SSC/Stability/BiPolytropes/RedGiantToPN/Pt3 - Revision history

Joel2: /* Yabushita68-Motivated Analysis */

2026-01-06T18:13:02Z

Yabushita68-Motivated Analysis

@@ Line 118: / Line 118: @@
 m^2 - 3m
 + (n+1)( m - \alpha )  Q
 ~\biggr\} \eta^{-m - 2}
 \, .

Joel2: /* Yabushita68-Motivated Analysis */

2026-01-06T18:07:19Z

Yabushita68-Motivated Analysis

@@ Line 88: / Line 88: @@
 + (n+1) \biggl[  - \alpha Q\biggr]
 ~\biggr\} \eta^{-m - 2}
 </math>
    </td>

Joel2: /* Yabushita68-Motivated Analysis */

2026-01-06T17:57:06Z

Yabushita68-Motivated Analysis

← Older revision		Revision as of 13:57, 6 January 2026
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	in which case we find that,		in which case we find that,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

	<tr>		<tr>
	<td align="right">		<td align="right">
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	</td>		</td>
	</tr>		</tr>
	</~~table~~>
			<tr>
			<td align="right">

			</td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>
			\biggl\{~
			m(m+1)
			- \biggl[ 4 - (n+1) Q \biggr] \biggl[ m \biggr]
			+ (n+1) \biggl[ - \alpha Q\biggr]
			~\biggr\} \eta^{-m - 2}
			</math>
			</td>
	</tr>		</tr>
	</table>		</table>

Joel2: /* Yabushita68-Motivated Analysis */

2026-01-06T17:48:36Z

Yabushita68-Motivated Analysis

@@ Line 34: / Line 34: @@
 </tr>
 <td align="center">
-<math>C_0 = \frac{d^2x}{d\xi^2} + \biggl[ 4 - (n+1) Q \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi}
+<math>C_0 = \frac{d^2x}{d\eta^2} + \biggl[ 4 - (n+1) Q \biggr] \frac{1}{\eta} \cdot \frac{dx}{d\eta}
-+ (n+1) \biggl[ \biggl( \frac{\sigma_c^2}{6\gamma_g } \biggr)  \frac{\xi^2}{\theta}
++ (n+1) \biggl[ \biggl( \frac{\sigma_c^2}{6\gamma_g } \biggr)  \frac{\eta^2}{\theta}
-- \alpha Q\biggr]  \frac{x}{\xi^2} </math>
+- \alpha Q\biggr]  \frac{x}{\eta^2} </math>
 </td>
 </tr>
 </table>

Joel2: /* Yabushita68-Motivated Analysis */

2026-01-06T17:26:03Z

Yabushita68-Motivated Analysis

← Older revision		Revision as of 13:26, 6 January 2026
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	{{ Math/EQ_RadialPulsation02 }}		{{ Math/EQ_RadialPulsation02 }}
	</div>		</div>
			Motivated by [[SSC/Stability/Isothermal#Yabushita_(1968)\|the derivation presented by]] {{ Yabushita68full }}, let's now insert an integration constant, <math>C_0</math>, into this 2<sup>nd</sup>-order ODE to obtain what we henceforth will refer to as the,
	~~Let~~'s now insert an ~~additive~~ constant, <math>C_0</math>, to this 2<sup>nd</sup>-order ODE as ~~follows:~~
	<table border=0 cellpadding=2 align="center">		<table border=0 cellpadding=2 align="center">
	<tr>		<tr>
	<td align="~~right~~">		<td align="center" colspan="1">
	~~[[Image:LSUkey.png\|25px\|link~~=~~Appendix~~/~~EquationTemplates#Stability:_Radial_Pulsation]]~~		<font color="maroon"><b>Yabushita68-Motivated Polytropic LAWE</b></font><br />
	</td>		</td>
			</tr>
	<td align="center">		<td align="center">
	<math>C_0 = \frac{d^2x}{d\xi^2} + \biggl[ 4 - (n+1) Q \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi}		<math>C_0 = \frac{d^2x}{d\xi^2} + \biggl[ 4 - (n+1) Q \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi}
Line 38:		Line 38:
	- \alpha Q\biggr] \frac{x}{\xi^2} </math>		- \alpha Q\biggr] \frac{x}{\xi^2} </math>
	</td>		</td>
	~~</tr>~~
	~~<tr>~~
	~~<td align="center" colspan="2">~~
	where:    <math>Q(\xi) \equiv - \frac{d\ln\theta}{d\ln\xi} \, ,</math>    <math>\sigma_c^2 \equiv \frac{3\omega^2}{2\pi G\rho_c} \, ,</math>     and,     <math>\alpha \equiv \biggl(3 - \frac{4}{\gamma_\mathrm{g}}\biggr)</math>
	~~</td>~~
	</tr>		</tr>
	</table>		</table>

