User:Tohline/StabilityVariationalPrincipal - Revision history

Tohline: /* Expectations */

2017-06-16T23:26:02Z

Expectations

← Older revision		Revision as of 23:26, 16 June 2017
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	<tr>		<tr>
	<td align="right">Virial ~~equlibrium~~:</td>		<td align="right">Virial equilibrium:</td>
	<td align="center">     </td>		<td align="center">     </td>
	<td align="right">		<td align="right">

Tohline: /* The Ledoux Variational Principle */

2017-06-06T22:57:02Z

The Ledoux Variational Principle

← Older revision		Revision as of 22:57, 6 June 2017
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	<math>~		<math>~
	- (3\gamma_\mathrm{g} - 4)\biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \int_0^R x^2 \biggl[\frac{R_\mathrm{norm}R^2}{GM^2}\biggr] \biggl( \frac{r_0}{R}\biggr)^3 \rho_0 \biggl(\frac{GM_r R^2}{r^2_0} \biggr) \frac{dr_0}{R}		- (3\gamma_\mathrm{g} - 4)\biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \int_0^R x^2 \biggl[\frac{R_\mathrm{norm}R^2}{GM^2}\biggr] \biggl( \frac{r_0}{R}\biggr)^3 \rho_0 \biggl(\frac{GM_r R^2}{r^2_0} \biggr) \frac{dr_0}{R}
	- 3 \gamma_\mathrm{g} x^2 \biggl[ \frac{R^3 }{ R_\mathrm{norm}^3} ~~\biggl( \frac{r_0}{R}\biggr)^3~~ \frac{P_e}{P_\mathrm{norm}}\biggr]		- 3 \gamma_\mathrm{g} x_\mathrm{surface}^2 \biggl[ \frac{R^3 }{ R_\mathrm{norm}^3} \frac{P_e}{P_\mathrm{norm}}\biggr]
	</math>		</math>
	</td>		</td>
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	- (3\gamma_\mathrm{g} - 4) \biggl[ \biggl( \frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho }\biggr] \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}		- (3\gamma_\mathrm{g} - 4) \biggl[ \biggl( \frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho }\biggr] \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}
	\int_0^R x^2 \biggl(\frac{\rho_0}{\rho_c} \biggr) \biggl( \frac{r_0}{R}\biggr) \biggl(\frac{M_r}{M} \biggr) \frac{dr_0}{R}		\int_0^R x^2 \biggl(\frac{\rho_0}{\rho_c} \biggr) \biggl( \frac{r_0}{R}\biggr) \biggl(\frac{M_r}{M} \biggr) \frac{dr_0}{R}
	~- ~3 \gamma_\mathrm{g} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3 x^2 \biggl[ ~~\biggl( \frac{r_0}{R}\biggr)^3~~ \frac{P_e}{P_\mathrm{norm}}\biggr] \, .		~- ~3 \gamma_\mathrm{g} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3 x_\mathrm{surface}^2 \biggl[ \frac{P_e}{P_\mathrm{norm}}\biggr] \, .
	</math>		</math>
	</td>		</td>
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	- (3\gamma_\mathrm{g} - 4) \biggl( \frac{3}{4\pi}\biggr) \biggl[ \frac{\rho_c}{\bar\rho }\biggr]^2 \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}		- (3\gamma_\mathrm{g} - 4) \biggl( \frac{3}{4\pi}\biggr) \biggl[ \frac{\rho_c}{\bar\rho }\biggr]^2 \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}
	\int_0^R x^2 \biggl(\frac{\rho_0}{\rho_c} \biggr) \biggl( \frac{r_0}{R}\biggr) \biggl\{ \int_0^R 3 \biggl(\frac{r_0}{R}\biggr)^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \frac{dr_0}{R} \bigg\} \frac{dr_0}{R}		\int_0^R x^2 \biggl(\frac{\rho_0}{\rho_c} \biggr) \biggl( \frac{r_0}{R}\biggr) \biggl\{ \int_0^R 3 \biggl(\frac{r_0}{R}\biggr)^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \frac{dr_0}{R} \bigg\} \frac{dr_0}{R}
	~- ~3 \gamma_\mathrm{g} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3 x^2 \biggl[ \biggl( \frac{r_0}{R}\biggr)^3 \frac{P_e}{P_\mathrm{norm}}\biggr] \, .		~- ~3 \gamma_\mathrm{g} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3 x_\mathrm{surface}^2 \biggl[ \frac{P_e}{P_\mathrm{norm}}\biggr] \, .
			</math>
			</td>
			</tr>
			</table>
			</div>

