User:Tohline/SSC/Synopsis StyleSheet - Revision history

Tohline: /* Bipolytropes */

2019-02-04T23:02:29Z

Bipolytropes

← Older revision		Revision as of 23:02, 4 February 2019
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	</table>		</table>
	</div>		</div>
	! style="vertical-align:top; text-align:left;" rowspan="5"\|		! style="vertical-align:top; text-align:left;" rowspan="3"\|
	As we have detailed in an [[User:Tohline/SSC/BipolytropeGeneralization#Free_Energy_and_Its_Derivatives\|accompanying discussion]], the first derivative of the relevant free-energy expression is,		As we have detailed in an [[User:Tohline/SSC/BipolytropeGeneralization#Free_Energy_and_Its_Derivatives\|accompanying discussion]], the first derivative of the relevant free-energy expression is,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	3^2 P_i V_\mathrm{core}(\gamma_c - \gamma_e)		3^2 P_i V_\mathrm{core}(\gamma_c - \gamma_e)
	+ (4-3\gamma_c ) W_\mathrm{core}		+ (4-3\gamma_c ) W_\mathrm{core}
	+ (4-3\gamma_e)W_\mathrm{env}		+ (4-3\gamma_e)W_\mathrm{env} \, .
	</math>		</math>
	</td>		</td>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\~~omega~~^2 \int_0^R r^2 dM_r</math>		<math>~\biggl( \frac{2\pi}{3}\biggr)\sigma_c^2 \int_0^R r^2 dM_r</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	<td align="left">		<td align="left">
	<math>~		<math>~
	(4- 3\~~gamma~~) W_\mathrm{~~grav~~}+ ~~3^2 \gamma P_eV \, .~~		(4- 3\gamma_c) W_\mathrm{core}+ (4- 3\gamma_e) W_\mathrm{env}+ 3^2 (\gamma_c - \gamma_e) P_i V_\mathrm{core} \, .
	~~</math>~~
	~~</td>~~
	~~</tr>~~
	~~</table>~~
	~~</div>~~

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	~~! style="text-align:center; background-color:#ffff99;" width="50%" \|<b><font color="maroon" size="+1">⑥</font></b>   <b>Approximation:   Homologous Expansion/Contraction</b>~~

	\|-
	~~! style="vertical-align:top; text-align:left;" \|~~
	If we ''guess'' that radial oscillations about the equilibrium state involve purely homologous expansion/contraction, then the radial-displacement eigenfunction is, <math>~x</math> = constant, and the governing variational relation gives,

	~~<div align="center">~~
	~~<table border="0" cellpadding="5" align="center">~~
	~~<tr>~~
	~~<td align="right">~~
	~~<math>~\omega^2 \int_0^R r^2 dM_r</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~\leq</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	(4- 3\~~gamma~~) W_\mathrm{~~grav~~}+ 3^2 \~~gamma~~ ~~P_eV~~ \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Bipolytropes */

2019-02-04T22:56:46Z

Bipolytropes

← Older revision		Revision as of 22:56, 4 February 2019
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	! style="text-align:center; background-color:#ffff99;" width="50%" \|<b><font color="maroon" size="+1">&#~~x2464~~;</font></b>   <b>~~Variational Principle~~</b>		! style="text-align:center; background-color:#ffff99;" width="50%" \|<b><font color="maroon" size="+1">⑨</font></b>   <b>Approximation:   Homologous Expansion/Contraction</b>

	\|-		\|-
	! style="vertical-align:top; text-align:left;" \|		! style="vertical-align:top; text-align:left;" \|
			If we ''guess'' that radial oscillations about the equilibrium state involve purely homologous expansion/contraction, then the radial-displacement eigenfunction is, <math>~x</math> = constant, and the governing variational relation gives,

	<div align="center">		<div align="center">
	~~<font color="#770000">'''Governing Variational Relation</font><br />~~
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

