A Statistical Study of Pulsar Proper Motions

UNIVERSITY BIELEFELD

FACULTY OF PHYSICS

Author: Timo Kaja Supervisor: Joris Verbiest 2nd referee: Golam Shaifullah

November 12, 2015

Declaration: I hereby afﬁrm that this bachelor thesis represents my own work and has not been previously submitted to any examination ofﬁce. All resources used have been referenced.

Abstract

This thesis is about a possible statistical bias in the measurement of pulsar proper motion. This bias is similar to the Lutz & Kelker bias in parallax measurements. To investigate this bias, a likelihood analysis was conducted concerning the pulsar proper motion. The likelihood distribution was constraint using the measured parallax, upper and lower distance limits and radio flux of a pulsar as well as the distribution of pulsar three-dimensional velocity, obtained by Hobbs et al. [2005]. The resulting likelihood distribution was analysed, using self-written python methods. For this two different samples of pulsars were used. In summary the results were, that for a single pulsar the bias is insignificant for all pulsars out of the samples, since either the measurement precision of the pulsar proper motion is to high for the bias to occur or the measurement was insignificant to make a statement on the bias. Alternatively, for a larger sample of proper motion measurements, the measured proper motion tend to be underestimated. This is rather surprising, since a distance measurements of a pulsar tend to be underestimated as well. Another result from the analysis is, that for a large value with a significance of two to four σ, there is the possibility, that the measured proper motion is inconsistent with the corrected biased value obtained by this analysis. Then again we find no evidence in the data that such in a case exists in the data available.

Table of Contents

1 Introduction 9 1.1 Stellar Evolution and the Formation of Neutron Stars ...... 10 1.1.1 Stellar Evolution ...... 10 1.1.2 Neutron Star Structure ...... 12 1.1.3 Pulsar Properties ...... 12 1.1.4 Pulsar Timing ...... 13 1.2 Astronomical Coordinate Systems ...... 15 1.2.1 Equatorial Coordinates ...... 15 1.2.2 Galactic Coordinates ...... 15 1.2.3 Converting Equatorial to Galactic Coordinates ...... 16 1.3 Pulsar Velocities ...... 17 1.3.1 Pulsar Proper Motions ...... 17 1.3.2 Pulsar Distances and the Lutz-Kelker Bias ...... 18 1.3.3 Pulsar Velocities ...... 22

2 Application of the Lutz-Kelker Bias to Pulsar Proper Motions 24 2.1 Theoretical Derivation ...... 24 2.2 Uncertainty Estimation ...... 26 2.3 Examples ...... 29

3 Results 32 3.1 First sample ...... 32 3.2 Second sample ...... 35

4 Conclusion and Future Research 42

5 Appendix 44 5.1 Functions ...... 44 5.2 The Geterrorbars Method ...... 45 5.3 The Sepint Method ...... 46 5.3.1 Graphics ...... 47

1 Introduction

The pulsar phenomenon is probably one of the most interesting subjects in modern physics. Shortly after its discovery it was proposed that the origin of these pulses were rotating magnetized neutron stars. Neutron stars up to this point were considered, if they were existing, not to be detectable. A neutron star is an exotic object and lies beyond our imagination. Hence they are a rich source of effects to test the theories abstracted from our environment in their limits. The range of theories needed to understand and describe the pulsar phenomenon ranges from general relativity, solid state physics and particle physics to the physics of exotic matter. And yet they are a riddle to solve. Neither is the mechanics that generates their characteristic emission known, nor their internal structure or their origin understood. In general pulsars can be considered as clocks. This gives rise to applications by which not only the object itself but also its surrounding can be understood. With the help of pulsars the structure of the galaxy can be illuminated. As a tool, they can also be used to test a missing piece from one of the best tested theories in physics, which refuses to this date to be directly observed: gravitational waves. The ﬁrst indirect proof of gravitational waves was indeed provided by the observation of a system of two pulsars in a close orbit. Since the measurement of an observable is a probability statement, this gives rise to systematic errors, if not all statistical effects are considered for this measurement. In this thesis statistical effects will be studied, which could result in a bias in the measured velocity of the objects. This type of bias was ﬁrst described by Lutz & Kelker for parallax measurements and hence is referred to as the Lutz & Kelker bias. The measurement of the velocity of pulsars provides insights in the dynamical processes they are formed in. Pulsars are in general high velocity objects. The additional speed compared to their ancestral population, pulsars gain in supernovae. A complete description of the velocities of the pulsar population constrains the mechanisms responsible for the velocity and hence leads to a better understanding of supernovae. 10 1 INTRODUCTION

1.1 Stellar Evolution and the Formation of Neutron Stars This chapter provides a brief overview of the concepts related to the topic of neutron stars and pulsars in particular, without going into more detail than required in the context of this thesis. For the parts on stellar evolution §(1.1.1), neutron star structure §(1.1.2) and pulsar properties §(1.1.3) the book of Irvine [1978] was mainly used with additional input from the book of Lorimer and Kramer [2012] for the Pulsar properties part.

1.1.1 Stellar Evolution The force responsible for forming structures in the universe with its inﬁnitely long range is gravity. Gravity is an attractive force. So for nature to form stable or quasi-stable objects, a repulsive interaction is needed to counter gravity, otherwise black holes would be the only existing objects. Sources of repulsive interactions are kinetic effects or fundamental forces like electromagnetic and nuclear forces. Examples of kinetic effects are angular momentum, hydrodynamic pressure, radiation pressure in main sequence stars and neutrino radiation pressure in white dwarf and neutron stars. Most kinetic effects lead only to quasi-stable states since angular momentum can be lost through the action of normal hydrodynamic viscous forces, or be radiated away gravitationally or, in case of a hot system, the free energy will be radiated away. Eventually all sources of free energy will become exhausted. Another particularly important kinetic effect for the stability of white dwarfs and neutrons stars is the Fermi pressure. Since matter fundamentally consists of fermions, which obey the Pauli exclusion principle, no two fermions can occupy the same state. For a gas of non-interacting fermions, the Fermi energy is the highest energy state fermions can occupy at zero temperature. As a consequence, even at zero temperature, there exists a distribution of non-zero momentum states for the fermions. This leads to the Fermi pressure, which counters the gravitational pull:

2 ~ 2 5 P = (3π)) 3 ρ 3 (1) f 5m where ρ is the density, compare eq.(12.22) and eq.(12.26) from Ford [2013]. This equation holds true if the temperature of the system is considered cold, compared to the Fermi tempera- 5 ture. It is important to note that the Fermi pressure is proportional to ρ 3 and proportional to the inverse mass m of the particle.

Due to the gravitational pull, a large cloud in space will collapse under its own mass, if the gravitational binding energy exceeds the thermal energy of the gas. As the cloud collapses, the temperature rises and so does the pressure provided by the temperature. How the pressure rises as a function of temperature and other thermodynamic variables is described by the equation of state of the matter. As the temperature rises, the cloud begins to radiate, which not only provides an additional source of pressure to stabilize the cloud, but also cools it as the kinetic energy is radiated into the cooler space surrounding the cloud. Due to the cooling, the cloud will collapse further. If the cloud is sufﬁciently massive, eventually the density and temperature at the core of the cloud will be large enough to sustain nuclear fusion processes.

When a collapsed cloud starts nuclear fusion, it is referred to as a star. In nuclear fusion processes the mass of the product is slightly smaller than the combined mass of the constituents. 1.1 Stellar Evolution and the Formation of Neutron Stars 11

So the fusion processes provide a source of temperature and hence pressure to stabilize the star. The initial cloud consisted mostly of hydrogen. So the first nuclear reaction in the core would be the hydrogen burning where two essentially four proton (hydrogen cores) are fused to a helium-4 core. For the longest span of a stars lifetime, hydrogen burning is the main source of energy, so a hydrogen burning star is referred to as a main-sequence star. Since the density and temperature is the highest in the centre of the star, the fusion rate is also the highest in the star’s centre. This leads to the formation of a helium core in the star’s centre. The helium-rich core can provide the nuclear fuel for the fusion of more massive nuclei. For this even higher temper- atures and densities are needed, to overcome the stronger Coulomb repulsion. In a massive star, shell burning could be initiated, where the more massive nuclei are fused towards the centre and the lighter elements toward the outer layers, leading to an onion-like structure. The highest element fused inside a star with a positive energy balance is Fe56. Adding a nucleon to Fe56 will cost energy, since the binding energy per nucleon is the highest for Fe56. So nuclear fusion as a source of energy and hence pressure, to counter the gravitational pull, is finally exhausted for the inner core when the core of the star is finished creating Fe56. If the source of energy is exhausted, either due to the ending Fe56 burning or due to the lack of density and temperature to start the fusion of the elements present in the core, the inner core region cools down. This leads to a decrease in pressure. So the inner core collapses until it is sustained by the Fermi pressure of the electron, since it is proportional to the density, or if the mass of the core exceeds a certain limit, it collapses further. A white dwarf star is the remnant of such a core collapse, where the core is supported against further collapse by the electron Fermion pressure. The mass limit of a white dwarf is given by the Chandrasekhar mass. If the mass of the object is larger, so is the density. The increasing density causing the energy of the electron to increase as well, since the Fermi energy depends on the density. If the energy of the electrons exceed the mass difference between protons and neutrons, the inverse beta-decay − e + p → n + µe (2) becomes possible. The mass at which the electron energy exceeds the proton electron mass difference is called the Chandrasekhar mass, which also depends on the equation of state. This reaction is, like the fusion of light elements, exothermic due to electrons and protons having kinetic energy in addition to the neutron proton mass difference. This leads to a increased neutron density. Since the Fermi pressure scales with the inverse mass of the particle the neutron Fermi pressure is smaller than the electron Fermi pressure, with the decrease of the electron density leading to a core collapse once again until the neutron density has increased sufficiently to provide enough Fermi pressure to counter the gravitational pull. In addition to the Fermi pressure, the radiation pressure of the neutrinos, which are generated in the inverse beta-decay and are absorbed by the stellar material, restrict the collapse by causing the temperature to rise. The process of beta-decay is extremely rapid since the density and energy of the electrons is very high.

If the exact dynamical process in which neutrons stars are formed is still neglected, the simple scenario is the following. It starts with a sufﬁciently massive star that performs shell-burning until its fuel in the core is completely used. Since density and temperature are the highest in the inner core, the exhaustion will occur ﬁrst in the inner core region causing the core to cool down and eventually collapse. When the core collapses, the outer layers follow and as the core collapse is stopped by the neutron Fermi pressure, the outer layers collide with the core matter. 12 1 INTRODUCTION

The kinetic energy of the outer layers will be transformed into heat at the shock front. Addi- tional heat is provided by the neutrino radiation from the core leading to a dramatic increase in pressure and eventually causing the star to blow up. This event is called a supernova. The remnants of supernovae typically show signs of asymmetry which is not surprising for such a violent process.

