Appendix/Ramblings/BiPolytropeStability - Revision history

Jet53man: /* Begin Our Analysis */

2022-04-12T21:16:41Z

Begin Our Analysis

Jet53man: /* Envelope Layers With n = 1 */

2022-04-12T21:11:21Z

Envelope Layers With n = 1

← Older revision		Revision as of 17:11, 12 April 2022
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	In a [[~~User:Tohline/~~SSC/Stability/InstabilityOnsetOverview#Configurations_Having_an_Index_Less_Than_Three\|separate chapter]], we explain that the analytically defined eigenfunction that satisfies this LAWE — when <math>~\omega_k^2 = 0</math> and <math>~\alpha_\mathrm{env} = +1</math> — is,		In a [[SSC/Stability/InstabilityOnsetOverview#Configurations_Having_an_Index_Less_Than_Three\|separate chapter]], we explain that the analytically defined eigenfunction that satisfies this LAWE — when <math>\omega_k^2 = 0</math> and <math>\alpha_\mathrm{env} = +1</math> — is,
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	<td align="right">		<td align="right">
	<math>~x_P\biggr\|_{n=1}</math>		<math>x_P\biggr\|_{n=1}</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>
	\frac{3b_e}{\eta^2}\biggl[ 1- \eta \cot\eta \biggr] \, .		\frac{3b_e}{\eta^2}\biggl[ 1- \eta \cot\eta \biggr] \, .
	</math>		</math>

Jet53man: /* Begin Our Analysis */

2022-04-12T21:10:09Z

Begin Our Analysis

Show changes

Jet53man: /* Envelope: */

2022-04-12T21:06:01Z

Envelope:

← Older revision		Revision as of 17:06, 12 April 2022
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	Given that,		Given that,
	<div align="center">		<div align="center">
	<math>~\biggl( \frac{\mu_e}{\mu_c} \biggr) \biggl(\frac{2\pi}{3}\biggr)^{1 / 2}~r^* = \eta \, ,</math>		<math>\biggl( \frac{\mu_e}{\mu_c} \biggr) \biggl(\frac{2\pi}{3}\biggr)^{1 / 2}~r^* = \eta \, ,</math>
	</div>		</div>
	let's multiply the LAWE through by <math>~(3/2\pi)( \mu_e/\mu_c)^{-2} </math>. This gives,		let's multiply the LAWE through by <math>(3/2\pi)( \mu_e/\mu_c)^{-2} </math>. This gives,

