SSC/Stability/Yabushita75 - Revision history

Jet53man: /* Equilibrium Structure */

2023-11-10T22:37:27Z

Equilibrium Structure

@@ Line 720: / Line 720: @@
 \frac{5K_e}{8\pi G}  \biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{5 / 6}
 \biggl( \frac{c_s^2}{K_e} \biggr)^{- 1 / 2} e^{- \psi_i/3} \biggr] \eta_s^2
 </math>
    </td>

Jet53man: /* Equilibrium Structure */

2023-11-10T22:02:17Z

Equilibrium Structure

@@ Line 672: / Line 672: @@
    <td align="left">
 <math>
-\biggl(\frac{\mu_e}{\mu_c}\biggr)^{- 5 / 2} \biggl[ \frac{c_s^2}{K_e} \biggr]^{3 / 2}
+\biggl(\frac{\mu_e}{\mu_c}\biggr)^{- 5 / 2} \biggl[ \frac{c_s^2}{K_e} \biggr]^{3 / 2} e^{\psi_i}
 \, .
 </math>
@@ Line 679: / Line 678: @@
 </tr>
 </table>
-This is identical to Eq. (3.1) of {{ Yabushita75 }} if we assume (as did {{ Yabushita75hereafter }}) that <math>\mu_e/\mu_c = 1</math>.
+This is identical to Eq. (3.1) of {{ Yabushita75 }} if we assume (as did {{ Yabushita75hereafter }}) that <math>\mu_e/\mu_c = 1</math>.  Given that, following {{ Yabushita75hereafter }}, we have set <math>\rho_e = \rho_c</math>, the radius of the equilibrium configuration is given by the expression,
 =See Also=

Jet53man: /* Equilibrium Structure */

2023-11-10T21:43:13Z

Equilibrium Structure

@@ Line 649: / Line 649: @@
 </tr>
 </table>
 =See Also=

Jet53man: /* Steps 2 & 3 */

2023-11-10T20:56:10Z

Steps 2 & 3

@@ Line 24: / Line 24: @@
 &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; <math>\xi = 0</math>.
 </div>
 The solution, <math>\psi(\xi)</math>, extends to infinity, so the interface between the core and the envelope can be positioned anywhere within the range, <math>0 < \xi_i < \infty</math>.

Jet53man: /* Step 8: Throughout the Envelope */

2023-11-09T19:55:28Z

Step 8: Throughout the Envelope

← Older revision		Revision as of 15:55, 9 November 2023
Line 592:		Line 592:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>\rho_e \~~phi~~^{~~n_e~~}</math>		<math>\rho_e \theta^{3 / 2}</math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>K_e \rho_e^{~~1+1~~/~~n_e~~} \~~phi~~^{~~n_e + 1~~}</math>		<math>K_e \rho_e^{5 / 3} \theta^{5 / 2}</math>
	</td>		</td>
	</tr>		</tr>
Line 616:		Line 616:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>\biggl[ \frac{(~~n_e + 1~~)K_e}{4\pi G} \biggr]^{1/2} \rho_e^{(1~~-n_e)~~/~~(2n_e)~~} \eta</math>		<math>\biggl[ \frac{(5/2)K_e}{4\pi G} \biggr]^{1/2} \rho_e^{- 1 / 6} \eta</math>
	</td>		</td>
	</tr>		</tr>
Line 628:		Line 628:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>4\pi \biggl[ \frac{(~~n_e + 1~~)K_e}{4\pi G} \biggr]^{3/2} \rho_e^{~~(3-n_e)~~/~~(2n_e)~~} \biggl(-\eta^2 \frac{d\~~phi~~}{d\eta} \biggr)</math>		<math>4\pi \biggl[ \frac{(5/2)K_e}{4\pi G} \biggr]^{3/2} \rho_e^{1 / 2} \biggl(-\eta^2 \frac{d\theta}{d\eta} \biggr)</math>
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Step 5: Interface Conditions */