1.1.2 Neutron Star Structure As the core of the star collapses, it eventually forms a neutron star. Due to the conservation of angular momentum, the neutron star is spun up to high frequencies. Like the angular momentum, the magnetic flux is conserved. If the collapsing star happens to have a magnetic field, the magnetic field strength increases inversely to the square of the radius leading to extremely strong magnetic fields. The lower mass limit for neutron stars is determined by the Chan- drasekhar mass and the upper mass limit is determined by the Oppenheimer-Tolman-Volkoff equation, where in contrast to the Chandrasekhar limit general relativistic effects are taken into account. If the neutron star’s mass exceeds the Oppenheimer-Tolman-Volkoff mass limit then either it collapses into a black hole or a more exotic state of matter that provides enough Fermi pressure to stabilise the star, like possibly quark matter. Both mass limits depend on the exact equation of state for a neutron star which is not fully understood yet. For the equation of state, the structure of a neutron star is important. Even in the simplest models the star does not con- sist purely of neutrons. Since neutrons, as well as protons, are just phases of nuclear matter, there are always some protons and electrons in the neutron matter. If the temperature of the neutron star is low enough, two neutrons can couple in pairs similar to Cooper-pairs. So the neutron matter forms a superfluid. Currents of superfluid neutrons dragging charged particles along with them, could provide a mechanism to sustain the strong magnetic fields in a neutron star. It is likely that, at the neutron star’s surface, a solid crust will be formed that behaves like a metal. So the crust consists of a lattice of nuclei formed in the star or in the supernova and an electron gas. Towards the centre it is possible that the neutron star consists of a more exotic form of matter like quark matter, for example Another interesting topic is the magnetosphere of a neutron star with such an extreme magnetic field. Even for a Neutron Star in vacuum, the rotating magnetic field generates an electric field that drives charged particles out of the star’s surface. So the neutron star is surrounded by a plasma which is coupled to the magnetic field.

1.1.3 Pulsar Properties Pulsars are apparently pulsating radio sources with short pulse periodicity. Rotating neutron stars with strong magnetic fields give a potential explanation for the pulsar phenomenon. The pulses can be explained by a rotating neutron star, where the magnetic dipole field is off by an angle with respect to the rotation axis. The emission is concentrated by the magnetic field, which leads to the so-called lighthouse effect. The emission is generated by the coupling of the magnetic field to the plasma of the magnetosphere. This coupling leads, due to the acceleration of charged particles, to synchroton radiation, which is observed as radio pulses. Due to the emission of radiation the pulsar loses rotational energy. This leads to a spin down of the pulsar. To some extent the spin-down velocity can be approximated by a simple model. The rotation of the dipole moment M with the rotational period p, leads to the emission of radiation energy at 1.1 Stellar Evolution and the Formation of Neutron Stars 13 the rate dE 32π2 M 2sin2α = − (3) dt 3c3 p4 see eq. (2.14) of Irvine [1978]. If this is the only mechanism that slows the pulsar down, the rate at which the rotational period slows down can be calculated. For this, the mass of the pulsar is estimated to be ≈ 1033 g and the radius ≈ 106 cm. The resulting moment of inertia for a homogenous and spherical rigid body is I ≈ 1045 g cm2. Hence the spin down is

dp 8M 2sin2α p = (4) dt 3Ic3 (based on eq. (2.16) of Irvine [1978]).From this a characteristic (i.e theoretically estimated) age can be obtained: p p p˙ −1 τ = ' 15.8Myr , (5) c 2p ˙ s 10−15 see eq. (3.6) of Lorimer and Kramer [2012].Similarly, the magnetic ﬁeld strength can be shown to be: 1 2 1 p p˙ p 2 B = 3.2 ∗ 1019G pp˙ ' 1012G (6) 10−15 s (from eq. (3.15) of Lorimer and Kramer [2012]).

1.1.4 Pulsar Timing The most powerful method to investigate pulsars, is so-called "pulsar timing". Pulsar timing is based on the measurement of the arrival times of pulsar’s pulses. Since pulsars are in general faint sources, the signals have to be accumulated to create a integrated pulse proﬁle. For this technique the times of arrival of the pulses have to be predicted. The integrated pulse pro- ﬁles provide information on the emission process and are different for every pulsar. With this method the parallax and the proper motion of a pulsar can be measured. Another constraint on the distance of a pulsar are the H1 distance limits can also be obtained by pulsar timing. The parallax can be obtained by plotting the time of arrival of the pulsar’s pulses over a year. Since the wave front of a pulse is curved, the arriving time of the pulses is delayed when the Earth is not aligned with the Sun and the pulsar. This delay is zero if the Sun, the pulsar and the earth are aligned. A quarter of a year later the delay is maximal. The maximal delay relates to the curvature of the wave front which relates to the parallax of the pulsar. For this method the angle of the pulsar with respect of the Earth’s orbit has to be taken into account to correct it for projection effects. If the pulsar lies orthogonal to the Earth’s orbit the parallax cannot be measured with this method. The parallax can also be measured using interferometry. For measurement the time of arrival for two different telescopes are compared. The method is essentially the same as described before. The delay in the time of arrival is used to determine the parallax for the observed pulsar.

The HI limits are obtained by measuring the Doppler shift of the closest hydrogen clouds associated with the pulsar. The radiation of the pulsar stimulates a transition in the hydrogen clouds. A cloud that lies in the line of sight with the pulsar will absorb part of its emission, while cloud behind the pulsar 14 1 INTRODUCTION will emit radiation at a sharp wavelength due to the relaxation of the transition. Since the matter in the Galaxy rotate at a certain speed as a function of distance to the galactic centre that can be modelled the absorption line of the hydrogen cloud observed is shifted. From that the velocity of the cloud with respect to the Solar System barycentre can be obtained and by that through the rotation model also the distance. 1.2 Astronomical Coordinate Systems 15

1.2 Astronomical Coordinate Systems

This chapter provides a quick overview of the two astronomical coordinate systems needed for this thesis.

1.2.1 Equatorial Coordinates

Equatorial coordinates are widely used system in astronomy. In a celestial coordinate system the position of an object that is seen in the sky is projected onto the celestial sphere, so the distance to that object is neglected. The celestial sphere is an imag- inary sphere with an arbitrary radius and which is concentric with the Earth’s centre. The equatorial coordinate system is a spherical coordinates system, where the position of an object on the celestial sphere is given by two angles. The polar angle lies in the plane of the Earth’s equator and the azimuth angle is measured perpendicular to the earth’s equator. In equatorial coordinates, the polar angle is called right ascension and the azimuth angle is called declination. The point of zero right ascension and zero declination is the vernal equinox. The right ascension is measured in hours Fig. 1: Despiction of the equatorial coordinate sys- minutes seconds etc. since the earth needs tem. Source: Tfr000 [2012] 24 hour to rotate around. The vernal equinox is the point on the celestial sphere, where the Sun is seen to cross the equator at the start of spring. Since the ecliptic changes due to the Earth’s precession, the coordinate system is not ﬁxed. To correct for that drift, different reference epochs are introduced referring to the vernal equinox at that particular time. For pulsars it is common to give their position in equatorial coordinates referring either to the Besselian epoch (1950.0) or for pulsars discovered after 1990 to the Julian epoch (2000.0). In summary the equatorial coordinate system is deﬁned by the vernal equinox, the equatorial plane and the right-handed convention for the right ascension.

1.2.2 Galactic Coordinates

The Galactic coordinate system is similar to the Equatorial coordinate system. It is a celestial coordinate system using spherical coordinates. In contrast to the equatorial coordinates it is heliocentric which means it is originated from the Sun instead of the Earth’s centre. For large distances, as for pulsar distances, the difference is negligible. The position of an object is given in galactic longitude l and galactic latitude b which are usually measured in degrees. 16 1 INTRODUCTION

Fig. 2: Artist’s depiction of the Milky Way galaxy, showing the galactic longitude relative to the galactic center (Hurt [2008])

The plane referring to zero galactic latitude is the galactic plane, where the direction of zero galactic longitude is deﬁned by the direction towards the galactic center as it is displayed in Fig. (2).

1.2.3 Converting Equatorial to Galactic Coordinates

It is useful to know how to convert equatorial coordinates to galactic coordinates. This is done by the following equations, where l is the galactic longitude, b is the galactic latitude, α is the right ascension and δ is the declination. It is necessary to express the right ascension in units of degrees beforehand. The following transformation refers to the Besselian epoch B1950.

◦ ◦ sin(192. 25 − α) l = 303 − arctan ◦ ◦ ◦ (7) cos(192. 25 − α)sin27. 4 − tanδcos27. 4

◦ ◦ ◦ sinb = sinδsin27. 4 + cosδcos27. 4cos(192. 25 − α) (8)

(equations 13.7 and 13.8 from Meeus [1998]). 1.3 Pulsar Velocities 17

1.3 Pulsar Velocities

1.3.1 Pulsar Proper Motions

Fig. 3: Sketch of the concept of proper motion. Source:ohare [2008]

Traditionally Proper motion is the astronomical measure of the change of the observed position of an object in the sky with respect to the ﬁxed stellar background over time. It is measured separately as a change in right ascension and declination as a function of time. The combined value is called total proper motion µ and is measured typically in units of milliarcseconds per year. Proper motion doesn’t take the radial velocity into account which could be measured independently for example via Doppler shift of spectral lines. In order to measure the true velocity of an astronomical object, measurements of µ as well as the distance and the radial velocity are needed. Proper motion is not entirely intrinsic to the object due to the motion of the Solar System itself. It follows that proper motion is a two-dimensional vector and is thus deﬁned by its position angle and its magnitude. The magnitude is given by

v µ = (9) d

where µ is the magnitude of the proper motion, d is the distance of the object to the barycentre of the Solar System and v is the magnitude of the relative velocity projected in the plane of the sky. Figure (4) visualises the proper motion for some pulsars in the milky way. 18 1 INTRODUCTION

Fig. 4: Hammer-Aitoff projection showing the proper motion vectors of pulsars in Galactic coordinates. The current position of each Pulsar is shown by the open circle. Fig. (2.2.) [Lorimer and Kramer, 2012]

The proper motion of a pulsar can be measured by various techniques. An important one is the measurement of proper motion by pulsar timing. If the position angle of a pulsar is constant, the measurement of the time of arrival of the pulse reveals a sine or cosine with a periodicity of one year, due to the Earth’s motion around the Solar System barycentre. As the position angle changes in time, the amplitude and the phase of the sine changes as well in time. This method measures the proper motion with respect to the Solar System barycentre, which is essentially the same as the previous deﬁnition, since the Earth is bound to the Solar System barycentre.

1.3.2 Pulsar Distances and the Lutz-Kelker Bias

Thomas E. Lutz and Douglas H. Kelker stated in their work on the use of trigonometric parallaxes for the calibration of luminosity systems (Lutz and Kelker [1973]), that the measured parallaxes of a sample of stars is biased compared to the true parallax values. Assuming that stars are uniformly distributed in space, the number of stars at the distance between r and r+dr is given by N(r)dr = 4πr2dr (10)

dr 2 dω Since the distance r is proportional to the reciprocal parallax ω ( dω = −ω → dr = − ω2 ), the number of stars between ω and ω+dω is 4πdω N(ω)dω = (11) ω4 Thus the number of stars per interval of parallax varies as 1/ω4. In a sample of stars with the observed parallax ω’, due to the measurement uncertainty, some of them will have a larger and some a smaller true parallax. Since we know that the number of stars increases while the parallax decreases, it is more likely for the stars in the sample to have smaller true parallaxes. This bias is a consequence of both the error of measurement and the increasing volume and thus number of stars as the parallax decreases. Lutz and Kelker have shown analytically that this systematic error occur for all stars and it depends on the ratio σ/ωmeas not on the size of ωmeas. It is useful to assume that the distribution for the observed parallax 1.3 Pulsar Velocities 19

ωmeas given the true parallax ω is a Gaussian

2 1 (ωmeas − ω) p(ωmeas|ω) = √ exp − (12) 2πσ 2σ2 where σ is the standard deviation of the measurement. The resulting bias-corrected distribution can be obtained, if the knowledge about the number density of stars in space per unit parallax eq.(11) is taken into account. So the Gaussian distribution is multiplied by the number density term. Since only the behaviour of the resulting distribution is of interest the result is proportional to p(ωmeas|ω). So it is useful as well to introduce a dimensionless parallax z= ω . Then ωmeas

1 (1 − z)2 p(ω|ωmeas) ∝ P (z) ≡ 4 exp − 2 (13) z 2(σ/ωmeas) is the resulting distribution. This distribution is normalised in a way that the value for P(z) is 1 for z=1. However it cannot be normalised via an integral since it diverges at z=0. Given a measurement for the observed parallax ωmeas with the measurement uncertainty σ, this distribution provides knowledge on the true parallax. In conclusion the size of the error does not depend on the value of parallax alone but rather on the ratio σ/ωmeas.

