User:Tohline/SSC/FreeEnergy/Equilibrium Sequence Instabilities - Revision history

Tohline: /* Analysis by Yabushita (1975) */

2017-03-02T04:05:56Z

Analysis by Yabushita (1975)

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	* Understands that a strong analogy can be drawn between the behavior of pressure-truncated and bipolytropic sequences —  "<font color="darkgreen">This will enable one to see the effect of an envelope on the stability of isothermal gas spheres ([http://adsabs.harvard.edu/abs/1968MNRAS.140..109Y Yabushita 1968]) which otherwise will extend to infinity.</font>"		* Understands that a strong analogy can be drawn between the behavior of pressure-truncated and bipolytropic sequences —  "<font color="darkgreen">This will enable one to see the effect of an envelope on the stability of isothermal gas spheres ([http://adsabs.harvard.edu/abs/1968MNRAS.140..109Y Yabushita 1968]) which otherwise will extend to infinity.</font>"

			It might be worth looking at the following publications:
			* [http://adsabs.harvard.edu/abs/1967AnAp...30..975G M. Gabriel &amp P. Ledoux (1967)]: Yabushita says,
			* [ S. Chandrasekhar (1972) ''General Relativity'']: Yabushita says, <font color="darkgreen">It is Chandrasekhar (1972) who first showed the analogy between neutron star models and a gaseous sphere with an isothermal core and an envelope with constant density.</font>

	==Close Binary Stars==		==Close Binary Stars==

Tohline: /* Analysis by Yabushita (1975) */

2017-03-02T03:26:47Z

Analysis by Yabushita (1975)

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	Then we see, from the article's ~~opening~~ introductory ~~paragraph — immediately following~~ &~~#8212~~; ~~that Yabushita understands~~ that it is useful to analyze stability from both a free-energy (virial theorem) standpoint and a linear stability (radial oscillations) analysis.		Then we see, from the article's introductory paragraphs that Yabushita …
			* Understands that it is useful to analyze stability from both a free-energy (virial theorem) standpoint and a linear stability (radial oscillations) analysis — see the following reproduction of the first introductory paragraph.
	<div align="center">		<div align="center">
	[[File:Yabushita1975Intro1.png\|700px\|Introductory paragraph from Yabushita (1975)]]		[[File:Yabushita1975Intro1.png\|700px\|Introductory paragraph from Yabushita (1975)]]
	</div>		</div>
			* Understands that a strong analogy can be drawn between the behavior of pressure-truncated and bipolytropic sequences —  "<font color="darkgreen">This will enable one to see the effect of an envelope on the stability of isothermal gas spheres ([http://adsabs.harvard.edu/abs/1968MNRAS.140..109Y Yabushita 1968]) which otherwise will extend to infinity.</font>"

	==Close Binary Stars==		==Close Binary Stars==

Tohline: /* Analysis by Yabushita (1975) */

2017-03-02T03:11:59Z

Analysis by Yabushita (1975)

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	==Analysis by Yabushita (1975)==		==Analysis by Yabushita (1975)==
	It appears as though the first analysis that asked the same set of questions we have asked regarding the type of stability that is associated with turning points is the one published by [http://adsabs.harvard.edu/abs/1975MNRAS.172..441Y S. Yabushita (1975)]		It appears as though the first analysis that asked the same set of questions we have asked regarding the type of stability that is associated with turning points is the one published by [http://adsabs.harvard.edu/abs/1975MNRAS.172..441Y S. Yabushita (1975)] and titled, ''On the Structure and Stability of a Polytrope with an Isothermal Core''.

			From the article's summary (abstract) alone — immediately following — we note the following key points;
			* Yabushita (1975) builds an equilibrium sequence for <math>~(n_c, n_e) = (\infty, \tfrac{3}{2})</math> bipolytropes and identifies a maximum-mass turning point along the sequence.
			* A solution of the relevant LAWE shows that the "dynamical instability sets in precisely at the mass peak," if the adopted adiabatic index is the same as the <math>~\gamma = d\ln\rho/d\ln r</math> calculated by the equation(s) of state governing the unperturbed state.

