Jet53man at 23:25, 27 July 2021

2021-07-27T23:25:04Z

← Older revision		Revision as of 19:25, 27 July 2021
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	=Bipolytrope Generalization=		=Bipolytrope Generalization=

	On <font color="red">26 August 2014</font>, Tohline finished rewriting the chapter titled "Bipolytrope Generalization" in a very concise manner ([[~~User:Tohline/~~SSC/~~BipolytropeGeneralization_Version2~~#Bipolytrope_Generalization\|go here for this Version2 chapter]]) then set this chapter aside to provide a collection of older attempts at the derivations. While much of what follows is technically correct, it is overly detailed and cumbersome. Because it likely also contains some misguided steps, we label it in entirety as Work in Progress.		On <font color="red">26 August 2014</font>, Tohline finished rewriting the chapter titled "Bipolytrope Generalization" in a very concise manner ([[SSC/BipolytropeGeneralizationVersion2#Bipolytrope_Generalization\|go here for this Version2 chapter]]) then set this chapter aside to provide a collection of older attempts at the derivations. While much of what follows is technically correct, it is overly detailed and cumbersome. Because it likely also contains some misguided steps, we label it in entirety as Work in Progress.

	{{ SGFworkInProgress }}		{{ SGFworkInProgress }}
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	where the subscript "<math>i</math>" means "at the interface," and <math>~\mathfrak{f}_{WM},</math> <math>~s_\mathrm{core},</math> and <math>~s_\mathrm{env}</math> are dimensionless functions of order unity (all three functions to be determined) akin to the [[~~User:Tohline/SphericallySymmetricConfigurations~~/Virial#Structural_Form_Factors\|structural form factors]] used in our examination of isolated polytropes.		where the subscript "<math>i</math>" means "at the interface," and <math>~\mathfrak{f}_{WM},</math> <math>~s_\mathrm{core},</math> and <math>~s_\mathrm{env}</math> are dimensionless functions of order unity (all three functions to be determined) akin to the [[SSCpt1/Virial#Structural_Form_Factors\|structural form factors]] used in our examination of isolated polytropes.

	While exploring how the free-energy function varies across parameter space, we choose to hold <math>~M_\mathrm{tot}</math> and <math>~K_c</math> fixed. By dimensional analysis, it is therefore reasonable to normalize all energies, length scales, densities and pressures by, respectively,		While exploring how the free-energy function varies across parameter space, we choose to hold <math>~M_\mathrm{tot}</math> and <math>~K_c</math> fixed. By dimensional analysis, it is therefore reasonable to normalize all energies, length scales, densities and pressures by, respectively,
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	As is detailed below — [[~~User:Tohline/SSC/BipolytropeGeneralization~~#Detailed_Derivations\|first, here]], and via [[~~User:Tohline/SSC/BipolytropeGeneralization~~#Another_Derivation_of_Free_Energy\|an independent derivation, here]] — quite generally the expression for the normalized free energy is,		As is detailed below — [[#Detailed_Derivations\|first, here]], and via [[#Another_Derivation_of_Free_Energy\|an independent derivation, here]] — quite generally the expression for the normalized free energy is,
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	As is [[~~User:Tohline/SSC/BipolytropeGeneralization~~#pressureMatch\|further detailed below]], the second expression for the coefficient, <math>~\mathcal{C}</math>, ensures that the pressure at the "surface" of the core matches the pressure at the "base" of the envelope; but it should only be employed ''after an equilibrium radius'', <math>~\chi_\mathrm{eq}</math>, ''has been identified by locating an extremum in the free energy.''		As is [[#pressureMatch\|further detailed below]], the second expression for the coefficient, <math>~\mathcal{C}</math>, ensures that the pressure at the "surface" of the core matches the pressure at the "base" of the envelope; but it should only be employed ''after an equilibrium radius'', <math>~\chi_\mathrm{eq}</math>, ''has been identified by locating an extremum in the free energy.''

