User:Tohline/SSC/Structure/BiPolytropes/Analytic5 1 - Revision history

Tohline: /* Stability Condition */

2019-05-04T20:32:01Z

Stability Condition

← Older revision		Revision as of 20:32, 4 May 2019
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	Otherwise, they will be dynamically unstable toward radial perturbations. For various values of the <math>~\mu_e/\mu_c</math> ratio, Table 1 identifies the value of <math>~\xi_i</math> — and the corresponding values of <math>~q</math> and <math>~\nu</math> — at which the left-hand side of this stability relation equals the right-hand side. The locus of points provided by Table 1 defines the curve that separates stable from unstable regions of the <math>~q-\nu</math> parameter space. The red-dashed curve drawn in Figure 3 graphically depicts this demarcation: the region below the curve identifies bipolytrope models that are dynamically stable while the region above the curve identifies unstable models.		Otherwise, they will be dynamically unstable toward radial perturbations. For various values of the <math>~\mu_e/\mu_c</math> ratio, Table 3 identifies the value of <math>~\xi_i</math> — and the corresponding values of <math>~q</math> and <math>~\nu</math> — at which the left-hand side of this stability relation equals the right-hand side. The locus of points provided by Table 3 defines the curve that separates stable from unstable regions of the <math>~q-\nu</math> parameter space. The red-dashed curve drawn in Figure 3 graphically depicts this demarcation: the region below the curve identifies bipolytrope models that are dynamically stable while the region above the curve identifies unstable models.

	<div align="center">		<div align="center">
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	<tr>		<tr>
	<th align="center" colspan="9">		<th align="center" colspan="9">
	Table 1: Points Defining Stability Curve		Table 3: Points Defining Stability Curve
	</th>		</th>
	</tr>		</tr>

Tohline: /* Parameter Values */

2019-05-02T21:47:04Z

Parameter Values

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	[[File:BiPolytropeParametersV01.xml\|Accompanying spreadsheet with parameter values]]		[[File:BiPolytropeParametersV01.xml\|Accompanying spreadsheet with parameter values]]

	<table border="1" cellpadding="5">		<table border="1" cellpadding="5" width="80%">
	<tr>		<tr>
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	3.0		3.0
			</td>
			<td rowspan="20">
			[[File:Bipolytrope51Boundaries02.png\|500px\|Examples]]<br />
			For bipolytropic models having <math>~\mu_e/\mu_c = 1.0</math>, this figure shows how the interface location, <math>~\eta_i</math> (solid purple curve), the surface radius, <math>~\eta_s</math> (green circular markers), and the parameter, <math>~\tan^{-1}\Lambda_i</math> (orange circular markers), vary with <math>~\xi_i</math> (ordinate) over the range, <math>0 \le \xi_i \le 12</math>. The three horizontal, red-dashed line segments identify the values of <math>~\xi_i</math> for which numerical values of these (and other) parameters have been listed in the table shown here on the left.
	</td>		</td>
	</tr>		</tr>

Tohline: /* Parameter Values */

2019-04-30T22:51:26Z

Parameter Values

← Older revision		Revision as of 22:51, 30 April 2019
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	Also,		It must be understood, therefore, that the interface location is restricted to the range,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~0</math>
			</td>
			<td align="center">
			<math>~\le \eta_i \le</math>
			</td>
			<td align="left">
			<math>~\frac{3}{2}\biggl(\frac{\mu_e}{\mu_c} \biggr)\, ,</math>
			</td>
			</tr>
			</table>

			and that this upper limit on <math>~\eta_i</math> is associated with a model whose core radius is, <math>~\xi_i = \sqrt{3}</math>. Also,

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

Tohline: /* Parameter Values */

2019-04-30T22:42:17Z

Parameter Values

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	<math>~\sqrt{3}		<math>~\sqrt{3}
			\biggl(\frac{\mu_e}{\mu_c} \biggr) \frac{3}{2\eta_i } \biggl\{ 1 \pm
			\sqrt{1 - \biggl[ \biggl(\frac{\mu_e}{\mu_c} \biggr)^{-1} \frac{2 \eta_i }{3}\biggr]^2 }
			\biggr\} \, .
			</math>
			</td>
			</tr>
			</table>

			Also,

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

			<tr>
			<td align="right">
			<math>~\Lambda_i </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{1}{\eta_i} -
	\biggl(\frac{\mu_e}{\mu_c} \biggr) \frac{3}{2\eta_i } \biggl\{ 1 \pm		\biggl(\frac{\mu_e}{\mu_c} \biggr) \frac{3}{2\eta_i } \biggl\{ 1 \pm
	\sqrt{1 - \biggl[ \biggl(\frac{\mu_e}{\mu_c} \biggr)^{-1} \frac{2 \eta_i }{3}\biggr]^2 }		\sqrt{1 - \biggl[ \biggl(\frac{\mu_e}{\mu_c} \biggr)^{-1} \frac{2 \eta_i }{3}\biggr]^2 }

Tohline: /* Parameter Values */

2019-04-30T22:36:40Z

Parameter Values

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	<!-- END TABLE OF PARAMETERS --->		<!-- END TABLE OF PARAMETERS --->


