SSC/Structure/OtherAnalyticRamblings - Revision history

Jet53man: /* Exploration2 */

2021-10-15T21:50:27Z

Exploration2

← Older revision		Revision as of 17:50, 15 October 2021
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	==Exploration2==		==Exploration2==

	Let's begin with the LAWE written in the following form (see, for example, the [[~~User:Tohline/~~MathProjects/EigenvalueProblemN1#Context\|related context discussion]]):		Let's begin with the LAWE written in the following form (see, for example, the [[MathProjects/EigenvalueProblemN1#Context\|related context discussion]]):
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	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</tr>		</tr>
	<tr>		<tr>
	<td align="center">[[~~User:Tohline~~/~~SSC~~/UniformDensity#The_Stability_of_Uniform-Density_Spheres\|Uniform-density]]</td>		<td align="center">[[SSC/Stability/UniformDensity#The_Stability_of_Uniform-Density_Spheres\|Uniform-density]]</td>
	<td align="center"><math>~1</math>		<td align="center"><math>~1</math>
	<td align="center"><math>~1 - x^2</math>		<td align="center"><math>~1 - x^2</math>
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	</tr>		</tr>
	<tr>		<tr>
	<td align="center">[[~~User:Tohline/~~SSC/Structure/~~Other_Analytic_Models~~#Linear_Density_Distribution\|Linear]]</td>		<td align="center">[[SSC/Structure/OtherAnalyticModels#Linear_Density_Distribution\|Linear]]</td>
	<td align="center"><math>~1-x</math>		<td align="center"><math>~1-x</math>
	<td align="center"><math>~(1-x)(1 + 2x - \tfrac{9}{5}x^2)</math>		<td align="center"><math>~(1-x)(1 + 2x - \tfrac{9}{5}x^2)</math>
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	</tr>		</tr>
	<tr>		<tr>
	<td align="center">[[~~User:Tohline/~~SSC/Structure/~~Other_Analytic_Models~~#Parabolic_Density_Distribution\|Parabolic]]</td>		<td align="center">[[SSC/Structure/OtherAnalyticModels#Parabolic_Density_Distribution\|Parabolic]]</td>
	<td align="center"><math>~1-x^2</math>		<td align="center"><math>~1-x^2</math>
	<td align="center"><math>~(1-x^2)(1 - \tfrac{1}{2} x^2)</math>		<td align="center"><math>~(1-x^2)(1 - \tfrac{1}{2} x^2)</math>
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	</tr>		</tr>
	<tr>		<tr>
	<td align="center">[[~~User:Tohline/~~SSC/Stability/Polytropes#n_.3D_1_Polytrope\|<math>~n=1</math> Polytrope]]</td>		<td align="center">[[SSC/Stability/Polytropes#n_.3D_1_Polytrope\|<math>~n=1</math> Polytrope]]</td>
	<td align="center"><math>~\frac{\sin }{ x}</math>		<td align="center"><math>~\frac{\sin }{ x}</math>
	<td align="center"><math>~\frac{\sin x}{x}</math>		<td align="center"><math>~\frac{\sin x}{x}</math>
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	===Another Trial===		===Another Trial===

	Start with a form of the LAWE found midway through the [[~~User:Tohline/~~SSC/Structure/~~Other_Analytic_Ramblings~~#New_Form_of_LAWE\|above derivation]], namely,		Start with a form of the LAWE found midway through the [[SSC/Structure/OtherAnalyticRamblings#New_Form_of_LAWE\|above derivation]], namely,

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Jet53man: /* Exploration */

2021-10-15T21:47:49Z

Exploration

← Older revision		Revision as of 17:47, 15 October 2021
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	====Compare LAWE to Hydrostatic Balance Condition====		====Compare LAWE to Hydrostatic Balance Condition====

	Returning to the [[~~User:Tohline/~~SSC/Structure/~~Other_Analytic_Models~~#Generic\|generic formulation derived earlier]], we have,		Returning to the [[SSC/Structure/OtherAnalyticModels#Generic\|generic formulation derived earlier]], we have,

