Some exact analytic models of time dependent collisionless stellar systems.

Abstract

This thesis began with the general theme of studying the time-dependent behavior of collisionless self-gravitating systems. It is rather hard to obtain exact, general results by analytical methods (see Mathur 1986 for an attempt in the linearized case). The traditional route chosen has been numerical, and considerable progress has recently been made in this direction (see, e.g., Barnes 1989). This approach will certainly allow the study of cases without special symmetry or other simplifying assumptions, at least when enough computing power is available to handle three space dimensions and a large (?10?) number of particles. The analytical studies that gave so much insight into steady-state systems (see, e.g., BT) are all too rare in the time-dependent case. To our knowledge, Kalnajs’ (1973) model was the only one before this work wherein time-dependent behavior was studied analytically without approximation. The work reported in this thesis has now added four distinct families of exact, analytic, time-dependent models (described in detail in Chapter 3, Chapter 4, and Appendix A). The shell models of Appendix A are cold, occupying a four-dimensional surface in six-dimensional phase space. Time-dependent generalizations of Freeman’s spheroid also occupy a four-dimensional region, while the time-dependent generalizations of Polyachenko’s “hot” spheroid occupy a five-dimensional region. It is not clear whether these models will be stable to small disturbances. In any case, it seems hard to find situations in which such models could form, and they should basically be regarded as illustrations of behavior allowed by the collisionless Boltzmann equation (CBE). Viewed as such, they demonstrate two interesting possibilities: (i) the existence of oscillating spheres with nonuniform density. This possibility was raised by Louis and Gerhard (1988) in their numerical work. (ii) the chaotic behavior of the spheroid axes. This is very reminiscent of anisotropic cosmological models, which again reduce to Hamiltonian systems with a finite number of degrees of freedom (Misner 1969). There are also two hot families, occupying a nonzero phase volume. The hot oscillating spheres have “inverted” phase-space distribution functions and uniform real-space density. In both these respects, they differ from realistic galaxy models, which have higher phase density at low energy and a strong decrease in real-space density outside a central core. It is not clear whether the oscillations of uniform spheres found here will turn out to be stable and hence applicable to a wider class of models. This possibility is worth examining, perhaps in future numerical work. An undamped or weakly damped oscillation of a dark matter halo (for example) could be an energy input for gas flow in the time-dependent potential. The second hot system studied in this thesis is the generalized Freeman disc. Again, there are significant differences between these models and the phase-space structure of more realistic bars. The stability is again an open question, which will probably have to be answered by numerical studies. It is interesting that these model bars are stable under external harmonic potentials (these keep the model within the class of generalized Freeman discs). A more general stability analysis would be of great interest because bars are abundantly found in the real world. Finally, the analytical study of a particular tidal encounter model in this thesis has allowed a convenient exploration of the validity of the impulse approximation, the effect of rotation, and resonance effects between internal and orbital time scales. It will again be of interest to see how well this tractable model is able to mirror more realistic situations. In brief, the new analytic models of time-dependent stellar systems presented here have interesting properties in their own right and may also point to directions that need systematic explorations by numerical methods.