Appendix/Ramblings/51BiPolytropeStability/NoAnalytic - Revision history

Jet53man: /* Renormalized LAWE */

2022-08-08T22:41:30Z

Renormalized LAWE

← Older revision		Revision as of 18:41, 8 August 2022
Line 761:		Line 761:
	</tr>		</tr>
	</table>		</table>

			----


			<font color="red">CAUTION:</font>

	==Numerical Integration==		==Numerical Integration==

Jet53man: /* Renormalized LAWE */

2022-08-08T21:06:12Z

Renormalized LAWE

← Older revision		Revision as of 17:06, 8 August 2022
Line 701:		Line 701:
	<math>		<math>
	\frac{d^2x}{d\tilde{r}^2}		\frac{d^2x}{d\tilde{r}^2}
	+ \~~underbrace~~{\biggl\{ 4 - \biggl(\frac{\tilde\rho}{\tilde{P}}\biggr)\frac{ \tilde{M}_r}{\tilde{r}}\biggr\}}_{\tilde\mathcal{H}}		+ \overbrace{\biggl\{ 4 - \biggl(\frac{\tilde\rho}{\tilde{P}}\biggr)\frac{ \tilde{M}_r}{\tilde{r}}\biggr\}}^{\tilde\mathcal{H}}
	\frac{1}{\tilde{r}}\frac{dx}{d\tilde{r}}		\frac{1}{\tilde{r}}\frac{dx}{d\tilde{r}}
	+ \underbrace{\frac{\tilde\rho}{\tilde{P}}		+ \underbrace{\frac{\tilde\rho}{\tilde{P}}

Jet53man: /* Renormalized LAWE */

2022-08-08T20:51:49Z

Renormalized LAWE

@@ Line 752: / Line 752: @@
    <td align="left">
 <math>
-\frac{\eta_s^2}{3A\theta_i^5}
+\frac{\eta_s^2}{3A\theta_i^5} \, .
 </math>
    </td>
@@ Line 758: / Line 758: @@
 </table>
    </td>
 </tr>

Jet53man: /* General Approach */

2022-08-08T18:44:00Z

General Approach

← Older revision		Revision as of 14:44, 8 August 2022
Line 1,022:		Line 1,022:
	In what follows we will also find it useful to rewrite <math>~\mathcal{K}</math> in the form,		In what follows we will also find it useful to rewrite <math>~\mathcal{K}</math> in the form,
	<div align="center">		<div align="center">
	<math>\tilde\mathcal{K} ~\rightarrow ~\biggl(\frac{\~~sigma_c~~^2}{\gamma_\mathrm{g}}\biggr) \tilde\mathcal{K}_1		<math>\tilde\mathcal{K} ~\rightarrow ~\biggl(\frac{\tilde\sigma^2}{\gamma_\mathrm{g}}\biggr) \tilde\mathcal{K}_1
	- \alpha_\mathrm{g} \tilde\mathcal{K}_2 \, .</math>		- \alpha_\mathrm{g} \tilde\mathcal{K}_2 \, .</math>
	</div>		</div>
Line 1,050:		Line 1,050:
	<td align="left">		<td align="left">
	<math>		<math>
	\frac{2\pi }{3}\biggl(\frac{\tilde\rho}{ \tilde{P} } \biggr) ~~\biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-10} \mathcal{m}^5_\mathrm{surf} \biggr]~~		\frac{2\pi }{3}\biggl(\frac{\tilde\rho}{ \tilde{P} } \biggr)
	</math>		</math>
	</td>		</td>

Jet53man: /* Renormalized LAWE */

2022-08-08T18:36:09Z

Renormalized LAWE

← Older revision		Revision as of 14:36, 8 August 2022
Line 642:		Line 642:
	</tr>		</tr>
	</table>		</table>
			That is, the renormalized LAWE becomes,