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	</div>		</div>

	Now, let's step aside from the LAWE and look directly at the differential relationship between the mass-density and the pressure, as dictated by combining the [[~~User:Tohline/SphericallySymmetricConfigurations~~/SolutionStrategies#Spherically_Symmetric_Configurations_.28Part_II.29\|two principal governing relations]], the		Now, let's step aside from the LAWE and look directly at the differential relationship between the mass-density and the pressure, as dictated by combining the [[SSCpt2/SolutionStrategies#Spherically_Symmetric_Configurations_.28Part_II.29\|two principal governing relations]], the

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	====Dabbling with Equilibrium Condition====		====Dabbling with Equilibrium Condition====
	In the meantime, I've found it instructive to play with the first of these two expressions to see how it might be restructured in order to most directly confirm that it is satisfied by the expressions presented in [[~~User:Tohline/~~SSC/Structure/~~Other_Analytic_Models~~#Examples\|Table 1]]. Adopting the shorthand notation,		In the meantime, I've found it instructive to play with the first of these two expressions to see how it might be restructured in order to most directly confirm that it is satisfied by the expressions presented in [[SSC/Structure/OtherAnalyticModels#Examples\|Table 1]]. Adopting the shorthand notation,
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	<math>~\Gamma \equiv 4\pi G\rho_c \tau_\mathrm{SSC}^2</math>		<math>~\Gamma \equiv 4\pi G\rho_c \tau_\mathrm{SSC}^2</math>
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	<math>~\tau_\mathrm{SSC}^2 = \frac{15}{4\pi G \rho_c} \, .</math>		<math>~\tau_\mathrm{SSC}^2 = \frac{15}{4\pi G \rho_c} \, .</math>
	</div>		</div>
	This has all worked satisfactorily because, [[~~User:Tohline/~~SSC/Structure/~~Other_Analytic_Models~~#Stabililty_2\|as presented above]], this is the correct value of <math>~\tau_\mathrm{SSC}^2</math> in the case of the parabolic density distribution.		This has all worked satisfactorily because, [[SSC/Structure/OtherAnalyticModels#Stabililty_2\|as presented above]], this is the correct value of <math>~\tau_\mathrm{SSC}^2</math> in the case of the parabolic density distribution.


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	<math>~\tau_\mathrm{SSC}^2 = \frac{2^2\cdot 3^2}{5\pi G \rho_c} \, .</math>		<math>~\tau_\mathrm{SSC}^2 = \frac{2^2\cdot 3^2}{5\pi G \rho_c} \, .</math>
	</div>		</div>
	This has all worked satisfactorily because, [[~~User:Tohline/~~SSC/Structure/~~Other_Analytic_Models~~#Lagrangian_Approach\|as presented above]], this is the correct value of <math>~\tau_\mathrm{SSC}^2</math> in the case of the linear density distribution.		This has all worked satisfactorily because, [[SSC/Structure/OtherAnalyticModels#Lagrangian_Approach\|as presented above]], this is the correct value of <math>~\tau_\mathrm{SSC}^2</math> in the case of the linear density distribution.


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	<math>~\tau_\mathrm{SSC}^2 = \frac{2\pi^2}{4\pi G \rho_c} = \frac{\pi}{2G \rho_c} \, .</math>		<math>~\tau_\mathrm{SSC}^2 = \frac{2\pi^2}{4\pi G \rho_c} = \frac{\pi}{2G \rho_c} \, .</math>
	</div>		</div>
	This has all worked satisfactorily because, [[~~User:Tohline/~~SSC/Stability/Polytropes#Setup\|as presented in a separate chapter discussion]], this is the correct value of <math>~\tau_\mathrm{SSC}^2</math> in the case of an n = 1 polytropic configuration.		This has all worked satisfactorily because, [[SSC/Stability/Polytropes#Setup\|as presented in a separate chapter discussion]], this is the correct value of <math>~\tau_\mathrm{SSC}^2</math> in the case of an n = 1 polytropic configuration.

