Oscillations in Neutron Stars

o n o ^ 5 t NTNU Trondheim f j t Norwegian University of £ s Science and Technology </i s S/T Dr. scient. thesis IP Faculty of Physics, Informatics and Mathematics P Department of Physics B'oo M CZ> Cl ct> £ p £ O po CO C/3 (Z> r-h r-f P sP * tzi CD EC Q H CD Oscillations in Neutron Stars by Gudrun Kristine Høye Dr. scient. thesis Department of Physics Faculty of Physics, Informatics and Mathematics NTNU 1999 Acknowledgements. The work in this thesis has been performed during the years 1995-99 at the Norwegian Uni- versity of Science and Technology (NTNU), Trondheim. The field of astrophysics has always fascinated me, and I am grateful for having been given the opportunity to do research in this field. I would like to give a special thank to my supervisor Professor Erlend Østgaard, who initiated this work, and who has been always helpful, supportive, and encouraging during the work with my thesis. I would also like to thank Professor Kåre Olaussen, who accepted the role as my formal supervisor during the first part of my thesis, when this was required due to funding reasons. Finally, I would like to thank my present employer, the Norwegian Defence Research Establishment (FFI), who has been helpful and kind enough to let me spend some of their time to complete and finish this work. This thesis has been funded by the Research Council of Norway (NFR), and financial support has also been received from the Department of Physics, NTNU. Kjeller, October 1999 Gudrun K. Høye Abstract We have studied radial and nonradial oscillations in neutron stars, both in a general relativistic and non-relativistic frame, for several different equilibrium models. Different equations of state were combined, and our results show that it is possible to distinguish between the models based on their oscillation periods. We have particularly focused on the p-, /-, and g-modes. We find oscillation periods of II ~ 0.1 ms for the p-modes, II ~ 0.1 — 0.8 ms for the /-modes, and II ~ 10 — 400 ms for the g-modes. For high-order (I > 4) /-modes we were also able to derive a formula that determines IIi+i from II; and IIi_i to an accuracy of 0.1%. Further, for the radial /-mode we find that the oscillation period goes to infinity as the maximum mass of the star is approached. Both p-, /-, and g-modes are sensitive to changes in the central baryon number density nc, while the g-modes are also sensitive to variations in the surface temperature. The p-modes are concentrated in the surface layer, while p- and /-modes can be found in all parts of the star. The effects of general relativity were studied, and we find that these are important at high central baryon number densities, especially for the p- and /-modes. General relativistic effects can therefore not be neglected when studying oscillations in neutron stars. We have further developed an improved Cowling approximation in the non-relativistic frame, which eliminates about half of the gap in the oscillation periods that results from use of the ordinary Cowling approximation. We suggest to develop an improved Cowling approximation also in the general relativistic frame. Contents 1 Introduction. 1 2 General theory. 3 2.1 Neutron stars 3 2.1.1 Supernova; the birth of a neutron star 3 2.1.2 The equilibrium state 5 2.1.3 Pulsars 8 2.2 Oscillations in neutron stars 9 2.2.1 The oscillation spectrum 10 2.2.2 Observations 10 3 Radial and nonradial oscillations; the Newtonian limit. 13 3.1 The Eulerian and Lagrangian formalism 14 3.2 The basic equations of the problem 16 3.2.1 The gravitational potential 16 3.2.2 Conservation of mass 17 3.2.3 Conservation of momentum 18 3.2.4 Conservation of energy. 18 3.3 Radial oscillations 21 3.3.1 The radial oscillation equation 21 3.3.2 Boundary conditions 24 3.3.3 The eigenvalue nature of the problem 25 3.4 Nonradial oscillations 27 3.4.1 The nonradial oscillation equations 27 3.4.2 Boundary conditions 31 3.4.3 The Cowling approximation 32 i Contents 3.4.4 The improved Cowling approximation 32 3.4.5 The eigenvalue nature of the problem 35 3.4.6 Classification of modes 35 3.4.7 Local analysis 41 4 Radial and nonradial oscillations; general relativistic treatment. 47 4.1 General relativity; a short introduction 48 4.2 The gravitational field of a spherical symmetric body; the Schwarzschild metric. 