			In an effort to help identify the various terms in this expression as well as the relationship between the entire expression and our unperturbed free energy expression, let's ignore all terms involving the radial eigenfunction, <math>~x</math>, and its derivative. In this case, we have,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{L_{\{\}} }{E_\mathrm{norm}}\biggr\|_\mathrm{unperturbed}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \biggl[\biggl( \frac{3}{4\pi }\biggr)\frac{\rho_c}{\bar\rho} \biggr]^2 \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl[\frac{\omega^2}{G\rho_c}\biggr] \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}
			\int_0^R \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl( \frac{r_0}{R}\biggr)^4 \frac{dr_0}{R}
			~- ~\frac{\gamma_\mathrm{g} (\gamma_\mathrm{g} -1)}{4\pi} \biggl\{\frac{4\pi}{3(\gamma_\mathrm{g} -1)} \biggl[\biggl( \frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{\gamma} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{3-3\gamma}
			\int_0^R 3\biggl(\frac{P_0}{P_c}\biggr) \biggl(\frac{r_0}{R}\biggr)^2 \frac{dr_0}{R} \biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
			+ (3\gamma_\mathrm{g} - 4) \biggl( \frac{1}{4\pi}\biggr)\biggl\{- \frac{3}{5} \biggl[ \frac{\rho_c}{\bar\rho }\biggr]^2 \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}
			\int_0^R 5\biggl(\frac{\rho_0}{\rho_c} \biggr) \biggl( \frac{r_0}{R}\biggr) \biggl[ \int_0^R 3 \biggl(\frac{r_0}{R}\biggr)^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \frac{dr_0}{R} \bigg] \frac{dr_0}{R} \biggr\}
			~- ~\frac{3^2 \gamma_\mathrm{g}}{4\pi}\biggl\{\frac{4\pi}{3} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3 \biggl[ \frac{P_e}{P_\mathrm{norm}}\biggr]\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~ \frac{4\pi L_{\{\}} }{E_\mathrm{norm}}\biggr\|_\mathrm{unperturbed}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \biggl[\biggl( \frac{3}{4\pi }\biggr)\frac{\rho_c}{\bar\rho} \biggr]^2 \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl[\frac{4\pi \omega^2}{G\rho_c}\biggr] \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}
			\int_0^R \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl( \frac{r_0}{R}\biggr)^4 \frac{dr_0}{R}
			~- ~\gamma_\mathrm{g} (\gamma_\mathrm{g} -1) \biggl\{ \frac{U_\mathrm{int}}{E_\mathrm{norm}} \biggr\}
			+ (3\gamma_\mathrm{g} - 4) \biggl\{ \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr\}
			~- ~3^2 \gamma_\mathrm{g} \biggl\{ \frac{P_e V}{E_\mathrm{norm}} \biggr\} \, .
	</math>		</math>
	</td>		</td>

Tohline: /* The Ledoux Variational Principle */

2017-06-06T22:14:04Z

The Ledoux Variational Principle

← Older revision		Revision as of 22:14, 6 June 2017
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	<math>~\frac{P_c}{P_\mathrm{norm}} = \biggl[\biggl( \frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{\gamma} \biggl( \frac{R}{R_\mathrm{norm}}\biggr)^{-3\gamma} \, ,</math>		<math>~\frac{P_c}{P_\mathrm{norm}} = \biggl[\biggl( \frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{\gamma} \biggl( \frac{R}{R_\mathrm{norm}}\biggr)^{-3\gamma} \, ,</math>
	</div>		</div>
			and,
			<div align="center">
			<math>~\frac{M_r}{M} = \biggl(\frac{\rho_c}{\bar\rho} \biggr) \int_0^R 3 \biggl(\frac{r_0}{R}\biggr)^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \frac{dr_0}{R} \, ,</math>
			</div>