	====Dabbling with LAWE====		====Dabbling with LAWE====
	Now, let's experiment with the [[~~User:Tohline/~~SSC/Structure/~~Other_Analytic_Models~~#Compare\|LAWE as presented above]], that is,		Now, let's experiment with the [[SSC/Structure/OtherAnalyticModels#Compare\|LAWE as presented above]], that is,

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	</div>		</div>

	(We could have, perhaps, obtained this expression in a more direct fashion had we started directly from the form of the [[~~User:Tohline/~~SSC/Structure/~~Other_Analytic_Models~~#LAWE\|LAWE derived earlier]].)		(We could have, perhaps, obtained this expression in a more direct fashion had we started directly from the form of the [[SSC/Structure/OtherAnalyticModels#LAWE\|LAWE derived earlier]].)

	=====Specific Case Attempts=====		=====Specific Case Attempts=====
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	</table>		</table>
	</div>		</div>
	where, [[~~User:Tohline/~~SSC/Structure/~~Other_Analytic_Models~~#Stabililty_2\|as defined above]],		where, [[SSC/Structure/OtherAnalyticModels#Stabililty_2\|as defined above]],
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	<math>~\alpha \equiv 3 - \frac{4}{\gamma_g} \, .</math>		<math>~\alpha \equiv 3 - \frac{4}{\gamma_g} \, .</math>
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	</table>		</table>
	</div>		</div>
	where, following [[~~User:Tohline~~/~~SSC~~/UniformDensity#Setup_as_Presented_by_Sterne_.281937.29\|Sterne's (1937) presentation]],		where, following [[SSC/Stability/UniformDensity#Setup_as_Presented_by_Sterne_.281937.29\|Sterne's (1937) presentation]],
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	<math>~\mathfrak{F} \equiv \sigma^2 - 2 \alpha \, .</math>		<math>~\mathfrak{F} \equiv \sigma^2 - 2 \alpha \, .</math>
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	</table>		</table>
	</div>		</div>


	==Exploration2==		==Exploration2==

Jet53man: /* Conjecture */

2021-10-15T21:43:49Z

Conjecture

← Older revision		Revision as of 17:43, 15 October 2021
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	===Conjecture===		===Conjecture===

	Returning to the [[~~User:Tohline/~~SSC/Structure/~~Other_Analytic_Models~~#Generic\|generic formulation derived earlier]], we have,		Returning to the [[SSC/Structure/OtherAnalyticModels#Generic\|generic formulation derived earlier]], we have,

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Jet53man: /* Independent Investigation of Parabolic Distribution */

2021-10-15T21:42:52Z

Independent Investigation of Parabolic Distribution

← Older revision		Revision as of 17:42, 15 October 2021
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	</div>		</div>

	The expressions for the coefficients presented in [[~~User:Tohline/~~SSC/Structure/~~Other_Analytic_Models~~#SecondTable\|this table]] exactly match the entire set of expressions [[~~User:Tohline/~~SSC/Structure/~~Other_Analytic_Models~~#FirstTable\|derived earlier]], except for the adopted sign convention — every term in this second derivation has the opposite sign to the corresponding term in the earlier derivation.		The expressions for the coefficients presented in [[SSC/Structure/OtherAnalyticModels#SecondTable\|this table]] exactly match the entire set of expressions [[SSC/Structure/OtherAnalyticModels#FirstTable\|derived earlier]], except for the adopted sign convention — every term in this second derivation has the opposite sign to the corresponding term in the earlier derivation.

	===Conjecture===		===Conjecture===

Jet53man: /* Parabolic Density Distribution */

2021-10-15T21:41:30Z

Parabolic Density Distribution

← Older revision		Revision as of 17:41, 15 October 2021
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	===Parabolic Density Distribution===		===Parabolic Density Distribution===

	In the case of a [[~~User:Tohline/~~SSC/Structure/~~Other_Analytic_Models~~#Parabolic_Density_Distribution\|parabolic density distribution]], the governing equation is,		In the case of a [[SSC/Structure/OtherAnalyticModels#Parabolic_Density_Distribution\|parabolic density distribution]], the governing equation is,

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Jet53man: /* Uniform Density */

2021-10-15T21:40:13Z

Uniform Density

← Older revision		Revision as of 17:40, 15 October 2021
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	</div>		</div>

	The following individual mode analyses should be compared with the results found in [[~~User:Tohline~~/~~SSC~~/UniformDensity#Sterne.27s_General_Solution\|our discussion of Sterne's general solution]].		The following individual mode analyses should be compared with the results found in [[SSC/Stability/UniformDensity#Sterne.27s_General_Solution\|our discussion of Sterne's general solution]].