	~~<tr>~~
	~~<td align="right">~~
	~~<math>~\biggl( \frac{2\pi}{3}\biggr)\sigma_c^2 \int_0^{R^} (x r^)^2 dM_r^* </math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	~~\gamma_c (\gamma_c-1) \int_0^{r^_\mathrm{core}} x^2~\biggl( \frac{d\ln x}{d\ln r^} \biggr)^2 dU^*_\mathrm{int}~~
	~~- (3\gamma_c - 4) \int_0^{r^_\mathrm{core}} x^2 dW^_\mathrm{grav}~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right">~~
	~~ ~~
	~~</td>~~
	~~<td align="center">~~
	~~ ~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	~~+ ~\gamma_e (\gamma_e-1) \int_{r^_\mathrm{core}}^{R^} x^2~\biggl( \frac{d\ln x}{d\ln r^} \biggr)^2 dU^_\mathrm{int}~~
	~~- (3\gamma_e - 4) \int_{r^_\mathrm{core}}^{R^} x^2 dW^*_\mathrm{grav}~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~

	<tr>		<tr>
	<td align="right">		<td align="right">
	~~ ~~		<math>~\omega^2 \int_0^R r^2 dM_r</math>
	</td>		</td>
	<td align="center">		<td align="center">
	~~ ~~		<math>~\leq</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>~
	~~+ ~3^2~~(~~\gamma_c~~ - \~~gamma_e~~) ~~x_i^2 P_i^* V_~~\mathrm{~~core~~}^* \, .		(4- 3\gamma) W_\mathrm{grav}+ 3^2 \gamma P_eV \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Bipolytropes */

2019-02-04T20:33:01Z

Bipolytropes

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Tohline: /* Bipolytropes */

2019-02-04T05:27:31Z

Bipolytropes

← Older revision		Revision as of 05:27, 4 February 2019
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	----		----

	If we make the reasonable assumption that, in equilibrium, the statements,		If — based for example on <b><font color="maroon" size="+1">⑦</font></b> — we make the reasonable assumption that, in equilibrium, the statements,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
	<tr>		<tr>

Tohline: /* Bipolytropes */

2019-02-04T05:15:48Z

Bipolytropes

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Tohline: /* Bipolytropes */

2019-02-04T05:04:15Z

Bipolytropes

← Older revision		Revision as of 05:04, 4 February 2019
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	</table>		</table>

	Note the similarity with <b><font color="maroon" size="+1">⑧</font></b>.		Note the similarity with <b><font color="maroon" size="+1">⑧</font></b> — temporarily, see [[User:Tohline/SSC/Stability/BiPolytropes#Revised_Free-Energy_Analysis\|this discussion]].

	----		----

Tohline: /* Bipolytropes */

2019-02-04T04:58:10Z

Bipolytropes

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Tohline: /* Stability */

2019-02-04T03:42:48Z

Stability

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Tohline: /* Tabular Overview */

2019-02-04T03:18:09Z

Tabular Overview

← Older revision		Revision as of 03:18, 4 February 2019
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	</tr>		</tr>
	</table>		</table>
			Note the similarity with <b><font color="maroon" size="+1">⑥</font></b>.
	----		----

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	</tr>		</tr>
	</table>		</table>
	the ~~condition~~ for virial equilibrium may be written as,		the conditions for virial equilibrium and stability, may be written respectively as,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	<td align="left">		<td align="left">
	<math>~		<math>~
	4W_\mathrm{grav} + 3\gamma ~~(2S_~~\mathrm{therm}) \, .		4W_\mathrm{grav} + 6\gamma S_\mathrm{therm} \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Tabular Overview */

2019-02-03T20:24:51Z

Tabular Overview

← Older revision		Revision as of 20:24, 3 February 2019
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	</tr>		</tr>
	</table>		</table>

			----


			Alternatively, recalling that,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~3(\gamma - 1)U_\mathrm{int}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~2S_\mathrm{therm} \, ,
			</math>
			</td>
			</tr>
			</table>
			the condition for virial equilibrium may be written as,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~3P_e V</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ 2S_\mathrm{therm}+ W_\mathrm{grav} </math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~ R^2 \biggl[\frac{\partial^2\mathfrak{G}}{\partial R^2}\biggr]_\mathrm{equil}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			2W_\mathrm{grav} - 2(2-3\gamma)S_\mathrm{therm} + 2 \biggl[ 2S_\mathrm{therm}+ W_\mathrm{grav} \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			4W_\mathrm{grav} + 3\gamma (2S_\mathrm{therm}) \, .
			</math>
			</td>
			</tr>
			</table>


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