The Structure, Stability, and Dynamics
of Self-Gravitating Systems

Joel E. Tohline
tohline@rouge.phys.lsu.edu

Elsewhere in this H_Book, we have detailed the steps that one must normally follow in order to construct equilibrium models of rotating, self-gravitating systems. As Tassoul (1978) points out,¹ however, not every equilibrium model can actually occur in nature: the models must not only obey a time-independent (steady-state) form of the principal governing equations, they must also be stable. Thus, an acceptable model must sustain itself against natural fluctuations to which any physical body is subject, i.e., whenever such disturbances arise, they should decrease with time. If the fluctuations do not die down, the system is then said to be unstable, for it progressively departs from its initial state of equilibrium.

A sharpened pencil that one attempts to carefully balance vertically on its point is a readily visualizable example of an unstable equilibrium system. In theory, one should be able to stand the pencil up on its point with sufficient precision that all the external forces acting upon it balance (i.e., the pencil feels no net acceleration). Hence, a sharpened pencil standing vertically on its point is a perfectly good theoretical model of an equilibrium system. However, we all know that, in practice, the pencil will soon fall over. Even the slightest perturbation away from a perfectly vertical alignment will cause the system to "progressively depart from its initial state of equilibrium." We should not be surprised to find the same thing to be true of some of our theoretically modeled, self-gravitating equilibrium configurations.

Throughout the accompanying sections of this H_Book, we will explain and demonstrate how one goes about examining the relative stability of self-gravitating equilibrium systems. Our objective is to provide the reader with both analytical and numerical tools that may be used to examine the stability of a wide variety of equilibrium structures, such as the ones that have been presented in the "structure" sections of this H_Book. Before focusing in on the specific stability analysis tools that have proven to be most effective when studying particular types of models, however, we will introduce the reader to a variety of techniques by which researchers in the field have examined the stability of equilibrium, self-gravitating systems. The techniques may be divided broadly into the following three categories.

As Tassoul (1978) has eloquently explained, the mathematical investigation of the stability of a known model generally proceeds as follows. An arbitrary perturbation is superposed on the system, and the system's subsequent behavior under the natural forces is studied by means of the time-dependent, principal governing equations. For convenience, we then assume the fluctuations to be small enough so that we can linearize the nonlinear (principal governing) equations expressing the conservation of mass, linear momenum, and energy. In other words, we neglect all products and powers (higher than the first) of the disturbances and retain only terms that are linear in them. It is usually assumed that the solutions of these linearized equations approximate closely enough the actual solutions to reveal the general trend of the motion in the immediate vicinity of the equilibrium state. In most simple cases, energetic considerations usually provide a physical interpretation of the results. Further insight into the problem necessarily requires an investigation of finite-amplitude motions, which we pursue in the "dynamics" sections of this H_Book.

Here we take each of the principal governing equations in turn and, following the instructions outlined above, "perturb" each physical variable away from its equilibrium value. More specifically, we will replace each physical variable as follows:

Euler's Equation: Although we may begin from any one of its numerous forms and ultimately achieve the same outcome, here we choose to work from the Lagrangian representation of Euler's equation that defines the time variation of the momentum density [Equation I.A.5], and immediately replace the source term involving a pressure gradient by an equivalent term involving the enthalpy as prescribed by equation [III.F.16],

Via the standard expression which relates the Lagrangian time-derivative to its Eulerian counterpart, we derive the equivalent Eulerian form of Euler's equation defining the time variation of the momentum density,

Replacing each variable by its "perturbed" value as prescribed above [IV.A.1],