	====Mode 0====		====Mode 0====
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	</table>		</table>
	</div>		</div>



	===Parabolic Density Distribution===		===Parabolic Density Distribution===

Jet53man: /* Examine Structural Pressure-Density Relation */

2021-10-15T21:37:42Z

Examine Structural Pressure-Density Relation

← Older revision		Revision as of 17:37, 15 October 2021
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	====Example1====		====Example1====

	First, let's set <math>~b=0</math>, but leave <math>~a</math> general. As described in the [[~~User:Tohline/~~SSC/Structure/~~Other_Analytic_Models~~#Surface\|above ASIDE]], this means that <math>~z_s=a^{-1}</math>. So the pressure distribution is,		First, let's set <math>~b=0</math>, but leave <math>~a</math> general. As described in the [[SSC/Structure/OtherAnalyticModels#Surface\|above ASIDE]], this means that <math>~z_s=a^{-1}</math>. So the pressure distribution is,

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	====Example2====		====Example2====

	First, let's set <math>~a=0</math>, but leave <math>~b</math> general. As described in the [[~~User:Tohline/~~SSC/Structure/~~Other_Analytic_Models~~#Surface\|above ASIDE]], this means that <math>~z_s=b^{-1/2}</math>. So the pressure distribution is,		First, let's set <math>~a=0</math>, but leave <math>~b</math> general. As described in the [[SSC/Structure/OtherAnalyticModels#Surface\|above ASIDE]], this means that <math>~z_s=b^{-1/2}</math>. So the pressure distribution is,

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	</div>		</div>

	The question that remains to be answered is: Is this expression for the pressure distribution "evenly divisible" by the square (or even the ''first'' power) of the normalized density distribution which, [[~~User:Tohline/~~SSC/Structure/~~Other_Analytic_Models~~#Derivation\|as defined above]] for the general quadratic case, is,		The question that remains to be answered is: Is this expression for the pressure distribution "evenly divisible" by the square (or even the ''first'' power) of the normalized density distribution which, [[SSC/Structure/OtherAnalyticModels#Derivation\|as defined above]] for the general quadratic case, is,
	<div align="center">		<div align="center">
	<math>\frac{\rho}{\rho_c} = 1 - az - bz^2 = 1 - (az_s)\zeta - (bz_s^2)\zeta^2 \, .</math>		<math>\frac{\rho}{\rho_c} = 1 - az - bz^2 = 1 - (az_s)\zeta - (bz_s^2)\zeta^2 \, .</math>
	</div>		</div>

	In attempting to answer this question, it may prove advantageous to refer back to the [[~~User:Tohline/~~SSC/Structure/~~Other_Analytic_Models~~#Surface\|above ASIDE discussion]] of the roots of this quadratic function and, in particular, that,		In attempting to answer this question, it may prove advantageous to refer back to the [[SSC/Structure/OtherAnalyticModels#Surface\|above ASIDE discussion]] of the roots of this quadratic function and, in particular, that,
	<div align="center">		<div align="center">
	<math>~az_s = \frac{2}{\ell^2}\biggl[\biggl(1+\ell^2\biggr)^{1/2} - 1\biggr]</math>		<math>~az_s = \frac{2}{\ell^2}\biggl[\biggl(1+\ell^2\biggr)^{1/2} - 1\biggr]</math>
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	</tr>		</tr>
	<tr>		<tr>
	<td align="center">[[~~User:Tohline~~/~~SSC~~/UniformDensity#The_Stability_of_Uniform-Density_Spheres\|Uniform-density]]</td>		<td align="center">[[SSC/Stability/UniformDensity#The_Stability_of_Uniform-Density_Spheres\|Uniform-density]]</td>
	<td align="center"><math>~0</math></td>		<td align="center"><math>~0</math></td>
	<td align="center"><math>~0</math></td>		<td align="center"><math>~0</math></td>
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	</tr>		</tr>
	<tr>		<tr>
	<td align="center">[[~~User:Tohline/~~SSC/Structure/~~Other_Analytic_Models~~#Linear_Density_Distribution\|Linear]]</td>		<td align="center">[[SSC/Structure/OtherAnalyticModels#Linear_Density_Distribution\|Linear]]</td>
	<td align="center"><math>~1</math></td>		<td align="center"><math>~1</math></td>
	<td align="center"><math>~0</math></td>		<td align="center"><math>~0</math></td>
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	</tr>		</tr>
	<tr>		<tr>
	<td align="center">[[~~User:Tohline/~~SSC/Structure/~~Other_Analytic_Models~~#Parabolic_Density_Distribution\|Parabolic]]</td>		<td align="center">[[SSC/Structure/OtherAnalyticModels#Parabolic_Density_Distribution\|Parabolic]]</td>
	<td align="center"><math>~0</math></td>		<td align="center"><math>~0</math></td>
	<td align="center"><math>~1</math></td>		<td align="center"><math>~1</math></td>