Bayes’ theorem is a statement of probability theory. It describes the probability of an event, based on knowledge about the circumstance prior to a measurement. Suppose that y is a vector (y1,...,yn) of n observations whose probability distribution p(y|Θ) depends on the values of k parameters Θ = (Θ1,...,Θk). Furthermore suppose that Θ itself has a probability distribution p(Θ). It follows that

p(Θ|y)p(y) p(y|Θ)p(Θ) = p(Θ|y)p(y) p(y|Θ) = (14) _ p(Θ) p(Θ) is called the prior since it contains knowledge about Θ without data. p(Θ|y) tells us what is known about Θ given data y so it is called the posterior. p(y|Θ) is a function of Θ for given data y. It is called the likelihood function of Θ. The likelihood function is the function through which the data y modiﬁes prior knowledge of Θ. So Bayes’ theorem states that the probability distribution for Θ posterior to the measurement data y is proportional to the product of the distribution for Θ prior to the measurement and the likelihood for Θ given y.

In their work on pulsar distance measurements and their uncertainties Verbiest et al. [2012] take a Bayesian approach on the subject of the Lutz-Kelker bias. While Lutz and Kelker were only interested in the systematic error in parallax measurements of stars in general, Verbiest et al. are interested in the statistical bias in distance and parallax measurements of pulsars in particular. It is the same approach, since the parallaxes and the distances of stars are closely related. However, instead of an uniform distribution of stars in the sky, a model distribution is used based on the measured positions of pulsars in the Galaxy. Measurements of the following quantities provide prior information on a distance measurement:

• parallax measurement ωmeas

• lower HI distance limit, Dlow

• upper HI distance limit, Dup • pulsar radio ﬂux, S

• galactic position, l b 20 1 INTRODUCTION

The likelihood for a star’s true parallax given a parallax measurement is eq.(12). Since the parallax is the reciprocal of distance, it is substituted as follows: p(D) = |∂ω/∂D|p(ω) ∝ p(ω)/D2. So eq.(12) becomes 1 1 ωmeas − 1/D p(D|ωmeas) ∝ 2 exp − (15) D 2 σω With the measurement of the HI limits, more constraints on the true distance of a pulsar are set. The HI limits are obtained by measuring the distance of the furthest absorbing respectively the nearest not absorbing HI gas. Assuming that this measurement is also a Gaussian, the likelihood for the true lower distance limit d is " 2# 1√ 1 Dlow − d p(d|Dlow) ∝ 2πσlowexp − . (16) 2 2 σlow If d is the true lower limit for the distance of the pulsar, D, the true distance, has to be larger than d: D ≥ d. Hence the likelihood of D for a given d is proportional to the Heaviside function: p(D|d) ∝ H(D − d). Hence the likelihood of the true distance given a measurement of the lower distance limit Dlow, σlow is obtained via integrating d.

∞ Z p(D|Dlow) = p(D|d)p(d|Dlow)dd 0 D Z = p(d|Dlow)dd 0 1 Dlow Dlow − D →p(D|Dlow) ∝ erf √ − erf √ (17) 2 2σlow 2σlow In the case of the upper distance limit the calculation is similar, except for the Heaviside function which for Dup is H(d − D). The resulting term is " ! # 1 Dup p(D|Dup) ∝ erf √ + 1 (18) 2 2σup

Instead of a uniform distribution, the distribution of pulsar density in the Galaxy was derived by Lorimer et al. [2006] and has the form N |z| R − R ρ(R, Φ, z) = ∝ RBexp − − C 0 kpc−3 (19) V E R0

The constants are determined by Lorimer et al. to be R0=8.5kpc, B=1.9, C=5 and E=330pc for common pulsars and E=500pc for millisecond pulsars. The distribution is given in cylindrical coordinates originating in the Galactic centre. It is more convenient to use a spherical coordinate system originating at the observer. So a coordinate transformation is performed. The transformation is q 2 2 R(D,Gl,Gb) = R0 + (DcosGb) − 2R0DcosGbcosGl

z(D,Gl,Gb) = DsinGb

Φ(D,Gl,Gb) = Gl

For a spherical coordinate system the Jacobi determinant is r2sin(θ). So the resulting infinitesimal volume is δV = D2δDδΩ (20) 1.3 Pulsar Velocities 21 where δΩ is an infinitesimal solid angle in spherical coordinates. Given the distribution eq.(19), is a probability density function, the number N of pulsars expected to be found in an arbitrary volume interval is the distribution integrated over the interval. Written in infinitesimals: 2 δN = ρ(D,Gl,Gb)D δDδΩ. (21) This expression could be used alternatively to obtain the likelihood of a pulsar being in the volume interval. If the direction of the pulsar is determined, the expression is reduced to a statement of the likelihood of a pulsar being at a certain distance. Therefore the solid angle is omitted. The likelihood for a pulsar to be at a certain distance becomes proportional to the density time D2, since the infinitesimal likelihood scales with the infinitesimal number of pulsars. 2 p(D|Gl,Gb) ∝ ρ(D,Gl,Gb)D (22) If the density distribution is applied, the final Galactic volume term is: 1.9 |z| R − R0 2 p(D|Gl,Gb) ∝ R exp − − C D . (23) E R0 The last constraint on the pulsar distance discussed by Verbiest et al. (2012) is derived from the radio flux S of the pulsar. The radio flux S is related to the luminosity L by SD2 = L, where L is an effective "pseudo-luminosity"1. The log-normal luminosity distribution of radio pulsars was derived by Faucher- Giguère and Kaspi [2006], where the mean is hλi = hlog(L)i = −1.1 and the standard deviation σλ = 0.9 where log is the ten-based logarithm and λ = log(L). " # 1 λ + 1.12 p(λ) ∝ exp − (24) 2 0.9

If L is substituted for SD2, eq.(24) can be treated as a likelihood function for the distance of a pulsar given a measured radio ﬂux S. Since λ = log(L) = log(S) + 2log(D) and ∂λ/∂D = 1/D the resulting term is: " # 1 1 logS + 2logD + 1.12 p(D|S) ∝ exp − (25) D 2 0.9 The resulting probability distribution for the distance based on these measurements is the multiplication of eq.(15), eq.(17), eq.(18), eq.(23) and eq.(25):

p(D|ωmeas,Dlow,Dup, S, Gl,Gb)

1Note that this "pseudo-luminosity" was deﬁned by Faucher-Giguère and Kaspi [2006]. It avoids the complex- ities of emission beam and viewing geometries, which is useful for our purposes. 22 1 INTRODUCTION

, assuming that these measurements are not correlated. Verbiest et al. ﬁnd, like Lutz and Kelker that for a sample of pulsars the distance measurement tend to be underestimated. There exists a webpage2 where this bias obtained by Verbiest et al. [2012] will be calculated.

1.3.3 Pulsar Velocities

This subsection is in the ﬁrst part about the causes of the high velocities of Pulsar and Neutron Stars in general. The ideas for this section can be found mainly in the work of Lai et al. [2001]. The second part is about the velocity distribution for the three-dimensional velocities of pulsars derived by Hobbs et al. [2005].

Neutron stars in general are high-velocity objects. In fact their three-dimensional mean velocities are much greater than the mean speed of the population of stars they evolve from. There is also a significant population of neutron stars, whose three-dimensional velocity exceeds 1000km s−1. This implies that there are mechanisms that are intrinsic to the nature or evolution of neutron stars that provide these velocities. As summarized by Lai et al. [2001] there are two classes of causes for kicks on the order of several 100km s−1 during the birth of a neutron star and one cause for a post-natal kick is discussed. Observations of supernova remnants show, that supernova explosions are not spherically symmetric. So these asymmetries in supernovae could lead to high velocities. One type of cause mentioned to be responsible not only for this asymmetry but also for a significantly large kick, are global hydrodynamical perturbations within the star during the core collapse and the supernova explosion. This leads to an asymmetric mass ejection, which leads to a kick. Another class of mechanisms results from strong magnetic fields with a magnitude on the order of 1015. Vilenkin [1995] showed that neutrinos in a magnetized medium are less likely to be scattered if their angular momentum and the magnetic field are aligned. This effect is a sequel of the parity violation in the weak interaction. It leads to an asymmetric flux of neutrinos during a supernova explosion in the direction of the magnetic field and thus to a kick. If the magnetic field itself is asymmetric the magnetic field strength dependency on the cross section for νe and ν¯e absorption by neutrons and protons can lead to an asymmetric flux as well (Arras and Lai [1999]). A strong dynamical magnetic field could lead to ’dark spots’ on the proto neutron star’s surface, where the neutrino flux is lower than average. In contrast to the natal kicks the electromagnetic rocket effect also discussed by Lai et al. [2001] provides a post-natal kick. Assuming the magnetic field of the pulsar to be a dipole, which is off-centred and the spin axis and the magnetic field axis inside the pulsar are not parallel. This leads to different rotation speeds for the points on the surface of the pulsar where the magnetic dipole emerges. Since a rotating dipole emits magnetic dipole radiation, depending on the rotation speed, the inequality in the speeds of the points where the magnetic dipole emerges from the neuton star’s surface leads to a inequality of the emitted radiation power and hence to a acceleration of the neutron star. Since all of these mechanisms only predict a limited range of velocities, a large sample of pulsar velocities could be used to test these mechanisms.

In their statistical study of 233 pulsar proper motions Hobbs et al. [2005] conclude that the distribution of pulsar proper motions is well described by a Maxwellian. Although different classes of pulsars have different proper motions the overall distribution is uni-modal. In the paper they distinguished between ﬁve types of pulsars: recycled, young, associated with globular clusters, associated with supernova remnants and normal pulsars. Each type of pulsar constrains the velocity regarding the sample. The velocities of recycled pulsars and those associated with globular clusters have to be lower than the escape velocities

2The webpage is: http://psrpop.phys.wvu.edu/LKbias/. For given parallax, HI limits or radio ﬂux measurements the bias can be calculated. 1.3 Pulsar Velocities 23 of those bound objects. The velocities of pulsars associated with supernova remnants are slow compared to the velocities of young pulsars not associated with supernova remnants, since a high-velocity object would leave the region of the supernova remnant within a short time scale. Therefore the relation between pulsar and supernova remnant would be more difﬁcult to identify. For their study they ended up ignoring pulsars associated with globular clusters, since their proper motion is dominated by the proper motion of the globular cluster.

All Normal Recycled SNR Young 1 Npsr 217 178 39 8 73 ¯ −1 V1 (kms ) 133(8) 152(10) 54(6) 150(42) 192(20)

2 Npsr 156 121 35 7 46 ¯ −1 V2 (kms ) 211(18) 246(22) 87(13) 227(85) 307(47)

1 Table 1: Taken from table 5 of Hobbs et al. [2005]. Npsr is the number of measured one- 2 ¯ dimensional velocity and Npsr for two-dimensional. V1 is the mean value of the measured one- ¯ dimensional velocity and V2 the mean for the two-dimensional case.