	this expression for the Lagrangian becomes,		this expression for the Lagrangian becomes,
	<div align="center">		<div align="center">
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	<td align="left">		<td align="left">
	<math>~		<math>~
	- (3\gamma_\mathrm{g} - 4) ~~\biggl[~~ \biggl( \frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho }\biggr] \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}		- (3\gamma_\mathrm{g} - 4) \biggl( \frac{3}{4\pi}\biggr) \biggl[ \frac{\rho_c}{\bar\rho }\biggr]^2 \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}
	\int_0^R x^2 \biggl(\frac{\rho_0}{\rho_c} \biggr) \biggl( \frac{r_0}{R}\biggr) \biggl(\frac{~~M_r~~}{M} \biggr) \frac{dr_0}{R}		\int_0^R x^2 \biggl(\frac{\rho_0}{\rho_c} \biggr) \biggl( \frac{r_0}{R}\biggr) \biggl\{ \int_0^R 3 \biggl(\frac{r_0}{R}\biggr)^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \frac{dr_0}{R} \bigg\} \frac{dr_0}{R}
	~- ~3 \gamma_\mathrm{g} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3 x^2 \biggl[ \biggl( \frac{r_0}{R}\biggr)^3 \frac{P_e}{P_\mathrm{norm}}\biggr] \, .		~- ~3 \gamma_\mathrm{g} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3 x^2 \biggl[ \biggl( \frac{r_0}{R}\biggr)^3 \frac{P_e}{P_\mathrm{norm}}\biggr] \, .
	</math>		</math>

Tohline: /* The Ledoux Variational Principle */

2017-06-06T22:02:13Z

The Ledoux Variational Principle

← Older revision		Revision as of 22:02, 6 June 2017
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	- (3\gamma_\mathrm{g} - 4) \biggl[ \biggl( \frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho }\biggr] \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}		- (3\gamma_\mathrm{g} - 4) \biggl[ \biggl( \frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho }\biggr] \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}
	\int_0^R x^2 \biggl(\frac{\rho_0}{\rho_c} \biggr) \biggl( \frac{r_0}{R}\biggr) \biggl(\frac{M_r}{M} \biggr) \frac{dr_0}{R}		\int_0^R x^2 \biggl(\frac{\rho_0}{\rho_c} \biggr) \biggl( \frac{r_0}{R}\biggr) \biggl(\frac{M_r}{M} \biggr) \frac{dr_0}{R}
	~- ~3 \gamma_\mathrm{g} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3 x^2 \biggl[ \biggl( \frac{r_0}{R}\biggr)^3 \frac{P_e}{P_\mathrm{norm}}\biggr]		~- ~3 \gamma_\mathrm{g} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3 x^2 \biggl[ \biggl( \frac{r_0}{R}\biggr)^3 \frac{P_e}{P_\mathrm{norm}}\biggr] \, .
			</math>
			</td>
			</tr>
			</table>
			</div>

			Given that,
			<div align="center">
			<math>~\frac{P_c}{P_\mathrm{norm}} = \biggl[\biggl( \frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{\gamma} \biggl( \frac{R}{R_\mathrm{norm}}\biggr)^{-3\gamma} \, ,</math>
			</div>
			this expression for the Lagrangian becomes,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{L_{\{\}} }{E_\mathrm{norm}}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \biggl[\biggl( \frac{3}{4\pi }\biggr)\frac{\rho_c}{\bar\rho} \biggr]^2 \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl[\frac{\omega^2}{G\rho_c}\biggr] \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}
			\int_0^R x^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl( \frac{r_0}{R}\biggr)^4 \frac{dr_0}{R}
			~- ~\gamma_\mathrm{g}\biggl[\biggl( \frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{\gamma} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{3-3\gamma}
			\int_0^R \biggl[ \biggl(\frac{r_0}{R}\biggr) \frac{\partial x}{\partial (r_0/R)}\biggr]^2 \biggl(\frac{P_0}{P_c}\biggr) \biggl(\frac{r_0}{R}\biggr)^2 \frac{dr_0}{R}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
			- (3\gamma_\mathrm{g} - 4) \biggl[ \biggl( \frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho }\biggr] \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}
			\int_0^R x^2 \biggl(\frac{\rho_0}{\rho_c} \biggr) \biggl( \frac{r_0}{R}\biggr) \biggl(\frac{M_r}{M} \biggr) \frac{dr_0}{R}
			~- ~3 \gamma_\mathrm{g} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3 x^2 \biggl[ \biggl( \frac{r_0}{R}\biggr)^3 \frac{P_e}{P_\mathrm{norm}}\biggr] \, .
	</math>		</math>
	</td>		</td>