	==Simplest Bipolytrope==		==Simplest Bipolytrope==

	===Familiar Setup===		===Familiar Setup===
	As has been shown in [[~~User:Tohline/SSC/BipolytropeGeneralization~~#.280.2C_0.29_Bipolytropes\|an accompanying presentation]], for an <math>~(n_c, n_e) = (0, 0)</math> bipolytrope,		As has been shown in [[#.280.2C_0.29_Bipolytropes\|an accompanying presentation]], for an <math>~(n_c, n_e) = (0, 0)</math> bipolytrope,
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	and where (see, for example, [[~~User:Tohline/~~SSC/Structure/BiPolytropes/~~Analytic0_0~~#Expression_for_Free_Energy\|in the context of its original definition]], or another, [[~~User:Tohline/~~SSC/Structure/BiPolytropes/~~Analytic0_0~~#LambdaDeff\|separate derivation]]),		and where (see, for example, [[SSC/Structure/BiPolytropes/Analytic00#Expression_for_Free_Energy\|in the context of its original definition]], or another, [[SSC/Structure/BiPolytropes/Analytic00#LambdaDeff\|separate derivation]]),
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	and where (see the [[~~User:Tohline/~~SSC/VirialStability#Expressions_for_Mass\|associated discussion of relevant mass integrals]]),		and where (see the [[SSC/VirialStability#Expressions_for_Mass\|associated discussion of relevant mass integrals]]),
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	<math>		<math>
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	where (see an [[~~User:Tohline/~~SSC/Structure/BiPolytropes/~~Analytic0_0~~#Gravitational_Potential_Energy\|associated discussion]] or the [[~~User:Tohline/~~SSC/VirialStability#Energy_Expressions\|original derivation]]),		where (see an [[SSC/Structure/BiPolytropes/Analytic00#Gravitational_Potential_Energy\|associated discussion]] or the [[SSC/VirialStability#Energy_Expressions\|original derivation]]),
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	<math>		<math>
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	This result precisely matches [[~~User:Tohline/~~SSC/Structure/BiPolytropes/~~Analytic0_0~~#CentralPressure\|the result obtained via the detailed force-balanced conditions]] imposed through hydrostatic equilibrium.		This result precisely matches [[SSC/Structure/BiPolytropes/Analytic00#CentralPressure\|the result obtained via the detailed force-balanced conditions]] imposed through hydrostatic equilibrium.

	Adopting our new variable normalizations and realizing, in particular, that,		Adopting our new variable normalizations and realizing, in particular, that,
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	* '''<font color="darkgreen">Step 1:</font>''' Guess a value of <math>~0 < q < 1</math>.		* '''<font color="darkgreen">Step 1:</font>''' Guess a value of <math>~0 < q < 1</math>.

	* '''<font color="darkgreen">Step 2:</font>''' Given the pair of parameter values, <math>~(\mathcal{A}, q)</math>, determine the interface-density ratio, <math>~\rho_e/\rho_c</math>, by finding the appropriate root of the expression that defines the function, <math>~\mathcal{A}(q, \rho_e/\rho_c)</math>. This can be straightforwardly accomplished because, [[~~User:Tohline/SSC/BipolytropeGeneralization~~#Explain_Logic\|as demonstrated below]], the relevant expression can be written as a quadratic function of <math>~(\rho_e/\rho_c)</math>.		* '''<font color="darkgreen">Step 2:</font>''' Given the pair of parameter values, <math>~(\mathcal{A}, q)</math>, determine the interface-density ratio, <math>~\rho_e/\rho_c</math>, by finding the appropriate root of the expression that defines the function, <math>~\mathcal{A}(q, \rho_e/\rho_c)</math>. This can be straightforwardly accomplished because, [[#Explain_Logic\|as demonstrated below]], the relevant expression can be written as a quadratic function of <math>~(\rho_e/\rho_c)</math>.