	~~----~~


	~~After multiplying through by~~
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~~\biggl[\biggl( \frac{G}{K_c} \biggr)^{3 / 2} M_\mathrm{tot}\rho_0^{1 / 5} \biggr]^{4}~~</math>		<math>0</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 659:		Line 654:
	<td align="left">		<td align="left">
	<math>		<math>
	\biggl[ \biggl(\frac{\~~mu_e~~}{\~~mu_c~~}\biggr)^{-2} \~~mathcal~~{m}_\~~mathrm~~{~~surf~~} \biggr]^4		\biggl[\biggl( \frac{G}{K_c} \biggr)^{3 / 2} M_\mathrm{tot}\rho_0^{1 / 5} \biggr]^{-4}\frac{d^2x}{d\tilde{r}^2}
			+ \biggl\{ 4 - \biggl(\frac{\tilde\rho}{\tilde{P}}\biggr)\frac{ \tilde{M}_r}{\tilde{r}}\biggr\}
			\biggl[\biggl( \frac{G}{K_c} \biggr)^{3 / 2} M_\mathrm{tot}\rho_0^{1 / 5} \biggr]^{-4}\frac{1}{\tilde{r}} \frac{dx}{d\tilde{r}}
			+ \frac{\tilde\rho}{\tilde{P}}\biggl[\biggl( \frac{G}{K_c} \biggr)^{3 / 2} M_\mathrm{tot}\rho_0^{1 / 5} \biggr]^{-4}
			\biggl\{ \frac{2\pi \tilde\sigma^2}{3\gamma_\mathrm{g}}
			-~\frac{\alpha_\mathrm{g} \tilde{M}_r}
			{(\tilde{r})^3}
			\biggr\} x \, .
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	~~we have,~~

			----


			After multiplying through by
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>0</math>		<math>\biggl[\biggl( \frac{G}{K_c} \biggr)^{3 / 2} M_\mathrm{tot}\rho_0^{1 / 5} \biggr]^{4}</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 676:		Line 683:
	<td align="left">		<td align="left">
	<math>		<math>
	~~\frac{d^2x}{d\tilde{r}^2}~~		\biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \mathcal{m}_\mathrm{surf} \biggr]^4
	+ \biggl~~\{ 4 -~~ \biggl(\frac{\~~tilde\rho~~}{\~~tilde{P}~~}\biggr)~~\frac{ \tilde{M}_r}{\tilde{r}}\biggr\}~~
	~~\frac{1}{\tilde{r}} \frac{dx}{d\tilde{r}}~~
	~~+ \frac{\tilde\rho}{\tilde{P}}~~
	~~\biggl\{ \frac~~{~~2\pi \sigma_c^~~2}~~{3\gamma_~~\~~mathrm~~{g}}~~\biggl[ \biggl(\frac{G}{K_c} \biggr)^{3 / 2} M_~~\mathrm{~~tot} \rho_0^{1/5~~} \biggr]^5
	~~-~\frac{\alpha_\mathrm{g} \tilde{M}_r}~~
	~~{(\tilde{r})^3}~~
	~~\biggr\} x~~
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
			</table>
			we have,
			<table border="0" cellpadding="5" align="center">

	<tr>		<tr>
	<td align="right">		<td align="right">
	~~ ~~		<math>0</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 701:		Line 704:
	\frac{1}{\tilde{r}}\frac{dx}{d\tilde{r}}		\frac{1}{\tilde{r}}\frac{dx}{d\tilde{r}}
	+ \underbrace{\frac{\tilde\rho}{\tilde{P}}		+ \underbrace{\frac{\tilde\rho}{\tilde{P}}
	\biggl\{ \frac{2\pi \~~sigma_c~~^2}{3\gamma_\mathrm{g}}~~\biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-10} \mathcal{m}^5_\mathrm{surf} \biggr]~~		\biggl\{ \frac{2\pi \tilde\sigma^2}{3\gamma_\mathrm{g}}
	-~\frac{\alpha_\mathrm{g} \tilde{M}_r}		-~\frac{\alpha_\mathrm{g} \tilde{M}_r}
	{(\tilde{r})^3}		{(\tilde{r})^3}

Jet53man: /* Renormalized LAWE */

2022-08-08T18:01:40Z

Renormalized LAWE

@@ Line 604: / Line 604: @@
 </tr>
 </table>
 After multiplying through by
 <table border="0" cellpadding="5" align="center">

Jet53man: /* Renormalized LAWE */

2022-08-07T22:12:47Z

Renormalized LAWE

@@ Line 803: / Line 803: @@
 </table>
    </td>
 </tr>

Jet53man: /* Renormalized LAWE */

2022-08-07T21:37:48Z

Renormalized LAWE

@@ Line 667: / Line 667: @@
 </table>
-<table border="1" align="center" cellpadding="5">
+<table border="1" align="center" cellpadding="8" width="80%">
 <tr>
    <td align="left">
-Also note that,
+From above, note that,
 <table border="0" align="center" cellpadding="5">
 <tr>
@@ Line 685: / Line 685: @@
    <td align="left">
 <math>
-\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-10} \biggl(\frac{2}{\pi}\biggr)^{5 / 2} \biggl[ \frac{A\eta_s}{\theta_i}\biggr]^5
+\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-10} \biggl(\frac{2}{\pi}\biggr)^{5 / 2} \biggl[ \frac{A\eta_s}{\theta_i}\biggr]^5 \, ;
 </math>
    </td>
 </tr>
 </table>
    </td>
 </tr>

