SSC/Stability/BiPolytropes/Pt3 - Revision history

Joel2: /* Foundation */

2026-01-19T17:29:42Z

Foundation

← Older revision		Revision as of 13:29, 19 January 2026
Line 122:		Line 122:

	Hence, multiplying the LAWE through by <math>~(K_c/G)\rho_c^{-4 / 5}</math> gives,		Hence, multiplying the LAWE through by <math>~(K_c/G)\rho_c^{-4 / 5}</math> gives,

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

	~~<tr>~~
	~~<td align="right">~~
	~~<math>~0</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	~~\frac{d^2x}{dr_0^2} + \biggl[\frac{4}{r_0} - \biggl(\frac{g_0 \rho_0}{P_0}\biggr) \biggr] \frac{dx}{dr_0}~~
	~~+ \biggl(\frac{\rho_0}{\gamma_\mathrm{g} P_0} \biggr)\biggl[\omega^2 + (4 - 3\gamma_\mathrm{g})\frac{g_0}{r_0} \biggr] x~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~
	~~</table>~~



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

Joel2: /* Try Again (December 2025) */

2025-12-11T18:28:45Z

Try Again (December 2025)

← Older revision		Revision as of 14:28, 11 December 2025
Line 1,718:		Line 1,718:
	From a numerical integration of the model for which <math>\mu_e/\mu_c =1</math>, we have found that,		From a numerical integration of the model for which <math>\mu_e/\mu_c =1</math>, we have found that,
	<math>~[ ~\xi_i = 1.6686460157,~x_i = +0.81437470~]</math>, which is consistent with the parabolic expression if we set, <math>x_0 = 1</math>. Also, from the parabolic expression, we deduce that <math>[d\ln x/d\ln\xi]_\mathrm{core} = - 0.455872</math>, which matches the numerically determined logarithmic slope.		<math>~[ ~\xi_i = 1.6686460157,~x_i = +0.81437470~]</math>, which is consistent with the parabolic expression if we set, <math>x_0 = 1</math>. Also, from the parabolic expression, we deduce that <math>[d\ln x/d\ln\xi]_\mathrm{core} = - 0.455872</math>, which matches the numerically determined logarithmic slope.
			<table border="1" align="center" width="80%" cellpadding="5"><tr><td align="left">
			<table border="0" align="left" width="25%" cellpadding="20">
			<tr><td align="center">
			[[File:Mod0MuRatio100.png\|250px\|Example eigenvector]]
			</td></tr>
			</table>
			Also note:   From [[SSC/Stability/BiPolytropes/Pt3#Equilibrium_Properties_of_Marginally_Unstable_Models\|Table 2, above]], we see that <math>R^_\mathrm{surf} = 2.139737</math>; and from a [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values\|separate Table 1]], <math>r^_\mathrm{core} = (3/2\pi)^{1 / 2} \xi_i = 1.153015</math>. This means that, <math>q \equiv r^_\mathrm{core}/R^_\mathrm{surf} = 0.538858</math>.

	~~<table border="1" align="center" width="80%" cellpadding="5"><tr><td align="left">~~
	Note: From [[SSC/Stability/BiPolytropes/Pt3#Equilibrium_Properties_of_Marginally_Unstable_Models\|Table 2, above]], we see that <math>R^_\mathrm{surf} = 2.139737</math>; and from a [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values\|separate Table 1]], <math>r^_\mathrm{core} = (3/2\pi)^{1 / 2} \xi_i = 1.153015</math>. This means that, <math>q \equiv r^_\mathrm{core}/R^_\mathrm{surf} = 0.538858</math>.
	</td></tr></table>		</td></tr></table>

Joel2: /* Try Again (December 2025) */

2025-12-11T18:17:14Z

Try Again (December 2025)

← Older revision		Revision as of 14:17, 11 December 2025
Line 1,723:		Line 1,723:
	</td></tr></table>		</td></tr></table>


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

	~~<tr>~~
	~~<th align="center">Figure 4</th>~~
	~~</tr>~~
	~~<tr>~~
	~~<td align="center" colspan="10">[[File:NEWCompositeDlabeled.png\|800px\|Marginally unstable models]]</td>~~
	~~</tr>~~
	~~</table>~~

	Next, following the [[#Interface\|above discussion of matching conditions at the interface]], we determined that, from the perspective of the envelope, the slope of the eigenfunction at the interface must therefore be,		Next, following the [[#Interface\|above discussion of matching conditions at the interface]], we determined that, from the perspective of the envelope, the slope of the eigenfunction at the interface must therefore be,

Joel2: /* Try Again (December 2025) */

2025-12-11T18:15:35Z

Try Again (December 2025)

