SSC/Structure/BiPolytropes/51RenormaizePart2 - Revision history

Jet53man: /* One of the Linearized Equations */

2023-10-26T22:59:28Z

One of the Linearized Equations

← Older revision		Revision as of 18:59, 26 October 2023
Line 592:		Line 592:
	<math>		<math>
	\omega^2 x \biggl[ K_c^{-15} G^{13} M_\mathrm{tot}^{10} \biggr]^{1 / 2}		\omega^2 x \biggl[ K_c^{-15} G^{13} M_\mathrm{tot}^{10} \biggr]^{1 / 2}
	\biggl[\mathcal{m}_\mathrm{surf}^{-2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{4} \biggl(\frac{3}{2\pi}\biggr)^{1/2} \xi\biggr]^3		\biggl[\mathcal{m}_\mathrm{surf}^{-6} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{12} \biggl(\frac{3}{2\pi}\biggr)^{3/2} \xi^3\biggr]
	\biggl\{ \mathcal{m}_\mathrm{surf~~}^{-1~~} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{2}		\biggl\{ \mathcal{m}_\mathrm{surf} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2}
	\biggl( \frac{2\cdot 3~~}{\pi~~ } \biggr)^{1/2} \biggl[ \xi^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \biggr]\biggr\}^{-1}		\biggl( \frac{\pi }{2\cdot 3} \biggr)^{1/2} \biggl[ \xi^{-3} \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{3/2} \biggr]\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right"> </td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>
			\biggl(\frac{3}{4\pi}\biggr)
			\omega^2 x \biggl[ K_c^{-15} G^{13} M_\mathrm{tot}^{10} \biggr]^{1 / 2}
			\biggl[\mathcal{m}_\mathrm{surf}^{-5} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{10} \biggr]
			\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{3/2} \, .
	</math>		</math>
	</td>		</td>

Jet53man: /* One of the Linearized Equations */

2023-10-26T22:48:58Z

One of the Linearized Equations

@@ Line 474: / Line 474: @@
    <td align="left">
 <math>
-\frac{1}{2\xi^2}\biggl( 1 + \frac{1}{3}\xi^2 \biggr)
+\frac{1}{2\xi}\biggl( 1 + \frac{1}{3}\xi^2 \biggr)
 \frac{d(p_P)}{d\xi} - (4x_P + p_P)
 </math>
@@ Line 485: / Line 485: @@
    <td align="left">
 <math>
-\frac{1}{2\xi^2}\biggl( 1 + \frac{1}{3}\xi^2 \biggr)
+\frac{1}{2\xi}\biggl( 1 + \frac{1}{3}\xi^2 \biggr)
 \frac{4\xi}{5}
 - 4\biggl[ 1 -  \frac{\xi^2}{15} \biggr]
@@ Line 498: / Line 498: @@
    <td align="left">
 <math>
-\frac{1}{\xi}\biggl( 1 + \frac{1}{3}\xi^2 \biggr)
+\biggl( 1 + \frac{1}{3}\xi^2 \biggr)
 \frac{2}{5} - 4 + \frac{4\xi^2}{15}
 + \frac{18}{5} - \frac{2\xi^2}{5}
@@ Line 510: / Line 510: @@
    <td align="left">
 <math>
-\frac{1}{\xi}\biggl( 1 + \frac{1}{3}\xi^2 \biggr)
+\biggl( 1 + \frac{1}{3}\xi^2 \biggr)
 \frac{2}{5}
 - \frac{2}{5} - \frac{2\xi^2}{15}
 </math>
    </td>

Jet53man: /* One of the Linearized Equations */

2023-10-26T22:13:06Z

One of the Linearized Equations

← Older revision		Revision as of 18:13, 26 October 2023
Line 277:		Line 277:
	\biggl\{		\biggl\{
	\tilde{P}		\tilde{P}
	\tilde{r}^2		\tilde{r}
	\tilde\rho^{-1}		\tilde\rho^{-1}
	\tilde{M}_r^{-1}		\tilde{M}_r^{-1}
	\biggr\}		\biggr\}
	\frac{dp}{d\~~tilde{r}~~} \, .		\frac{dp}{d\ln \xi} \, .
	</math>		</math>
	</td>		</td>
Line 306:		Line 306:
	\mathcal{m}_\mathrm{surf}^{-2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{4}		\mathcal{m}_\mathrm{surf}^{-2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{4}
	\biggl(\frac{3}{2\pi}\biggr)^{1/2} \xi		\biggl(\frac{3}{2\pi}\biggr)^{1/2} \xi
	\biggr]\frac{dp}{d\xi}		\biggr]\frac{dp}{d\ln \xi}
	</math>		</math>
	</td>		</td>
Line 323:		Line 323:
	\biggr\}		\biggr\}

