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Virial Equilibrium of Embedded Polytropic Spheres


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.

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) publication,