# User:Tohline/SSC/Virial/PolytropesEmbeddedOutline

# Virial Equilibrium of Embedded Polytropic Spheres

## Outline

### First Effort

My first attempt to analytically define the free energy, and then the virial equilibrium, of pressure-truncated (embedded) polytropic spheres was developed as a direct extension of my description of the virial equilibrium of *isolated* polytropes. An important outcome of this "first effort" was the unveiling of analytic expressions for the key structural form factors, both for *isolated* polytropes and, separately, for *pressure-truncated* polytropic structures.

I am very confident that the form-factor expressions presented for isolated polytropes are all correct because they have been cross-checked with expressions for closely related "integral" parameters discussed by Chandrasekhar [C67]. Although the form-factor expressions derived for pressure-truncated polytropes make some sense — they look very similar to the ones presented for isolated polytropes and seem to behave properly for models which, based on detailed force-balanced analysis, are known to be in equilibrium — I have much less confidence that they are correct. A couple of strategies were developed in an effort to demonstrate the validity and utility of these more general form-factor expressions, resulting in the derivation of a *concise virial equilibrium* relation,

<math>\Pi_\mathrm{ad} = \chi_\mathrm{ad}^{-3\gamma} - \chi_\mathrm{ad}^{-4} \, ,</math>

that incorporates the newly defined normalization parameters, <math>~R_\mathrm{ad}</math> and <math>~P_\mathrm{ad}</math>. But subsequent derivations aimed at more conclusively demonstrating the correctness of the more general form-factor expressions were messy and got bogged down.

### Second Effort

My second attempt to analytically define the free energy, and then the virial equilibrium, of pressure-truncated (embedded) polytropic spheres built upon my *first effort* and, for a couple of different polytropic indexes, focused on comparing the mass-radius relationship embodied in detailed force-balanced models against the mass-radius relationship implied by the virial theorem.

© 2014 - 2021 by Joel E. Tohline |