User:Tohline/SSC/Structure/BiPolytropes/Analytic0 0 - Revision history

Tohline: /* Stability Condition */ Insert sub-subsection to define omega_mathfrak{G}

2017-01-07T20:42:03Z

Stability Condition: Insert sub-subsection to define omega_mathfrak{G}

← Older revision		Revision as of 20:42, 7 January 2017
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	==Stability Condition==		==Stability Condition==

			===Second Derivative of Free Energy===
	The relative stability of an equilibrium configuration can be determined by the sign of the second derivative of the free energy evaluated at the equilibrium radius; if the second derivative is negative, the system is dynamically unstable. The second derivative of the free energy is,		The relative stability of an equilibrium configuration can be determined by the sign of the second derivative of the free energy evaluated at the equilibrium radius; if the second derivative is negative, the system is dynamically unstable. The second derivative of the free energy is,
	<div align="center">		<div align="center">
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	\frac{( \gamma_e - \frac{4}{3})}{(\gamma_e - \gamma_c)} f		\frac{( \gamma_e - \frac{4}{3})}{(\gamma_e - \gamma_c)} f
	-1 -\frac{5}{2} (g^2-1) \biggr] \, .		-1 -\frac{5}{2} (g^2-1) \biggr] \, .
			</math>
			</td>
			</tr>
			</table>
			</div>

			===Associated Oscillation Frequency===

			When a stable configuration is displaced (by a radial perturbation) from its equilibrium state, it should radially oscillate back and forth through its equilibrium radius at a frequency, <math>~\omega_\mathfrak{G}</math>, such that,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\omega_\mathfrak{G}^2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{1}{M_\mathrm{tot}} \cdot \frac{d^2\mathfrak{G}}{dR^2} \biggr\|_{R_\mathrm{eq}} \, .</math>
			</td>
			</tr>
			</table>
			</div>

			Recognizing, from [[#Thermal_Energy_Content\|above]], that
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~P_i </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~P_c - q^2\Pi </math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{3}{2^3\pi} \biggl(\frac{GM_\mathrm{tot}^2}{R_\mathrm{eq}^4} \biggr) \frac{\nu^2}{q^4} (g^2-1) \, ,</math>
			</td>
			</tr>
			</table>
			</div>
			and that,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\rho_c</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{3M_\mathrm{tot}}{4\pi G R_\mathrm{eq}^3} \cdot \frac{\nu}{q^3} \, ,</math>
			</td>
			</tr>
			</table>
			</div>
			we can write,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{P_i R_\mathrm{eq}}{M_\mathrm{tot}}</math>
			</td>
			<td align="center">
			<math>~=</math>
			<td align="left">
			<math>~\biggl(\frac{G \rho_c}{2} \biggr) \frac{\nu}{q} (g^2-1) \, .</math>
			</td>
			</tr>
			</table>
			</div>
			Hence, the characteristic oscillation frequency is,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\omega_\mathfrak{G}^2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			6\pi G \rho_c \cdot \frac{\nu}{q} (g^2-1)
			~(\gamma_e - \gamma_c) q^3 \Lambda \biggl[
			\frac{( \gamma_e - \frac{4}{3})}{(\gamma_e - \gamma_c)} f
			-1 -\frac{5}{2} (g^2-1) \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl( \frac{2\pi G \rho_c}{5}\biggr) \cdot q^2 \nu \biggl[
			2( 3\gamma_e - 4) f
			+ 3(\gamma_e - \gamma_c)(3 - 5 g^2) \biggr] \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Step 7: Surface Boundary Condition */

2017-01-06T22:25:03Z

Step 7: Surface Boundary Condition

← Older revision		Revision as of 22:25, 6 January 2017
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	</table>		</table>
	</div>		</div>
	<span id="gdefinition">It therefore seems prudent to define a function,<\span>		<span id="gdefinition">It therefore seems prudent to define a function,</span>
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Summary */ Add ID tag to stability figure

