User:Tohline/SSC/Synopsis - Revision history

Tohline: /* Spherically Symmetric Configurations Synopsis */

2017-06-18T18:47:25Z

Spherically Symmetric Configurations Synopsis

← Older revision		Revision as of 18:47, 18 June 2017
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	<td align="left" rowspan="3">		<td align="left" rowspan="5">
	The second derivative of the free-energy function is,		The second derivative of the free-energy function is,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</td>		</td>
	<!-- END 1ST LEFT STABILITY COLUMN -->		<!-- END 1ST LEFT STABILITY COLUMN -->
			</tr>
			<tr>
			<th align="center" width="50%">
			Approximation:   Homologous Expansion/Contraction
			</th>
			</tr>

			<!-- BEGIN ANOTHER MAJOR STABILITY ROW -->
			<tr>
			<!-- BEGIN 2ND LEFT STABILITY COLUMN -->
			<td 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>~\approx</math>
			</td>
			<td align="left">
			<math>~
			(4- 3\gamma) W_\mathrm{grav}+ 3^2 \gamma P_eV \, .
			</math>
			</td>
			</tr>
			</table>
			</div>
			</td>
	</tr>		</tr>
	</table>		</table>

Tohline: /* Spherically Symmetric Configurations Synopsis */

2017-06-18T18:36:17Z

Spherically Symmetric Configurations Synopsis

← Older revision		Revision as of 18:36, 18 June 2017
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	<tr>		<tr>
	<!-- FIRST COLUMN -->		<!-- FIRST COLUMN -->
	<td align="left~~" width="25%~~">		<td align="left">
	<table border="0" cellpadding="5" align="left">		<table border="0" cellpadding="5" align="left">
	<tr>		<tr>
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	<td align="left" rowspan="3">		<td align="left" rowspan="3">
			The second derivative of the free-energy function is,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~</math>		<math>~\frac{d^2 \mathfrak{G}}{dR^2}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~</math>		<math>~
			-2aR^{-3} + (3-3\gamma)(2-3\gamma)b R^{1-3\gamma} + 6cR
			</math>
			</td>
			</tr>
			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{1}{R^2}\biggl[
			2W_\mathrm{grav} - 3(\gamma-1)(2-3\gamma)U_\mathrm{int} + 6P_e V
			\biggr] \, .
			</math>
			</td>
			</tr>
			</table>
			Evaluating this second derivative for an equilibrium configuration — that is by calling upon the (virial) equilibrium condition to set the value of the internal energy — we have,
			<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>~3P_e V - W_\mathrm{grav} </math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~ R^2 \biggl[\frac{d^2\mathfrak{G}}{dR^2}\biggr]_\mathrm{equil}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~2W_\mathrm{grav} - (2-3\gamma)\biggl[3P_e V - W_\mathrm{grav} \biggr] + 6P_e V
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~(4-3\gamma)W_\mathrm{grav} + 3^2\gamma P_e V \, .
			</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Spherically Symmetric Configurations Synopsis */

2017-06-18T18:06:30Z

Spherically Symmetric Configurations Synopsis

← Older revision		Revision as of 18:06, 18 June 2017
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	<td align="left">		<td align="left">
	Multiply the LAWE through by <math>~x dr</math>, and integrate over the volume of the configuration gives the,		Multiply the LAWE through by <math>~4\pi x dr</math>, and integrate over the volume of the configuration gives the,
	<div align="center">		<div align="center">
	<font color="#770000">'''Governing Variational Relation</font><br />		<font color="#770000">'''Governing Variational Relation</font><br />
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\int_0^R r^4 \gamma P \biggl(\frac{dx}{dr}\biggr)^2 dr		\int_0^R 4\pi r^4 \gamma P \biggl(\frac{dx}{dr}\biggr)^2 dr
	- \int_0^R (3\gamma - 4) r^3 x^2 \biggl( \frac{dP}{dr} \biggr) dr		- \int_0^R 4\pi (3\gamma - 4) r^3 x^2 \biggl( \frac{dP}{dr} \biggr) dr
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>~		<math>~
	- \biggr[r^4 \gamma Px \biggl(\frac{dx}{dr}\biggr) \biggr]_0^R		- 4\pi \biggr[r^4 \gamma Px \biggl(\frac{dx}{dr}\biggr) \biggr]_0^R
	- \int_0^R \omega^2 \rho r^4 x^2 dr \, .		- \int_0^R 4\pi \omega^2 \rho r^4 x^2 dr \, .
			</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{d\ln x}{d\ln r}\biggr)^2 \gamma 4\pi r^2P dr
			- \int_0^R (3\gamma - 4)x^2 \biggl( - \frac{GM_r}{r} \biggr) 4\pi \rho r^2 dr
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

