0 C O W D ( C a l c . )

b 5 . 0 x 1 0 - 1 2 P

0 . 0

- 5 . 0 x 1 0 - 1 2

- 1 . 0 x 1 0 - 1 1 0 . 0 1 0 . 1 1

m c

Fig. 29: The P˙b calculations compared with the measured values plotted against the com- panion mass for binaries with double neutron stars, ultra-light companions and he- lium white dwarfs. To get a clearer plot the PSRs J2115+5448, J2322−2650 and J2019+2425 were removed from the plot.

38 6.9. Cluster 47 Tucanae Most of the pulsars with helium WD companions in globular clusters are from the cluster 47 Tucanae. In 47 Tucanae ionized gas was found by checking the correlation between the period derivative and the DM (Freire et al., 2001). An alternative method is to check the correlation between the orbital-period derivative and the DM. In Fig. 30 one can see a reproduction of Fig. 2 of Freire et al. (2001), where the measured DM values are also taken from Freire et al. (2001). The horizontal line and the vertical line represent the mean values. The result is nearly the same, the values for PSR U and H matches the most with the values from Freire et al. (2001). The value for PSR T is imprecise but matches, too. The values for PSR I and Q instead, are inaccurate. The reason could be the chaotic state of the cluster that affects the orbital-period derivative due to acceleration, caused by the gravity of the cluster.

2 4 . 5

Y

I 2 4 . 4

) T c p

3 - S H m

c R (

M U D 2 4 . 3

Q

2 4 . 2 - 6 - 4 - 2 0 2 4 6 ∆ - 1 7 - 1 ( P b / P b ) o b s ( 1 0 s )

Fig. 30: Measured DM plotted against (P˙b/Pb)obs for the pulsars of the globular cluster 47 Tucanae. The DM values were taken from Freire et al. (2001) and the P˙b values were calculated with the script, which was used in Section 6 (see Appendix A). The Pb values were taken from the ATNF Pulsar Catalogue (Catalog version: 1.60).

39 7. Conclusions

The aim of this thesis was to investigate the properties and behaviour of BWPs. BWPs are mostly known as unstable because some of the earliest investigated BWPs show unstable timing solutions. To check if the observation of PSR J0636+5128 provide an unstable timing solution, a timing solution was found using the psrchive-package and TEMPO2. With that timing solution the stability of the orbital period, projected semi-major axis and the epoch of ascending node was analysed. To investigate the assumption if PSR J0636+5128 could be stable, it was checked in Section 3.1.1 if the orbital period of PSR J0636+5128 looks stable for a longer timespan than a few month. The orbital period fluctuates partially, but it also have a more quite phase. There are also small fluctuations and slight peaks in the projected semi-major axis and the epoch of the ascending node. Future observations will show if there are more fluctuations. With new data, the analysis of the BWP J2051-0827 by Shaifullah et al. (2016) could be added by 5 more years of observations. This data allowed the investigation of the change of the orbital period, projected semi-major axis and the epoch of ascending node for a current period of time. While the change of the orbital parameters fluctuated at first, the new observations show that the pulsar is now in a more ”quiet” phase. But even the new data just suggests that the BWP is stable for now. Future observations will show if this ”quiet” phase continues. After measuring the orbital-period derivative for PSR J0636+5128 it was checked if the theoretically predicted value deviated from the measured value. To check if that behaviour is seen for other pulsars, the orbital-period derivatives for binary systems with different companion types were analyzed in Chapter 6. As seen in Section 6.4 the ˙ calculated orbital-period derivative deviates from the measured Pb for all pulsars with ultra-light companions that have published timing solutions. The calculated values also ˙ deviate from the measured Pb for main-sequence star companions. Only the double neutron-star binaries and carbon-oxygen WD companions have explained orbital-period derivatives. For pulsars with a helium WD companion that are in a globular cluster the orbital-period derivative is not explained, but if the system is in the Galactic field the orbital-period derivative is explained. This is due acceleration by the gravitational field ˙ of the globular clusters, which causes a change of Pb. Future observations of the orbital parameters of these PSRs will provide more precise calculations, which may impact the results.

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44 Statutory declaration

I declare that I have written this thesis independently, that I have not used other than the declared sources. I clearly marked and separately listed all of the literature and all of the other sources which I employed when producing this academic work, either literally or in content.