50 4.2.1 Einstein's field equations 51 4.2.2 Conservation of mass-energy, momentum, and baryon number 52 4.2.3 The equations of hydrostatic equilibrium 53 4.2.4 Linearizing the basic equations of the problem 55 4.3 Radial oscillations 57 4.3.1 The radial oscillation equations 57 4.3.2 Boundary conditions 60 4.3.3 The eigenvalue nature of the problem 61 4.4 Nonradial oscillations 61 4.4.1 The nonradial oscillation equations; the relativistic Cowling approximation 62 4.4.2 Boundary conditions 64 4.4.3 Local analysis 65 5 The equilibrium state. 69 5.1 The ideal neutron gas 71 5.2 The ideal Fermi-gas of quarks 74 5.3 The asymptotic MIT bag model for quark matter 76 5.4 The perturbative quantum chromodynamics (QCD) model for quark matter. 77 5.5 The Pandharipande-Smith equation of state (PS) 79 5.6 The Arntsen-Øvergård equation of state (A0-5) 80 5.7 The Wiringa-Fiks-Fabrocini equations of state (WFF) 82 5.8 The Baym-Bethe-Pethick equation of state (BBP) 84 5.9 The Haensel-Zdunik-Dobaczewski equation of state (HZD) 88 5.10 The ideal electron gas 90 5.11 The temperature dependent equation of state 92 ii Contents 5.11.1 The radiation pressure 92 5.11.2 The ion pressure 94 5.11.3 The electron pressure 96 6 Results. 101 6.1 The equilibrium state 101 6.2 The oscillations 109 6.2.1 The influence of general relativity on the oscillations 109 6.2.2 The Cowling approximation/improved Cowling approximation 125 6.2.3 General features of the oscillations (in a general relativistic frame). 134 7 Discussion. 159 7.1 The equilibrium state 159 7.2 The oscillations 163 7.2.1 The influence of general relativity on the calculations 163 7.2.2 The Cowling approximation/improved Cowling approximation 165 7.2.3 General features of the oscillations (in a general relativistic frame). 166 7.3 Accuracy of the calculations 172 7.4 Comparison with results in the literature 173 7.5 Observations 174 7.6 Topics for further study. 174 7.6.1 Rotation 175 7.6.2 Magnetic fields 175 7.6.3 Superfluidity. 176 7.6.4 Nonadiabatic effects 176 7.6.5 Non-linearity. 178 7.6.6 Other 178 8 Conclusions. 179 A Constants and units. 187 B The effective charge of the ions, {Zeg). 189 C The opacity. 193 iii Contents D The Taylor expansion of eigenfunctions. 197 D.l The Taylor expansion of the radial eigenfunction £ at the centre and at the surface of the star 197 D.2 The Taylor expansion of the nonradial eigenfunctions v, w, and <j> at the centre of the star 200 D.3 The Taylor expansion of the radial eigenfunctions £ and r) at the centre of the star 203 D.4 The Taylor expansion of the nonradial eigenfunctions V and W at the centre of the star 205 E Software. 209 iv Chapter 1 Introduction. Recently a new field of research called "helio- and astero-seismology" has opened up, in which the internal structure of the Sun and stars can be probed using their oscillations. Stars are like musical instruments in the way that they axe able to oscillate with variations in wave length or in frequency, and the modes of the oscillation differ from one star to another and change as the star evolves. Neutron stars, for instance, are believed to oscillate with oscillation periods from ~ 0.1 ms to ~ 500 ms corresponding to frequencies of 2 — 10,000 Hz. This is within the frequency range of the human ear, and if we travelled close by an oscillating neutron star we could listen to the "music" from the star (that is, if it was surrounded by a medium through which the sound waves could be transmitted). By studying the oscillations of a star, we can probe its internal structure, similarly to what is done on Earth by the use of seismic waves. Helio-seismology has been very successful in studying the internal structure of the Sun, and it is the hope that astero-seismology will prove equally successful in studying the interior of other stars. The theory of stellar pulsation was originally developed in order to explain the pulsations of classical variable stars such as the Cepheids and RR Lyrae Stars. These stars are believed to pulsate radially, and radial pulsations have therefore been studied for a while. The theoretical study of nonradial pulsations, however, has remained largely within academic circles until recently. We will give a brief overview of the history of the work in the field of stellar pulsations below. The theory of nonradial pulsations was described first by Lord Kelvin [51] in 1863, while the theory for radial pulsations came a few years later (Ritter [44] in 1879). In 1938 Pekeris [40] obtained the exact analytic solution for adiabatic nonradial oscillations in the homogeneous compressible model, and Cowling [14] extended the study to the polytropic model in 1941.

Load more