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

			</td>
			<td align="left">
			<math>~
			+ \biggr[\gamma 4\pi r^3 Px^2 \biggl(-\frac{d\ln x}{d\ln r}\biggr) \biggr]_0^R
			- \int_0^R 4\pi \omega^2 \rho r^4 x^2 dr \, .
	</math>		</math>
	</td>		</td>
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	</table>		</table>
	</div>		</div>
			Now, by setting <math>~(d\ln x/d\ln r)_{r=R} = -3</math>, we can ensure that the pressure fluctuation is zero and, hence, <math>~P = P_e</math> at the surface, in which case this relation becomes,

			<div align="center">
			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="right">
			<math>~\omega^2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{\gamma (\gamma -1) \int_0^R x^2 \bigl(\frac{d\ln x}{d\ln r}\bigr)^2 dU_\mathrm{int}
			- \int_0^R (3\gamma - 4)x^2 dW_\mathrm{grav}
			+ 3^2 \gamma x^2 P_eV}{ \int_0^R x^2 r^2 dM_r}
			</math>
			</td>
			</tr>
			</table>
			</div>

	</td>		</td>
	<!-- END 1ST LEFT STABILITY COLUMN -->		<!-- END 1ST LEFT STABILITY COLUMN -->

Tohline: /* Spherically Symmetric Configurations Synopsis */

2017-06-18T16:02:06Z

Spherically Symmetric Configurations Synopsis

← Older revision		Revision as of 16:02, 18 June 2017
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	<td align="left">		<td align="left" rowspan="3">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
	<tr>		<tr>
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	</td>		</td>
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			</tr>
			<tr>
			<th align="center" width="50%">
			Variational Principle
			</th>
			</tr>

			<!-- BEGIN ANOTHER MAJOR STABILITY ROW -->
			<tr>
			<!-- BEGIN 2ND LEFT STABILITY COLUMN -->
			<td align="left">
			Multiply the LAWE through by <math>~x dr</math>, and integrate over the volume of the configuration gives the,
			<div align="center">
			<font color="#770000">'''Governing Variational Relation</font><br />
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~0</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\int_0^R r^4 \gamma P \biggl(\frac{dx}{dr}\biggr)^2 dr
			- \int_0^R (3\gamma - 4) r^3 x^2 \biggl( \frac{dP}{dr} \biggr) dr
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

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

			</td>
			<td align="left">
			<math>~
			- \biggr[r^4 \gamma Px \biggl(\frac{dx}{dr}\biggr) \biggr]_0^R
			- \int_0^R \omega^2 \rho r^4 x^2 dr \, .
			</math>
			</td>
			</tr>
			</table>
			</div>
			</td>
			<!-- END 1ST LEFT STABILITY COLUMN -->
	</tr>		</tr>
	</table>		</table>

Tohline: /* Spherically Symmetric Configurations Synopsis */

2017-06-18T15:47:48Z

Spherically Symmetric Configurations Synopsis

← Older revision		Revision as of 15:47, 18 June 2017
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	{{LSU_HBook_header}}		{{LSU_HBook_header}}