Jet53man: /* Renormalized LAWE */

2022-08-07T18:33:46Z

Renormalized LAWE

@@ Line 663: / Line 663: @@
 \biggr\}}_{\tilde\mathcal{K}}  x \, .
 </math>
    </td>
 </tr>

Jet53man: /* General Approach */

2022-08-07T17:06:55Z

General Approach

← Older revision		Revision as of 13:06, 7 August 2022
Line 684:		Line 684:
	<td align="left">		<td align="left">
	<math>		<math>
	x'' + \frac{\mathcal{H}}{\tilde{r}} x' + \mathcal{K}x \, ,		x'' + \frac{\tilde\mathcal{H}}{\tilde{r}} x' + \tilde\mathcal{K}x \, ,
	</math>		</math>
	</td>		</td>
Line 756:		Line 756:
	<td align="left">		<td align="left">
	<math>		<math>
	- \frac{\mathcal{H}}{\tilde{r}} \biggl[ \frac{x_+ - x_-}{2\delta \tilde{r}} \biggr] ~-~ \mathcal{K}x_j		- \frac{\tilde\mathcal{H}}{\tilde{r}} \biggl[ \frac{x_+ - x_-}{2\delta \tilde{r}} \biggr] ~-~ \tilde\mathcal{K}x_j
	</math>		</math>
	</td>		</td>
Line 770:		Line 770:
	<td align="left">		<td align="left">
	<math>		<math>
	- \frac{\delta \tilde{r}}{2\tilde{r}} \biggl[ x_+ - x_- \biggr]\mathcal{H} ~-~ (\delta \tilde{r})^2\mathcal{K}x_j		- \frac{\delta \tilde{r}}{2\tilde{r}} \biggl[ x_+ - x_- \biggr]\tilde\mathcal{H} ~-~ (\delta \tilde{r})^2 \tilde\mathcal{K}x_j
	</math>		</math>
	</td>		</td>
Line 777:		Line 777:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>\Rightarrow ~~~ x_{j+1} \biggl[1 + \biggl( \frac{\delta \tilde{r}}{2\tilde{r}}\biggr) \mathcal{H} \biggr]</math>		<math>\Rightarrow ~~~ x_{j+1} \biggl[1 + \biggl( \frac{\delta \tilde{r}}{2\tilde{r}}\biggr) \tilde\mathcal{H} \biggr]</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 784:		Line 784:
	<td align="left">		<td align="left">
	<math>		<math>
	\biggl[ 2 - (\delta \tilde{r})^2\mathcal{K}\biggr] x_j ~		\biggl[ 2 - (\delta \tilde{r})^2 \tilde\mathcal{K}\biggr] x_j ~
	-~\biggl[ 1 - \biggl( \frac{\delta \tilde{r}}{2\tilde{r}} \biggr) \mathcal{H} \biggr]x_{j-1} \, .		-~\biggl[ 1 - \biggl( \frac{\delta \tilde{r}}{2\tilde{r}} \biggr) \tilde\mathcal{H} \biggr]x_{j-1} \, .
	</math>		</math>
	</td>		</td>
Line 793:		Line 793:
	In what follows we will also find it useful to rewrite <math>~\mathcal{K}</math> in the form,		In what follows we will also find it useful to rewrite <math>~\mathcal{K}</math> in the form,
	<div align="center">		<div align="center">
	<math>~\mathcal{K} ~\rightarrow ~\biggl(\frac{\sigma_c^2}{\gamma_\mathrm{g}}\biggr) \mathcal{K}_1 - \alpha_\mathrm{g} \mathcal{K}_2 \, .</math>		<math>\tilde\mathcal{K} ~\rightarrow ~\biggl(\frac{\sigma_c^2}{\gamma_\mathrm{g}}\biggr) \tilde\mathcal{K}_1
			- \alpha_\mathrm{g} \tilde\mathcal{K}_2 \, .</math>
	</div>		</div>

Line 801:		Line 802:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>\mathcal{H}</math>		<math>\tilde\mathcal{H}</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 813:		Line 814:
	<td align="center">      ,       </td>		<td align="center">      ,       </td>
	<td align="right">		<td align="right">
	<math>\mathcal{K}_1</math>		<math>\tilde\mathcal{K}_1</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 825:		Line 826:
	<td align="center">      and       </td>		<td align="center">      and       </td>
	<td align="right">		<td align="right">
	<math>\mathcal{K}_2</math>		<math>\tilde\mathcal{K}_2</math>
	</td>		</td>
	<td align="center">		<td align="center">