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	<td align="right">		<td align="right">
	<math>~0</math>		<math>0</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>
	\frac{d^2x}{d\eta^2} + \biggl\{ 4 -\biggl(\frac{\rho^}{P^}\biggr)\frac{ M_r^}{(r^)}\biggr\}\frac{1}{\eta} \cdot \frac{dx}{d\eta}		\frac{d^2x}{d\eta^2} + \biggl\{ 4 -\biggl(\frac{\rho^}{P^}\biggr)\frac{ M_r^}{(r^)}\biggr\}\frac{1}{\eta} \cdot \frac{dx}{d\eta}
	+ \frac{3}{2\pi} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \biggl(\frac{\rho^}{ P^ } \biggr)\biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}} ~-~\frac{\alpha_\mathrm{g} M_r^}{(r^)^3}\biggr\} x \, .		+ \frac{3}{2\pi} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \biggl(\frac{\rho^}{ P^ } \biggr)\biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}} ~-~\frac{\alpha_\mathrm{g} M_r^}{(r^)^3}\biggr\} x \, .
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\frac{x_+ -2x_j + x_-}{(\delta \eta)^2}</math>		<math>\frac{x_+ -2x_j + x_-}{(\delta \eta)^2}</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>
	-~ \frac{\mathcal{H}}{\eta} \biggl[ \frac{x_+ - x_-}{2\delta \eta} \biggr] ~-~ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl[ \frac{3\mathcal{K}}{2\pi} \biggr]x_j		-~ \frac{\mathcal{H}}{\eta} \biggl[ \frac{x_+ - x_-}{2\delta \eta} \biggr] ~-~ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl[ \frac{3\mathcal{K}}{2\pi} \biggr]x_j
	</math>		</math>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\Rightarrow ~~~ x_+ -2x_j + x_-</math>		<math>\Rightarrow ~~~ x_+ -2x_j + x_-</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>
	-~ \frac{\delta \eta}{2\eta} \biggl[ x_+ - x_- \biggr]\mathcal{H} ~-~ (\delta \eta)^2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl[ \frac{3\mathcal{K}}{2\pi} \biggr] x_j		-~ \frac{\delta \eta}{2\eta} \biggl[ x_+ - x_- \biggr]\mathcal{H} ~-~ (\delta \eta)^2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl[ \frac{3\mathcal{K}}{2\pi} \biggr] x_j
	</math>		</math>
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	<td align="right">		<td align="right">
	<math>~\Rightarrow ~~~ x_{j+1} \biggl[1 + \biggl( \frac{\delta \eta}{2\eta}\biggr) \mathcal{H} \biggr]</math>		<math>\Rightarrow ~~~ x_{j+1} \biggl[1 + \biggl( \frac{\delta \eta}{2\eta}\biggr) \mathcal{H} \biggr]</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>
	\biggl[ 2 - (\delta \eta)^2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \biggl( \frac{3\mathcal{K}}{2\pi} \biggr) \biggr] x_j ~-~\biggl[ 1 - \biggl( \frac{\delta \eta}{2\eta} \biggr) \mathcal{H} \biggr]x_{j-1} \, .		\biggl[ 2 - (\delta \eta)^2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \biggl( \frac{3\mathcal{K}}{2\pi} \biggr) \biggr] x_j ~-~\biggl[ 1 - \biggl( \frac{\delta \eta}{2\eta} \biggr) \mathcal{H} \biggr]x_{j-1} \, .
	</math>		</math>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~x_{i+1}		<math>x_{i+1}
	</math>		</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>
	\biggl[ 1 - \tfrac{1}{2}(\delta \eta)^2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \biggl( \frac{3\mathcal{K}}{2\pi} \biggr)\biggr] x_i		\biggl[ 1 - \tfrac{1}{2}(\delta \eta)^2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \biggl( \frac{3\mathcal{K}}{2\pi} \biggr)\biggr] x_i
	~+~		~+~

Jet53man: /* Special Handling at the Center */

2022-04-12T21:03:44Z

Special Handling at the Center

← Older revision		Revision as of 17:03, 12 April 2022
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	===Special Handling at the Center===		===Special Handling at the Center===
	In order to kick-start the integration, we set the displacement function value to <math>~x_1 = 1</math> at the center of the configuration <math>~(\xi_1 = 0)</math>, then draw on the [[~~User:Tohline/~~Appendix/Ramblings/PowerSeriesExpressions#PolytropicDisplacement\|derived power-series expression]] to determine the value of the displacement function at the first radial grid line, <math>~\xi_2 = \delta\xi</math>, away from the center. Specifically, we set,		In order to kick-start the integration, we set the displacement function value to <math>~x_1 = 1</math> at the center of the configuration <math>(\xi_1 = 0)</math>, then draw on the [[Appendix/Ramblings/PowerSeriesExpressions#PolytropicDisplacement\|derived power-series expression]] to determine the value of the displacement function at the first radial grid line, <math>\xi_2 = \delta\xi</math>, away from the center. Specifically, we set,
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Jet53man: /* General Approach */