2023-11-09T19:49:38Z

Step 5: Interface Conditions

@@ Line 568: / Line 568: @@
 </tr>
 </table>
    </td>
 </tr>

Jet53man: /* Step 5: Interface Conditions */

2023-11-09T19:25:21Z

Step 5: Interface Conditions

← Older revision		Revision as of 15:25, 9 November 2023
Line 181:		Line 181:
	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left">		<td align="left">
	~~[[File:CommentButton02.png\|right\|100px\|Comment by J. E. Tohline: There is a type-setting mistake in Yabushita75's Eq. (2.9); ρ<sub>e</sub> appears as ρ<sub>0</sub>.]]~~
	<math>		<math>
	\biggl(\frac{\mu_e}{\mu_c}\biggr)e^{-\psi_i} \theta_i^{-3 / 2}		\biggl(\frac{\mu_e}{\mu_c}\biggr)e^{-\psi_i} \theta_i^{-3 / 2}
Line 187:		Line 186:
	</math>		</math>
	</td>		</td>
	~~<td align="right" width="5%">(2.9)</td>~~
	</tr>		</tr>

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	</math>		</math>
	</td>		</td>
	~~<td align="right" width="5%"> </td>~~
	</tr>		</tr>

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	</math>		</math>
	</td>		</td>
	~~<td align="right" width="5%"> </td>~~
	</tr>		</tr>

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	</math>		</math>
	</td>		</td>
	~~<td align="right" width="5%">(2.11)</td>~~
	</tr>		</tr>

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	</math>		</math>
	</td>		</td>
	~~<td align="right" width="5%"> </td>~~
	</tr>		</tr>

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	</math>		</math>
	</td>		</td>
	~~<td align="right" width="5%"> </td>~~
	</tr>		</tr>

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	</math>		</math>
	</td>		</td>
	~~<td align="right" width="5%"> </td>~~
	</tr>		</tr>

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	</math>		</math>
	</td>		</td>
	~~<td align="right" width="5%">(2.12)</td>~~
	</tr>		</tr>
	</table>		</table>
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	</math>		</math>
	</td>		</td>
	~~<td align="right" width="5%"> </td>~~
	</tr>		</tr>

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	</math>		</math>
	</td>		</td>
	~~<td align="right" width="5%"> </td>~~
	</tr>		</tr>

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	</math>		</math>
	</td>		</td>
	~~<td align="right" width="5%"> </td>~~
	</tr>		</tr>

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	</math>		</math>
	</td>		</td>
	~~<td align="right" width="5%"> </td>~~
	</tr>		</tr>

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	</math>		</math>
	</td>		</td>
	~~<td align="right" width="5%"> </td>~~
	</tr>		</tr>

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	</math>		</math>
	</td>		</td>
	~~<td align="right" width="5%"> </td>~~
	</tr>		</tr>
	</table>		</table>

Jet53man: /* Step 5: Interface Conditions */

2023-11-09T19:18:15Z

Step 5: Interface Conditions

← Older revision		Revision as of 15:18, 9 November 2023
Line 391:		Line 391:
	<tr>		<tr>
	<td align="center" colspan="1">Our Derivations</td>		<td align="center" colspan="1">Our Derivations</td>
	<td align="center" colspan="1">After setting <math>(\~~mu_e/\mu_c)~~ = ~~1</math> and <math>~~\~~theta_i = e^{-2\psi_i/ 3}~~</math></td>		<td align="center" colspan="1">After setting <math>\rho_e = \rho_0</math></td>
	<td align="center" colspan="1">Presented by {{ Yabushita75 }}</td>		<td align="center" colspan="1">Presented by {{ Yabushita75 }}<br />(after setting <math>\mu_e/\mu_c = 1</math>)</td>
	</tr>		</tr>

