Jet53man at 17:12, 12 October 2022

2022-10-12T17:12:44Z

← Older revision		Revision as of 13:12, 12 October 2022
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	=BiPolytrope with <~~math~~>~~n_c = 0~~</~~math~~> ~~and~~ <~~math~~>~~n_e=0~~</~~math~~>=		=BiPolytrope with (n<sub>c</sub>, n<sub>e</sub>) = (0, 0)=

	Here we construct a [[SSC/Structure/BiPolytropes#BiPolytropes\|bipolytrope]] in which both the core and the envelope have uniform densities, that is, the structure of both the core and the envelope will be modeled using an <math>n = 0</math> polytropic index. It should be possible for the entire structure to be described by closed-form, analytic expressions. Generally, we will follow the [[SSC/Structure/BiPolytropes#Solution_Steps\|general solution steps for constructing a bipolytrope]] that we have outlined elsewhere. [On '''<font color="red">1 February 2014</font>''', J. E. Tohline wrote: This particular system became of interest to me during discussions with Kundan Kadam about the relative stability of bipolytropes.]		Here we construct a [[SSC/Structure/BiPolytropes#BiPolytropes\|bipolytrope]] in which both the core and the envelope have uniform densities, that is, the structure of both the core and the envelope will be modeled using an <math>n = 0</math> polytropic index. It should be possible for the entire structure to be described by closed-form, analytic expressions. Generally, we will follow the [[SSC/Structure/BiPolytropes#Solution_Steps\|general solution steps for constructing a bipolytrope]] that we have outlined elsewhere. [On '''<font color="red">1 February 2014</font>''', J. E. Tohline wrote: This particular system became of interest to me during discussions with Kundan Kadam about the relative stability of bipolytropes.]

	==Step 4: Throughout the core (~~<math>~~0 \le \chi \le ~~\chi_i~~</~~math~~>)==		==Step 4: Throughout the core (0 ≤ χ ≤ χ<sub>i</sub>)==
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	==Step 6: Envelope Solution (<~~math>~\chi~~ > ~~\chi_i~~</~~math~~>)==		==Step 6: Envelope Solution (χ ≥ χ<sub>i</sub>)==
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Jet53man at 16:50, 15 October 2021

2021-10-15T16:50:27Z

← Older revision		Revision as of 12:50, 15 October 2021
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	=BiPolytrope with <math>n_c = 0</math> and <math>n_e=0</math>=		=BiPolytrope with <math>n_c = 0</math> and <math>n_e=0</math>=

	Here we construct a [[~~User:Tohline/~~SSC/Structure/BiPolytropes#BiPolytropes\|bipolytrope]] in which both the core and the envelope have uniform densities, that is, the structure of both the core and the envelope will be modeled using an <math>n = 0</math> polytropic index. It should be possible for the entire structure to be described by closed-form, analytic expressions. Generally, we will follow the [[~~User:Tohline/~~SSC/Structure/BiPolytropes#Solution_Steps\|general solution steps for constructing a bipolytrope]] that we have outlined elsewhere. [On '''<font color="red">1 February 2014</font>''', J. E. Tohline wrote: This particular system became of interest to me during discussions with Kundan Kadam about the relative stability of bipolytropes.]		Here we construct a [[SSC/Structure/BiPolytropes#BiPolytropes\|bipolytrope]] in which both the core and the envelope have uniform densities, that is, the structure of both the core and the envelope will be modeled using an <math>n = 0</math> polytropic index. It should be possible for the entire structure to be described by closed-form, analytic expressions. Generally, we will follow the [[SSC/Structure/BiPolytropes#Solution_Steps\|general solution steps for constructing a bipolytrope]] that we have outlined elsewhere. [On '''<font color="red">1 February 2014</font>''', J. E. Tohline wrote: This particular system became of interest to me during discussions with Kundan Kadam about the relative stability of bipolytropes.]

	==Step 4: Throughout the core (<math>0 \le \chi \le \chi_i</math>)==		==Step 4: Throughout the core (<math>0 \le \chi \le \chi_i</math>)==
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	=Gravitational Potential Energy=		=Gravitational Potential Energy=
	In a [[~~User:Tohline/~~SSC/VirialStability#Expressions_for_Mass\|separate discussion]], we derived the following expression for the total gravitational potential energy of an <math>~(n_c, n_e) = (0, 0)</math> bipolytrope:		In a [[SSC/VirialStability#Expressions_for_Mass\|separate discussion]], we derived the following expression for the total gravitational potential energy of an <math>~(n_c, n_e) = (0, 0)</math> bipolytrope:

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	Hooray! This precisely matches the expression for the central pressure, <math>~P_0</math>, that was obtained above [[~~User:Tohline/SSC/Structure/BiPolytropes/Analytic0_0~~#Step_7:__Surface_Boundary_Condition\|at the end of step #7 of the detailed force balance derivation]]. It is very satisfying to see that the central pressure can be derived from demanding a balance of the thermal and gravitational potential energies via the virial equilibrium expression.		Hooray! This precisely matches the expression for the central pressure, <math>~P_0</math>, that was obtained above [[#Step_7:__Surface_Boundary_Condition\|at the end of step #7 of the detailed force balance derivation]]. It is very satisfying to see that the central pressure can be derived from demanding a balance of the thermal and gravitational potential energies via the virial equilibrium expression.

