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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. A lot of reasonable results seem to have arisen from a discussion of models (done numerically using Excel) with <math>~n=4</math> polytropic index. And there are some nice aspects of models with an <math>~n=5</math> index, but these models raise some serious concerns related to the fact that two of our "derived" form-factor expressions involve division by the factor, <math>~(5-n)</math>, that is, division by zero.

Third Effort

In an attempt to answer the serious concern(s) raised during our first two efforts, we finally buckled down and performed the integrals necessary to determine expressions for key structural form factors in the cases where the internal structure is known analytically, specifically, for indexes <math>~n=5</math> and <math>~n=1</math>. The result is that the individual expressions derived by direct integration for <math>~\mathfrak{f}_W</math> and for <math>~\mathfrak{f}_A</math> do not match the general form-factor expressions that were rather cavalierly "derived" during our first effort. Oddly enough, as we discovered while fiddling around with the new results, the ratio of these form factors appears to be the same as before, namely,

<math>~\frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_A - \tilde\theta^{n+1}}</math>


<math>~ \biggl[ \frac{3\cdot 5}{(n+1) \tilde\xi^2 } \biggr] \, . </math>

It is worth noting that, as a result of this more thorough "third effort" examination, we have confirmed that the third key form factor,

<math>~\mathfrak{f}_M = \frac{\bar\rho}{\rho_c} = \biggl[- \frac{3\tilde\theta^'}{\tilde\xi}\biggr] \, ,</math>

which is the same as before and the same as for isolated polytropes. We also have determined that,

<math>~\biggl(\frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\mathfrak{f}_M} = \biggl(\frac{\tilde\xi^2 \tilde\theta^'}{\xi_1 \theta^'_1} \biggr)\biggl[- \frac{\tilde\xi}{3\tilde\theta^'}\biggr] = - \frac{\tilde\xi^3 }{3\xi_1 \theta^'_1} \, , </math>

except in the case of <math>~n=5</math> structures, for which we have determined,

<math>~\biggl[\biggl(\frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\mathfrak{f}_M} \biggr]_{n=5} = \ell^3 = \biggl( \frac{\tilde\xi^2}{3} \biggr)^{3/2} \, . </math>

First Effort, Second Time Around

In an accompanying chapter, we reproduce the discussion associated with our "First Effort", as referenced above, but correct expressions for \mathfrak{f}_W and \mathfrak{f}_A, as identified in our "Third Effort" and, accordingly, re-derive various affected expressions that follow.

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,