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	\biggl(\frac{\mu_e}{\mu_c}\biggr) e^{-\psi_i/ 2}		\biggl(\frac{\mu_e}{\mu_c}\biggr) e^{-\psi_i/ 2}
	\theta_i^{-1 / 4}		\theta_i^{-1 / 4}
	\, .		\, ;
	</math>		</math>
	</td>		</td>
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	<tr>		<tr>
	<td align="right"><math>\~~frac~~{~~\rho_e}{\rho_0~~}</math></td>		<td align="right"><math>\theta_i^{3 / 2}</math></td>
	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left">		<td align="left">
	<math>		<math>
	1		\biggl(\frac{\mu_e}{\mu_c}\biggr)e^{-\psi_i}
	\, ;		\, ;
	</math>		</math>
Line 475:		Line 475:

	<tr>		<tr>
	<td align="right"><math>\frac{c_s^2}{K_e}		<td align="right"><math>\biggl[ \frac{c_s^2}{K_e} \biggr]^{3}
	</math></td>		</math></td>
	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left">		<td align="left">
	<math>		<math>
	\biggl[		\biggl(\frac{\mu_e}{\mu_c}\biggr)^{5 } \biggl[\rho_0e^{-\psi_i} \biggr]^{2}
	\rho_0e^{-\psi_i}
	\biggr]^{2 ~~/ 3~~}
	\, ;		\, ;
	</math>		</math>
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	<td align="left">		<td align="left">
	<math>		<math>
	\biggl(\frac{2}{5}\biggr)^{1 / 2} e^{-\psi_i/ 2} \biggl[ e^{-2 \psi_i ~~/ 3~~ }\biggr]^{-1 / 4}		\biggl(\frac{2}{5}\biggr)^{1 / 2}
			\biggl(\frac{\mu_e}{\mu_c}\biggr) e^{-\psi_i/ 2}
			\biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)e^{-\psi_i} \biggr]^{-1 / 6}
	=		=
	\biggl(\frac{2}{5}\biggr)^{1 / 2} e^{-\psi_i/ 3}		\biggl(\frac{2}{5}\biggr)^{1 / 2}
			\biggl(\frac{\mu_e}{\mu_c}\biggr)^{5 / 6} e^{-\psi_i/ 3}
	\, ;		\, ;
	</math>		</math>
Line 511:		Line 512:
	<math>		<math>
	\biggl(\frac{2}{5}\biggr)^{1 / 2}		\biggl(\frac{2}{5}\biggr)^{1 / 2}
	~~e^{+ \psi_i / 2} \biggl[ e^{-2 \psi_i / 3} \biggr]^{5 / 4}~~
	\biggl( \frac{d\psi}{d\xi} \biggr)_i		\biggl( \frac{d\psi}{d\xi} \biggr)_i
			e^{+ \psi_i / 2} \biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)e^{-\psi_i} \biggr]^{5 / 6}
	=		=
	\biggl(\frac{2}{5}\biggr)^{1 / 2}		\biggl(\frac{2}{5}\biggr)^{1 / 2}
	~~e^{- \psi_i / 3}~~
	\biggl( \frac{d\psi}{d\xi} \biggr)_i		\biggl( \frac{d\psi}{d\xi} \biggr)_i
			e^{- \psi_i / 3} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{5 / 6}
	\, .		\, .
	</math>		</math>

Jet53man: /* Step 5: Interface Conditions */

2023-11-09T18:57:50Z

Step 5: Interface Conditions

← Older revision		Revision as of 14:57, 9 November 2023
Line 391:		Line 391:
	<tr>		<tr>
	<td align="center" colspan="1">Our Derivations</td>		<td align="center" colspan="1">Our Derivations</td>
	<td align="center" colspan="1">After setting <math>(\mu_e/\mu_c) = 1</math> and <math>\theta_i = 1</math></td>		<td align="center" colspan="1">After setting <math>(\mu_e/\mu_c) = 1</math> and <math>\theta_i = e^{-2\psi_i/ 3}</math></td>
	<td align="center" colspan="1">Presented by {{ Yabushita75 }}</td>		<td align="center" colspan="1">Presented by {{ Yabushita75 }}</td>
	</tr>		</tr>
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	</math>		</math>
	</td>		</td>
	~~<td align="right" width="5%">(2.9)</td>~~
	</tr>		</tr>