Tohline: /* The Ledoux Variational Principle */

2017-06-06T21:21:35Z

The Ledoux Variational Principle

← Older revision		Revision as of 21:21, 6 June 2017
Line 1,634:		Line 1,634:
	<td align="left">		<td align="left">
	<math>~		<math>~
	- (3\gamma_\mathrm{g} - 4)\biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \int_0^R x^2 \biggl[\frac{R_\~~mathrm{norm}R^2~~}{~~GM^2~~}\biggr] \biggl( \frac{r_0}{R}\biggr)^3		- (3\gamma_\mathrm{g} - 4) \biggl[ \biggl( \frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho }\biggr] \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}
	~~\rho_0~~ \biggl(\frac{~~GM_r R^2~~}{~~r^2_0~~} \biggr) \frac{dr_0}{R}		\int_0^R x^2 \biggl(\frac{\rho_0}{\rho_c} \biggr) \biggl( \frac{r_0}{R}\biggr) \biggl(\frac{M_r}{M} \biggr) \frac{dr_0}{R}
	~- ~3 \gamma_\mathrm{g} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3 x^2 \biggl[ \biggl( \frac{r_0}{R}\biggr)^3 \frac{P_e}{P_\mathrm{norm}}\biggr]		~- ~3 \gamma_\mathrm{g} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3 x^2 \biggl[ \biggl( \frac{r_0}{R}\biggr)^3 \frac{P_e}{P_\mathrm{norm}}\biggr]
	</math>		</math>

