Difference between revisions of "User:Tohline/SSC/Virial/PolytropesEmbeddedOutline"

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(New chapter that simply outlines the work to be found in our various detailed discussions of the free energy associated with embedded polytropes)
 
(→‎Virial Equilibrium of Embedded Polytropic Spheres: Finish brief descriptions of 2nd and 3rd efforts)
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===Second Effort===
===Second Effort===
My [[User:Tohline/SSC/Virial/PolytropesSummary#Virial_Equilibrium_of_Adiabatic_Spheres_.28Summary.29|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.   
My [[User:Tohline/SSC/Virial/PolytropesSummary#Virial_Equilibrium_of_Adiabatic_Spheres_.28Summary.29|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 [[User:Tohline/SSC/Virial/PolytropesSummary#Relating_and_Reconciling_Two_Mass-Radius_Relationships_for_n_.3D_4_Polytropes|models (done numerically using Excel) with <math>~n=4</math> polytropic index]].  And there are some nice aspects of [[User:Tohline/SSC/Virial/PolytropesSummary#Relating_and_Reconciling_Two_Mass-Radius_Relationships_for_n_.3D_5_Polytropes|models with an <math>~n=5</math> index]], but these models raise some [[User:Tohline/SSC/Virial/PolytropesSummary#Serious_Concern|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 [[User:Tohline/SSC/Virial/FormFactors#Structural_Form_Factors|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 appears to be that the expressions derived by direct integration ''do not match'' the general form-factor expressions "derived" during our first effort.
 


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Revision as of 18:44, 9 January 2015


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. 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 appears to be that the expressions derived by direct integration do not match the general form-factor expressions "derived" during our first effort.


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) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation