UNIVERSITY BIELEFELD FACULTY OF PHYSICS A Statistical Study of Pulsar Proper Motions 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 9 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 magne- tized 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 emis- sion 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 ob- jects. 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 ex- isting objects. Sources of repulsive interactions are kinetic effects or fundamental forces like electromagnetic and nuclear forces. Examples of kinetic effects are angular momentum, hydro- dynamic 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 an- gular 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 pro- cesses 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.