	Note that, because <math>~P_0 = P_i + \Pi q^2</math>, we may alternatively write the solution in terms of the pressure at the interface, that is,		Note that, because <math>~P_0 = P_i + \Pi q^2</math>, we may alternatively write the solution in terms of the pressure at the interface, that is,
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	Now, according to our [[~~User:Tohline/SSC/Structure/BiPolytropes/Analytic0_0~~#Thermal_Energy_Content\|earlier derivation of the total thermal energy content]] of this <math>~(n_c, n_e) = (0, 0)</math> bipolytrope, the right-hand side of this last expression is <math>~S_\mathrm{tot}</math>. Hence the equilibrium condition just derived from the first derivative of the free energy is precisely the [[~~User:Tohline/SSC/Structure/BiPolytropes/Analytic0_0~~#Virial_Equilibrium\|virial equilibrium condition discussed above]], namely,		Now, according to our [[#Thermal_Energy_Content\|earlier derivation of the total thermal energy content]] of this <math>~(n_c, n_e) = (0, 0)</math> bipolytrope, the right-hand side of this last expression is <math>~S_\mathrm{tot}</math>. Hence the equilibrium condition just derived from the first derivative of the free energy is precisely the [[#Virial_Equilibrium\|virial equilibrium condition discussed above]], namely,
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	This, in turn, means that from the free energy expression we are able to deduce the equilibrium values of the dimensionless radius, mass, and central pressure that were [[~~User:Tohline/SSC/Structure/BiPolytropes/Analytic0_0~~#Step_7:__Surface_Boundary_Conditio\|derived earlier from the hydrostatic balance equation]], namely,		This, in turn, means that from the free energy expression we are able to deduce the equilibrium values of the dimensionless radius, mass, and central pressure that were [[#Step_7:__Surface_Boundary_Conditio\|derived earlier from the hydrostatic balance equation]], namely,
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	=Bipolytrope Generalization=		=Bipolytrope Generalization=
	[[~~User:Tohline/~~SSC/BipolytropeGeneralization#Bipolytrope_Generalization\|This subsection has been moved to another location]] because it more naturally stands on its own, that is, it should not be tied only to the discussion of the structure of <math>~(n_c, n_e) = (0, 0)</math> bipolytropes.		[[SSC/BipolytropeGeneralization#Bipolytrope_Generalization\|This subsection has been moved to another location]] because it more naturally stands on its own, that is, it should not be tied only to the discussion of the structure of <math>~(n_c, n_e) = (0, 0)</math> bipolytropes.

	{{ SGFworkInProgress }}		{{ SGFworkInProgress }}

	=Plotting Free-Energy Surfaces=		=Plotting Free-Energy Surfaces=
	Thus far, we have been able to determine what the equilibrium radius of the bipolytrope is for any specified set of parameters, <math>~(M_\mathrm{tot}, \nu, q)</math>. Now let's examine how the free energy varies across the two-dimensional plane defined by the parameters, <math>~(q, \nu)</math>. We begin by copying an informative segment from the [[~~User:Tohline/SSC/Structure/BiPolytropes/Analytic0_0~~#Expression_for_Free_Energy\|above subsection where we developed an expression for the free energy]].		Thus far, we have been able to determine what the equilibrium radius of the bipolytrope is for any specified set of parameters, <math>~(M_\mathrm{tot}, \nu, q)</math>. Now let's examine how the free energy varies across the two-dimensional plane defined by the parameters, <math>~(q, \nu)</math>. We begin by copying an informative segment from the [[#Expression_for_Free_Energy\|above subsection where we developed an expression for the free energy]].

	The free energy is,		The free energy is,
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	=Related Discussions=		=Related Discussions=
	* [[~~User:Tohline/~~SSC/Structure/BiPolytropes/~~Analytic5_1~~#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|Analytic solution]] with <math>~n_c = 5</math> and <math>~n_e=1</math>.		* [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|Analytic solution]] with <math>~n_c = 5</math> and <math>~n_e=1</math>.


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Jet53man: Created page with "FORCETOC =BiPolytrope with $n_c = 0$ and $n_e=0$ = Here we construct a User:Tohline/SSC/Structure/BiPolytrop..."

2021-10-15T16:44:49Z

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SSC/Structure/BiPolytropes/Analytic00 - Revision history

Jet53man at 17:12, 12 October 2022

Jet53man at 16:50, 15 October 2021

Jet53man: Created page with "__FORCETOC__ =BiPolytrope with n_c = 0 and n_e=0= Here we construct a User:Tohline/SSC/Structure/BiPolytrop..."

Jet53man: Created page with "FORCETOC =BiPolytrope with $n_c = 0$ and $n_e=0$ = Here we construct a User:Tohline/SSC/Structure/BiPolytrop..."