Bielefeld,

45 A. P˙b-contribution script

#Calculations of Pb dot by Philipp Krause

#import libraries import numpy as np import os

# Defining Constants G=6.673e−11 #mˆ3 kgˆ−1 sˆ−2 c=2.997925e8 #m sˆ−1 M 0=1.98892e30 #kg R 0=8.4 #kpc Theta 0=254. #km sˆ−1 T=4.9257929e−6

#define input variable to get data from psr catalogue x=raw input(”Enter Pulsar JName ”)

#define variable for integration from catalogue with entered pu ls a r name y1=”ps rc a t −d b file ˜/Downloads/psrcat tar/psrcat.db −nohead −x −c \” PB DIST DM PMRA PMDEC \” %s >Text.txt” % (x) y2=”ps rc a t −d b file ˜/Downloads/psrcat tar/psrcat.db −nohead −x −c \” Gl Gb ZZ \” %s >Text1.txt” % (x) y3=”ps rc a t −d b file ˜/Downloads/psrcat tar/psrcat.db −nohead −x −c \” ECC M2 \” %s >Text2.txt” % (x) y4=”ps rc a t −d b file ˜/Downloads/psrcat tar/psrcat.db −nohead −x −c \” PBDOT \” %s >Text3.txt” % (x)

#integrate the values from psr catalogue import os os.system(y1) os.system(y2) os.system(y3) os.system(y4)

#assign values to the variables Pb=np.genfromtxt(”Text.txt”, usecols=(0)) dPb=np.genfromtxt(”Text.txt”, usecols=(1)) d=np.genfromtxt(”Text.txt”, usecols=(2)) dd=0.05

46 mu RA=np.genfromtxt(”Text.txt”, usecols=(3)) dmu RA=np.genfromtxt(”Text.txt”, usecols=(4)) mu DEC=np.genfromtxt(”Text.txt”, usecols=(5)) dmu DEC=np.genfromtxt(”Text.txt”, usecols=(6)) l=np.genfromtxt(”Text1.txt”, usecols=(0)) b=np.genfromtxt(”Text1.txt”, usecols=(1)) z=d∗np . s i n (b∗(np.pi/180.)) dz=dd∗np . s i n (b∗(np.pi/180.)) e=np.genfromtxt(”Text2.txt”, usecols=(0)) de=np.genfromtxt(”Text2.txt”, usecols=(1)) mc=np.genfromtxt(”Text2.txt”, usecols=(2)) dmc=np.genfromtxt(”Text2.txt”, usecols=(3)) mp=1.4 dmp=0.1 PBDOT=np.genfromtxt(”Text3. txt”, usecols=(0)) dPBDOT=np. genfromtxt(”Text3. txt”, usecols=(1)) dR 0=0.6 dTheta 0 =16.

#calculation of the contributions and their uncertainties #proper motion contribution def Pb mu(mu RA, mu DEC, d , Pb) : g l o b a l Pb mu2 Pb mu2=Pb∗d ∗3.086 e19 ∗np.power(4.847e −9 ,2.) ∗(mu RA∗∗2. + mu DEC∗ ∗ 2 . ) /( c ∗ ( 3 6 5 . 2 5 ∗ 2 4 . ∗ 3 6 0 0 . ∗ 3 6 5 . 2 5 ) ) p r i n t ”Pb mu=”,Pb mu2 return Pb mu2

#proper motion error value def dPb mu(mu RA, mu DEC, Pb , d ,dmu RA, dmu DEC, dd,dPb): g l o b a l dPb mu2 dPb mu2=np. sqrt ((dPb∗d ∗3.086 e19 ∗np.power(4.847e −9 ,2.) ∗( mu RA∗∗2. +mu DEC∗ ∗ 2 . ) /( c ∗ ( 3 6 5 . 2 5 ∗ 2 4 . ∗ 3 6 0 0 . ∗ 3 6 5 . 2 5 ) ) ) ∗∗2 +(Pb∗dd ∗3.086 e19 ∗np.power(4.847e −9 ,2.) ∗(mu RA ∗∗2. +mu DEC∗ ∗ 2 . ) /( c ∗ ( 3 6 5 . 2 5 ∗ 2 4 . ∗ 3 6 0 0 . ∗ 3 6 5 . 2 5 ) ) ) ∗∗2 +(Pb∗d ∗3.086 e19 ∗np.power(4.847e −9 ,2.) ∗(dmu RA∗ ∗ 2 . ) /( c ∗ ( 3 6 5 . 2 5 ∗ 2 4 . ∗ 3 6 0 0 . ∗ 3 6 5 . 2 5 ) ) ) ∗∗2 +(Pb∗d ∗3.086 e19 ∗np . power(4.847e −9 ,2.) ∗(dmu DEC∗ ∗ 2 . ) /( c ∗ ( 3 6 5 . 2 5 ∗ 2 4 . ∗ 3 6 0 0 . ∗ 3 6 5 . 2 5 ) ) ) ∗∗2) p r i n t ”dPb mu=”, dPb mu2 return dPb mu2

