SSC/Stability/BiPolytropes/RedGiantToPN

Joel2 — Sat, 11 Apr 2026 00:41:54 GMT

Temporary Excel Interpolations

SSC/Structure/Polytropes/VirialSummary

Joel2 — Sun, 05 Apr 2026 00:50:28 GMT

Fix External Pressure While Varying Mass

← Older revision		Revision as of 20:50, 4 April 2026
Line 564:		Line 564:
	</table>		</table>

	(As has been demonstrated in an [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2#modNormalizations\|accompanying discussion]], we could alternatively have obtained this same algebraic relation by shifting to a different pair of mass- and radius-normalizations — namely, <math>M_\mathrm{mod}</math> and <math>R_\mathrm{mod}</math> — instead of choosing these specific values of <math>\mathcal{A}</math> and <math>\mathcal{B}.</math>)		<span id="StahlerSchematic">(As has been demonstrated</span> in an [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2#modNormalizations\|accompanying discussion]], we could alternatively have obtained this same algebraic relation by shifting to a different pair of mass- and radius-normalizations — namely, <math>M_\mathrm{mod}</math> and <math>R_\mathrm{mod}</math> — instead of choosing these specific values of <math>\mathcal{A}</math> and <math>\mathcal{B}.</math>)

	<table border="1" align="center" cellpadding="8">		<table border="1" align="center" cellpadding="8">

SSC/Stability/BiPolytropes/RedGiantToPN

Joel2 — Sat, 04 Apr 2026 18:11:18 GMT

Preface

@@ Line 19: / Line 19: @@
 ==Preface==
 <table border="0" cellpadding="8" align="right">
@@ Line 31: / Line 44: @@
 </tr>
 </table>
-As has been detailed in an [[SSC/Stability/BiPolytropes#Overview|accompanying chapter]], we have [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|successfully analyzed the relative stability of pressure-truncated polytopes]].  The curves shown here on the right in Figure 1 graphically present the mass-radius relationship for pressure-truncated model sequences having a variety of polytropic indexes, as labeled, over the range <math>1 \le n \le 6</math>.  ([[SSC/Stability/InstabilityOnsetOverview#Turning_Points_along_Sequences_of_Pressure-Truncated_Polytropes|Another version of this figure]] includes the isothermal sequence.)  On each sequence for which <math>n \ge 3</math>, the green filled circle identifies the model with the largest mass.  We have shown ''analytically'' that the oscillation frequency of the fundamental-mode of radial oscillation is precisely zero for each one of these maximum-mass models.  As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.
+As has been detailed in an [[SSC/Stability/BiPolytropes#Overview|accompanying chapter]], we have [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|successfully analyzed the relative stability of pressure-truncated polytopes]].  The curves shown here on the right in Figure 1 graphically present the mass-radius relationship for pressure-truncated model sequences having a variety of polytropic indexes, as labeled, over the range <math>1 \le n \le 6</math>.  ([[SSC/Stability/InstabilityOnsetOverview#Turning_Points_along_Sequences_of_Pressure-Truncated_Polytropes|Another version of this figure]] includes the isothermal sequence.)
 <font color="red">'''Key Realization:'''</font>  ''Along sequences of pressure-truncated polytropes, the maximum-mass models identify precisely where the onset of dynamical instability occurs.''

JETohlineWiki - Recent changes [en]

SSC/Stability/BiPolytropes/RedGiantToPN

SSC/Structure/Polytropes/VirialSummary

SSC/Stability/BiPolytropes/RedGiantToPN