SSC/StabilityConjecture/Bipolytrope51 - Revision history

Jet53man: /* Trial Core Eigenvector */

2021-08-14T23:18:14Z

Trial Core Eigenvector

← Older revision		Revision as of 19:18, 14 August 2021
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	</tr>		</tr>
	</table>		</table>
	where, <math>\ell_i \equiv \xi_i/\sqrt{3}</math>.		where, <math>\ell_i \equiv \xi_i/\sqrt{3}</math>. For example, if <math>\xi_i = 9.014959755</math>, then <math>\ell_i = 5.204789446</math> and <math>[m^*_\mathrm{core}]^{2 / 3} = 2.893199794</math>.

	=See Also=		=See Also=

2021-08-14T23:08:26Z

Trial Core Eigenvector

@@ Line 1,962: / Line 1,962: @@
 </table>
 Now, from [[#Expressions_for_q_and_.CE.BD|above]], we know that,
 =See Also=

2021-08-14T22:55:30Z

Trial Core Eigenvector

← Older revision		Revision as of 18:55, 14 August 2021
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	</tr>		</tr>
	</table>		</table>
			Now, from [[#Expressions_for_q_and_.CE.BD\|above]], we know that,

	=See Also=		=See Also=

2021-08-14T22:47:07Z

Trial Core Eigenvector

@@ Line 1,868: / Line 1,868: @@
 </table>
 In order for this RHS to be zero, we therefore need, <math>B = -A/15</math>.  Okay.  This matches our earlier derivation.  In order to compare this trial eigenfunction with the "core" portion of the eigenfunction that we obtained empirically via the B-KB74 conjecture, we need to know how <math>x_\mathrm{trial}</math> varies with <math>M_r^*</math>.
 =See Also=

2021-08-14T22:16:15Z

Trial Core Eigenvector

← Older revision		Revision as of 18:16, 14 August 2021
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	</tr>		</tr>
	</table>		</table>
	In order or this RHS to be zero, we therefore need, <math>B = -A/15</math>.		In order for this RHS to be zero, we therefore need, <math>B = -A/15</math>. Okay. This matches our earlier derivation. In order to compare this trial eigenfunction with the "core" portion of the eigenfunction that we obtained empirically via the B-KB74 conjecture, we need to know how <math>x_\mathrm{trial}</math> varies with <math>M_r^*</math>.

	=See Also=		=See Also=

2021-08-14T21:55:03Z

Trial Core Eigenvector

@@ Line 1,826: / Line 1,826: @@
 (2B)\biggl[ 2 \xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \biggr]
++
 \frac{2}{3} \biggl[ \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \biggr]
-\biggr\} \biggl[A + B\xi^2 \biggr]
+\biggl[A + B\xi^2 \biggr]
 </math>
    </td>
 </tr>
 </table>
 =See Also=

2021-08-14T21:44:15Z

LAWE for (nc, ne) = (5, 1) Models

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 ==Core==
 From the equilibrium structure of n = 5 polytropes, we know that throughout the core,
 <table border="0" cellpadding="5" align="center">
@@ Line 1,683: / Line 1,685: @@
    </td>
    <td align="left">
-<math>4 -
+<math>
-\xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \, ,
+- 2 \xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \, ,
 </math>
    </td>
@@ Line 1,707: / Line 1,709: @@
    </td>
    <td align="left">
-<math>\frac{4\pi}{3}
+<math>
-\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \, ;
+\frac{4\pi}{3} \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \, ;
 </math>
    </td>
@@ Line 1,738: / Line 1,740: @@
 <math>
 \frac{1}{\xi}\biggl(\frac{2\pi}{3 }\biggr)\frac{dx}{d\xi} \, .
 </math>
    </td>

2021-08-14T20:53:28Z

LAWE for (nc, ne) = (5, 1) Models

← Older revision		Revision as of 16:53, 14 August 2021
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	<td align="center">      and       </td>		<td align="center">      and       </td>
	<td align="right">		<td align="right">
	<math>x'~~' \equiv~~ \frac{~~d^2x~~}{d(r^*)^2}</math>		<math>\frac{x'}{r^} = \frac{1}{r^}\frac{dx}{dr^*}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	<td align="left">		<td align="left">
	<math>		<math>
	\biggl(\frac{2\pi}{3}\biggr)\frac{~~d^2x~~}{d\xi^2} \, ,		\frac{1}{\xi}\biggl(\frac{2\pi}{3 }\biggr)\frac{dx}{d\xi} \, .
	</math>		</math>
	</td>		</td>

2021-08-14T20:46:17Z

LAWE for (nc, ne) = (5, 1) Models

@@ Line 1,707: / Line 1,707: @@
    </td>
    <td align="left">
-<math>\frac{2\pi}{3\xi^2}\cdot
+<math>\frac{4\pi}{3}
-\xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \, .
+\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \, ;
 </math>
    </td>

2021-08-14T20:22:20Z

LAWE for (nc, ne) = (5, 1) Models

@@ Line 1,623: / Line 1,623: @@
 <math>
 \biggl(\frac{\rho^*}{ P^* } \biggr)\frac{M_r^*}{(r^*)^3} \, .
 </math>
    </td>