#gravitational wave emission contribution def Pb GW(Pb, e, mp, mc):

47 g l o b a l Pb GW2 Pb GW2 = ( −192.∗np . pi ∗np.cbrt(np.power(T, 5.)) ∗ (1.+(73./24.) ∗e ∗∗2. + (37./96.) ∗e ∗ ∗ 4 . ) ∗mp∗mc) / ( 5 . ∗ np. cbrt(np.power(Pb∗ 2 4 . ∗ 3 6 0 0 . / ( 2 . ∗ np.pi), 5.)) ∗np . sqrt(np.power((1. −( e ∗ ∗ 2 . ) ) , 7 . ) ) ∗np. cbrt (mp+mc)) p r i n t ”Pb GW=”,Pb GW2 return Pb GW2

#gravitational wave emission error value def dPb GW(Pb,dPb,e, de,mp, dmp,mc, dmc): g l o b a l dPb GW2 dPb GW2 = np.sqrt((((( −192.∗ np . pi ∗np. cbrt(np.power(T, 5 . ) ) ∗mp∗mc) / ( 5 . ∗ np. cbrt(np.power(Pb∗ 2 4 . ∗ 3 6 0 0 . / ( 2 . ∗ np . pi ) , 5 . ) ) ∗np. cbrt(mp+mc))) ∗ ( ( 7 . ∗ e ∗ ( 3 7 . / 9 6 . ∗e ∗∗4.+ 7 3 . / 2 4 . ∗e ∗∗2. +1.) ∗(1. − e ∗ ∗ 2 . ) ∗ ∗ 6.)/(np.sqrt(((1. − e ∗ ∗ 2 . ) ∗ ∗ 7 . ) ∗ ∗ 3.)) + (37./24. ∗e ∗∗3. +73.∗ e / 1 2 . ) /np . s q r t ((1. − e ∗ ∗ 2 . ) ∗ ∗ 7 . ) ) ) ∗de ) ∗∗2. + (((( −192.∗ np . pi ∗np .cbrt(np.power(T, 5.)) ∗ (1.+(73./24.) ∗e ∗∗2. + ( 3 7 . / 9 6 . ) ∗e ∗ ∗ 4 . ) ∗mp∗mc) / ( 5 . ∗ np. sqrt(np.power((1. −( e ∗ ∗ 2 . ) ) , 7 . ) ) ∗np. cbrt(mp+mc)) ) ∗(( −10.∗np.cbrt(4.) ∗np .cbrt(np.pi ∗ ∗ 5 . ) ) / ( 3 . ∗ np . cbrt (Pb∗ ∗ 8 . ) ) ) ) ∗dPb) ∗∗2. +(((( −192.∗np . pi ∗np.cbrt(np.power(T, 5.)) ∗ (1.+(73./24.) ∗e ∗∗2. + (37./96.) ∗e ∗ ∗ 4 . ) ∗mc) / ( 5 . ∗ np . cbrt(np.power(Pb∗ 2 4 . ∗ 3 6 0 0 . / ( 2 . ∗ np.pi), 5.)) ∗np . s q r t ( np.power((1. −( e ∗ ∗ 2.)), 7.)))) ∗ ( ( 2 . ∗mp+3.∗mc) / ( 3 . ∗ np . cbrt ( (mp+mc) ∗ ∗ 4 . ) ) ) ) ∗dmp) ∗∗2. +(((( −192.∗np . pi ∗np . cbrt(np.power(T, 5.)) ∗ (1.+(73./24.) ∗e ∗∗2. + ( 3 7 . / 9 6 . ) ∗e ∗ ∗ 4 . ) ∗mp) / ( 5 . ∗ np. cbrt(np.power(Pb ∗ 2 4 . ∗ 3 6 0 0 . / ( 2 . ∗ np.pi), 5.)) ∗np. sqrt(np.power((1. −( e ∗ ∗ 2.)), 7.))) ) ∗ ( ( 2 . ∗mc+3.∗mp) / ( 3 . ∗ np. cbrt ((mc+mp) ∗ ∗ 4 . ) ) ) ) ∗dmc) ∗∗2. ) p r i n t ”dPb GW=”,dPb GW2 return dPb GW2