@@ Line 1,718: / Line 1,718: @@
 From a  numerical integration of the model for which <math>\mu_e/\mu_c =1</math>, we have found that,
 <math>~[ ~\xi_i = 1.6686460157,~x_i = +0.81437470~]</math>, which is consistent with the parabolic expression if we set, <math>x_0 = 1</math>. Also, from the parabolic expression, we deduce that <math>[d\ln x/d\ln\xi]_\mathrm{core} = - 0.455872</math>, which matches the numerically determined logarithmic slope.
 Next, following the [[#Interface|above discussion of matching conditions at the interface]], we determined that, from the perspective of the envelope, the slope of the eigenfunction at the interface must therefore be,
@@ Line 1,788: / Line 1,803: @@
 </tr>
 </table>
 ===Other Modes===

Joel2: /* Try Again (December 2025) */

2025-12-11T18:06:37Z

Try Again (December 2025)

@@ Line 1,788: / Line 1,788: @@
 </tr>
 </table>
 ===Other Modes===

Joel2: /* Try Again (December 2025) */

2025-12-10T21:50:02Z

Try Again (December 2025)

← Older revision		Revision as of 17:50, 10 December 2025
Line 1,784:		Line 1,784:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>-\biggl[ \frac{18}{(15 - \xi^2)} \biggr] = - 1.473523</math>		<math>-\biggl[ \frac{18}{(15 - \xi^2)} \biggr] = - 1.473523</math>    <font color="red"><b>Yes!</b></font>
	</td>		</td>
	</tr>		</tr>

Joel2: /* Try Again (December 2025) */

2025-12-10T21:47:45Z

Try Again (December 2025)

@@ Line 1,718: / Line 1,718: @@
 From a  numerical integration of the model for which <math>\mu_e/\mu_c =1</math>, we have found that,
 <math>~[ ~\xi_i = 1.6686460157,~x_i = +0.81437470~]</math>, which is consistent with the parabolic expression if we set, <math>x_0 = 1</math>. Also, from the parabolic expression, we deduce that <math>[d\ln x/d\ln\xi]_\mathrm{core} = - 0.455872</math>, which matches the numerically determined logarithmic slope.
 ===Other Modes===

Joel2: /* Try Again (December 2025) */

2025-12-10T19:00:32Z

Try Again (December 2025)

← Older revision		Revision as of 15:00, 10 December 2025
Line 1,717:		Line 1,717:

	From a numerical integration of the model for which <math>\mu_e/\mu_c =1</math>, we have found that,		From a numerical integration of the model for which <math>\mu_e/\mu_c =1</math>, we have found that,
	<math>~[ ~\xi_i = 1.6686460157,~x_i = +0.81437470~]</math>, which is consistent with the parabolic expression if we set, <math>x_0 = 1</math>. Also, from the parabolic expression, we deduce that <math>d\ln x/d\ln\xi = - 0.455872</math>, which matches the numerically determined logarithmic slope.		<math>~[ ~\xi_i = 1.6686460157,~x_i = +0.81437470~]</math>, which is consistent with the parabolic expression if we set, <math>x_0 = 1</math>. Also, from the parabolic expression, we deduce that <math>[d\ln x/d\ln\xi]_\mathrm{core} = - 0.455872</math>, which matches the numerically determined logarithmic slope.

	===Other Modes===		===Other Modes===

Joel2: /* Try Again (December 2025) */

2025-12-10T18:57:30Z

Try Again (December 2025)

← Older revision		Revision as of 14:57, 10 December 2025
Line 1,717:		Line 1,717:

	From a numerical integration of the model for which <math>\mu_e/\mu_c =1</math>, we have found that,		From a numerical integration of the model for which <math>\mu_e/\mu_c =1</math>, we have found that,
	<math>~[ ~\xi_i = 1.6686460157,~x_i = +0.81437470~]</math>, which is consistent with the parabolic expression if we set, <math>x_0 = 1</math>.		<math>~[ ~\xi_i = 1.6686460157,~x_i = +0.81437470~]</math>, which is consistent with the parabolic expression if we set, <math>x_0 = 1</math>. Also, from the parabolic expression, we deduce that <math>d\ln x/d\ln\xi = - 0.455872</math>, which matches the numerically determined logarithmic slope.

	===Other Modes===		===Other Modes===

Joel2: /* Try Again (December 2025) */

2025-12-10T18:37:22Z

Try Again (December 2025)

← Older revision		Revision as of 14:37, 10 December 2025
Line 1,717:		Line 1,717:

	From a numerical integration of the model for which <math>\mu_e/\mu_c =1</math>, we have found that,		From a numerical integration of the model for which <math>\mu_e/\mu_c =1</math>, we have found that,
	<math>~[ ~\xi_i = 1.6686460157 ~]</math>		<math>~[ ~\xi_i = 1.6686460157,~x_i = +0.81437470~]</math>, which is consistent with the parabolic expression if we set, <math>x_0 = 1</math>.

	===Other Modes===		===Other Modes===