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	</math>		</math>
	</td>		</td>
	~~<td align="right" width="5%">(2.11)</td>~~
	</tr>		</tr>

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	</math>		</math>
	</td>		</td>
	~~<td align="right" width="5%">(2.12)</td>~~
	</tr>		</tr>

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	</math>		</math>
	</td>		</td>
	~~<td align="right" width="5%"> </td>~~
	</tr>		</tr>
	</table>		</table>
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	<td align="left">		<td align="left">
	<math>		<math>
	~~e^{-\psi_i} \theta_i^{-3 / 2}~~		1
	\, ;		\, ;
	</math>		</math>
	</td>		</td>
	~~<td align="right" width="5%">(2.9)</td>~~
	</tr>		</tr>

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	</math>		</math>
	</td>		</td>
	~~<td align="right" width="5%">(2.11)</td>~~
	</tr>		</tr>

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	<td align="left">		<td align="left">
	<math>		<math>
	\biggl(\frac{2}{5}\biggr)^{1 / 2}		\biggl(\frac{2}{5}\biggr)^{1 / 2} e^{-\psi_i/ 2} \biggl[ e^{-2 \psi_i / 3 }\biggr]^{-1 / 4}
	e^{-\psi_i/ 2}		=
	\~~theta_i~~^{-1 / 4}		\biggl(\frac{2}{5}\biggr)^{1 / 2} e^{-\psi_i/ 3}
	\, .		\, ;
	</math>		</math>
	</td>		</td>
	~~<td align="right" width="5%">(2.12)</td>~~
	</tr>		</tr>

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	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left">		<td align="left">
	<math>\biggl(\frac{2}{5}\biggr)^{1 / 2}		<math>
	e^{+ \psi_i / 2} \~~theta_i~~^{5 / 4}		\biggl(\frac{2}{5}\biggr)^{1 / 2}
			e^{+ \psi_i / 2} \biggl[ e^{-2 \psi_i / 3} \biggr]^{5 / 4}
			\biggl( \frac{d\psi}{d\xi} \biggr)_i
			=
			\biggl(\frac{2}{5}\biggr)^{1 / 2}
			e^{- \psi_i / 3}
	\biggl( \frac{d\psi}{d\xi} \biggr)_i		\biggl( \frac{d\psi}{d\xi} \biggr)_i
	\, .		\, .
	</math>		</math>
	</td>		</td>
	~~<td align="right" width="5%"> </td>~~
	</tr>		</tr>
	</table>		</table>
Line 533:		Line 530:

	<tr>		<tr>
	<td align="right"><math>~~[ ~\lambda_e ~]~~ \theta_i^{3 / 2}</math></td>		<td align="right"><math>\theta_i^{3 / 2}</math></td>
	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left">		<td align="left">
	<math>		<math>
	~~\lambda_c e^{-\psi_i}~~		e^{-\psi_i}
	~~~~\Rightarrow ~~ \theta_i^{1 / 4} = \biggl[\frac{\lambda_c}{[~\lambda_e~]}~~ e^{-\psi_i~~}\biggr]^{1 / 6~~}
	\, ;		\, ;
	</math>		</math>
Line 563:		Line 559:
	<td align="left">		<td align="left">
	<math>		<math>
	~~\biggl(\frac{2}{5}\biggr)^{1 / 2}e^{-\psi_i/ 2}~~		\biggl(\frac{2}{5}\biggr)^{1 / 2}e^{-\psi_i/ 3}
	~~\theta_i^{-1 / 4}~~
	~~= \biggl(\frac{2}{5}\biggr)^{1 / 2}e^{-\psi_i/ 2}\biggl[\frac{\lambda_c}{[~\lambda_e~]} e^{-\psi_i}\biggr]^{-1 / 6}~~
	= \biggl(\frac{2}{5}\biggr)^{1 / 2}e^{-\psi_i/ 3~~}\biggl[\frac{[~\lambda_e~]}{\lambda_c}\biggr]^{1 / 6~~}
	\, ;		\, ;
	</math>		</math>
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	<td align="right">		<td align="right">
	<math>		<math>
	\biggl(-\frac{d\theta}{d\eta} \biggr)_i		\biggl(\frac{d\theta}{d\eta} \biggr)_i
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>		<math>
	\biggl(\frac{2}{5}\biggr)^{1 / 2~~} e^{+ \psi_i / 2} \theta_i^{5 / 4} \biggl( \frac{d\psi}{d\xi} \biggr)_i~~		-\biggl(\frac{2}{5}\biggr)^{1 / 2} \biggl( \frac{d\psi}{d\xi} \biggr)_i e^{- \psi_i / 3}
	=
	~~\biggl(\frac{2}{5}\biggr)^{1 / 2} e^{+ \psi_i / 2} \biggl[\frac{\lambda_c}{[~\lambda_e~]} e^{-\psi_i}\biggr]^{5 / 6~~} \biggl( \frac{d\psi}{d\xi} \biggr)_i
	=
	~~\biggl(\frac{2}{5}\biggr)^{1 / 2}~~ e^{- \psi_i / 3} ~~\biggl[\frac{\lambda_c}{[~\lambda_e~]} \biggr]^{5 / 6} \biggl( \frac{d\psi}{d\xi} \biggr)_i~~
	\, .		\, .
	</math>		</math>
	</td>		</td>
	<td align="right" width="5%">~~ ~~</td>		<td align="right" width="5%">(2.13)</td>
	</tr>		</tr>
	</table>		</table>

Jet53man: /* Step 5: Interface Conditions */

2023-11-09T18:19:14Z

Step 5: Interface Conditions

← Older revision		Revision as of 14:19, 9 November 2023
Line 538:		Line 538:
	<math>		<math>
	\lambda_c e^{-\psi_i}		\lambda_c e^{-\psi_i}
			~~\Rightarrow ~~ \theta_i^{1 / 4} = \biggl[\frac{\lambda_c}{[~\lambda_e~]} e^{-\psi_i}\biggr]^{1 / 6}
	\, ;		\, ;
	</math>		</math>
Line 562:		Line 563:
	<td align="left">		<td align="left">
	<math>		<math>
	\biggl(\frac{2}{5}\biggr)^{1 / 2}		\biggl(\frac{2}{5}\biggr)^{1 / 2}e^{-\psi_i/ 2}
	e^{-\psi_i/ 2}		\theta_i^{-1 / 4}
	\theta_i^{-1 / 4}		= \biggl(\frac{2}{5}\biggr)^{1 / 2}e^{-\psi_i/ 2}\biggl[\frac{\lambda_c}{[~\lambda_e~]} e^{-\psi_i}\biggr]^{-1 / 6}
	\, .		= \biggl(\frac{2}{5}\biggr)^{1 / 2}e^{-\psi_i/ 3}\biggl[\frac{[~\lambda_e~]}{\lambda_c}\biggr]^{1 / 6}
			\, ;
	</math>		</math>
	</td>		</td>
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	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left">		<td align="left">
	<math>\biggl(\frac{2}{5}\biggr)^{1 / 2}		<math>
	e^{+ \psi_i / 2} \theta_i^{5 / 4}		\biggl(\frac{2}{5}\biggr)^{1 / 2} e^{+ \psi_i / 2} \theta_i^{5 / 4} \biggl( \frac{d\psi}{d\xi} \biggr)_i
	\biggl( \frac{d\psi}{d\xi} \biggr)_i		=
			\biggl(\frac{2}{5}\biggr)^{1 / 2} e^{+ \psi_i / 2} \biggl[\frac{\lambda_c}{[~\lambda_e~]} e^{-\psi_i}\biggr]^{5 / 6} \biggl( \frac{d\psi}{d\xi} \biggr)_i
			=
			\biggl(\frac{2}{5}\biggr)^{1 / 2} e^{- \psi_i / 3} \biggl[\frac{\lambda_c}{[~\lambda_e~]} \biggr]^{5 / 6} \biggl( \frac{d\psi}{d\xi} \biggr)_i
	\, .		\, .
	</math>		</math>