#differential galactic rotation contribution def Pb DGR(Pb,l ,b,d): g l o b a l Pb DGR2 Pb DGR2 = −Pb∗24.∗3600.∗ np . cos (b∗(np.pi/180.)) ∗(( Theta 0 ∗1000.) ∗∗2./( c∗R 0 ∗3.086 e19 ) ) ∗(np . cos ( l ∗(np . pi/180.))+(((d ∗3.086e19)/(R 0 ∗3.086 e19 ) ) ∗np . cos (b∗( np.pi/180.))−np . cos ( l ∗(np.pi/180.)))/((np.sin(l ∗(np . pi / 1 8 0 . ) ) ) ∗∗2. +(((d ∗3.086e19)/(R 0 ∗3.086 e19 ) ) ∗np . cos (b∗(np.pi/180.))−np . cos ( l ∗(np.pi/180.))) ∗ ∗ 2 . ) ) p r i n t ”Pb DGR=”,Pb DGR2

48 return Pb DGR2

#gravitational galactic rotation error value def dPb DGR(Pb,dPb,l ,b,d,dd): g l o b a l dPb DGR2 dPb DGR2 = np.sqrt((−dPb∗24.∗3600.∗ np . cos (b∗(np . pi / 1 8 0 . ) ) ∗(( Theta 0 ∗1000.) ∗∗2./( c∗R 0 ∗3.086 e19 ) ) ∗(np . cos ( l ∗(np.pi/180.))+(((d ∗3.086e19)/(R 0 ∗3.086 e19 ) ) ∗ np . cos (b∗(np.pi/180.))−np . cos ( l ∗(np.pi/180.)))/((np. s i n ( l ∗(np.pi/180.))) ∗∗2. +(((d ∗3.086e19)/(R 0 ∗3.086 e19 ) ) ∗np . cos (b∗(np.pi/180.))−np . cos ( l ∗(np.pi/180.))) ∗ ∗ 2 . ) ) ) ∗∗2. +(( −Pb∗24.∗3600.∗ np . cos (b∗(np.pi/180.)) ∗(( Theta 0 ∗1000.) ∗∗2./( c∗R 0 ∗3.086 e19 ) ) ∗( np . cos (b ∗(np.pi/180.)))/(R 0 ∗3.086 e19 ∗(( d ∗3.086 e19 ∗np . cos ( b∗(np.pi/180.))/(R 0 ∗3.086 e19 ) −np . cos ( l ∗(np . pi / 1 8 0 . ) ) ) ∗∗2. +(np.sin(l ∗(np.pi/180.))) ∗ ∗ 2 . ) ) −(2.∗np . cos (b∗(np.pi/180.)) ∗(d ∗3.086 e19 ∗np . cos (b∗(np . pi / 1 8 0 . ) ) /( R 0 ∗3.086 e19 ) −np . cos ( l ∗(np.pi/180.))) ∗∗2) /( R 0 ∗3.086 e19 ∗(( d ∗3.086 e19 ∗np.cos(b)/(R 0 ∗3.086 e19 ) −np . cos ( l ∗(np.pi/180.))) ∗∗2 +(np.sin(l ∗( np.pi/180.))) ∗∗2) ∗∗2) ) ∗dd ) ∗∗2. +((−Pb∗24.∗3600.∗ np . cos (b∗(np.pi/180.)) ∗ ( 1 . / ( c∗R 0 ∗3.086 e19 ) ) ∗(np . cos ( l ∗(np.pi/180.))+(((d ∗3.086e19)/(R 0 ∗3.086 e19 ) ) ∗np . cos (b∗(np.pi/180.))−np . cos ( l ∗(np.pi/180.)))/((np.sin(l ∗(np.pi/180.))) ∗∗2. +(((d ∗3.086e19)/(R 0 ∗3.086 e19 ) ) ∗ np . cos (b∗(np.pi/180.))−np . cos ( l ∗(np.pi/180.))) ∗ ∗ 2 . ) )) ∗dR 0 ) ∗∗2 +((Pb∗24.∗3600.∗ np . cos (b∗(np.pi/180.)) ∗(( Theta 0 ∗1000.) ∗∗2./( c ∗(R 0 ∗3.086 e19 ) ∗∗2) ) ∗(np . cos ( l ∗(np.pi/180.))+(((d ∗3.086e19)/(R 0 ∗3.086 e19 ) ) ∗np . cos (b∗(np.pi/180.))−np . cos ( l ∗(np.pi/180.)))/((np.sin ( l ∗(np.pi/180.))) ∗∗2. +(((d ∗3.086e19)/(R 0 ∗3.086 e19 ) ) ∗np . cos (b∗(np.pi/180.))−np . cos ( l ∗(np.pi/180.))) ∗ ∗ 2 . ) ) ) ∗dTheta 0 ) ∗∗2) p r i n t ”dPb DGR=”,dPb DGR2 return dPb DGR2