An important conclusion is that the velocity vector is isotropic. To test this Hobbs et all. compared the ratio of the mean value of the one-dimensional and the two-dimensional velocity. They find it to be in statistical agreement with the ratio of the mean value of a one-dimensional and a two-dimensional vector projected from an isotropically distributed vector in three dimensions. The mean values for the two- and one-dimensional case are given in Tab.(1). With simple calculation described before the given data one can show, that the velocity vector is in fact isotropic. The derivation for the case of a three-dimensional vector projected on a plane is shown in eq.(32). Another important conclusion is that although the sample is split, they find no evidence that the resulting distribution is bi-modal. As a result they find the pulsar velocity distribution to be a Maxwellian with rms σ=265km/s. 24 2 APPLICATION OF THE LUTZ-KELKER BIAS TO PULSAR PROPER MOTIONS

2 Application of the Lutz-Kelker Bias to Pulsar Proper Mo- tions

2.1 Theoretical Derivation

Using both the knowledge about the bias in the distance measurement and the distribution of pulsar velocities as prior information, one could make a Bayesian ansatz to obtain knowledge on the nature of a possible bias in measurements of proper motion. Since proper motion is proportional to the inverse distance of the pulsar (cf. eq.(9)), the underestimation in the pulsar distance measurement could lead to a overestimation in the measurement of pulsars proper motions. To address this issue the same approach is taken as in the paper of Verbiest et al. [2012]. Their resulting prior (eq.(26)) and the Maxwellian distribution obtained by Hobbs et al. [2005] are used to investigate a possible bias in proper motion measurements. The Bayesian ansatz is:

p(µobs|µ)p(µ) p(µ|µobs) = (27) p(µobs)

where p(µobs) = 1 and p(µobs|µ) is the likelihood function obtained by the measurement of the proper motion, which is assumed to be gaussianly distributed. The interesting part is the prior p(µ). Since the proper motion depends on the distance D, as well as the transverse velocity v of that object, the probability distribution is in fact two-dimensional. To remove the second dimension, p(µ) is assumed to depend on another parameter η. To express p(µ, η) in terms of the velocity v and the distance D, µ and η have to be substituted by v and D. This can be made in two equivalent ways, either η = D or η = v. To preserve the structure of the distance prior (eq.(26)), η is chosen to be equal to D.

∂µ ∂µ ∂µ ∂µ ∂v ∂D ∂v ∂D p(µ, η) = p(v, D) ∂η ∂η = p(v)p(D) (28) ∂v ∂D 0 1

Assuming that D and v are not correlated, p(v, D) can be written as the multiplication of the two proba- bilities. To obtain a probability distribution that only depends on µ, η has to be integrated over, while µ and η were substituted by vD and D respectively:

Z Z Z ∂µ 1 p(µ) = p(µ, η)dη = pv(v)pD(D) dD = pv(µD)pD(D) dD. (29) ∂v D

So the resulting probability distribution for µ is with the use of the prior from Verbiest et al. [2012] and the Maxwellian obtained by Hobbs et al. [2005] as a likelihood function for the transverse velocity of an 2.1 Theoretical Derivation 25 object:

p(µ) Z " 2# 1 1 ωmeas − 1/D ∝ 2 exp − D 2 σω 1 D D − D × erf √ low − erf √low 2 2σlow 2ωlow " !# 1 D − D × 1 + erf √up 2 2ωup |DsinG | R − R × R1.9D2exp − b − 5 0 E R " # 1 1 logS + 2logD + 1.12 × exp − D 2 0.9 1 µ2D2 × µ2D2exp − dD. (30) D 2a2

To obtain this equation, eq.(15), eq.(17), eq.(18), eq.(23) and eq.(25) were used. This result is the base of all the following calculations. The last term of eq.(30) is the Maxwellian distribution. To make sure the results come out right, the units have to be checked. Since the rms velocity for the Maxwellian is σ=265km/s, the units we use in the calculation are kpc and mas/year, the units need to be corrected. An additional factor is needed, since the rms σ refers to the magnitude of a three-dimensional where the proper motion is only two-dimensional. So four factors are needed:

• a1 converts seconds to years

• a2 converts km to kpc

◦ • a3 converts to mas

• a4 projects a three-dimensional vector onto two dimensions

The third factor is not that intuitive, it takes care of the unit mas, so that in fact the argument of the ex- ponential function is unit-less. The explanation is following: two right-angled triangles are constructed. In the ﬁrst one the magnitude of the adjacent with respect to the angle α is the proper motion µ times the distance D times a time interval ∆t. The opposite has the magnitude D. For small angles the angle α is calculated via the ratio of µ∆tD divided by D. The resulting angle α comes in units of milliarcseconds. We can construct the same triangle with the adjacent velocity v times ∆t. This time the angle α is in units of radians if the velocity is expressed in units of kpc per time. So we need a third factor that converts this angle in units of mas. As the traverse speed is just a two-dimensional vector but the distribution concerns three dimensional velocity vector, a fourth factor is needed. It is equal to the factor used by Hobbs et al. [2005]. If v is the magnitude of a three-dimensional vector and ~v is isotropically distributed via f(~v), it follows that f(~v) becomes f(v). For isotropic systems it is useful to use spherical coordinates in which the projection of a three-dimensional vector ~v onto a two-dimensional vector ~v2 the magnitude becomes v2 = vsin(ϑ). The mean value of v is:

R vf(~v)d~v 3 < v >= R . (31) R f(~v)d~v R3 26 2 APPLICATION OF THE LUTZ-KELKER BIAS TO PULSAR PROPER MOTIONS

If this is used to calculate the mean value of the two-dimensional vector divided by the mean of the three-dimensional vector integrated over a whole sphere the result is:

2π π R dϕ R dϑsin2(ϑ) < v > 1 π π 2 = 0 0 = [ϑ − sin(ϑ)cos(ϑ) ] = , (32) 2π π < v > 4 0 4 R dϕ R dϑsin(ϑ) 0 0 since the v dependency is reduced. So the complete factor for a from eq.(30) is

a = σ · a1 · a2 · a3 · a4 = 265 · 365 · 24 · 3600/3.0856776e7/4.848 · π/4. (33)

The complete posterior is the prior multiplied by the likelihood function given by the measurement of the proper motion, which is assumed to be a Gaussian.

2.2 Uncertainty Estimation For this thesis, all calculations were made via python programs. For this purpose a distance and a proper- motion interval are deﬁned. In the code they are represented as arrays with a certain step size. A package that is useful when working with arrays is the numpy package. It provides the possibility to let functions act on the complete arrays element-wise without establishing a loop and furthermore includes tools to perform matrix multiplication. This is very useful for a number of calculations in the present work. The step size of the proper-motion interval was chosen to be 0.01 mas/year and its range is from (0,250]. Since the output of the Bayesian analysis is a probability distribution, it is necessary to obtain some key values out of it, like the most likeliest value and its uncertainty. The most likeliest value is the largest value of the distribution. For the uncertainties a method was scripted as follows. For a Gaussian distribution the standard deviation of the mean value is rather simple to determine. In contrast, for an asymmetric uni-modal probability distribution, the one-σ uncertainties are obtained by comparing the integral limited by the uncertainties to the total integral of the probability distribution with respect to the most likeliest value.

b Z Z a, b are one-σuncertainties ⇔ p(x)dx = 0.683 · p(x)dx (34)

a R One way to do this is to choose arbitrarily a and b, to calculate the resulting integral and to compare it to the condition. In fact one can see that this is wrong. One condition is missing. p(a) has to be equal to p(b). So the uncertainties are defined by the two points of intersection of a constant and the graph of the probability distribution. To do this, an iteration is constructed, using the second condition as well. The first step of the iteration is to start with a value of p and to determine both x values for that p. If the left-hand side of eq.(34) is smaller than the right-hand side, using the x values as limits of the integration, a smaller value of p is used for the next step, otherwise a larger value of p is used. It is necessary to introduce the precision at which the condition must hold true, since the condition eq.(34) cannot fit exactly due to the discontinuity of the representation in the arrays. Another way of obtaining the uncertainties is by integrating the distribution starting at the maximum. For this method it is necessary to separate the distribution at the maximum into two functions. These two functions can be inverted. This is done by the function sepint. The code of the sepint method is displayed in §(5.3). In figure (5) the method sepint is represented by a flow chart. It starts with the input x,y and max where x and y are arrays corresponding to the domain respectively the codomain of 2.2 Uncertainty Estimation 27

the probability distribution, where each point (xi,yi) is connected via the same indices and max is also an array, representing the two coordinates (xmax,ymax) and the index of the maxmima of the probability distribution.

Fig. 5: Flow chart of the sepint method

In the next step the arrays x1,y1,x2 and y2 are declared. It follows a loop. In principle the loop runs through the whole x array with the index i. If the value of x[i] is smaller than the x value where the probability distribution is maximal then x[i] is add to the y1 array and y[i] is added to the x1 array. To make sure that the two new functions are integrated easily both start at the origin of ordinates and the (x1,y1) one is ﬂipped. This is done by the append statement. The result of the sepint method is shown in Fig.(6) 28 2 APPLICATION OF THE LUTZ-KELKER BIAS TO PULSAR PROPER MOTIONS

Fig. 6: A probability distribution before (top plot) and after (bottom plot) the sepint method. The blue graph shows the inverted right-hand side and the red graph the inverted left-hand side of the black graph split at the maximum.

In Fig.(6), the effect of the sepint method is displayed. The black probability distribution can easily be integrated from the top to the bottom using the blue and the red graph. The next step for obtaining the one-sigma uncertainties is to integrate both functions simultaneously and add both integrals up until eq.(34) is satisﬁed. It is important, to make sure that the second condition is true too, because of the different step width of the new x1 and x2 intervals. When eq.(34) is satisﬁed the resulting uncertainties are the values out of the arrays x1,x2 with the index i1,i2. 2.3 Examples 29

Fig. 7: Flow chart of the complete geterrorbars method

In Fig.(7) the complete method geterrorbars, that is used to obtain the one-σ uncertainty and the most likeliest value, is presented by a ﬂow chart.

