SSC/Stability/NeutralMode - Revision history

Jet53man: /* Equilibrium Structure */

2021-10-20T21:55:47Z

Equilibrium Structure

← Older revision		Revision as of 17:55, 20 October 2021
Line 2,555:		Line 2,555:
	</table>		</table>

			[[File:EntropyDistribution245.png\|right\|400px]]
	The slope is zero when,		The slope is zero when,

2021-10-20T17:11:08Z

Equilibrium Structure

@@ Line 2,434: / Line 2,434: @@
 </div>
-For purposes of discussion (later), we find from [[Appendix/Ramblings/PatrickMotl#Tying_Expressions_into_H_Book_Context|a separate examination of entropy distributions]], <math>s_0(r_0)</math>, that,
+<table border="1" width="90%" cellpadding="8" align="center"><tr><td align="left">
 <table border="0" cellpadding="5" align="center">
@@ Line 2,493: / Line 2,497: @@
 </tr>
 </table>
 ==Some Relevant Structural Derivatives==

2021-10-19T18:38:08Z

Equilibrium Structure

@@ Line 2,421: / Line 2,421: @@
    <td align="left">
 <math>
-\frac{4\pi G\rho_c^2 R^2}{15} \biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^2
+\frac{4\pi G\rho_c^2 R^2}{15}
 \biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]
 \, ,</math>
@@ Line 2,432: / Line 2,433: @@
 <math>P_c = \frac{4\pi}{15} G\rho_c^2 R^2 \, .</math>
 </div>
 ==Some Relevant Structural Derivatives==

2021-10-19T00:47:13Z

Some Relevant Structural Derivatives

@@ Line 2,541: / Line 2,541: @@
 </tr>
 </table>
 Hence,

2021-10-18T01:53:05Z

Second Attempt

@@ Line 3,512: / Line 3,512: @@
 \biggr]
 +~ \biggl( \frac{5\sigma_c^2}{\gamma_g}\biggr) \biggl(1 - x^2\biggr)
 </math>
 </td>

2021-10-18T01:38:07Z

Second Attempt

@@ Line 3,492: / Line 3,492: @@
 \biggl(5 - \frac{15a}{b}\biggr) + x^2 \biggl(-3 + \frac{15a}{b}  \biggr) \biggr] \biggl(1 - x^2\biggr)
 \, .
 </math>
 </td>

2021-10-18T01:28:08Z

Second Attempt

@@ Line 3,431: / Line 3,431: @@
 \biggl(1 - \frac{3}{5} x^2\biggr)
 - \frac{3a}{b} \biggl(1-x^2\biggr)\biggr\} \biggl(1 - x^2\biggr)
 </math>
 </td>

2021-10-18T00:46:31Z

Second Attempt

@@ Line 3,356: / Line 3,356: @@
 - \frac{3a}{b}\biggl(1 - \frac{3}{5}x^2\biggr)^{-1} \biggl(1-x^2\biggr)\biggr\} \biggl(1 - x^2\biggr)^{-1} \, .
 </math>
 </td>

2021-10-18T00:31:31Z

Second Attempt

← Older revision		Revision as of 20:31, 17 October 2021
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	<math>		<math>
	+ \biggl[\frac{d\ln P_0}{d\ln r_0} \biggr]		+ \biggl[\frac{d\ln P_0}{d\ln r_0} \biggr]
	\biggl[\frac{a\alpha}{b} - \frac{4}{15} x^2 \biggl(1 - x^2\biggr)^{-2} - \alpha \biggl( \~~Delta~~ \biggr)\biggr]		\biggl[\frac{a\alpha}{b} - \frac{4}{15} x^2 \biggl(1 - x^2\biggr)^{-2}
			- \frac{\alpha}{3} \biggl(1 - \frac{3}{5} x^2\biggr)\biggl(1 - x^2\biggr)^{-1}\biggr]
	</math>		</math>
	</td>		</td>
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	+ \biggl[\frac{d\ln P_0}{d\ln r_0}\biggr]\biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{		+ \biggl[\frac{d\ln P_0}{d\ln r_0}\biggr]\biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{
	1		1
	- \frac{a}{b}\biggl[\frac{1}{\~~Delta~~} \biggr] \biggr\} \~~frac~~{~~\rho_c~~}{\~~rho_0}~~		- \frac{3a}{b}\biggl(1 - \frac{3}{5}x^2\biggr)^{-1} \biggl(1-x^2\biggr)\biggr\} \biggl(1 - x^2\biggr)^{-1} \, .
	</math>		</math>
	</td>		</td>

2021-10-18T00:22:56Z

Second Attempt

← Older revision		Revision as of 20:22, 17 October 2021
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	</tr>		</tr>

			<tr>
			<td align="right"> </td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>
			- \frac{4}{15} x^2 \biggl(1 - x^2\biggr)^{-3}\biggl( 1 + 3x^2\biggr)
			- \frac{16}{15} x^2 \biggl(1 - x^2\biggr)^{-2}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right"> </td>
			<td align="center"> </td>
			<td align="left">
			<math>
			+ \biggl[\frac{d\ln P_0}{d\ln r_0} \biggr]
			\biggl[\frac{a\alpha}{b} - \frac{4}{15} x^2 \biggl(1 - x^2\biggr)^{-2} - \alpha \biggl( \Delta \biggr)\biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right"> </td>
			<td align="center"> </td>
			<td align="left">
			<math>
			+ \biggl[\frac{d\ln P_0}{d\ln r_0}\biggr]\biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{
			1
			- \frac{a}{b}\biggl[\frac{1}{\Delta} \biggr] \biggr\} \frac{\rho_c}{\rho_0}
			</math>
			</td>
			</tr>
	</table>		</table>