#acceleration to the galactic plane contribution def Pb AGP(Pb, b, z): g l o b a l Pb AGP2 Pb AGP2= Pb∗ 2 4 .∗ 3 6 0 0 .∗ 1 . 0 8 e−19 ∗np . s i n (b∗(np.pi/180.)) ∗(1.25∗ z/(np.sqrt(z ∗∗2. +0.0324)) +0.58∗ z ) p r i n t ”Pb AGP=”,Pb AGP2 return Pb AGP2

49 #acceleration to the galactic plane error value def dPb AGP(Pb,dPb, b,z, dz): g l o b a l dPb AGP2 dPb AGP2= np. sqrt ((dPb ∗ 2 4 . ∗ 3 6 0 0 .∗ 1 . 0 8 e−19 ∗np . s i n (b∗(np . pi / 1 8 0 . ) ) ∗(1.25∗ z/(np.sqrt(z ∗∗2. +0.0324)) +0.58∗ z ) ) ∗∗2. +((Pb∗ 2 4 . ∗ 3 6 0 0 .∗ 1 . 0 8 e−19 ∗np . s i n (b∗(np . pi / 1 8 0 . ) ) ∗ (1.25/(np.sqrt(z ∗∗2. +0.0324)) +0.58 −(1.25∗ z ∗ ∗ 2.)/np.sqrt((z ∗∗2. +0.0324) ∗ ∗ 3 . ) ) ) ∗dz ) ∗ ∗ 2 . ) p r i n t ”dPb AGP=”,dPb AGP2 return dPb AGP2

#function that prints out the used values def Values(Pb, d, mu RA, mu DEC, l, b, z, e, mc, mp): print ”Pb=”, Pb print ”d=”, d p r i n t ”mu RA=”, mu RA p r i n t ”mu DEC=”, mu DEC print ”l=”, l print ”b=”, b print ”z=”, z print ”e=”, e print ”mc=”, mc print ”mp=”, mp print ”dPb=”, dPb print ”dd=”, dd p r i n t ”dmu RA=”, dmu RA p r i n t ”dmu DEC=”, dmu DEC print ”dz”, dz print ”de”, de p r i n t ”dmc” , dmc p r i n t ”dmp” , dmp

#print the corresponding contributions print ”Pb contributions for”,x,”:” Pb GW(Pb, e, mp, mc) Pb mu(mu RA, mu DEC, d , Pb) Pb DGR(Pb,l ,b,d) Pb AGP(Pb, b, z)

#print the corresponding error values of the contributions

50 print ”Error values:” dPb mu(mu RA, mu DEC, Pb , d ,dmu RA, dmu DEC, dd,dPb) dPb GW(Pb,dPb,e, de,mp, dmp,mc, dmc) dPb DGR(Pb,dPb,l ,b,d,dd) dPb AGP(Pb, dPb, b,z, dz)

#print the Measured Value, given by the psrcatalogue print ”Measured Value for PBDOT=”,PBDOT, ”+−”,dPBDOT

#add all the contributions and uncertainties print ”Sum of all contributions=”,(Pb mu2+ Pb GW2 + Pb DGR2 + Pb AGP2) print ”Sum of all Error values=”, (dPb mu2+ dPb GW2 + dPb DGR2 + dPb AGP2)

#print the used parameters to check for typos print ”Used Parameters:” Values(Pb, d, mu RA, mu DEC, l, b, z, e, mc, mp)

51