2.3 Examples In this section, we illustrate how the code works. For this two artificial pulsars are introduced, to display the effect of the analysis in detail. In Fig.(8) and Fig.(9) on the left-hand side the various contributions of eq.(26) are shown multiplied by the Maxwellian eq.(30) and integrated over the distance. On the right- hand side, the resulting total posterior, the measurement likelihood distribution and the complete prior are shown where the posterior is just the multiplication of the prior and the likelihood distribution. The first thing to note is that the prior seems to be dominated by the parallax term. The modification of the value for luminosity S1400 changes the distribution not visibly. This still holds true for more extreme values of S1400. The distribution for the luminosity and the upper distance limit seem to not differ much from the behaviour of the Maxwellian. The behaviour of the galactic and lower distance term are pretty much the same. They peak at zero Proper motion, which is due to the fact that, for both distance distributions, larger distances are more likely. One the right-hand side, the effect of the prior 30 2 APPLICATION OF THE LUTZ-KELKER BIAS TO PULSAR PROPER MOTIONS

Fig. 8: J1700+2900 parallax = 1.5±0.1 Dlow = 0.5±0.5 Dup = 2±0.1 S1400 = 2 µtotal = 42.4±10 on the measurement is shown. The peak of the total distribution lies between those of the prior and the measurement likelihood, as expected

Fig. 10: Prior = 9.7 mas/yr measurement = 65.9 mas/yr, intensity plot of the measured proper motion plotted vs the corrected proper motion both in mas/year

Fig. (10) shows for an artificial pulsar, the behaviour of the posterior as a function of the measurement uncertainty. The lines indicate the one-sigma confidence interval of the measurement. In the plot one can see that the corrected proper motion may lie outside the one-σ confidence interval of the measurement. The grey area is the calculated probability distribution from eq.(30). For this particular pulsar, 2.3 Examples 31

Fig. 9: J1700-2900 parallax = 0.5±0.1 Dlow = 1±0.1 Dup = 1±0.5 S1400 = 0.1 µtotal = 42.4±10 the probability distribution does not converge directly to the measured value. In fact, if the significance of the measurement is between one and three, the probability distribution is not even consistent with the one-sigma confidence interval of the measurement. This effect indicates that there can be pulsar proper motion measurements that differ significantly from the prediction of this analysis. This issue is investigated further in the following chapter. This shows clearly, that the bias depends not only on the uncertainty, but also of the ratio of the measurement divided by its uncertainty, as predicted for parallax measurements by Lutz & Kelker. 32 3 RESULTS

3 Results

The sample of pulsar proper motion measurements in §3.1 and §3.2 are chosen not to be correct for stellar and galactic motion effects, since a precise measurement is not of interest, rather than an estimation of the nature of the bias in proper motion due to statistical effects. Since for the calculation only the magnitude of the proper motion measurement is of interest, the magnitude of the proper motion vector is referred to as µtotal. The errors for the total proper motion µtotal were obtained through Gaussian error propagation:

s 2 2 (µraσµra) + (µdecσµdec) σµtotal = 2 2 . (35) µra + µdec

Since we are interested in whether the corrected value is consistent with the measured value, we introduce µ − µ ∆ = corr meas . (36) σmeas ∆ is a measure of how strongly the corrected value is biased from the measured value. The sign of ∆ determines also, whether the measured proper motion is underestimated (+) or overestimated (-). If the magnitude of ∆ exceeds 1, the corrected value inconsistent with the measured value.

3.1 First sample To evaluate the presence of biases in published proper motion values, we analysed two sets of pulsars with known proper motion measurements. The ﬁrst sample is extracted from Verbiest et al. [2012], im- plying also distance information is available. The paper from Verbiest et al. [2012] provided a sample pulsars with distance information, especially the lower respectively upper HI distance are corrected. So for every pulsar out of this sample at least an upper respectively lower distance limit exists. Further constraint for the distance for every pulsar was set by the pulsar radio ﬂux density and a parallax measurement if available. All of those values were adopted from the table 1 out of Verbiest et al. [2012], where also a two-dimensional proper motion measurement was available in the ATNF Pulsar catalogue3 Manchester et al. [2005]. The sample consists of 26 pulsars. Nine pulsars out of this sample have a parallax measurement and two pulsar are millisecond pulsars with a rotation period smaller than 30 ms, which has to be considered concerning the different scale hight of the distribution of pulsars in the Galaxy.

In this sample there are no corrected proper motion measurement that are inconsistent with the measured values. The maximal ∆ is 0.94, and the minimal is -0.47. There are four entries in Tab.(2) with a ∆ larger than 0.7. For those, the significance of the measured total proper motion is smaller than two and in most cases deceeds one. What we expected to see from Fig.(10) is, that there exist proper motion measurements that are inconsistent within a significance ranging form one to four. In this sample, the corrected values that are the most biased are in fact non-measurements.For PSR J1823+0550 for example, the measurement uncertainty is twice as large as the measured value. The pulsars that also have a parallax measurement also have very precise proper motion measurements. Therefore their ∆ is also close to zero, with one exception. PSR J1939+2134 has a ∆ = −0.4. So it is slightly biased towards a smaller value, where the significance of the measurement is extremely high. In Fig.(11) PSR J1824−1945 and J1932+1059 are excluded because of the large uncertainties respectively for reasons of clarity. In this plot it becomes clear that for a population of pulsars, the proper motion tend to be underestimated, since most of the corrected values lie above the line of identity. From this

3The data is available online. See http://www.atnf.csiro.au/people/pulsar/psrcat/ 3.1 First sample 33

Fig. 11: Measured proper motion plotted against the corrected proper motion. In this plot two pulsars were excluded. The complete plot can be found in §5.3.1. 34 3 RESULTS

Pulsar name $meas Dlow Dup S1400 µra µdec µtotal µcorr ∆ J2000 mas kpc kpc mJy mas/yr mas/yr mas/yr mas/yr +0.34 J0332+5434 0.94±(0.11) 1.7±(0.7) 2.0±(0.8) 203 17.0±(0.3) -9.5±(0.4) 19.47±(0.33) 19.48−0.33 0.02 +0.3 J0358+5413 0.91±(0.16) 1.4±(0.7) 2.2±(0.9) 23 9.20±(0.18) 8.17±(0.39) 12.3±(0.29) 12.32−0.3 0.05 +1.62 J0738-4042 ...... 2.1±(0.6) ...... 80 -14.0±(1.2) 13±(2) 19.1±(1.62) 19.29−1.61 0.11 +2.01 J0742-2822 ...... 2.0±(0.6) 6.9±(0.8) 15 -29±(2) 4±(2) 29.27±(2.0) 29.37−2.0 0.05 +2.92 J0837-4135 ...... 1.8±(0.8) 6.0±(0.7) 16 -2.3±(1.8) -18±(3) 18.15±(2.98) 18.86−2.91 0.24 +0.89 J1453-6413 ...... 2.5±(0.5) ...... 14 -16±(1) -21.3±(0.8) 26.64±(0.88) 26.63−0.88 -0.01 +0.06 J1559-4438 0.384±(0.081) 2.0±(0.5) ...... 40 1.52±(0.14) 13.15±(0.05) 13.24±(0.05) 13.24−0.06 0.05 +2.64 J1745-3040 ...... 5.5±(0.6) 13 6±(3) 4±(2.6) 7.21±(2.88) 8.99−2.55 0.62 +4.49 J1752-2806 ...... 0.125±(0.025) ...... 18 -4±(6) -5±(5) 6.4±(5.41) 11.3−4.13 0.9 +2.09 J1803-2137 ...... 4.0±(0.6) 4.9±(0.3) 7.6 11.6±(1.8) 14.8±(2.3) 18.8±(2.12) 18.65−2.08 -0.07 +3.35 J1807-0847 ...... 1.5±(0.7) ...... 15 -5±(4) 1±(4) 5.1±(4.0) 8.58−3.09 0.87 +7.48 J1823+0550 ...... 1.6±(0.5) ...... 1.7 5±(11) -2±(4) 5.39±(10.32) 15.06−6.38 0.94 +12.12 J1824-1945 ...... 3.2±(0.5) ...... 4.9 -12±(14) -100±(220) 100.72±(218.44) 17.26−8.64 -0.38 +8.06 J1825-0935 ...... 1.9±(0.4) 12 -13±(11) -9±(5) 15.81±(9.48) 22.59−7.58 0.71 +6.08 J1832-0827 ...... 4.7±(0.3) 5.8±(0.3) 2.1 -4±(4) 20±(15) 20.4±(14.73) 13.49−5.38 -0.47 +0.05 J1857+0943 1.1±(0.2) 1.6±(0.5) 2.0±(0.4) 5 -2.64±(0.03) -5.46±(0.04) 6.06±(0.04) 6.07−0.05 0.14 +1.43 J1917+1353 ...... 4.8±(1.0) 5.7±(1.7) 1.9 0±(12) -6±(1.5) 6.0±(1.5) 6.52−1.41 0.35 +5.49 J1921+2153 ...... 2.8±(1.2) 6 17±(4) 32±(6) 36.24±(5.62) 37.22−5.46 0.18 +0.13 J1932+1059 2.77±(0.07) ...... 1.6±(0.5) 36 94.09±(0.11) 42.99±(0.16) 103.45±(0.12) 103.45−0.13 0.03 +0.17 J1935+1616 0.22±(0.1) 5.2±(1.7) ...... 42 1.13±(0.13) -16.09±(0.15) 16.13±(0.15) 16.13−0.16 0.0 +0.02 J1939+2134 0.13±(0.07) 4.6±(1.9) 14.8±(0.9) 10 0.072±(0.002) -0.415±(0.003) 0.423±(0.003) 0.42−0.01 -0.4 +3.62 J1946+1805 ...... 1.9±(0.7) 10 1±(5) -9±(4) 9.06±(4.01) 11.69−3.47 0.66 +0.92 J1952+3252 ...... 3.1±(2.0) ...... 1 -28.8±(.9) -14.7±(0.9) 32.33±(0.9) 32.35−0.91 0.02 +0.39 J2018+2839 1.03±(0.10) 3.2±(2.1) ...... 30 -2.6±(0.2) -6.2±(0.4) 6.72±(0.38) 6.76−0.38 0.1 +0.32 J2022+2854 0.37±(0.12) 3.1±(2.1) ...... 38 -4.4±(0.5) -23.6±(0.3) 24.01±(0.31) 24.01−0.32 0.01 +14.07 J2321+6024 ...... 2.6±(0.6) ...... 12 -17±(22) -7±(19) 18.38±(21.59) 23.48−10.73 0.24 Table 2: First sample of pulsars including parallax measurements, lower distance limit, upper distance limit, radio ﬂux and Proper motion measurements. The data was provided by Verbiest et al. [2012] as well as the ATNF pulsar catalogue Manchester et al. [2005] . observation we can conclude, that for this sample of pulsar proper motion measurements the measured values tends to be underestimated, which is in fact surprising, since we expected it to be overestimated compare §2.1. 3.2 Second sample 35

Fig. 12: Histogram of the number of pulsars from Tab.(3) with respect to ∆. In the green bins are the pulsars with a measurement signiﬁcance σ < 4. In the red bins are the pulsars with 2 < σ < 4. The blue bins show all the pulsars from Tab.(3). The total number of measurements is 202, where 66 are in the green and 21 in the red bins

3.2 Second sample To investigate the bias further, a larger sample is introduced, containing all the pulsars with a measured total proper motion and radio flux. The data were provided by the ATNF pulsar catalogue. In the histogram Fig.(12) the results of Tab.(3) are summarized. First to mention, there are corrected pulsar proper motions that are inconsistent with the measured proper motion. Another important result is that the corrected value is in most cases larger than the measured value. So for this sample the measured values are underestimated. This histogram shows, along with Tab.(4), that the measurements that are the most biased have very small measurement significance. In Tab.(4) all except three measurements are in fact non-measurements. PSR J1752-2806 and PSR J1807-0847 are in fact not that significant, with a significance little larger than one. PSR J1713+0747 and any similar measurements with extremely high precision and measurement significance, cannot be adequately evaluated by our analysis since their measurement precision is smaller than our step size (which is limited reasonable bounds on computing time). We therefore ignore these pulsars in the interpretation of our results. 36 3 RESULTS