Joel2 at 17:06, 6 January 2026

2026-01-06T17:06:52Z

← Older revision		Revision as of 13:06, 6 January 2026
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	<div align="center">		<div align="center">
	<font color="maroon"><b>~~Yabushita68-Motivated~~ Polytropic LAWE</b></font><br />		<font color="maroon"><b>Polytropic LAWE</b></font><br />
	{{ Math/EQ_RadialPulsation02 }}		{{ Math/EQ_RadialPulsation02 }}
	</div>		</div>

			Let's now insert an additive constant, <math>C_0</math>, to this 2<sup>nd</sup>-order ODE as follows:
			<table border=0 cellpadding=2 align="center">
			<tr>
			<td align="right">
			[[Image:LSUkey.png\|25px\|link=Appendix/EquationTemplates#Stability:_Radial_Pulsation]]
			</td>
			<td align="center">
			<math>C_0 = \frac{d^2x}{d\xi^2} + \biggl[ 4 - (n+1) Q \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi}
			+ (n+1) \biggl[ \biggl( \frac{\sigma_c^2}{6\gamma_g } \biggr) \frac{\xi^2}{\theta}
			- \alpha Q\biggr] \frac{x}{\xi^2} </math>
			</td>
			</tr>
			<tr>
			<td align="center" colspan="2">
			where:    <math>Q(\xi) \equiv - \frac{d\ln\theta}{d\ln\xi} \, ,</math>    <math>\sigma_c^2 \equiv \frac{3\omega^2}{2\pi G\rho_c} \, ,</math>     and,     <math>\alpha \equiv \biggl(3 - \frac{4}{\gamma_\mathrm{g}}\biggr)</math>
			</td>
			</tr>
			</table>

	=Related Discussions=		=Related Discussions=

	{{ SGFfooter }}		{{ SGFfooter }}

Joel2: /* Main Sequence to Red Giant to Planetary Nebula (Part 3) */

2026-01-06T17:01:01Z

Main Sequence to Red Giant to Planetary Nebula (Part 3)

← Older revision		Revision as of 13:01, 6 January 2026
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	{{ Math/EQ_RadialPulsation02 }}		{{ Math/EQ_RadialPulsation02 }}
	</div>		</div>


	=Related Discussions=		=Related Discussions=

	{{ SGFfooter }}		{{ SGFfooter }}

Joel2 at 17:00, 6 January 2026

2026-01-06T17:00:25Z

← Older revision		Revision as of 13:00, 6 January 2026
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	__FORCETOC__		__FORCETOC__
	=Main Sequence to Red Giant to Planetary Nebula (Part 2)=		=Main Sequence to Red Giant to Planetary Nebula (Part 3)=
	<table border="1" align="center" width="100%" colspan="8">		<table border="1" align="center" width="100%" colspan="8">
	<tr>		<tr>

Joel2 at 17:00, 6 January 2026

2026-01-06T17:00:01Z

← Older revision		Revision as of 13:00, 6 January 2026
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	</td>		</td>
	<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt3\|~~Part III: ~~ ]]		<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt3\|Yabushita68-Motivated Analysis]]

	</td>		</td>
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	==Yabushita68-Motivated Analysis==		==Yabushita68-Motivated Analysis==
	In an [[SSC/Perturbations#2ndOrderODE\|accompanying discussion]], we derived the so-called,		In an [[SSC/Perturbations#2ndOrderODE\|accompanying discussion]], we derived the so-called,


			<div align="center">
			<font color="maroon"><b>Yabushita68-Motivated Polytropic LAWE</b></font><br />
			{{ Math/EQ_RadialPulsation02 }}
			</div>


	=Related Discussions=		=Related Discussions=

	{{ SGFfooter }}		{{ SGFfooter }}

Joel2: Created page with "FORCETOC =Main Sequence to Red Giant to Planetary Nebula (Part 2)=

Part I: Background & Objective
Part II:
2026-01-06T16:49:32Z

Created page with "FORCETOC =Main Sequence to Red Giant to Planetary Nebula (Part 2)= <table border="1" align="center" width="100%" colspan="8"> <tr> <td align="center" bgcolor="lightblue" width="25%"><br />Part I: Background & Objective </td> <td align="center" bgcolor="lightblue" width="25%"><br />Part II: </td> <td align="center" bgcolor="lightblue" wi..."

New page
FORCETOC
=Main Sequence to Red Giant to Planetary Nebula (Part 2)=
<table border="1" align="center" width="100%" colspan="8">
<tr>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN|Part I:  Background & Objective]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt2|Part II:  ]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt3|Part III:  ]]
 
</td>
<td align="center" bgcolor="lightblue"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt4|Part IV: ]]
 
</td>
</tr>
</table>

==Yabushita68-Motivated Analysis==
In an [[SSC/Perturbations#2ndOrderODE|accompanying discussion]], we derived the so-called,

=Related Discussions=

{{ SGFfooter }}