			<div align="center">
			[[File:Yabushita1975Summary.png\|Summary (Abstract) from Yabushita (1975)]]
			</div>

			Then we see, from the article's opening introductory paragraph — immediately following — that Yabushita understands that it is useful to analyze stability from both a free-energy (virial theorem) standpoint and a linear stability (radial oscillations) analysis.
			<div align="center">
			[[File:Yabushita1975Intro1.png\|700px\|Introductory paragraph from Yabushita (1975)]]
			</div>

	==Close Binary Stars==		==Close Binary Stars==

Tohline: /* Instabilities Associated with Equilibrium Sequence Turning Points */ Begin subsection to outline Yabushita's (1975) work

2017-03-02T02:36:25Z

Instabilities Associated with Equilibrium Sequence Turning Points: Begin subsection to outline Yabushita's (1975) work

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			==Analysis by Yabushita (1975)==
			It appears as though the first analysis that asked the same set of questions we have asked regarding the type of stability that is associated with turning points is the one published by [http://adsabs.harvard.edu/abs/1975MNRAS.172..441Y S. Yabushita (1975)]

	==Close Binary Stars==		==Close Binary Stars==

Tohline: /* Statement of the Problem */ Begin succinct "statement of the problem"

2016-12-04T21:01:39Z

Statement of the Problem: Begin succinct "statement of the problem"

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	==Statement of the Problem==		==Statement of the Problem==
			We can construct detailed force-balance (DFB) models of spherically symmetric, ''isolated'' polytropes over the entire range of polytropic indexes: <math>~\infty \ge n \ge 0</math>. Given that, <math>~\gamma = (n+1)/n</math>, this corresponds to the range of adiabatic exponents, <math>~1 \le \gamma \le \infty</math>. These configurations are ''known'' to be dynamically stable if <math>~n < 3</math>; that is, if <math>~\gamma > 4/3</math>. How do we know the answer to this stability question?
			* An ''analytic'' free-energy analysis tells us that <math>~\partial^2 \mathfrak{G}/\partial R^2 > 0</math> (i.e., the system is dynamically unstable) for models having polytropic indexes, <math>~n > 3</math>;
			* Linear stability analyses, performed numerically, tell us that the square of the system's fundamental-mode eigenfrequency is negative for models having <math>~\gamma < 4/3</math>.


	{{LSU_WorkInProgress}}		{{LSU_WorkInProgress}}

	==Example Equilibrium Sequences==		==Example Equilibrium Sequences==

Tohline: Insert new subsection "Statement of the Problem"; and insert LSU_WorkInProgress template

2016-12-04T20:35:00Z

Insert new subsection "Statement of the Problem"; and insert LSU_WorkInProgress template

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	{{LSU_HBook_header}}		{{LSU_HBook_header}}
			==Statement of the Problem==


			{{LSU_WorkInProgress}}
	==Example Equilibrium Sequences==		==Example Equilibrium Sequences==

Tohline: /* Instabilities Associated with Equilibrium Sequence Turning Points */

2016-10-20T23:35:04Z

Instabilities Associated with Equilibrium Sequence Turning Points

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	=Instabilities Associated with Equilibrium Sequence Turning Points=		=Instabilities Associated with Equilibrium Sequence Turning Points=

			Detailed derivations and discussions that underpin this chapter can be found in [[User:Tohline/Appendix/Ramblings/Turning_Points#Turning_Points\|a chapter of our ''Ramblings Appendix'']].

	{{LSU_HBook_header}}		{{LSU_HBook_header}}

	==~~Introduction~~==		==Example Equilibrium Sequences==

			Consider a binary star system (or a planet/satellite system) in which both components are uniform in density, but in which one of the components — the ''secondary'' of mass <math>~M^'</math> — is much more compact <math>~(R^' \ll R)</math> and perhaps, also, much more massive than the other component — the ''primary'' of mass <math>~M</math>. The tidal stresses exerted by the secondary on the primary will deform the primary into a nonspherical shape that, to a reasonably high degree of satisfaction, can be described as an ellipsoid having semi-axes <math>~a_1 > a_2 > a_3</math> and a volume-preserving "radius," <math>~R \equiv (a_1 a_2 a_3)^{1 / 3} = 3M/(4\pi)</math>.