	* '''<font color="darkgreen">Step 3:</font>''' Given the pair of parameter values, <math>~(q, \rho_e/\rho_c)</math>, determine the value of the core-to-total mass ratio, <math>~\nu</math>, from the expression that was obtained from an integration over the mass, namely,		* '''<font color="darkgreen">Step 3:</font>''' Given the pair of parameter values, <math>~(q, \rho_e/\rho_c)</math>, determine the value of the core-to-total mass ratio, <math>~\nu</math>, from the expression that was obtained from an integration over the mass, namely,
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	==(0, 0) Bipolytropes==		==(0, 0) Bipolytropes==
	===Review===		===Review===
	In an [[~~User:Tohline/~~SSC/Structure/BiPolytropes/~~Analytic0_0~~#BiPolytrope_with_nc_.3D_0_and_ne_.3D_0\|accompanying discussion]] we have derived analytic expressions describing the equilibrium structure and the stability of bipolytropes in which both the core and the envelope have uniform densities, that is, bipolytropes with <math>~(n_c, n_e) = (0, 0)</math>. From this work, we find that integrals over the mass and pressure distributions give:		In an [[SSC/Structure/BiPolytropes/Analytic00#BiPolytrope_with_nc_.3D_0_and_ne_.3D_0\|accompanying discussion]] we have derived analytic expressions describing the equilibrium structure and the stability of bipolytropes in which both the core and the envelope have uniform densities, that is, bipolytropes with <math>~(n_c, n_e) = (0, 0)</math>. From this work, we find that integrals over the mass and pressure distributions give:
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	===Renormalize===		===Renormalize===
	Let's renormalize these energy terms in order to more readily relate them to the [[~~User:Tohline/SSC/BipolytropeGeneralization~~#Setup\|generalized expressions derived above]].		Let's renormalize these energy terms in order to more readily relate them to the [[#Setup\|generalized expressions derived above]].
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	==(5, 1) Bipolytropes==		==(5, 1) Bipolytropes==
	In another [[~~User:Tohline/~~SSC/Structure/BiPolytropes/~~Analytic5_1~~#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|accompanying discussion]] we have derived analytic expressions describing the equilibrium structure of bipolytropes with <math>~(n_c, n_e) = (5, 1)</math>. Can we perform a similar stability analysis of these configurations?		In another [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|accompanying discussion]] we have derived analytic expressions describing the equilibrium structure of bipolytropes with <math>(n_c, n_e) = (5, 1)</math>. Can we perform a similar stability analysis of these configurations?
	Work in progress!		Work in progress!

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	where (see, for example, [[~~User:Tohline/~~SSC/Structure/BiPolytropes/~~Analytic0_0~~#Expression_for_Free_Energy\|in the context of its original definition]]),		where (see, for example, [[SSC/Structure/BiPolytropes/Analytic00#Expression_for_Free_Energy\|in the context of its original definition]]),
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	Fortunately, this precisely matches our [[~~User:Tohline/~~SSC/Structure/BiPolytropes/~~Analytic0_0~~#LambdaDeff\|earlier derivation]], which states that,		Fortunately, this precisely matches our [[SSC/Structure/BiPolytropes/Analytic00#LambdaDeff\|earlier derivation]], which states that,
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Jet53man: Created page with "FORCETOC =Bipolytrope Generalization= On 26 August 2014, Tohline finished rewriting the chapter titled "Bipolytrope Generalization" in a very con..."

2021-07-27T23:16:04Z

Created page with "__FORCETOC__ =Bipolytrope Generalization= On <font color="red">26 August 2014</font>, Tohline finished rewriting the chapter titled "Bipolytrope Generalization" in a very con..."

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SSC/BipolytropeGeneralization - Revision history

Jet53man at 23:25, 27 July 2021

Jet53man: Created page with "__FORCETOC__ =Bipolytrope Generalization= On 26 August 2014, Tohline finished rewriting the chapter titled "Bipolytrope Generalization" in a very con..."

Jet53man: Created page with "FORCETOC =Bipolytrope Generalization= On 26 August 2014, Tohline finished rewriting the chapter titled "Bipolytrope Generalization" in a very con..."