2017-01-06T20:57:20Z

Summary: Add ID tag to stability figure

← Older revision		Revision as of 20:57, 6 January 2017
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	The "Dynamical Stability" figure below shows a large segment of the <math>~(q,\nu)</math> model parameter space. The figure displays stability boundaries for four different choices of the core/envelope adiabatic pairs <math>~(\gamma_c, \gamma_e)</math>, as detailed in the accompanying "Stability Boundaries" table. (The corresponding polytropic indexes are also listed, where, <math>~\gamma = 1+1/n</math>.) Two bipolytrope model sequences are also drawn: One shows how the core-to-total mass ratio, <math>~\nu</math>, varies with increasing <math>~q</math> when the density ratio at the core-envelope interface is <math>~\rho_e/\rho_c = 1/2</math> (purple crosses); the other shows the same information, but for <math>~\rho_e/\rho_c = 1/4</math> (purple asterisks). For a given choice of adiabatic indexes, equilibrium configurations that lie below the relevant stability boundary, along either sequence, are stable while configurations that lie above the relevant boundary are dynamically unstable.		The "Dynamical Stability" figure below shows a large segment of the <math>~(q,\nu)</math> model parameter space. The figure displays stability boundaries for four different choices of the core/envelope adiabatic pairs <math>~(\gamma_c, \gamma_e)</math>, as detailed in the accompanying "Stability Boundaries" table. (The corresponding polytropic indexes are also listed, where, <math>~\gamma = 1+1/n</math>.) Two bipolytrope model sequences are also drawn: One shows how the core-to-total mass ratio, <math>~\nu</math>, varies with increasing <math>~q</math> when the density ratio at the core-envelope interface is <math>~\rho_e/\rho_c = 1/2</math> (purple crosses); the other shows the same information, but for <math>~\rho_e/\rho_c = 1/4</math> (purple asterisks). For a given choice of adiabatic indexes, equilibrium configurations that lie below the relevant stability boundary, along either sequence, are stable while configurations that lie above the relevant boundary are dynamically unstable.

			<div align="center" id="Illustration">
	<table border="0" align="center">		<table border="0" align="center">
	<tr>		<tr>
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	</tr>		</tr>
	</table>		</table>
			</div>

	It is important to emphasize that the stability analysis presented above is an approximation in the following sense: The underlying equilibrium structure of each model is that of an <math>~(n_c, n_e) = (0, 0)</math> bipolytrope, that is, both the core and the envelope have uniform (but different) densities; the gravitational potential energy, <math>~W</math> and the thermal energy content of the component volumes, <math>~S_\mathrm{core}</math> and <math>~S_\mathrm{env}</math>, are determined exactly. However, the relative stability of a configuration is determined by assuming that, when the bipolytropic configuration is perturbed from its equilibriums state, its core and envelope heat up (or cool) by evolving along <math>~(\gamma_c, \gamma_e)</math> adiabats.		It is important to emphasize that the stability analysis presented above is an approximation in the following sense: The underlying equilibrium structure of each model is that of an <math>~(n_c, n_e) = (0, 0)</math> bipolytrope, that is, both the core and the envelope have uniform (but different) densities; the gravitational potential energy, <math>~W</math> and the thermal energy content of the component volumes, <math>~S_\mathrm{core}</math> and <math>~S_\mathrm{env}</math>, are determined exactly. However, the relative stability of a configuration is determined by assuming that, when the bipolytropic configuration is perturbed from its equilibriums state, its core and envelope heat up (or cool) by evolving along <math>~(\gamma_c, \gamma_e)</math> adiabats.

Tohline: /* Step 7: Surface Boundary Condition */

2016-11-30T21:03:16Z

Step 7: Surface Boundary Condition

← Older revision		Revision as of 21:03, 30 November 2016
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	</div>		</div>

	in which case the expressions for the equilibrium radius and equilibrium total mass are, respectively,		<span id="MassRadius">in which case the expressions for the equilibrium radius and equilibrium total mass are, respectively,</span>
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5">		<table border="0" cellpadding="5">

Tohline: /* Step 7: Surface Boundary Condition */ Insert additional (simplified) expression for chi_i

2016-11-23T23:11:04Z

Step 7: Surface Boundary Condition: Insert additional (simplified) expression for chi_i

← Older revision		Revision as of 23:11, 23 November 2016
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	</tr>		</tr>
	</table>		</table>
			</div>
			Note that this means that,
			<div align="center">
			<math>
			\chi_i^2 = \biggl( \frac{3}{2\pi}\biggr) \frac{1}{g^2} \, .
			</math>
	</div>		</div>
	<span id="CentralPressure">We can also combine these two expressions and eliminate direct reference to the central density,</span> <math>\rho_0</math>, obtaining,		<span id="CentralPressure">We can also combine these two expressions and eliminate direct reference to the central density,</span> <math>\rho_0</math>, obtaining,

