3.6: Summary

In this chapter we have explored the results of applying a specific analytical technique to the virial theorem. As in other chapters, we began with the simple and moved to the more complex. Having discussed the implications of the variational approach to the virial theorem we moved to develop the explicit form for the simple scalar theorem appropriate for self-gravitating systems. We recreated the pulsational formula [Equation \ref{3.2.19}] originally due to Ledoux. One implication of this result is that the fundamental mode of oscillation depends only on the square root of the density and when coupled with the stability criterion in section 5, leads immediately to the Jean's stability criterion. This is not surprising as both results have as the derivational origin the same concept (i.e., the Equations of motion). However, it is reassuring when a different approach yields results already well accepted.

In section 3 we expanded the variational approach to include the effects of magnetic fields and rotation. In spite of many distractions dealing with the variation of magnetic fields, etc., the influence of these added features on the pulsation frequency and hence stability became clear. Rotation can either enhance or reduce the stability of a configuration depending on whether or not the value of \(\gamma\) for the gas permits net energy to be fed to the pulsation. The influence of an internal magnetic field is to destabilize the star for all realistic values of \(\gamma\). However, the effect of a surface field proves to be more complex. Here the result depends critically on the geometry of the field. In the last section we dealt briefly with the overall question of stability and showed explicitly how the virial theorem provided an excellent basis for a linear stability analysis of a symmetric system. Throughout the chapter we confined ourselves to spherically symmetric systems exhibiting radial pulsations only. As mentioned, this is inappropriate when considering either rotation or magnetic fields as neither can exhibit spherical symmetry and thus one would expect non-radial oscillation to be excited. However, unless the field energies become quite large one would expect the pulsational frequencies not to differ greatly from the purely radial theory.

This line of reasoning becomes particularly dangerous when one turns to a discussion of stability. First the interesting situations of marginal stability are liable to involve substantial magnetic fields or rapid rotation. If these aspherical properties are large, the departure from spherical systems of the mass distribution will also be large invalidating every aspect of the analysis. In addition, for the stability analysis to be valid all possible modes of perturbation must be included. Limiting oneself to only the radial modes is to invite a misleading result. Fortunately the techniques for dealing with these problems exist and have been developed here as well as the literature. The tensor virial theorem as is presented in Chapter II, section 1 allows one to follow perturbations in independent spatial coordinates. In principle, a complete variational analysis of perturbations to all independent spatial coordinates will allow one to compute the non-radial as well as radial modes of oscillation and thereby obtain a much more secure analysis of the system's stability.