2022-04-12T21:03:07Z

General Approach

← Older revision		Revision as of 17:03, 12 April 2022
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	</tr>		</tr>
	</table>		</table>
	Adopting the same approach [[~~User:Tohline/~~SSC/Stability/Polytropes#Numerical_Integration_from_the_Center.2C_Outward\|as before when we integrated the LAWE for pressure-truncated polytropes]], we will enlist the finite-difference approximations,		Adopting the same approach [[SSC/Stability/Polytropes#Numerical_Integration_from_the_Center.2C_Outward\|as before when we integrated the LAWE for pressure-truncated polytropes]], we will enlist the finite-difference approximations,
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Jet53man: /* Model 10 */

2022-04-12T21:01:45Z

Model 10

← Older revision		Revision as of 17:01, 12 April 2022
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	==Model 10==		==Model 10==
	As we have [[~~User:Tohline/~~SSC/Structure/BiPolytropes/~~Analytic1_5~~#Murphy_and_Fiedler_.281985.29\|reviewed in an accompanying discussion]], equilibrium Model 10 from [http://adsabs.harvard.edu/abs/1985PASAu...6..219M Murphy & Fiedler (1985, Proc. Astr. Soc. of Australia, 6, 219)] is defined by setting <math>~(\xi_i, m) = (2.5646, 1)</math>. Drawing directly from [[~~User:Tohline/~~SSC/Structure/BiPolytropes/~~Analytic1_5~~#Murphy_and_Fiedler_.281985.29\|our reproduction of their Table 1]], we see that a few relevant structural parameters of Model 10 are,		As we have [[SSC/Structure/BiPolytropes/Analytic15#Murphy_and_Fiedler_.281985.29\|reviewed in an accompanying discussion]], equilibrium Model 10 from [http://adsabs.harvard.edu/abs/1985PASAu...6..219M Murphy & Fiedler (1985, Proc. Astr. Soc. of Australia, 6, 219)] is defined by setting <math>(\xi_i, m) = (2.5646, 1)</math>. Drawing directly from [[SSC/Structure/BiPolytropes/Analytic15#Murphy_and_Fiedler_.281985.29\|our reproduction of their Table 1]], we see that a few relevant structural parameters of Model 10 are,
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Jet53man: /* Profile */

2022-04-12T21:00:48Z

Profile

← Older revision		Revision as of 17:00, 12 April 2022
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	==Profile==		==Profile==

	Now, referencing the [[~~User:Tohline/~~SSC/Structure/BiPolytropes/~~Analytic1_5~~#Profile\|derived bipolytropic model profile]], we should incorporate the following relations:		Now, referencing the [[SSC/Structure/BiPolytropes/Analytic15#Profile\|derived bipolytropic model profile]], we should incorporate the following relations:

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	<td align="left" colspan="6">		<td align="left" colspan="6">
	<sup>&dagger;</sup>In order to obtain the various envelope profiles, it is necessary to evaluate <math>~\phi(\eta)</math> and its first derivative using the information [[~~User:Tohline/~~SSC/Structure/BiPolytropes/~~Analytic1_5~~#Step_6:__Envelope_Solution\|presented in Step 6 of our accompanying discussion]].		<sup>&dagger;</sup>In order to obtain the various envelope profiles, it is necessary to evaluate <math>~\phi(\eta)</math> and its first derivative using the information [[SSC/Structure/BiPolytropes/Analytic15#Step_6:__Envelope_Solution\|presented in Step 6 of our accompanying discussion]].
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Foundation */

2022-04-12T20:59:39Z

Foundation

← Older revision		Revision as of 16:59, 12 April 2022
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	=Regroup=		=Regroup=
	==Foundation==		==Foundation==
	In an [[~~User:Tohline/~~SSC/Perturbations#2ndOrderODE\|accompanying discussion]], we derived the so-called,		In an [[SSC/Perturbations#2ndOrderODE\|accompanying discussion]], we derived the so-called,

	<div align="center" id="2ndOrderODE">		<div align="center" id="2ndOrderODE">
	<font color="#770000">'''Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br />		<font color="#770000">'''Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br />

	{{~~User:Tohline/~~Math/EQ_RadialPulsation01}}		{{Math/EQ_RadialPulsation01}}
	</div>		</div>