Tohline: /* Thermal Energy Content */

2016-07-26T20:02:35Z

Thermal Energy Content

← Older revision		Revision as of 20:02, 26 July 2016
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	</div>		</div>

	where,		<span id="PiDefinition">where,</span>
	<div align="center">		<div align="center">
	<math>		<math>

Tohline: /* Step 7: Surface Boundary Condition */ Add span tag to central pressure equation

2014-06-22T20:51:27Z

Step 7: Surface Boundary Condition: Add span tag to central pressure equation

← Older revision		Revision as of 20:51, 22 June 2014
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	</table>		</table>
	</div>		</div>
	We can also combine these two expressions and eliminate direct reference to the central density, <math>\rho_0</math>, obtaining,		<span id="CentralPressure">We can also combine these two expressions and eliminate direct reference to the central density,</span> <math>\rho_0</math>, obtaining,
	<div align="center">		<div align="center">
	<table border="0">		<table border="0">

Tohline: /* Virial Equilibrium */ Add simpler expression for P_i/\Pi ratio, in terms of function g

2014-06-18T16:09:07Z

Virial Equilibrium: Add simpler expression for P_i/\Pi ratio, in terms of function g

← Older revision		Revision as of 16:09, 18 June 2014
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	<td align="left">		<td align="left">
	<math>		<math>
	2\biggl( \frac{\rho_e}{\rho_0} \biggr) q^2(1-q) + \biggl( \frac{\rho_e}{\rho_0} \biggr)^2 ( 1 - 3q^2 + 2q^3) \, .		2\biggl( \frac{\rho_e}{\rho_0} \biggr) q^2(1-q) + \biggl( \frac{\rho_e}{\rho_0} \biggr)^2 ( 1 - 3q^2 + 2q^3)
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=~</math>
			</td>
			<td align="left">
			<math>
			q^2(g^2-1) \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Step 7: Surface Boundary Condition */ Rewrite function g and insert span id tag

2014-06-18T15:57:54Z

Step 7: Surface Boundary Condition: Rewrite function g and insert span id tag

← Older revision		Revision as of 15:57, 18 June 2014
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	</table>		</table>
	</div>		</div>
	It therefore seems prudent to define a function,		<span id="gdefinition">It therefore seems prudent to define a function,<\span>
	<div align="center">		<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~g(\nu,q)</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
	<math>		<math>
	~~g(\nu,q) \equiv~~ \biggl\{ 1 +		\biggl\{ 1 + \biggl(\frac{\rho_e}{\rho_0}\biggr) \biggl[ 2 \biggl(1 - \frac{\rho_e}{\rho_0} \biggr) \biggl( 1-q \biggr) +
	\biggl(\frac{\rho_e}{\rho_0}\biggr) \biggl[ 2 \biggl(1 - \frac{\rho_e}{\rho_0} \biggr) \biggl( 1-q \biggr) +		\frac{\rho_e}{\rho_0} \biggl(\frac{1}{q^2} - 1\biggr) \biggr] \biggr\}^{1/2}
	\frac{\rho_e}{\rho_0} \biggl(\frac{1}{q^2} - 1\biggr) \biggr] \biggr\}^{1/2}~~\, ,~~
	</math>		</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>
			\Rightarrow ~~~~ q^2 g^2
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>
			q^2 + \biggl(\frac{\rho_e}{\rho_0}\biggr) \biggl[
			2q^2(1-q) + \biggl(\frac{\rho_e}{\rho_0}\biggr) (1-3q^2 + 2q^3)
			\biggr] \, ,
			</math>
			</td>
			</tr>
			</table>
	</div>		</div>

	in which case the expressions for the equilibrium radius and equilibrium total mass are, respectively,		in which case the expressions for the equilibrium radius and equilibrium total mass are, respectively,
	<div align="center">		<div align="center">

Tohline: /* Equilibrium Condition */ Add LambdaDeff span tag to critical equation

2014-06-02T20:12:41Z

Equilibrium Condition: Add LambdaDeff span tag to critical equation

← Older revision		Revision as of 20:12, 2 June 2014
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	</table>		</table>
	</div>		</div>
	Hence, the dimensionless structural parameter that appears in our free energy expression takes on the value,		<span id="LambdaDeff">Hence, the dimensionless structural parameter that appears in our free energy expression takes on the value,</span>
	<div align="center">		<div align="center">
	<table border="0">		<table border="0">