	\frac{dp}{d\xi}		\frac{dp}{d\ln \xi}
	</math>		</math>
	</td>		</td>
Line 334:		Line 334:
	<math>		<math>
	\frac{1}{2\xi^2}\biggl( 1 + \frac{1}{3}\xi^2 \biggr)		\frac{1}{2\xi^2}\biggl( 1 + \frac{1}{3}\xi^2 \biggr)
	\frac{dp}{d\xi} \, .		\frac{dp}{d\ln \xi} \, .
	</math>		</math>
	</td>		</td>

Jet53man: /* One of the Linearized Equations */

2023-10-26T16:13:51Z

One of the Linearized Equations

← Older revision		Revision as of 12:13, 26 October 2023
Line 233:		Line 233:
	<math>		<math>
	\biggl\{		\biggl\{
	\frac{P_0 r_0^2}{\rho_0 G M_r}		\frac{P_0 r_0}{\rho_0 G M_r}
	\biggr\}		\biggr\}
	\frac{dp}{~~dr_0~~}		\frac{dp}{d\ln r_0}
	</math>		</math>
	</td>		</td>
Line 246:		Line 246:
	<math>		<math>
	\tilde{P}\biggl[K_c^{-10} G^{9} M_\mathrm{tot}^{6} \biggr]^{-1}		\tilde{P}\biggl[K_c^{-10} G^{9} M_\mathrm{tot}^{6} \biggr]^{-1}
	\tilde{r}^2\biggl[\biggl( \frac{K_c}{G} \biggr)^{5 / 2} M_\mathrm{tot}^{-2} \biggr]^{-2}		\tilde{r}\biggl[\biggl( \frac{K_c}{G} \biggr)^{5 / 2} M_\mathrm{tot}^{-2} \biggr]^{-1}
	\frac{1}{\tilde\rho}\biggl[\biggl( \frac{K_c}{G} \biggr)^{3 / 2} \frac{1}{M_\mathrm{tot}} \biggr]^{-5}		\frac{1}{\tilde\rho}\biggl[\biggl( \frac{K_c}{G} \biggr)^{3 / 2} \frac{1}{M_\mathrm{tot}} \biggr]^{-5}
	\biggl\{		\biggl\{
	\frac{1}{G M_\mathrm{tot}\tilde{M}_r}		\frac{1}{G M_\mathrm{tot}\tilde{M}_r}
	\biggr\}~~\biggl[\biggl( \frac{K_c}{G} \biggr)^{5 / 2} M_\mathrm{tot}^{-2}\biggr]~~		\biggr\}
	\frac{dp}{d\~~tilde{r}~~}		\frac{dp}{d\ln \xi}
	</math>		</math>
	</td>		</td>
Line 262:		Line 262:
	<math>		<math>
	\tilde{P}\biggl[K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]		\tilde{P}\biggl[K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
	\tilde{r}^2\biggl[K_c^{-5}G^5 M_\mathrm{tot}^{4} \biggr]		\tilde{r}\biggl[K_c^{-5 / 2}G^{5 / 2} M_\mathrm{tot}^{2} \biggr]
	\frac{1}{\tilde\rho}\biggl[K_c^{- 15 / 2} G^{15 / 2} M_\mathrm{tot}^5 \biggr]		\frac{1}{\tilde\rho}\biggl[K_c^{- 15 / 2} G^{15 / 2} M_\mathrm{tot}^5 \biggr]
	G^{-1} M_\mathrm{tot}^{-1} \tilde{M}_r^{-1}		\biggl[ G^{-1} M_\mathrm{tot}^{-1}\biggr] \tilde{M}_r^{-1}
	~~\biggl[K_c^{5 / 2} G^{- 5 / 2} M_\mathrm{tot}^{-2}\biggr]~~		\frac{dp}{d\ln \xi}
	\frac{dp}{d\~~tilde{r}~~}
	</math>		</math>
	</td>		</td>