	<table border="1" cellpadding="8" width="85%" align="center">		<table border="1" cellpadding="8" width="85%" align="center">
	<tr>		<tr>
	<td align="center" colspan="2">		<td align="center" colspan="2" bgcolor="lightgreen">
	<b>Spherically Symmetric Configurations that undergo Adiabatic Compression/Expansion</b> — adiabatic index, <math>~\gamma</math>		<font size="+1"><b>Spherically Symmetric Configurations that undergo Adiabatic Compression/Expansion</b></font> — adiabatic index, <math>~\gamma</math>
	</td>		</td>
	</tr>		</tr>
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	</tr>		</tr>
	</table>		</table>
			</td>
			</tr>
			<tr>
			<td align="center" colspan="2" bgcolor="lightgreen">
			<font size="+1"><b>Equilibrium Structure</b></font>
	</td>		</td>
	</tr>		</tr>
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	</table>		</table>
	</td>		</td>
			</tr>
			<tr>
			<td align="center" colspan="2" bgcolor="lightgreen">
			<font size="+1"><b>Stability Analysis</b></font>
			</td>
			</tr>
			<tr>
			<th align="center" width="50%">
			Perturbation Theory
			</th>
			<th align="center">
			Free-Energy Analysis
			</th>
			</tr>
			<!-- BEGIN MAJOR STABILITY ROW -->
			<tr>
			<!-- BEGIN 1ST LEFT STABILITY COLUMN -->
			<td align="left">
			Given the radial profile of the density and pressure in the equilibrium configuration, solve the [[User:Tohline/SSC/VariationalPrinciple#Ledoux_and_Pekeris_.281941.29\|eigenvalue problem defined]] by the,
			<div align="center">
			<font color="#770000">'''LAWE:   Linear Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br />
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~0</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{d}{dr}\biggl[ r^4 \gamma P ~\frac{dx}{dr} \biggr]
			+\biggl[ \omega^2 \rho r^4 + (3\gamma - 4) r^3 \frac{dP}{dr} \biggr] x
			</math>
			</td>
			</tr>
			</table>
			</div>
			to find one or more radially dependent, radial-displacement eigenvectors, <math>~x \equiv \delta r/r</math>, along with (the square of) the corresponding oscillation eigenfrequency, <math>~\omega^2</math>.
			</td>
			<!-- END 1ST LEFT STABILITY COLUMN -->
			<!-- BEGIN 1ST RIGHT STABILITY COLUMN -->
			<td align="left">
			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="right">
			<math>~</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~</math>
			</td>
			</tr>
			</table>
			</td>
			<!-- END 1ST RIGHT STABILITY COLUMN -->
	</tr>		</tr>
	</table>		</table>

Tohline: /* Spherically Symmetric Configurations Synopsis */

2017-06-18T04:34:46Z

Spherically Symmetric Configurations Synopsis

← Older revision		Revision as of 04:34, 18 June 2017
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~dV = 4\pi r^2 dr~~</math>    and     <math>~dM_r~~</math>		<math>~dV = 4\pi r^2 dr</math>
	</td>		</td>
	<td align="center">		<td align="center">
	~~<math>~=~</math>~~		and
	</td>		</td>
	<td align="left">		<td align="left">
	<math>\rho dV ~~~\Rightarrow ~~~M_r = 4\pi \int_0^r \rho r^2 dr</math>		<math>~dM_r = \rho dV ~~~\Rightarrow ~~~M_r = 4\pi \int_0^r \rho r^2 dr</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Spherically Symmetric Configurations Synopsis */

2017-06-18T04:28:26Z

Spherically Symmetric Configurations Synopsis

Show changes

Tohline: /* Spherically Symmetric Configurations Synopsis */

2017-06-18T00:05:39Z

Spherically Symmetric Configurations Synopsis

← Older revision		Revision as of 00:05, 18 June 2017
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	<tr>		<tr>
	<td align="center" colspan="4">		<td align="center" colspan="4">
	<b>Spherically Symmetric Configurations</b>		<b>Spherically Symmetric Configurations that undergo Adiabatic Compression/Expansion</b> — adiabatic index, <math>~\gamma</math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~-\int_0^R 4\pi r^~~3 dP~~ + W_\mathrm{grav}</math>		<math>~-\int_0^R\biggl[ \frac{d}{dr}\biggl( 4\pi r^3P \biggr) - 12\pi r^2 P\biggr] dr + W_\mathrm{grav}</math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	~~<math>~\int_0^R 4\pi \biggl[ 3r^2 P dr - d(r^3P)\biggr] + W_\mathrm{grav}</math>~~		<math>~\int_0^R 3\biggl[ 4\pi r^2 P \biggr]dr - \int_0^R \biggl[ d(3PV)\biggr] + W_\mathrm{grav}</math>
	~~</td>~~
	~~</tr>~~
	~~<tr>~~
	~~<td align="right">~~
	~~ ~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	<math>~\int_0^R 3\biggl[ 4\pi r^2 P dr \biggr] - \int_0^R \biggl[ d(3PV)\biggr] + W_\mathrm{grav}</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Spherically Symmetric Configurations Synopsis */

