Dynamics of Self-Gravitating Disks

University of Cambridge May, 1994 Dynamics of self-gravitating disks Christophe Pichon Clare College and Institute of Astronomy A dissertation submitted to the University of Cambridge in accordance with the regulations for admission to the degree of Doctor of Philosophy ii Summary A fair fraction of observed galactic disks present a bar corresponding to elongated isophotes beyond the central core. Numerical simulations also suggest that bar instabilities are quite common. Why should some galaxies have bars and others not? This question is addressed here by investigating the stability of a self gravitating disk with respect to instabilities induced by the adiabatic re-alignment of resonant orbits. It is shown that the dynamical equations can be recast in terms of the interaction of resonant flows. Concentrating on the stars belonging to the inner Lindblad resonance, it is argued that this interaction yields an instability of Jeans’ type. The Jeans instability traps stars moving in phase with respect to a growing potential; here the azimuthal instability corresponds to a rotating growing potential that captures the lobe of orbital streams of resonant stars. The precession rate of this growing potential is identified with the pattern speed of the bar. The instability criterion puts constraints on the relative dispersion in angular frequency of the underlying distribution function. The outcome of orbital instability is investigated while constructing distribution functions induced by the adiabatic re-alignment of inner Lindblad orbits. These distribution functions maximise entropy while preserving the underlying axisymmetric component of the galaxy and the constraints of angular momentum, total energy, and detailed circulations conservation. Below a given temperature, the disk prefers a barred configuration with a given pattern speed. A analysis of the thermodynamics of these systems is presented when the interaction between the orbits is independent of their shapes. Axisymmetric distribution functions characterise the equilibrium state of a galaxy. Their formal expression in terms of the invariants of the motion is desirable both from a theoretical and observational point of view: they constrain observed radial and azimuthal velocity distributions measured from the absorption lines in order to asses the gravitational nature of the equilibrium; they correspond to the core of all numerical and linear analyses studying non-axisymmetric features to describe the unperturbed axisymmetric initial state. A general inversion method is presented here leading to the construction of distribution functions chosen to match either a given potential-density pair and a given temperature profile, or alternatively to account for detailed observed kinematics. Complete sequences of new analytic solutions of Einstein’s equations which describe thin super massive disks are constructed. These solutions are derived geometrically. The identification of points across two symmetrical cuts through a vacuum solution of Einstein’s equations defines the gradient discontinuity from which the properties of the disk can be deduced. The subset of possible cuts which lead to physical solutions is presented. At large distances, all these disks become Newtonian, but in their central regions they exhibit relativistic features such as velocities close that of light, and large redshifts. For static metrics, the vacuum solution is derived via line superposition of fictitious sources placed symmetrically on each side of the disk by analogy with the classical method of images. The corresponding solutions describe two counter-rotating stellar streams. Sections with zero extrinsic curvature yield cold disks. Curved sections may induce disks which are stable against radial instability. The general counter rotating flat disk with planar pressure tensor is found. Owing to gravomagnetic forces, there is no systematic method of constructing vacuum stationary fields for which the non-diagonal component of the metric is a free parameter. However, all static vacuum solutions may be extended to fully stationary fields via simple algebraic transformations. Such disks can generate a great variety of different metrics including Kerr’s metric with any ratio of a to m. A simple inversion formula is given which yields all distribution functions compatible with the characteristics of the flow, providing formally a complete description of the stellar dynamics of flattened relativistic disks. Cambridge February 1994 iv Summary This dissertation is the outcome of my own work, except where explicit reference has been made to the work of others. Section 2 of chapter 2 has already been published as Pichon, C. “Orbital instabilities in galaxies”, contribution to the “Ecole de Physique des Houches” on: Transport phenomena in Astrophysics, Plasma Physics, and Nuclear Physics. In press. The idea of applying thermodynamics to study the evolution of bars within galaxies was suggested to me by Jim Collett. Most of the material presented in chapter 4 was published as B`ıcák, J., Lynden-Bell, D & Pichon C. “Relativistic discs and flat galaxy models”, Monthly Notices of the Royal Astronomical Society (1993) 265 126-144, where the basic method was suggested by Lynden-Bell, and carried out independently B`ıcák, Lynden-Bell and myself. The material of chapter 5 has been submitted to Physical Review D for publication as: Pichon, C. & Lynden-Bell, D. “New sources of Kerr and other metrics: Rapidly rotating relativistic disks with pressure support”, and the main ideas developped here are presented in Pichon, C. & Lynden-Bell, D.“Quasar engines and relativistic disk models”, contribution to the conference: N-body problems and gravitational dynamics, CNRS Editions. I hereby declare that this dissertation is not substantially the same as any that I have submitted for a degree or diploma or other qualification at any other University, and no part of it has been or is submitted for any such degree, diploma or any other qualification. This dissertation does not exceed 60,000 words in length. Christophe Pichon Cambridge February 1994 vi vii Dedicated to the memory of John Treherne viii ix Un de mes amis definissait l’observateur comme celui qui ne comprend riena ` la théorie, et le théoricien comme un exalté qui ne comprend riena ` rien. L. Boltzmann (1890) x Contents xii Contents 1 Prologue Various aspects of the dynamics of self-gravitating disks are presented here. Chapter 2 ad- dresses the stability of thin axisymmetric galactic stellar disks with respect to non-axisymmetric perturbations. Models of such disks accounting for observed or parameterised kinematics are constructed in chapter 3. Chapters 4 and 5 present a derivation of the relativistic stellar dynamics of flattened systems applied to models of rotating and counter-rotating super-massive disks. The dynamics of disks has been the subject of many excellent reviews in recent years, of which those of Sellwood (1993) [101] for bars, and Tremaine (1989) [109] are particularly worthy of mention for their scope and detail. In view of this, given that each chapter has its own introduction, a different perspective is taken in this prologue: the main common features and hypotheses, such as that of an axisymmetric infinitely thin disk, are sketched. These assumptions provide the mathematical simplicity which allows more detailed investigation without resorting to sophisticated numerical analysis. The infinitely thin disk approximation is used throughout as it reduces the number of dimensions involved, and is essential in constructing solutions of Einstein’s equation geometrically. As in electromagnetism, the properties of the distribution of sources is deduced in this context from the jump conditions at the boundary of the discontinuity. This approximation allows tremendous simplification in comparison with the fully three dimensional set of Einstein’s equations. This assumption together with that of axisymmetry makes the distribution function inversion problem a degenerate case (two degrees of functional freedom in the distribution function varying with energy and angular momentum, only one functional constraint while requiring that the distribution adds up to the right surface density). In effect, the thin disk approximation corresponds to an extra explicit integral of the motion which allows the inversion while also requiring that the solution satisfies all observed or prescribed kinematical properties. The infinitely thin disk approximation turns out to be useful also when studying the stability of the disk as it reduces the number of phase space dimensions required to describe the dynamics to twice the number of available invariants for an axisymmetric disk. This in turn implies that the motion of stars is regular (not chaotic), and suggests labelling the trajectories by combinations of these invariants called action variables. These variables have two attractive properties: the conjugate variable varies uniformly between zero and 2π as the star describes one oscillation of the corresponding degree of freedom1; actions may behave adiabatically, i.e. remain approxi- mately constant, as the slow perturbation is affecting the symmetry on which the conservation of these actions relies. All disks are assumed to have initially reached a stationary equilibrium. The energy of each 1 The formalism of angle and action variables is presented in appendix A. Chapter 1 Prologue 1 star, or fluid element, is therefore an invariant of the motion corresponding to the fact that the

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