Jet53man: /* Polytropic Configurations */

2021-10-15T21:35:04Z

Polytropic Configurations

← Older revision		Revision as of 17:35, 15 October 2021
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	===Polytropic Configurations===		===Polytropic Configurations===
	Let's compare this presentation of the LAWE to the form of the LAWE that has been derived [[~~User:Tohline/~~SSC/Stability/Polytropes#Adiabatic_.28Polytropic.29_Wave_Equation\|specifically for polytropic equilibrium configurations]], namely,		Let's compare this presentation of the LAWE to the form of the LAWE that has been derived [[SSC/Stability/Polytropes#Adiabatic_.28Polytropic.29_Wave_Equation\|specifically for polytropic equilibrium configurations]], namely,
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	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</table>		</table>
	</div>		</div>
	[Note that <math>~\theta_c = 1</math> and, therefore for all practical purposes, it can be dropped. This notation was introduced in our [[~~User:Tohline/~~SSC/Stability/Polytropes#Adiabatic_.28Polytropic.29_Wave_Equation\|separate discussion of the polytropic LAWE]] in order to make it clear how our derivations have overlapped earlier published work.] Regrouping terms, we have,		[Note that <math>~\theta_c = 1</math> and, therefore for all practical purposes, it can be dropped. This notation was introduced in our [[SSC/Stability/Polytropes#Adiabatic_.28Polytropic.29_Wave_Equation\|separate discussion of the polytropic LAWE]] in order to make it clear how our derivations have overlapped earlier published work.] Regrouping terms, we have,
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	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</table>		</table>
	</div>		</div>
	which precisely matches the general expression for the LAWE presented at the end of our [[~~User:Tohline/~~SSC/Structure/~~Other_Analytic_Models~~#Generic_Setup\|generic setup, directly above]].		which precisely matches the general expression for the LAWE presented at the end of our [[SSC/Structure/OtherAnalyticModels#Generic_Setup\|generic setup, directly above]].

	This seems to be a particularly insightful way to write the LAWE, as the only structural functions that appear explicitly are <math>~P_0(\chi_0)</math> and <math>~\rho_0(\chi_0)</math>. It appears as though the eigenfunctions that describe ''adiabatic'' radial pulsations do not explicitly depend ''a priori'' on the radial dependence of the equilibrium gravitational acceleration.		This seems to be a particularly insightful way to write the LAWE, as the only structural functions that appear explicitly are <math>~P_0(\chi_0)</math> and <math>~\rho_0(\chi_0)</math>. It appears as though the eigenfunctions that describe ''adiabatic'' radial pulsations do not explicitly depend ''a priori'' on the radial dependence of the equilibrium gravitational acceleration.