2017-06-17T22:52:56Z

Spherically Symmetric Configurations Synopsis

← Older revision		Revision as of 22:52, 17 June 2017
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~- \int_0^R \biggl(\frac{GM_r}{r}\biggr) dM_r</math>		<math>~- \int_0^R \biggl(\frac{GM_r}{r}\biggr) dM_r ~~ \propto ~~ R^{-1}</math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{1}{(\gamma -1)} \int_0^R 4\pi r^2 P dr</math>		<math>~\frac{1}{(\gamma -1)} \int_0^R 4\pi r^2 P dr ~~ \propto ~~ R^{3-3\gamma}</math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left" colspan="2">		<td align="left" colspan="2">
	<table border="5" cellpadding="5" align="left">		<table border="0" cellpadding="5" align="left">
	<tr>		<tr>
	<td align="left">		<td align="left">
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	<td align="left">		<td align="left">
	<math>~\int_0^R 4\pi \biggl[ 3r^2 P dr - d(r^3P)\biggr] + W_\mathrm{grav}</math>		<math>~\int_0^R 4\pi \biggl[ 3r^2 P dr - d(r^3P)\biggr] + W_\mathrm{grav}</math>
			</td>
			</tr>
			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\int_0^R 3\biggl[ 4\pi r^2 P dr \biggr] - \int_0^R \biggl[ d(3PV)\biggr] + W_\mathrm{grav}</math>
			</td>
			</tr>
			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~3(\gamma-1)U_\mathrm{int} + W_\mathrm{grav} - \biggl[ 3PV \biggr]_0^R \, .</math>
	</td>		</td>
	</tr>		</tr>
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	<tr>		<tr>
	<td align="left">		<td align="left">
	~~Given a barotropic equation of state~~, <math>~P(\~~rho)~~</math>, ~~solve the equation of~~		The Free-Energy is,
	<~~div~~ align="center">		<table border="0" cellpadding="5" align="center">
	<~~font color~~="~~maroon~~"><b>~~Hydrostatic Balance~~</b></~~font~~><br />		<tr>
	{{ ~~User:Tohline~~/~~Math~~/~~EQ_SShydrostaticBalance01~~ }}		<td align="right">
	</~~div~~>		<math>~\mathfrak{G}</math>
	~~for~~ the ~~radial density distribution~~, <math>~\~~rho~~(r)</math>.		</td>
	</td>		<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~W_\mathrm{grav} + U_\mathrm{int} + P_eV</math>
			</td>
			</tr>
			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~-a R^{-1} + bR^{3-3\gamma}+ cR^3 \, .</math>
			</td>
			</tr>
			</table>
			Therefore, also,
			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="right">
			<math>~\frac{d\mathfrak{G}}{dR}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~aR^{-2} +(3-3\gamma)bR^{2-3\gamma} + 3cR^2</math>
			</td>
			</tr>
			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{1}{R}\biggl[ -W_\mathrm{grav} - 3(\gamma-1)U_\mathrm{int} + 3P_eV\biggr]</math>
			</td>
			</tr>
			</table>
			</td>
			</tr>
			</table>
			Equilibrium configurations exist at extrema of the free-energy function, that is, they are identified by setting <math>~d\mathfrak{G}/dR = 0</math>. Hence, equilibria are defined by the condition,
			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="right">
			<math>~0</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~W_\mathrm{grav} + 3(\gamma-1)U_\mathrm{int} - 3P_eV\, .</math>
			</td>
	</tr>		</tr>
	</table>		</table>

Tohline: /* Spherically Symmetric Configurations Synopsis */

2017-06-17T21:48:53Z

Spherically Symmetric Configurations Synopsis

← Older revision		Revision as of 21:48, 17 June 2017
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	<tr>		<tr>
	<td align="center" colspan="4">		<td align="center" colspan="4">
	~~For~~ Spherically Symmetric Configurations:		<b>Spherically Symmetric Configurations</b>
			</td>
			</tr>
			<tr>
			<td align="center" colspan="4">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
	<tr>		<tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\int_0^R r\biggl(\frac{dP}{dr}\biggr)dV + \int_0^R r\biggl(\frac{GM_r}{r^2}\biggr)dV</math>		<math>~-\int_0^R r\biggl(\frac{dP}{dr}\biggr)dV - \int_0^R r\biggl(\frac{GM_r \rho}{r^2}\biggr)dV</math>
			</td>
			</tr>
			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~-\int_0^R 4\pi r^3 \biggl(\frac{dP}{dr}\biggr) dr - \int_0^R \biggl(\frac{GM_r}{r}\biggr)dM_r</math>
			</td>
			</tr>
			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~-\int_0^R 4\pi r^3 dP + W_\mathrm{grav}</math>
			</td>
			</tr>
			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\int_0^R 4\pi \biggl[ 3r^2 P dr - d(r^3P)\biggr] + W_\mathrm{grav}</math>
	</td>		</td>
	</tr>		</tr>