Tohline: /* The Ledoux Variational Principle */

2017-06-06T21:08:47Z

The Ledoux Variational Principle

← Older revision		Revision as of 21:08, 6 June 2017
Line 1,530:		Line 1,530:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~ 2\pi e^{2i\omega t} \biggl\{		<math>~
			2\pi e^{2i\omega t} \biggl\{
	- \int_0^R \rho_0 \omega^2 r_0^4 x^2 dr_0		- \int_0^R \rho_0 \omega^2 r_0^4 x^2 dr_0
	- \int_0^R \gamma_\mathrm{g} P_0 r_0^4\biggl( \frac{\partial x}{\partial r_0}\biggr)^2 dr_0		- \int_0^R \gamma_\mathrm{g} P_0 r_0^4\biggl( \frac{\partial x}{\partial r_0}\biggr)^2 dr_0
Line 1,557:		Line 1,558:
	- \int_0^R \biggl[\frac{R_\mathrm{norm}}{GM_\mathrm{tot}^2}\biggr] \rho_0 \omega^2 r_0^4 x^2 dr_0		- \int_0^R \biggl[\frac{R_\mathrm{norm}}{GM_\mathrm{tot}^2}\biggr] \rho_0 \omega^2 r_0^4 x^2 dr_0
	- \gamma_\mathrm{g} \int_0^R \biggl[\frac{1}{P_\mathrm{norm} R_\mathrm{norm}^3}\biggr] P_0 r_0^4\biggl( \frac{\partial x}{\partial r_0}\biggr)^2 dr_0		- \gamma_\mathrm{g} \int_0^R \biggl[\frac{1}{P_\mathrm{norm} R_\mathrm{norm}^3}\biggr] P_0 r_0^4\biggl( \frac{\partial x}{\partial r_0}\biggr)^2 dr_0
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
	- (3\gamma_\mathrm{g} - 4) \int_0^R \biggl[\frac{R_\mathrm{norm}}{GM_\mathrm{tot}^2}\biggr] r_0^3 \rho_0 x^2 \biggl(- \frac{1}{\rho_0} \frac{dP_0}{dr_0} \biggr) dr_0		- (3\gamma_\mathrm{g} - 4) \int_0^R \biggl[\frac{R_\mathrm{norm}}{GM_\mathrm{tot}^2}\biggr] r_0^3 \rho_0 x^2 \biggl(- \frac{1}{\rho_0} \frac{dP_0}{dr_0} \biggr) dr_0
	-\biggl[ \frac{3 \gamma_\mathrm{g} }{P_\mathrm{norm} R_\mathrm{norm}^3} ~r_0^3 x^2 P_0\biggr]_0^{R}		-\biggl[ \frac{3 \gamma_\mathrm{g} }{P_\mathrm{norm} R_\mathrm{norm}^3} ~r_0^3 x^2 P_0\biggr]_0^{R}
Line 1,572:		Line 1,586:
	<td align="left">		<td align="left">
	<math>~		<math>~
	- \int_0^R x^2 \biggl[\frac{~~\omega^2 R^5~~ R_\mathrm{norm}}{G\rho_c}\biggr] \biggl[ \frac{3\rho_c}{4\pi \bar\rho R^3} \biggr]^{2} \frac{\rho_0}{\rho_c} \biggl(\frac{ r_0}{R}\biggr)^4 \frac{dr_0}{R}		- \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2\int_0^R x^2 \biggl[\frac{R_\mathrm{norm} R^5}{G\rho_c}\biggr]
			\biggl[ \frac{3\rho_c}{4\pi \bar\rho R^3}\biggr]^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \omega^2 \biggl( \frac{r_0}{R}\biggr)^4 \frac{dr_0}{R}
	- \gamma_\mathrm{g} \int_0^R \biggl[\frac{1}{P_\mathrm{norm} R_\mathrm{norm}^3}\biggr] P_0 r_0^4\biggl( \frac{\partial x}{\partial r_0}\biggr)^2 dr_0		- \gamma_\mathrm{g} \int_0^R \biggl[\frac{1}{P_\mathrm{norm} R_\mathrm{norm}^3}\biggr] P_0 r_0^4\biggl( \frac{\partial x}{\partial r_0}\biggr)^2 dr_0
	</math>		</math>
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	<td align="left">		<td align="left">
	<math>~		<math>~
	- (3\gamma_\mathrm{g} - 4) \int_0^R \biggl[\frac{R_\mathrm{norm}}{~~GM_\mathrm{tot}~~^2}\biggr] r_0^3 \rho_0 x^2 \biggl(- \frac{1}{\rho_0} \frac{~~dP_0~~}{~~dr_0~~} \biggr) dr_0		- (3\gamma_\mathrm{g} - 4)\biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \int_0^R x^2 \biggl[\frac{R_\mathrm{norm}R^2}{GM^2}\biggr] \biggl( \frac{r_0}{R}\biggr)^3 \rho_0 \biggl(\frac{GM_r R^2}{r^2_0} \biggr) \frac{dr_0}{R}
	-\biggl[ \frac{3 \gamma_\mathrm{g} }{P_\mathrm{~~norm~~} R_\mathrm{norm}^3} ~~~r_0~~^3 x^2 ~~P_0~~\biggr~~]_0~~^{R}		- 3 \gamma_\mathrm{g} x^2 \biggl[ \frac{R^3 }{ R_\mathrm{norm}^3} \biggl( \frac{r_0}{R}\biggr)^3 \frac{P_e}{P_\mathrm{norm}}\biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \biggl[\biggl( \frac{3}{4\pi }\biggr)\frac{\rho_c}{\bar\rho} \biggr]^2 \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl[\frac{\omega^2}{G\rho_c}\biggr] \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}
			\int_0^R x^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl( \frac{r_0}{R}\biggr)^4 \frac{dr_0}{R}
			~- ~\gamma_\mathrm{g}\biggl[\frac{P_c }{P_\mathrm{norm} }\biggr] \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3
			\int_0^R \biggl[ \biggl(\frac{r_0}{R}\biggr) \frac{\partial x}{\partial (r_0/R)}\biggr]^2 \biggl(\frac{P_0}{P_c}\biggr) \biggl(\frac{r_0}{R}\biggr)^2 \frac{dr_0}{R}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
			- (3\gamma_\mathrm{g} - 4)\biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \int_0^R x^2 \biggl[\frac{R_\mathrm{norm}R^2}{GM^2}\biggr] \biggl( \frac{r_0}{R}\biggr)^3
			\rho_0 \biggl(\frac{GM_r R^2}{r^2_0} \biggr) \frac{dr_0}{R}
			~- ~3 \gamma_\mathrm{g} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3 x^2 \biggl[ \biggl( \frac{r_0}{R}\biggr)^3 \frac{P_e}{P_\mathrm{norm}}\biggr]
	</math>		</math>
	</td>		</td>

