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SSC/Stability/BiPolytropes/RedGiantToPN/TryAgain

Joel2 — Sat, 31 Oct 2026 00:33:52 GMT

Review

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	=~~BiPolytrope~~ with n<sub>c</sub> = 5 and n<sub>e</sub> = 1 ~~(Pt 1)~~=		=BiPolytropes with n<sub>c</sub> = 5 and n<sub>e</sub> = 1=

			==Review==

			(October 30, 2026): A significant portion of the information that we have gathered regarding the stability of (n<sub>c</sub>, n<sub>e</sub>) = (5, 1) bipolytropes is [[SSC/Stability/BiPolytropes/51Models\|presented on the following page]] ...

			The most important result is associated with the case of <math>\mu_e/\mu_c = 1</math>. With our first numerical integration of the LAWE, we found that the interface location for which <math>\sigma_c^2 = 0</math> is <math>\xi = 1.6686460157</math>; a separate numerical integration gave the result: <font color="darkgreen"><math>\xi_i = 1.6639103365</math></font>.

			Separately, we asked where along the equilibrium sequence we will find that <math>d M_\mathrm{tot}/d\xi = 0</math>. For the case of unity molecular-weight ratio, we found analytically that <math>\xi_i = 1.6684629814</math>. These are virtually identical coordinate positions; it seems likely that they should be identically the same.

			=See Also=

	~~=Related Discussions=~~
	* [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres\|Polytropes emdeded in an external medium]]
	* [[SSC/Structure/BiPolytropes#BiPolytropes\|Constructing BiPolytropes]]
	* [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|Bonnor-Ebert spheres]]
	** [http://en.wikipedia.org/wiki/Bonnor-Ebert_mass Bonnor-Ebert Mass] according to Wikipedia
	** <font color="red">Link has disappeared:</font> [http://www.astro.umd.edu/~cychen/MATLAB/ASTR320/matlabFrom320spring2011/Bonnor-EbertSphere/html/BonnorEbert.html A MATLAB script to determine the Bonnor-Ebert Mass coefficient] developed by [http://www.astro.umd.edu/people/cychen.html Che-Yu Chen] as a graduate student in the University of Maryland Department of Astronomy
	* [[SSC/Structure/LimitingMasses#Sch.C3.B6nberg-Chandrasekhar_Mass\|Schönberg-Chandrasekhar limiting mass]]
	* [[SSC/Structure/LimitingMasses#Relationship_Between_the_Bonnor-Ebert_and_Sch.C3.B6nberg-Chandrasekhar_Critical_Masses\|Relationship between Bonnor-Ebert and Schönberg-Chandrasekhar limiting masses]]

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SSC/Stability/BiPolytropes/RedGiantToPN/TryAgain

Joel2 — Fri, 30 Oct 2026 23:31:38 GMT

Created page with "__FORCETOC__ =BiPolytrope with nc = 5 and ne = 1 (Pt 1)= =Related Discussions= * Polytropes emdeded in an external medium * Constructing BiPolytropes * Bonnor-Ebert spheres ** [http://en.wikipedia.org/wiki/Bonnor-Ebert_mass Bonnor-Ebert Mass] according to Wikipedia ** <font c..."

New page

__FORCETOC__
=BiPolytrope with nc = 5 and ne = 1 (Pt 1)=

=Related Discussions=
* [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres|Polytropes emdeded in an external medium]]
* [[SSC/Structure/BiPolytropes#BiPolytropes|Constructing BiPolytropes]]
* [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere|Bonnor-Ebert spheres]]
** [http://en.wikipedia.org/wiki/Bonnor-Ebert_mass Bonnor-Ebert Mass] according to Wikipedia
** Link has disappeared: [http://www.astro.umd.edu/~cychen/MATLAB/ASTR320/matlabFrom320spring2011/Bonnor-EbertSphere/html/BonnorEbert.html A MATLAB script to determine the Bonnor-Ebert Mass coefficient] developed by [http://www.astro.umd.edu/people/cychen.html Che-Yu Chen] as a graduate student in the University of Maryland Department of Astronomy
* [[SSC/Structure/LimitingMasses#Sch.C3.B6nberg-Chandrasekhar_Mass|Schönberg-Chandrasekhar limiting mass]]
* [[SSC/Structure/LimitingMasses#Relationship_Between_the_Bonnor-Ebert_and_Sch.C3.B6nberg-Chandrasekhar_Critical_Masses|Relationship between Bonnor-Ebert and Schönberg-Chandrasekhar limiting masses]]

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Joel2 — Fri, 30 Oct 2026 23:23:55 GMT

Stability Analysis

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	! style="height: 150px; width: 150px; " \|[[SSC/Stability/BiPolytropes/RedGiantToPN\|Eureka!<br>(October 2025)]]		! style="height: 150px; width: 150px; border-right:2px solid black; " \|[[SSC/Stability/BiPolytropes/RedGiantToPN\|Eureka!<br>(October 2025)]]
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			! style="height: 150px; width: 150px; \|[[SSC/Stability/BiPolytropes/RedGiantToPN/TryAgain\|Try Again<br>(November 2026)]]<br /><br />(n<sub>c</sub>, n<sub>e</sub>) = (5, 1)
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Appendix/Ramblings/PowerSeriesExpressions

Joel2 — Thu, 15 Oct 2026 22:44:11 GMT

Power-Series Derivation

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Appendix/Ramblings/PowerSeriesExpressions

70.180.56.157 — Tue, 13 Oct 2026 22:58:59 GMT

Displacement Function for Isothermal LAWE

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 + \xi^4\biggl[-\frac{1}{2^2 \cdot 3^3\cdot 5} + \frac{1}{2^2\cdot 3^2\cdot 5}  -\frac{1}{3^2 \cdot 5\cdot 7} \biggr]
 + \xi^6 \biggl[\frac{1}{2^2\cdot 3^5\cdot 5\cdot 7} + \frac{1}{2^3 \cdot 3^3\cdot 5^2}  + \frac{61 }{2^3 \cdot 3^5 \cdot 5 \cdot 7}\biggr]
 </math>
    </td>

Appendix/Ramblings/PowerSeriesExpressions

Joel2 — Tue, 13 Oct 2026 22:50:12 GMT

Displacement Function for Isothermal LAWE

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