			Alternatively, if given <math>~\mu_e/\mu_c</math> and the value of the parameter, <math>~\eta_i</math>, then we have,
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			<tr>
			<td align="right">
			<math>~\biggl(\frac{\mu_e}{\mu_c} \biggr)^{-1}\eta_i</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{3^{3/2}\xi_i}{3 + \xi_i^2}</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~ 0</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\xi_i^2 - \biggl[ \biggl(\frac{\mu_e}{\mu_c} \biggr) \frac{3^{3/2}}{\eta_i }\biggr] \xi_i + 3</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~\xi_i </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\sqrt{3}
			\biggl(\frac{\mu_e}{\mu_c} \biggr) \frac{3}{2\eta_i } \biggl\{ 1 \pm
			\sqrt{1 - \biggl[ \biggl(\frac{\mu_e}{\mu_c} \biggr)^{-1} \frac{2 \eta_i }{3}\biggr]^2 }
			\biggr\} \, .
			</math>
			</td>
			</tr>
			</table>

	==Profile==		==Profile==

Tohline: /* In Parts */

2019-02-04T23:38:47Z

In Parts

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	<math>~2S_\mathrm{core} ~~= 3P_i V_~~\mathrm{core} ~~- W_~~\mathrm{core}</math>		<math>~2S_\mathrm{core} + W_\mathrm{core} = 3P_i V_\mathrm{core}</math>
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	<td align="center">		<td align="center">
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	<td align="left">		<td align="left">
	<math>~2S_\mathrm{env} = - 3P_i V_\mathrm{core} ~~- W_\mathrm{env}~~ \, ,</math>		<math>~2S_\mathrm{env} + W_\mathrm{env} = - 3P_i V_\mathrm{core} \, ,</math>
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			This is '''precisely''' the "extra term" that shows up (with opposite signs) in the above-derived expressions for the separate quantities, <math>~(2S + W)_\mathrm{core}</math> and <math>~(2S + W)_\mathrm{env}</math>. Hence our speculation has been shown to be correct, at least for the case of bipolytropes with, <math>~(\gamma_c, \gamma_e) = (\tfrac{6}{5}, 2)</math>.

	==Stability Condition==		==Stability Condition==

Tohline: /* In Parts */

2019-02-04T23:32:11Z

In Parts

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	<math>~		<math>~
	4\pi \biggl( 1 + \frac{\xi_i^2}{3} \biggr)^{- 3}\biggl(\frac{3}{2\pi}\biggr)^{3 / 2} \xi_i^3		4\pi \biggl( 1 + \frac{\xi_i^2}{3} \biggr)^{- 3}\biggl(\frac{3}{2\pi}\biggr)^{3 / 2} \xi_i^3
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl( 1 + \frac{\xi_i^2}{3} \biggr)^{- 3}\biggl(\frac{2 \cdot 3^3 }{\pi} \biggr)^{1 / 2} \xi_i^3 \, .
	</math>		</math>
	</td>		</td>

Tohline: /* In Parts */

2019-02-04T23:28:51Z

In Parts

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	<math>~		<math>~
	\biggl(\frac{3}{~~2\pi~~}\biggr)^{~~1 / 2~~} ~~\xi_i~~ \biggl( ~~1 +~~ \frac{~~\xi_i^2~~}{3} \biggr)^{~~- 1~~}		4\pi \biggl( 1 + \frac{\xi_i^2}{3} \biggr)^{- 3}\biggl(\frac{3}{2\pi}\biggr)^{3 / 2} \xi_i^3
	</math>		</math>
	</td>		</td>

Tohline: /* In Parts */

2019-02-04T23:27:28Z

In Parts

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	<math>~</math>		<math>~
			\biggl(\frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \biggl( 1 + \frac{\xi_i^2}{3} \biggr)^{- 1}
			</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Equilibrium Condition */

2019-02-04T23:25:26Z

Equilibrium Condition

← Older revision		Revision as of 23:25, 4 February 2019
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	==Equilibrium Condition==		==Equilibrium Condition==
			===Global===

	Recognizing from the above [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#Parameter_Values\|Table of Parameters]] that,		Recognizing from the above [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#Parameter_Values\|Table of Parameters]] that,
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	This demonstrates that the detailed force-balanced models of <math>~(n_c, n_e) = (5,1)</math> bipolytropes derived above are also all in virial equilibrium, as should be the case. More importantly, showing that these four separate energy integrals sum to zero helps provide confirmation that the four energy integrals have been derived correctly. This allows us to confidently proceed to an evaluation of the relative dynamical stability of the models.		This demonstrates that the detailed force-balanced models of <math>~(n_c, n_e) = (5,1)</math> bipolytropes derived above are also all in virial equilibrium, as should be the case. More importantly, showing that these four separate energy integrals sum to zero helps provide confirmation that the four energy integrals have been derived correctly. This allows us to confidently proceed to an evaluation of the relative dynamical stability of the models.

			===In Parts===
			In section <b><font color="maroon" size="+1">⑩</font></b> of our ''[[User:Tohline/SSC/Synopsis_StyleSheet#Bipolytropes\|Tabular Overview]]'', we speculated that, in bipolytropic equilibrium structures, the statements

			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="right">
			<math>~2S_\mathrm{core} = 3P_i V_\mathrm{core} - W_\mathrm{core}</math>
			</td>
			<td align="center">
			and
			</td>
			<td align="left">
			<math>~2S_\mathrm{env} = - 3P_i V_\mathrm{core} - W_\mathrm{env} \, ,</math>
			</td>
			</tr>
			</table>
			hold separately. Let's evaluate the "PV" term. We find that,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~
			3P_i V_\mathrm{core} = 4\pi P_i r_i^3
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~</math>
			</td>
			</tr>
			</table>

	==Stability Condition==		==Stability Condition==