Jet53man: /* Generic Setup */

2021-10-15T21:33:26Z

Generic Setup

← Older revision		Revision as of 17:33, 15 October 2021
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	</tr>		</tr>
	<tr>		<tr>
	<td align="center">[[~~User:Tohline~~/~~SSC~~/UniformDensity#The_Stability_of_Uniform-Density_Spheres\|Uniform-density]]</td>		<td align="center">[[SSC/Stability/UniformDensity#The_Stability_of_Uniform-Density_Spheres\|Uniform-density]]</td>
	<td align="center"><math>~1</math>		<td align="center"><math>~1</math>
	<td align="center"><math>~1 - \chi_0^2</math>		<td align="center"><math>~1 - \chi_0^2</math>
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	</tr>		</tr>
	<tr>		<tr>
	<td align="center">[[~~User:Tohline/~~SSC/Structure/~~Other_Analytic_Models~~#Linear_Density_Distribution\|Linear]]</td>		<td align="center">[[SSC/Structure/OtherAnalyticModels#Linear_Density_Distribution\|Linear]]</td>
	<td align="center"><math>~1-\chi_0</math>		<td align="center"><math>~1-\chi_0</math>
	<td align="center"><math>~\tfrac{1}{5} (5 -24 \chi_0^2 + 28\chi_0^3 - 9\chi_0^4)</math>		<td align="center"><math>~\tfrac{1}{5} (5 -24 \chi_0^2 + 28\chi_0^3 - 9\chi_0^4)</math>
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	</tr>		</tr>
	<tr>		<tr>
	<td align="center">[[~~User:Tohline/~~SSC/Structure/~~Other_Analytic_Models~~#Parabolic_Density_Distribution\|Parabolic]]</td>		<td align="center">[[SSC/Structure/OtherAnalyticModels#Parabolic_Density_Distribution\|Parabolic]]</td>
	<td align="center"><math>~1-\chi_0^2</math>		<td align="center"><math>~1-\chi_0^2</math>
	<td align="center"><math>~\tfrac{1}{2} (2 - 5\chi_0^2 + 4\chi_0^4 - \chi_0^6)</math>		<td align="center"><math>~\tfrac{1}{2} (2 - 5\chi_0^2 + 4\chi_0^4 - \chi_0^6)</math>
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	</tr>		</tr>
	<tr>		<tr>
	<td align="center">[[~~User:Tohline/~~SSC/Stability/Polytropes#n_.3D_1_Polytrope\|<math>~n=1</math> Polytrope]]</td>		<td align="center">[[SSC/Stability/Polytropes#n_.3D_1_Polytrope\|<math>~n=1</math> Polytrope]]</td>
	<td align="center"><math>~\frac{\sin(\pi\chi_0)}{\pi\chi_0}</math>		<td align="center"><math>~\frac{\sin(\pi\chi_0)}{\pi\chi_0}</math>
	<td align="center"><math>~\biggl[\frac{\sin(\pi\chi_0)}{\pi\chi_0}\biggr]^2</math>		<td align="center"><math>~\biggl[\frac{\sin(\pi\chi_0)}{\pi\chi_0}\biggr]^2</math>
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	</tr>		</tr>
	<tr>		<tr>
	<td align="center">[[~~User:Tohline~~/~~SSC~~/UniformDensity#The_Stability_of_Uniform-Density_Spheres\|Uniform-density]]</td>		<td align="center">[[SSC/Stability/UniformDensity#The_Stability_of_Uniform-Density_Spheres\|Uniform-density]]</td>
	<td align="center"><math>~1</math>		<td align="center"><math>~1</math>
	<td align="center"><math>~2\chi_0</math>		<td align="center"><math>~2\chi_0</math>
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	</tr>		</tr>
	<tr>		<tr>
	<td align="center">[[~~User:Tohline/~~SSC/Structure/~~Other_Analytic_Models~~#Linear_Density_Distribution\|Linear]]</td>		<td align="center">[[SSC/Structure/OtherAnalyticModels#Linear_Density_Distribution\|Linear]]</td>
	<td align="center"><math>~1-\chi_0</math>		<td align="center"><math>~1-\chi_0</math>
	<td align="center"><math>~\tfrac{48}{5}(\chi_0 - \tfrac{3}{4}\chi_0^2)</math>		<td align="center"><math>~\tfrac{48}{5}(\chi_0 - \tfrac{3}{4}\chi_0^2)</math>
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	</tr>		</tr>
	<tr>		<tr>
	<td align="center">[[~~User:Tohline/~~SSC/Structure/~~Other_Analytic_Models~~#Parabolic_Density_Distribution\|Parabolic]]</td>		<td align="center">[[SSC/Structure/OtherAnalyticModels#Parabolic_Density_Distribution\|Parabolic]]</td>
	<td align="center"><math>~1-\chi_0^2</math>		<td align="center"><math>~1-\chi_0^2</math>
	<td align="center"><math>~5\chi_0 - 3\chi_0^3</math>		<td align="center"><math>~5\chi_0 - 3\chi_0^3</math>
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	</tr>		</tr>
	<tr>		<tr>
	<td align="center">[[~~User:Tohline/~~SSC/Stability/Polytropes#n_.3D_1_Polytrope\|<math>~n=1</math> Polytrope]]</td>		<td align="center">[[SSC/Stability/Polytropes#n_.3D_1_Polytrope\|<math>~n=1</math> Polytrope]]</td>
	<td align="center"><math>~\frac{\sin(\pi\chi_0)}{\pi\chi_0}</math>		<td align="center"><math>~\frac{\sin(\pi\chi_0)}{\pi\chi_0}</math>
	<td align="center"><math>~\frac{2}{\pi\chi_0^2}\biggl[ \sin(\pi\chi_0) - \pi\chi_0 \cos(\pi\chi_0) \biggr]</math>		<td align="center"><math>~\frac{2}{\pi\chi_0^2}\biggl[ \sin(\pi\chi_0) - \pi\chi_0 \cos(\pi\chi_0) \biggr]</math>