Pulsar name S1400 µra µdec µtotal µcorr ∆ J2000 mJy mas/yr mas/yr mas/yr mas/yr +1.66 J0014+4746 3 19.3±1.8 -19.7±1.5 27.58±1.65 27.71−1.65 0.08 +0.48 J0024-7204 0.6 5.2±0.5 -3.4±0.4 6.21±0.47 6.28−0.47 0.14 +1.08 J0030+0451 0.6 -5.3±0.9 -2±2.0 5.66±1.1 6.06−1.07 0.36 +0.14 J0034-0721 11 10.37±0.08 -11.13±0.16 15.21±0.13 15.21−0.13 -0.02 +1.66 J0101-6422 0.28 10±1.0 -12±2.0 15.62±1.66 15.93−1.65 0.19 +0.09 J0139+5814 4.6 -19.11±0.07 -16.60±0.07 25.31±0.07 25.31−0.08 -0.04 +22.59 J0151-0635 1.3 15±47.0 -30±34.0 33.54±36.97 50.59−20.17 0.46 +2.0 J0152-1637 1.5 3.1±1.2 -27±2.0 27.18±1.99 27.4−1.98 0.11 +0.17 J0205+6449 0.045 -1.40±0.16 0.54±0.16 1.5±0.16 1.53−0.16 0.18 +23.9 J0206-4028 1.0 -10±25.0 75±35.0 75.66±34.85 68.1−22.67 -0.22 +0.09 J0218+4232 0.9 5.35±0.05 -3.74±0.12 6.53±0.08 6.53−0.09 0.03 +7.46 J0248+6021 13.7 48±10.0 48±4.0 67.88±7.62 67.49−7.44 -0.05 +13.69 J0255-5304 3 0±20.0 70±15.0 70.0±15.0 68.51−13.55 -0.1 +4.06 J0304+1932 3 6±7.0 -37±4.0 37.48±4.1 37.99−4.04 0.12 +5.13 J0323+3944 0.9 16±6.0 -30±5.0 34.0±5.24 34.8−5.1 0.15 +0.34 J0332+5434 203 17.0±0.3 -9.5±0.4 19.47±0.33 19.48−0.33 0.02 +0.3 J0358+5413 23 9.20±0.18 8.17±0.39 12.3±0.29 12.32−0.3 0.06 +0.08 J0437-4715 149 121.679±0.05 -71.820±0.09 141.29±0.06 141.29−0.07 -0.06 +2.01 J0452-1759 5.3 8.9±2.2 10.6±1.9 13.84±2.03 14.37−1.99 0.26 +0.08 J0454+5543 13 53.34±0.06 -17.56±0.14 56.16±0.07 56.16−0.08 0.05 +3.79 J0502+4654 2.5 -8±3.0 8±5.0 11.31±4.12 13.42−3.67 0.51 +6.66 J0525+1115 1.6 30±7.0 -4±5.0 30.27±6.97 32.02−6.57 0.25 +14.06 J0528+2200 9 -20±19.0 7±9.0 21.19±18.18 33.0−12.65 0.65 +0.81 J0534+2200 14 -14.7±0.8 2.0±0.8 14.84±0.8 14.91−0.8 0.09 +0.11 J0538+2817 1.9 -23.57±0.1 52.87±0.1 57.89±0.1 57.89−0.11 0.04 +6.75 J0543+2329 9 19±7.0 12±8.0 22.47±7.3 25.36−6.57 0.4 +7.02 J0601-0527 2.5 18±8.0 -16±7.0 24.08±7.57 26.97−6.83 0.38 +2.89 J0610-2100 0.4 7±3.0 11±3.0 13.04±3.0 14.12−2.84 0.36 +0.21 J0613-0200 2.3 1.84±0.08 -10.6±0.2 10.76±0.2 10.77−0.21 0.06 +4.6 J0614+2229 2.2 -4±5.0 -3±7.0 5.0±5.8 10.57−4.11 0.96 +0.32 J0621+1002 1.9 3.5±0.3 -0.3±0.9 3.51±0.31 3.57−0.32 0.18 +9.01 J0629+2415 3.2 -7±12.0 2±12.0 7.28±12.0 19.04−7.82 0.98 +0.94 J0630-2834 23 -46.30±0.99 21.26±0.52 50.95±0.93 50.96−0.94 0.01 +2.94 J0653+8051 0.4 19±3.0 -1±3.0 19.03±3.0 19.76−2.92 0.24 +0.64 J0659+1414 3.7 44.07±0.63 -2.40±0.29 44.14±0.63 44.14−0.63 0.01 +0.09 J0711-6830 3.2 -15.55±0.08 14.23±0.07 21.08±0.08 21.08−0.09 0.02 +0.56 J0737-3039 1.6 -3.82±0.62 2.13±0.23 4.37±0.55 4.5−0.54 0.23 Caption see Tab.(3) 3.2 Second sample 37

Pulsar name S1400 µra µdec µtotal µcorr ∆ J2000 mJy mas/yr mas/yr mas/yr mas/yr +0.56 J0737-3039 1.3 -3.82±0.62 2.13±0.23 4.37±0.55 4.5−0.55 0.23 +1.62 J0738-4042 80 -14.0±1.2 13±2.0 19.1±1.62 19.34−1.62 0.14 +2.01 J0742-2822 15.0 -29±2.0 4±2.0 29.27±2.0 29.44−2.0 0.08 +3.24 J0754+3231 0.6 -4±5.0 7±3.0 8.06±3.6 10.41−3.11 0.65 +4.61 J0758-1528 2.0 1±4.0 4±6.0 4.12±5.9 10.23−4.08 1.03 +0.37 J0814+7429 10 24.02±0.09 -44.0±0.4 50.13±0.35 50.13−0.36 0.0 +0.07 J0820-1350 7 21.64±0.09 -39.44±0.05 44.99±0.06 44.99−0.07 0.05 +8.11 J0823+0159 1.5 5±11.0 -1±8.0 5.1±10.9 17.06−7.05 1.1 +2.37 J0826+2637 10 61±3.0 -90±2.0 108.72±2.36 108.51−2.36 -0.09 +0.09 J0835-4510 1100 -49.68±0.06 29.9±0.1 57.98±0.07 57.98−0.08 -0.05 +3.0 J0837+0610 4 2±5.0 51±3.0 51.04±3.0 51.14−3.0 0.03 +2.93 J0837-4135 16.0 -2.3±1.8 -18±3.0 18.15±2.98 18.84−2.9 0.23 +30.17 J0846-3533 2.7 93±72.0 -15±65.0 94.2±71.83 65.15−28.83 -0.4 +10.34 J0908-1739 3.2 27±11.0 -40±11.0 48.26±11.0 49.63−10.22 0.13 +0.72 J0922+0638 4.2 18.8±0.9 86.4±0.7 88.42±0.71 88.41−0.72 -0.02 +12.58 J0943+1631 1.4 23±16.0 9±11.0 24.7±15.43 33.94−11.73 0.6 +11.58 J0944-1354 0.6 -1±32.0 -22±14.0 22.02±14.06 30.41−10.69 0.6 +0.09 J0953+0755 84 -2.09±0.08 29.46±0.07 29.53±0.07 29.53−0.08 -0.06 +0.03 J1012+5307 3 2.562±0.01 -25.61±0.02 25.74±0.02 25.74−0.03 0.11 +0.07 J1017-7156 1.00 -7.31±0.06 6.76±0.05 9.96±0.06 9.96−0.07 0.06 +0.28 J1024-0719 1.5 -35.3±0.2 -48.2±0.3 59.74±0.27 59.74−0.27 -0.01 +4.6 J1041-1942 4 -1±3.0 14±5.0 14.04±4.99 16.71−4.47 0.54 +0.22 J1045-4509 2.7 -6.0±0.2 5.3±0.2 8.01±0.2 8.01−0.21 0.02 +3.0 J1115+5030 3 22±3.0 -51±3.0 55.54±3.0 55.6−2.99 0.02 +13.75 J1116-4122 3 -1±5.0 7±20.0 7.07±19.81 26.8−11.55 1.0 +0.31 J1125-5825 0.86 -10.0±0.3 2.4±0.3 10.28±0.3 10.3−0.31 0.05 +1.18 J1239+2453 10 -104.5±1.1 49.4±1.4 115.59±1.16 115.53−1.17 -0.05 +0.08 J1300+1240 2 45.50±0.05 -84.70±0.07 96.15±0.07 96.15−0.08 0.04 +1.64 J1302-6350 1.70 -6.6±1.8 -4.4±1.4 7.93±1.69 8.42−1.61 0.29 +16.49 J1321+8323 0.9 -53±20.0 13±7.0 54.57±19.5 56.33−16.02 0.09 +18.65 J1328-4357 2 3±7.0 54±23.0 54.08±22.97 55.71−18.05 0.07 +4.63 J1337-6423 0.29 -6±6.0 -7±5.0 9.22±5.45 11.56−4.19 0.43 +6.73 J1405-4656 0.92 -44±6.0 20±10.0 48.33±6.85 48.63−6.72 0.04 +4.42 J1430-6623 8.0 -31±5.0 -21±3.0 37.44±4.47 37.85−4.41 0.09 +3.34 J1431-4715 0.73 -7±3.0 -8±4.0 10.63±3.6 12.25−3.23 0.45 +0.24 J1446-4701 0.40 -4.0±0.2 -2.0±0.3 4.47±0.22 4.49−0.23 0.08 +0.89 J1453-6413 14.0 -16±1.0 -21.3±0.8 26.64±0.88 26.67−0.88 0.03 +10.25 J1455-3330 1.2 5±6.0 24±12.0 24.52±11.81 29.87−9.66 0.45 Caption see Tab.(3) 38 3 RESULTS

Pulsar name S1400 µra µdec µtotal µcorr ∆ J2000 mJy mas/yr mas/yr mas/yr mas/yr +0.41 J1456-6843 80 -39.5±0.4 -12.3±0.3 41.37±0.39 41.37−0.4 -0.0 +11.78 J1502-6752 0.69 -6±9.0 -14±16.0 15.23±15.13 23.5−9.98 0.55 +0.09 J1509+5531 8 -73.64±0.05 -62.65±0.09 96.68±0.07 96.68−0.08 -0.06 +0.34 J1518+0204 0.039 4.67±0.14 -8.24±0.36 9.47±0.32 9.49−0.33 0.06 +0.05 J1518+4904 4 -0.67±0.04 -8.53±0.04 8.56±0.04 8.56−0.05 0.09 +0.03 J1537+1155 0.6 1.482±0.01 -25.285±0.01 25.33±0.01 25.33−0.03 0.13 +2.05 J1543-0620 2.0 -17±2.0 -4±3.0 17.46±2.06 17.88−2.05 0.2 +0.08 J1543+0929 5.9 -7.61±0.06 -2.87±0.07 8.13±0.06 8.13−0.07 -0.05 +1.61 J1543-5149 0.55 -4.3±1.4 -4±2.0 5.87±1.7 6.51−1.57 0.37 +0.38 J1550-5418 3.3 4.8±0.5 -7.9±0.3 9.24±0.36 9.26−0.37 0.04 +6.68 J1552-4937 0.14 -3±3.0 -13±8.0 13.34±7.82 14.88−5.88 0.2 +0.06 J1559-4438 40 1.52±0.14 13.15±0.05 13.24±0.05 13.24−0.06 0.05 +0.31 J1600-3053 2.5 -1.06±0.09 -7.1±0.3 7.18±0.3 7.2−0.3 0.07 +0.1 J1603-7202 3.1 -2.52±0.06 -7.42±0.09 7.84±0.09 7.84−0.1 0.04 +6.76 J1604-4909 5.5 -30±7.0 -1±3.0 30.02±7.0 31.15−6.68 0.16 +7.13 J1607-0032 5 -1±14.0 -7±9.0 7.07±9.13 16.24−6.37 1.0 +3.21 J1622-6617 0.60 -3±2.0 6±4.0 6.71±3.69 9.18−3.01 0.67 +4.35 J1623-2631 1.6 -13.4±1.0 -25±5.0 28.36±4.43 29.06−4.32 0.16 +0.08 J1640+2224 2 2.10±0.03 -11.20±0.07 11.4±0.07 11.4−0.08 0.07 +0.26 J1643-1224 4.8 5.99±0.1 4.1±0.4 7.26±0.24 7.27−0.25 0.05 +1.61 J1645-0317 21 -3.7±1.5 30.0±1.6 30.23±1.6 30.34−1.6 0.07 +1.25 J1708-3506 1.31 -5.3±0.8 -2±3.0 5.66±1.3 6.11−1.24 0.34 +6.56 J1709-1640 4 3±9.0 0±14.0 3.0±9.0 13.2−5.6 1.13 +0.89 J1709+2313 0.2 -3.2±0.7 -9.7±0.9 10.21±0.88 10.35−0.88 0.15 +0.03 J1713+0747 10.2 4.915±0.003 -3.914±0.01 6.283±0.001 6.28−0.02 -0.78 +1.93 J1719-1438 0.42 1.9±0.4 -11±2.0 11.16±1.97 11.66−1.92 0.25 +4.85 J1720-0212 1.0 -1±4.0 -26±5.0 26.02±5.0 27.06−4.81 0.21 +17.66 J1721-2457 0.58 1.8±1.8 -14±25.0 14.12±24.8 25.91−12.83 0.48 +20.82 J1722-3207 3.4 -1±5.0 -40±27.0 40.01±26.99 44.14−19.21 0.15 +0.72 J1727-2946 0.25 0.6±0.9 0±8.0 0.6±0.9 1.59−0.64 1.1 +2.56 J1731-1847 0.37 -1.7±0.3 -6±3.0 6.24±2.89 7.9−2.44 0.58 +0.24 J1732-5049 1.7 -0.51±0.11 -9.90±0.22 9.91±0.22 9.92−0.23 0.03 +2.98 J1735-0724 1.7 -2.4±1.7 28±3.0 28.1±2.99 28.41−2.97 0.1 +2.01 J1740+1311 3.9 -22±2.0 -20±2.0 29.73±2.0 29.9−2.0 0.08 +9.79 J1741-2054 0.16 -63±12.0 89±9.0 109.04±10.1 105.38−9.77 -0.36 +16.67 J1741-3927 4.7 20±15.0 -6±59.0 20.88±22.22 32.39−14.23 0.52 +0.05 J1744-1134 3.1 18.804±0.01 -9.40±0.06 21.02±0.03 21.02−0.04 -0.09 +2.49 J1745-0952 0.38 -21.2±1.1 11±5.0 23.88±2.5 24.06−2.49 0.07 +10.73 J1745-3040 13.0 6±3.0 4±26.0 7.21±14.64 19.78−8.79 0.86 +4.43 J1752-2806 18.0 -4±6.0 -5±5.0 6.4±5.41 10.75−4.02 0.8 Caption see Tab.(3) 3.2 Second sample 39