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	-->		-->

	whose solution gives eigenfunctions that describe various radial modes of oscillation in spherically symmetric, self-gravitating fluid configurations. Assuming that the underlying equilibrium structure is that of a [[~~User:Tohline/~~SSC/Structure/BiPolytropes/~~Analytic1_5~~#BiPolytrope_with_nc_.3D_1_and_ne_.3D_5\|bipolytrope having <math>~(n_c, n_e) = (1, 5)</math>]], it makes sense to adopt the [[~~User:Tohline/~~SSC/Structure/BiPolytropes/~~Analytic1_5~~#Normalization\|normalizations used when defining the equilibrium structure]], namely,		whose solution gives eigenfunctions that describe various radial modes of oscillation in spherically symmetric, self-gravitating fluid configurations. Assuming that the underlying equilibrium structure is that of a [[SSC/Structure/BiPolytropes/Analytic15#BiPolytrope_with_nc_.3D_1_and_ne_.3D_5\|bipolytrope having <math>~(n_c, n_e) = (1, 5)</math>]], it makes sense to adopt the [[SSC/Structure/BiPolytropes/Analytic15#Normalization\|normalizations used when defining the equilibrium structure]], namely,
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	</div>		</div>

	We [[~~User:Tohline/~~SSC/Stability/Polytropes#Groundwork\|note as well]] that,		We [[SSC/Stability/Polytropes#Groundwork\|note as well]] that,
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Jet53man: /* Our Numerical Integration */

2022-04-12T20:57:49Z

Our Numerical Integration

← Older revision		Revision as of 16:57, 12 April 2022
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	==Our Numerical Integration==		==Our Numerical Integration==

	Let's try to integrate this bipolytrope's LAWE from the center, outward, using as a guideline an [[~~User:Tohline/~~SSC/Stability/Polytropes#Numerical_Integration_from_the_Center.2C_Outward\|accompanying ''Numerical Integration'' outline]]. Generally, for any polytropic index, the relevant LAWE can be written in the form,		Let's try to integrate this bipolytrope's LAWE from the center, outward, using as a guideline an [[SSC/Stability/Polytropes#Numerical_Integration_from_the_Center.2C_Outward\|accompanying ''Numerical Integration'' outline]]. Generally, for any polytropic index, the relevant LAWE can be written in the form,

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	</table>		</table>

	In order to kick-start the integration, we will set the displacement function value to <math>~x_1 = 1</math> at the center of the configuration <math>~(\xi_1 = 0)</math>, then we will draw on the [[~~User:Tohline/~~Appendix/Ramblings/PowerSeriesExpressions#PolytropicDisplacement\|derived power-series expression]] to determine the value of the displacement function at the first radial grid line, <math>~\xi_2 = \delta\xi</math>, away from the center. Specifically, we will set,		In order to kick-start the integration, we will set the displacement function value to <math>~x_1 = 1</math> at the center of the configuration <math>~(\xi_1 = 0)</math>, then we will draw on the [[Appendix/Ramblings/PowerSeriesExpressions#PolytropicDisplacement\|derived power-series expression]] to determine the value of the displacement function at the first radial grid line, <math>~\xi_2 = \delta\xi</math>, away from the center. Specifically, we will set,
	<div align="center">		<div align="center">
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	===Interface===		===Interface===

	Keep in mind that, as has been [[~~User:Tohline/~~SSC/Structure/BiPolytropes/~~Analytic1_5~~#Parameter_Values\|detailed in the accompanying ''equilibrium structure'' chapter]], for <math>~(n_c, n_e) = (1, 5)</math> bipolytropes,		Keep in mind that, as has been [[SSC/Structure/BiPolytropes/Analytic15#Parameter_Values\|detailed in the accompanying ''equilibrium structure'' chapter]], for <math>~(n_c, n_e) = (1, 5)</math> bipolytropes,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">