Tohline: /* The Ledoux Variational Principle */

2017-06-06T16:44:16Z

The Ledoux Variational Principle

← Older revision		Revision as of 16:44, 6 June 2017
Line 1,556:		Line 1,556:
	<math>~		<math>~
	- \int_0^R \biggl[\frac{R_\mathrm{norm}}{GM_\mathrm{tot}^2}\biggr] \rho_0 \omega^2 r_0^4 x^2 dr_0		- \int_0^R \biggl[\frac{R_\mathrm{norm}}{GM_\mathrm{tot}^2}\biggr] \rho_0 \omega^2 r_0^4 x^2 dr_0
	- \gamma_\mathrm{g} \int_0^R \biggl[\frac{1}{P_\mathrm{norm} R_\mathrm{~~tot~~}^3}\biggr] P_0 r_0^4\biggl( \frac{\partial x}{\partial r_0}\biggr)^2 dr_0		- \gamma_\mathrm{g} \int_0^R \biggl[\frac{1}{P_\mathrm{norm} R_\mathrm{norm}^3}\biggr] P_0 r_0^4\biggl( \frac{\partial x}{\partial r_0}\biggr)^2 dr_0
	- (3\gamma_\mathrm{g} - 4) \int_0^R \biggl[\frac{R_\mathrm{norm}}{GM_\mathrm{tot}^2}\biggr] r_0^3 \rho_0 x^2 \biggl(- \frac{1}{\rho_0} \frac{dP_0}{dr_0} \biggr) dr_0		- (3\gamma_\mathrm{g} - 4) \int_0^R \biggl[\frac{R_\mathrm{norm}}{GM_\mathrm{tot}^2}\biggr] r_0^3 \rho_0 x^2 \biggl(- \frac{1}{\rho_0} \frac{dP_0}{dr_0} \biggr) dr_0
	-\biggl[ \frac{3 \gamma_\mathrm{g} }{P_\mathrm{norm} R_\mathrm{tot}^3} ~r_0^3 x^2 P_0\biggr]_0^{R}		-\biggl[ \frac{3 \gamma_\mathrm{g} }{P_\mathrm{norm} R_\mathrm{norm}^3} ~r_0^3 x^2 P_0\biggr]_0^{R}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \int_0^R x^2 \biggl[\frac{\omega^2 R^5 R_\mathrm{norm}}{G\rho_c}\biggr] \biggl[ \frac{3\rho_c}{4\pi \bar\rho R^3} \biggr]^{2} \frac{\rho_0}{\rho_c} \biggl(\frac{ r_0}{R}\biggr)^4 \frac{dr_0}{R}
			- \gamma_\mathrm{g} \int_0^R \biggl[\frac{1}{P_\mathrm{norm} R_\mathrm{norm}^3}\biggr] P_0 r_0^4\biggl( \frac{\partial x}{\partial r_0}\biggr)^2 dr_0
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
			- (3\gamma_\mathrm{g} - 4) \int_0^R \biggl[\frac{R_\mathrm{norm}}{GM_\mathrm{tot}^2}\biggr] r_0^3 \rho_0 x^2 \biggl(- \frac{1}{\rho_0} \frac{dP_0}{dr_0} \biggr) dr_0
			-\biggl[ \frac{3 \gamma_\mathrm{g} }{P_\mathrm{norm} R_\mathrm{norm}^3} ~r_0^3 x^2 P_0\biggr]_0^{R}
	</math>		</math>
	</td>		</td>
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	</table>		</table>
	</div>		</div>


	=See Also=		=See Also=

	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline at 16:20, 6 June 2017

2017-06-06T16:20:27Z

← Older revision		Revision as of 16:20, 6 June 2017
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	Q. E. D.		Q. E. D.