Pulsar name S1400 µra µdec µtotal µcorr ∆ J2000 mJy mas/yr mas/yr mas/yr mas/yr +0.1 J1756-2251 0.6 -2.42±0.08 0±20.0 2.42±0.08 2.42−0.09 0.0 +29.05 J1801-1417 0.17 -8±7.0 -23±45.0 24.35±42.56 23.96−14.16 -0.01 +7.31 J1801-2451 0.85 -11±9.0 -1±15.0 11.05±9.07 15.02−6.24 0.44 +6.91 J1801-3210 0.32 -8±2.0 -11±10.0 13.6±8.17 16.46−6.13 0.35 +0.04 J1802-2124 0.77 -0.85±0.1 4.8±0.0 4.87±0.02 4.87−0.03 -0.27 +2.12 J1803-2137 7.6 11.6±1.8 14.8±2.3 18.8±2.12 19.04−2.11 0.11 +3.35 J1807-0847 15.0 -5±4.0 1±4.0 5.1±4.0 8.46−3.07 0.84 +0.88 J1809-1943 4.5 -6.60±0.06 -11.7±1.0 13.43±0.87 13.5−0.88 0.08 +24.6 J1810-2005 1.33 0±2.0 17±37.0 17.0±37.0 29.61−16.57 0.34 +12.79 J1813-2621 0.65 -7.3±0.9 -22±16.0 23.18±15.19 26.52−11.14 0.22 +7.62 J1823+0550 1.7 5±11.0 -2±4.0 5.39±10.32 15.02−6.41 0.93 +32.77 J1824-1945 4.9 -12±14.0 -100±220.0 100.72±218.44 56.94−29.84 -0.2 +1.62 J1824-2452 2.0 -0.9±0.1 -4.6±1.8 4.69±1.77 5.68−1.57 0.56 +8.13 J1825-0935 12.0 -13±11.0 -9±5.0 15.81±9.48 20.67−7.51 0.51 +2.52 J1826-1334 2.1 23.0±2.5 -3.9±3.1 23.33±2.52 23.49−2.51 0.06 +32.26 J1829-1751 7.7 22±13.0 -150±130.0 151.6±128.64 64.06−30.99 -0.68 +12.3 J1832-0827 2.1 -4±4.0 20±15.0 20.4±14.73 24.73−10.87 0.29 +33.72 J1835-1106 2.2 27±46.0 56±190.0 62.17±172.31 51.64−29.35 -0.06 +32.66 J1836-1008 3.7 18±65.0 12±220.0 21.63±133.48 51.05−28.0 0.22 +3.45 J1840+5640 4 -30±4.0 -21±2.0 36.62±3.47 36.98−3.45 0.1 +0.22 J1843-1113 0.10 -2.17±0.07 -2.74±0.25 3.5±0.2 3.51−0.21 0.07 +9.36 J1843-1448 0.57 10.50±0.19 12±15.0 15.95±11.29 20.13−8.17 0.37 +6.13 J1844+1454 1.5 -9±10.0 45±6.0 45.89±6.2 46.14−6.11 0.04 +0.13 J1853+1303 0.4 -1.68±0.07 -2.94±0.12 3.39±0.11 3.39−0.12 0.04 +0.05 J1857+0943 5.0 -2.64±0.03 -5.46±0.04 6.06±0.04 6.07−0.05 0.14 +0.85 J1900-2600 13 -19.9±0.3 -47.3±0.9 51.32±0.84 51.32−0.84 0.01 +0.12 J1903+0327 1.3 -2.06±0.07 -5.21±0.12 5.6±0.11 5.61−0.12 0.07 +0.38 J1905+0400 0.050 -3.80±0.18 -7.3±0.4 8.23±0.36 8.24−0.37 0.03 +2.84 J1907+4002 1.8 11±4.0 11±1.0 15.56±2.92 16.38−2.82 0.28 +0.03 J1909-3744 2.1 -9.510±0.01 -35.859±0.02 37.1±0.02 37.1−0.03 0.07 +0.13 J1910+1256 0.5 0.21±0.1 -7.25±0.12 7.25±0.12 7.26−0.13 0.06 +0.07 J1910-5959 0.21 -3.08±0.06 -3.97±0.06 5.02±0.06 5.03−0.07 0.09 +2.02 J1910-5959 0.24 -4.1±1.7 -4.6±2.5 6.16±2.18 7.38−1.97 0.56 +10.93 J1911-1114 0.5 -6±4.0 -23±13.0 23.77±12.62 27.52−10.03 0.3 +8.98 J1913-0440 4.4 7±13.0 -5±9.0 8.6±11.8 18.51−7.71 0.84 +0.14 J1915+1606 0.9 -1.43±0.13 -0.70±0.13 1.59±0.13 1.61−0.13 0.14 +10.74 J1917+1353 1.9 0±12.0 -6±15.0 6.0±15.0 17.15−8.27 0.74 +0.22 J1918-0642 0.58 -7.20±0.1 -5.7±0.3 9.18±0.2 9.19−0.21 0.03 +17.43 J1919+0021 0.8 -2±30.0 -1±10.0 2.24±27.2 24.36−12.38 0.81 +5.53 J1921+2153 6 17±4.0 32±6.0 36.24±5.62 36.84−5.5 0.11 Caption see Tab.(3) 40 3 RESULTS

Pulsar name S1400 µra µdec µtotal µcorr ∆ J2000 mJy mas/yr mas/yr mas/yr mas/yr +0.13 J1932+1059 36 94.09±0.11 42.99±0.16 103.45±0.12 103.45−0.13 0.03 +0.17 J1935+1616 42 1.13±0.13 -16.09±0.15 16.13±0.15 16.13−0.16 0.0 +0.02 J1939+2134 13.2 0.072±0.0 -0.415±0.0 0.42±0.0 0.42−0.01 -0.4 +2.91 J1941-2602 3 12±2.0 -10±4.0 15.62±2.99 16.52−2.88 0.3 +3.6 J1946+1805 10 1±5.0 -9±4.0 9.06±4.01 11.39−3.44 0.58 +15.14 J1946-2913 0.8 19±9.0 -33±20.0 38.08±17.9 43.02−14.3 0.28 +0.61 J1948+3540 8.3 -12.6±0.6 0.7±0.6 12.62±0.6 12.67−0.61 0.08 +0.09 J1949+3106 0.23 -2.94±0.06 -5.17±0.08 5.95±0.08 5.95−0.09 0.03 +1.85 J1952+2630 0.085 -6±2.0 0±3.0 6.0±2.0 6.72−1.78 0.36 +0.91 J1952+3252 1.0 -28.8±0.9 -14.7±0.9 32.33±0.9 32.35−0.91 0.02 +11.83 J1954+2923 8 25±17.0 -36±10.0 43.83±12.71 45.42−11.64 0.13 +0.71 J1955+2527 0.28 -1.9±0.6 -2.4±0.8 3.06±0.73 3.35−0.69 0.4 +0.12 J1955+2908 1.1 -0.9±0.1 -4.1±0.1 4.2±0.1 4.2−0.11 0.02 +4.96 J1955+5059 4 -23±5.0 54±5.0 58.69±5.0 58.71−4.95 0.0 +0.59 J1959+2048 0.4 -16.0±0.5 -25.8±0.6 30.36±0.57 30.36−0.58 0.0 +1.83 J2013+3845 6.4 -32.1±1.7 -25±2.0 40.69±1.82 40.74−1.83 0.03 +0.39 J2018+2839 30 -2.6±0.2 -6.2±0.4 6.72±0.38 6.76−0.38 0.1 +0.32 J2022+2854 38 -4.4±0.5 -23.6±0.3 24.01±0.31 24.01−0.31 0.01 +0.3 J2022+5154 27 -5.23±0.17 11.5±0.3 12.63±0.28 12.64−0.29 0.02 +10.39 J2046-0421 1.7 9±16.0 -7±8.0 11.4±13.55 23.27−9.2 0.88 +5.27 J2046+1540 1.7 -13±6.0 3±4.0 13.34±5.92 16.7−5.04 0.57 +0.04 J2048-1616 13 113.16±0.02 -4.60±0.28 113.25±0.02 113.25−0.03 -0.15 +1.0 J2051-0827 2.8 5.3±1.0 0.3±3.0 5.31±1.01 5.66−0.98 0.35 +0.14 J2055+3630 2.6 1.04±0.04 -2.46±0.13 2.67±0.12 2.68−0.13 0.08 +1.23 J2108+4441 5.4 3.5±1.3 1.4±1.4 3.77±1.31 4.51−1.2 0.56 +2.88 J2113+2754 1.1 -23±2.0 -54±3.0 58.69±2.87 58.69−2.87 -0.0 +8.04 J2116+1414 0.8 8±15.0 -11±5.0 13.6±9.71 20.54−7.37 0.71 +0.26 J2124-3358 3.6 -14.12±0.13 -50.34±0.25 52.28±0.24 52.28−0.25 -0.01 +0.26 J2129+1210 0.2 -0.54±0.14 -4.33±0.25 4.36±0.25 4.39−0.25 0.11 +0.11 J2129-5721 1.1 9.35±0.1 -9.47±0.1 13.31±0.1 13.31−0.11 0.02 +0.6 J2144-3933 0.8 -57.89±0.88 -155.90±0.54 166.3±0.59 166.28−0.6 -0.04 +0.3 J2145-0750 8.9 -9.66±0.15 -8.9±0.4 13.13±0.29 13.15−0.3 0.05 +3.27 J2149+6329 2.9 14±3.0 10±4.0 17.2±3.37 18.15−3.24 0.28 +0.11 J2157+4017 17 16.13±0.1 4.12±0.12 16.65±0.1 16.65−0.11 0.02 +6.12 J2219+4754 3 -12±8.0 -30±6.0 32.31±6.31 33.47−6.06 0.18 +3.0 J2225+6535 2.0 144±3.0 112±3.0 182.43±3.0 181.68−3.0 -0.25 +3.84 J2229+2643 0.9 1±4.0 -17±4.0 17.03±4.0 18.46−3.78 0.36 +0.45 J2240+5832 2.7 -6.1±0.8 -21.0±0.4 21.87±0.44 21.88−0.45 0.03 +1.99 J2305+3100 2.2 2±2.0 -20±2.0 20.1±2.0 20.42−1.98 0.16 +6.93 J2308+5547 1.9 -15±8.0 0±27.0 15.0±8.0 19.4−6.47 0.55 Caption see Tab.(3) 3.2 Second sample 41