Jet53man: /* Derivation */

2021-10-15T21:25:43Z

Derivation

← Older revision		Revision as of 17:25, 15 October 2021
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	====Derivation====		====Derivation====

	One striking property exhibited by the [[~~User:Tohline/~~SSC/Structure/~~Other_Analytic_Models~~#Examples\|example configurations tabulated above]] is the ''structural'' relationship between the chosen function, <math>~\rho_0(\chi_0)</math>, and the corresponding radial pressure distribution, <math>~P_0(\chi_0)</math>, that is [[~~User:Tohline/SphericallySymmetricConfigurations~~/SolutionStrategies#Spherically_Symmetric_Configurations_.28Part_II.29\|dictated by]],		One striking property exhibited by the [[SSC/Structure/OtherAnalyticModels#Examples\|example configurations tabulated above]] is the ''structural'' relationship between the chosen function, <math>~\rho_0(\chi_0)</math>, and the corresponding radial pressure distribution, <math>~P_0(\chi_0)</math>, that is [[SSCpt2/SolutionStrategies#Spherically_Symmetric_Configurations_.28Part_II.29\|dictated by]],

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	As has been detailed in the last column of [[~~User:Tohline/~~SSC/Structure/~~Other_Analytic_Models~~#Table1b\|Table 1b]], in all four cases, the ratio, <math>~(P_0/P_c)(\rho_0/\rho_c)^{-2}</math>, is an analytically prescribed polynomial expression. That is, the pressure is "evenly divisible" by the square of the density. Let's examine how broadly reliable this behavior is. [Note that, for simplicity in typing, hereafter throughout this subsection we will drop the subscript zero and, rather than <math>~\chi_0</math>, we will use <math>~z \equiv r/R</math> to denote the dimensionless radial coordinate.]		As has been detailed in the last column of [[SSC/Structure/OtherAnalyticModels#Table1b\|Table 1b]], in all four cases, the ratio, <math>~(P_0/P_c)(\rho_0/\rho_c)^{-2}</math>, is an analytically prescribed polynomial expression. That is, the pressure is "evenly divisible" by the square of the density. Let's examine how broadly reliable this behavior is. [Note that, for simplicity in typing, hereafter throughout this subsection we will drop the subscript zero and, rather than <math>~\chi_0</math>, we will use <math>~z \equiv r/R</math> to denote the dimensionless radial coordinate.]


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	<math>~P_n \equiv 4\pi G \rho_c^2 R^2 \, .</math>		<math>~P_n \equiv 4\pi G \rho_c^2 R^2 \, .</math>
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	The integration constant — which also proves to be the normalized central pressure — is determined by ensuring that the pressure goes to zero at [[~~User:Tohline/~~SSC/Structure/~~Other_Analytic_Models~~#Surface\|the surface of the configuration]], <math>~z_s</math>. That is,		The integration constant — which also proves to be the normalized central pressure — is determined by ensuring that the pressure goes to zero at [[SSC/Structure/OtherAnalyticModels#Surface\|the surface of the configuration]], <math>~z_s</math>. That is,
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