			==The Ledoux Variational Principle==
			Drawing from a [[User:Tohline/SSC/VariationalPrinciple#Ledoux.27s_Expression\|separate presentation of Ledoux's variational principle]], let's normalize his Lagrangian using the same normalizations that have been used, above. His expression is …

			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~L </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ 2\pi e^{2i\omega t} \biggl\{
			- \int_0^R \rho_0 \omega^2 r_0^4 x^2 dr_0
			- \int_0^R \gamma_\mathrm{g} P_0 r_0^4\biggl( \frac{\partial x}{\partial r_0}\biggr)^2 dr_0
			+ \int_0^R r_0^3 x^2 \frac{d}{dr_0}\biggl[ (3\gamma_\mathrm{g} - 4)P_0\biggr]dr_0
			-\biggl[3 \gamma_\mathrm{g} r_0^3 x^2 P_0\biggr]_0^{R}
			\biggr\} \, .
			</math>
			</td>
			</tr>
			</table>
			</div>

			Our normalization produces,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{L_{\{\}} }{E_\mathrm{norm}}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \int_0^R \biggl[\frac{R_\mathrm{norm}}{GM_\mathrm{tot}^2}\biggr] \rho_0 \omega^2 r_0^4 x^2 dr_0
			- \gamma_\mathrm{g} \int_0^R \biggl[\frac{1}{P_\mathrm{norm} R_\mathrm{tot}^3}\biggr] P_0 r_0^4\biggl( \frac{\partial x}{\partial r_0}\biggr)^2 dr_0
			- (3\gamma_\mathrm{g} - 4) \int_0^R \biggl[\frac{R_\mathrm{norm}}{GM_\mathrm{tot}^2}\biggr] r_0^3 \rho_0 x^2 \biggl(- \frac{1}{\rho_0} \frac{dP_0}{dr_0} \biggr) dr_0
			-\biggl[ \frac{3 \gamma_\mathrm{g} }{P_\mathrm{norm} R_\mathrm{tot}^3} ~r_0^3 x^2 P_0\biggr]_0^{R}
			</math>
			</td>
			</tr>
			</table>
			</div>


	=See Also=		=See Also=

	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: /* Test Virial Equilibrium Condition */

2017-06-06T01:15:27Z

Test Virial Equilibrium Condition

← Older revision		Revision as of 01:15, 6 June 2017
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	</div>		</div>
	Let's see if this expression is valid when we plug in the expressions for the equilibrium ~~radius~~ that has been given by [[User:Tohline/SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation\|Horedt (1970)]], namely,		Let's see if this expression is valid when we plug in the expressions for the equilibrium parameter pair — <math>~R_\mathrm{eq}</math> and <math>~P_e</math> — that has been given by [[User:Tohline/SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation\|Horedt (1970)]], namely,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="3">		<table border="0" cellpadding="3">

Tohline: /* Test Virial Equilibrium Condition */

2017-06-05T19:59:21Z

Test Virial Equilibrium Condition

← Older revision		Revision as of 19:59, 5 June 2017
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	<td align="right">		<td align="right">
	<math>~\frac{R_\mathrm{Hoerdt}}{R_\mathrm{norm}} </math>		<math>~\biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{-(n-1)/(n-3)}\frac{R_\mathrm{Hoerdt}}{R_\mathrm{norm}} </math>
	</td>		</td>
	<td align="center">		<td align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~		<math>~\biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{2(n+1)/(n-3)}
	\frac{P_\mathrm{Hoerdt}}{P_\mathrm{norm}}		\frac{P_\mathrm{Hoerdt}}{P_\mathrm{norm}}
	</math>		</math>