Pulsar name S1400 µra µdec µtotal µcorr ∆ J2000 mJy mas/yr mas/yr mas/yr mas/yr +0.12 J2313+4253 15 24.15±0.1 5.95±0.13 24.87±0.1 24.87−0.11 -0.02 +2.78 J2317+1439 4 -1.7±1.5 7.4±3.1 7.59±3.04 9.48−2.68 0.62 +15.86 J2321+6024 12 -17±22.0 -7±19.0 18.38±21.59 33.0−13.77 0.68 +1.92 J2330-2005 3 74.7±1.9 5±3.0 74.87±1.91 74.82−1.91 -0.03 +12.34 J2337+6151 1.4 -1±18.0 -15±16.0 15.03±16.01 24.74−10.52 0.61 +2.92 J2354+6155 5 22±3.0 6±2.0 22.8±2.94 23.26−2.9 0.15

Table 3: Sample of 202 pulsar proper motion measurements and the S1400 radio ﬂux measurements. The data was provided by the ATNF pulsar catalogue Manchester et al. [2005].

Pulsar µtotal µcorr ∆ σ +4.6 J0614+2229 0.5±5.8 10.6−4.1 1 0.09 +9.0 J0629+2415 7.3±12 19.0−7.8 1 0.61 +4.6 J0758-1528 4.1±5.9 10.2−4.1 1 0.7 +8.1 J0823+0159 5.1±11.0 17.1−7.1 1.1 0.46 +13.8 J1116-4122 7.1±19.8 26.8−11.6 1 0.36 +7.1 J1607-0032 7.1±9.1 16.2−6.4 1 0.77 +6.6 J1709-1640 3±9 13.2−5.6 1.1 0.33 +0.7 J1727-2946 0.6±0.9 1.6 −0.6 1.1 0.67

+4.4 J1745-3040 7.2±14.6 19.8−4.0 0.9 0.49 +7.6 J1823+0550 5.4±10.3 15.4−6.4 0.9 0.52 +10.4 J2046-0421 11.4±13.6 23.3−9.2 0.9 0.84

+0.03 J1713+0747 6.283±0.001 6.28−0.02 0.8 6283 +4.4 J1752-2806 6.4±5.4 10.8−4.0 0.8 1.19 +3.4 J1807-0847 5.1±4.0 8.5−3.1 0.8 1.28 +9.0 J1913-0440 8.6±11.8 18.5−7.7 0.8 0.73 +17.4 J1919+0021 2.2±27.2 24.412.4 0.8 0.08

Table 4: Subset of pulsars from Tab.(3) with high values for ∆ (∆ > 0.7). σ = µtotal is the σtotal signiﬁcance of the measurement 42 4 CONCLUSION AND FUTURE RESEARCH

4 Conclusion and Future Research

First and foremost, we showed that there is the possibility that a proper motion measurement, for a large value with a uncertainty smaller than 4 sigma, is inconsistent with the prior information as seen in Fig.(10). However in the two samples used to investigate this issue,no clear evidence was found that such a bias exists in published data. In fact all measurements that are inconsistent with the corrected value are insignificant. The more precise measurements are all consistent with the analysis and are not strongly biased. So in conclusion, the effect is negligible for a pulsars with sufficiently precise measurements. For a sample of pulsars where not so precise measurements are included there is a bias towards higher values. So the measured Proper motion is underestimated. This is a contradiction to the statement, that since the magnitude of the proper motion is proportional to the distance, a underestimation in the distance measurement that was predicted by Lutz & Kelker leads to an overestimation of the proper motion measurement. If the proper motion of a sample of pulsars is underestimated, Pulsars could in fact be even faster objects than was assumed so far. This could lead to a rethinking of the mechanisms, that are responsible for the high velocities. Overall the bias in the proper motion measurements is not as significant as the bias in the distance respectively parallax of a pulsar.

Since the luminosity and galactic distribution used in this thesis are poorly constrained, their effect on this analysis is limited. In fact the prior is dominated by the parallax measurements. So a more constrained distribution for these priors could provide a better insight into the nature of this bias. More quality and quantity of data will provided more detailed and more constrained distributions for the radio ﬂux, the galactic density and the velocity of pulsars. Additional constraints on the velocity of a pulsar could also be useful, for instance the measurement of the radial velocity of the pulsar. The discovery of new pulsars will lead to a more detailed pulsar density distribution. Additional information of the bias can be obtained by tracking historic data, how data on pulsar proper motion converges with increasing measurement precision. This could provide insight in whether the proper motion measurement converges directly to the values we obtained for the corrected proper motion or if it ﬁrst converges to a different value compare Fig.(10). REFERENCES 43

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5 Appendix 5.1 Functions Here is the code displayed used to calculate the total posterior function. For this all contributions of the prior compare eq.(30) are implemented as well as a Gaussian distribution for the likelihood function of the measured proper motion. def Gaus(Pxm, Pxu, dist): #dist is a vector return np . exp ( − 1 . 0 / 2 . 0 ∗ ( ( Pxm−1.0/dist)/Pxu) ∗∗2) / d i s t ∗∗2 # exp applies elementwise to the vector def Gaussian(x_meas ,sigma ,x) return np . exp ( − 1 . 0 / 2 . 0 ∗ ( x_meas−x ) ∗ ∗ 2 . / sigma ∗ ∗ 2 . ) # r e a l g a u s s i a n def Dlow(Dl,Dlu, dist): return erf(Dl/(np.sqrt (2.) ∗Dlu ) )−e r f ( ( Dl−dist)/(np.sqrt(2.) ∗ Dlu ) ) def Dup(Du,Duu, dist ): return 1.+ e r f ( ( Du−dist)/(np.sqrt(2.) ∗Duu ) ) def Lum(S1400, dist ): return np . exp ( − 1 . / 2 . ∗ ((np.log10(S1400)+2.∗ np.log10(dist)+1.1) / 0 . 9 ) ∗ ∗ 2 . ) / d i s t def Maxwell(dist ,mu): a = 2 6 5 . ∗ (2./math.sqrt(math.pi/2.)) ∗(365∗24∗3600/3.0856776e7 / 4 . 8 4 8 ) #a is given as 265km/s; first factor is for the mean of the maxwellian, second is to get the units in mas/ yr return ( ( d i s t ∗mu) ∗∗2.∗ np . exp ( −( d i s t ∗mu / a ) ∗ ∗ 2 . / 2 . ) / a ∗ ∗ 3 . ) def Galactic(Gl,Gb, dist): R0=8.5 # kpc E=0.33 #kpc for common puslars R=np. sqrt (R0 ∗ ∗ 2 . + ( d i s t ∗np.cos(Gb)) ∗∗2. −2.∗R0∗ d i s t ∗np . cos (Gb) ∗ np.cos(Gl))

i f (data[line ][element]!=’ ’ and data[line][ element] != ’\t’ and data[line ][element] != ’ \ n ’ ) :

dummy += data[line ][element] 5.2 The Geterrorbars Method 45

pool = True

i f (data[line][element] == ’ ’ or data[line][ element] == ’\t’) and pool or data[line][ element] == ’\n’:

5.2 The Geterrorbars Method In this section the code of the geterrorbar method is displayed. It is used to calculate the one-σ uncertainties of an uni-modal probability distribution. Compare §2.2. def geterrorbars(pm_prob): i_max = np.argmax(pm_prob[1]) maxm = [pm_prob[0][i_max] ,pm_prob[1][i_max] ,i_max] area=inte (pm_prob[0][0] ,pm_prob[0][ − 1],pm_prob[0] ,pm_prob[1]) a r e a 1 =0. a r e a 2 =0. inverse=sepint (pm_prob[0] ,pm_prob[1] ,maxm) condition=True index1 =1 index2 =0 lowup=[0. ,0.] while c o n d i t i o n : # in principle a integration but for two seperate intervalls while inverse[0][ − index1] < inverse[2][index2]: # t a k e s care of the different step sizes width1=inverse[0][ − index1 −1]− inverse[0][ − index1 ] c1=inverse[1][ − index1 −1]− inverse[1][ − index1 ] area1+=width1 ∗ inverse[1][ − index1 ] −1/2∗ width1 ∗ c1 index1 +=1 while inverse[2][index2] < inverse[0][ − index1 ] : width2=inverse [2][index2]− inverse [2][index2 −1] c2=inverse [3][index2]− inverse [3][index2 −1] area2+=width2 ∗ inverse [3][index2] −1/2∗ width2 ∗ c2 index2 +=1 i f inverse[0][ − index1]==inverse [2][index2]: area1+=(inverse[0][ − index1 −1]− inverse[0][ − index1 ] ) ∗ inverse[1][ − index1 ] −1/2∗(inverse[0][ − index1 −1]− inverse[0][ − index1 ] ) ∗(inverse[1][ − index1 −1]− inverse[1][ − index1 ] ) area2+=(inverse [2][index2]− inverse [2][index2 −1]) ∗ inverse [3][index2] −1/2∗(inverse [2][index2]− i n v e r s e [ 2 ] [ index2 −1]) ∗(inverse [3][index2]− inverse[3][ index2 −1]) index1 +=1 index2 +=1 total_area=area1+area2+inverse [3][index2] ∗ ( maxm[1] − i n v e r s e [2][index2])+inverse[1][ − index1 ] ∗ ( maxm[1] − inverse[0][ − 46 5 APPENDIX

index1 ] ) condition=(total_area)/area < 0.683

return (maxm[0] , inverse[1][ − index1],inverse [3][index2])

5.3 The Sepint Method

This is the code used to implement the sepint method. In §2.2 it use and its concept is explained. The ﬂow chart of this code can be seen in Fig.(5).

def sepint(x,y, max): #inverts y(x) to x(y) and separates it in to arrays at the maxmimum y1,y2,x1,x2=[] ,[] ,[] ,[] f o r i in range ( l e n ( x ) ) : i f i < max [ 2 ] : y1 . append(−x [ i ]+max [ 0 ] ) x1 . append(−y [ i ]+max [ 1 ] ) e l s e : y2.append(x[i]−max [ 0 ] ) x2 . append (max[1] −y [ i ] )

return (x1,y1,x2,y2) 5.3 The Sepint Method 47

5.3.1 Graphics

Fig. 13: Actual scatterplot compare Fig. (11)