Jet53man: /* One of the Linearized Equations */

2023-10-25T21:38:51Z

One of the Linearized Equations

@@ Line 487: / Line 487: @@
 <math>
 \frac{1}{2\xi^2}\biggl( 1 + \frac{1}{3}\xi^2 \biggr)
-\frac{4\xi}{5} -
+\frac{4\xi}{5}
-\biggl[4x_P + p_P \biggr]
+- 4\biggl[ 1 -  \frac{\xi^2}{15} \biggr]
 </math>
    </td>

Jet53man: /* One of the Linearized Equations */

2023-10-25T20:40:58Z

One of the Linearized Equations

@@ Line 462: / Line 462: @@
 -\frac{18}{5}\biggl[1 - \frac{\xi^2}{9}\biggr]
 \, .
 </math>
    </td>

Jet53man: /* One of the Linearized Equations */

2023-10-25T20:35:12Z

One of the Linearized Equations

← Older revision		Revision as of 16:35, 25 October 2023
Line 379:		Line 379:

	</td></tr></table>		</td></tr></table>

			From the structural solution for [[SSC/Structure/Polytropes#Primary_E-Type_Solution_2\|equilibrium,]] <math>n=5</math> polytropes, we know that,

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

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

			<tr>
			<td align="right"><math>\Rightarrow ~~~ \frac{d\theta}{d\xi}</math></td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>
			- \frac{\xi}{3}\biggl(1 + \frac{\xi^2}{3}\biggr)^{-3 / 2}
			\, .
			</math>
			</td>
			</tr>
			</table>
			Therefore, for <math>n=5</math> structures,

			<table border="0" align="center" cellpadding="8">

			<tr>
			<td align="right"><math>x_P</math></td>
			<td align="center"><math>=</math></td>
			<td align="left"><math>
			\frac{6}{5}\biggl[1 - \frac{1}{2}\biggl( \frac{1}{\xi \theta^{5}}\biggr)
			\frac{\xi}{3}\biggl(1 + \frac{\xi^2}{3}\biggr)^{-3 / 2}\biggr] </math>
			</td>
			</tr>

			<tr>
			<td align="right"> </td>
			<td align="center"><math>=</math></td>
			<td align="left"><math>
			\frac{6}{5}\biggl[1 - \frac{1}{6}
			\biggl(1 + \frac{\xi^2}{3}\biggr)\biggr] </math>
			</td>
			</tr>

			<tr>
			<td align="right"> </td>
			<td align="center"><math>=</math></td>
			<td align="left"><math>
			1 - \frac{\xi^2}{15} \, ;
			</math>
			</td>
			</tr>

			<tr>
			<td align="right"><math>d_P</math></td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>
			-3 + 3 \biggl(1 + \frac{\xi^2}{3}\biggr)^3 \frac{\xi^2}{9}\biggl(1 + \frac{\xi^2}{3}\biggr)^{-3}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right"> </td>
			<td align="center"><math>=</math></td>
			<td align="left"><math>
			-3 + \frac{\xi^2}{3}
			\, ;
			</math>
			</td>
			</tr>

			<tr>
			<td align="right"><math>p_P</math></td>
			<td align="center"><math>=</math></td>
			<td align="left"><math>
			-\frac{18}{5}\biggl[1 - \frac{\xi^2}{9}\biggr]
			\, .
			</math>
			</td>
			</tr>
			</table>

	===Rescale for Bonnor-Ebert-Type Analysis===		===Rescale for Bonnor-Ebert-Type Analysis===

Jet53man: /* One of the Linearized Equations */

2023-10-25T18:21:27Z

One of the Linearized Equations

@@ Line 340: / Line 340: @@
 </tr>
 </table>
 ===Rescale for Bonnor-Ebert-Type Analysis===

Jet53man: /* One of the Linearized Equations */

2023-10-25T17:47:18Z

One of the Linearized Equations

@@ Line 325: / Line 325: @@
 \frac{dp}{d\xi}
 </math>
    </td>

Jet53man: /* One of the Linearized Equations */

2023-10-25T17:43:00Z

One of the Linearized Equations

@@ Line 308: / Line 308: @@
 \biggl(\frac{3}{2\pi}\biggr)^{1/2} \xi
 \biggr]\frac{dp}{d\xi}
 </math>
    </td>