			==Equation of State Considerations==

	Spherically symmetric, self-gravitating, equilibrium configurations can be constructed from gases exhibiting a wide variety of degrees of compressibility. When examining how the internal structure of such configurations varies with compressibility, or when examining the relative stability of such structures, it can be instructive to construct models using a [[User:Tohline/SR#Time-Independent_Problems\|polytropic equation of state]],		Spherically symmetric, self-gravitating, equilibrium configurations can be constructed from gases exhibiting a wide variety of degrees of compressibility. When examining how the internal structure of such configurations varies with compressibility, or when examining the relative stability of such structures, it can be instructive to construct models using a [[User:Tohline/SR#Time-Independent_Problems\|polytropic equation of state]],

Tohline: /* Incompressible Roche Ellipsoids (λ = 0) */ Finished table 3

2016-10-19T21:07:42Z

Incompressible Roche Ellipsoids (λ = 0): Finished table 3

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	<td align="center"> 2.4178 </td>		<td align="center"> 2.4178 </td>
	<td align="center"> 1.8399 </td>		<td align="center"> 1.8399 </td>
			</tr>
			<tr>
			<td align="center"> 72° </td>
			<td align="center"> 0.32254 </td>
			<td align="center"> 0.30902 </td>
			<td align="center"> 0.074648 </td>
			<td align="center"> 2.6140 </td>
			<td align="center"> 0.4636 </td>
			<td align="center"> 2.5679 </td>
			<td align="center"> 1.8598 </td>
			</tr>
			<tr>
			<td align="center"> 75° </td>
			<td align="center"> 0.26827 </td>
			<td align="center"> 0.25882 </td>
			<td align="center"> 0.064426 </td>
			<td align="center"> 2.7455 </td>
			<td align="center"> 0.4110 </td>
			<td align="center"> 3.1728 </td>
			<td align="center"> 1.9359 </td>
			</tr>
			<tr>
			<td align="center"> 79° </td>
			<td align="center"> 0.19569 </td>
			<td align="center"> 0.19081 </td>
			<td align="center"> 0.047111 </td>
			<td align="center"> 3.0475 </td>
			<td align="center"> 0.3342 </td>
			<td align="center"> 4.6471 </td>
			<td align="center"> 2.0951 </td>
	</tr>		</tr>
	<tr>		<tr>

Tohline: /* Incompressible Roche Ellipsoids (λ = 0) */ Extend Table 3

2016-10-19T20:49:58Z

Incompressible Roche Ellipsoids (λ = 0): Extend Table 3

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	<td align="center"> 1.4415 </td>		<td align="center"> 1.4415 </td>
	<td align="center"> 1.7206 </td>		<td align="center"> 1.7206 </td>
			</tr>
			<tr>
			<td align="center"> 60° </td>
			<td align="center"> 0.53013 </td>
			<td align="center"> 0.50000 </td>
			<td align="center"> 0.089946 </td>
			<td align="center"> 2.4565 </td>
			<td align="center"> 0.6424 </td>
			<td align="center"> 1.5523 </td>
			<td align="center"> 1.7286 </td>
			</tr>
			<tr>
			<td align="center"> 61° </td>
			<td align="center"> 0.51373 </td>
			<td align="center"> 0.48481 </td>
			<td align="center"> 0.090068 </td>
			<td align="center"> 2.4554 </td>
			<td align="center"> 0.6292 </td>
			<td align="center"> 1.5964 </td>
			<td align="center"> 1.7329 </td>
			</tr>
			<tr>
			<td align="center"> 62° </td>
			<td align="center"> 0.49714 </td>
			<td align="center"> 0.46947 </td>
			<td align="center"> 0.089977 </td>
			<td align="center"> 2.4562 </td>
			<td align="center"> 0.6157 </td>
			<td align="center"> 1.6450 </td>
			<td align="center"> 1.7382 </td>
			</tr>
			<tr>
			<td align="center"> 63° </td>
			<td align="center"> 0.48040 </td>
			<td align="center"> 0.45399 </td>
			<td align="center"> 0.089689 </td>
			<td align="center"> 2.4589 </td>
			<td align="center"> 0.6019 </td>
			<td align="center"> 1.6984 </td>
			<td align="center"> 1.7443 </td>
			</tr>
			<tr>
			<td align="center"> 66° </td>
			<td align="center"> 0.42898 </td>
			<td align="center"> 0.40674 </td>
			<td align="center"> 0.087201 </td>
			<td align="center"> 2.48202 </td>
			<td align="center"> 0.5588 </td>
			<td align="center"> 1.8959 </td>
			<td align="center"> 1.7694 </td>
			</tr>
			<tr>
			<td align="center"> 71° </td>
			<td align="center"> 0.34052 </td>
			<td align="center"> 0.32557 </td>
			<td align="center"> 0.077474 </td>
			<td align="center"> 2.5818 </td>
			<td align="center"> 0.4804 </td>
			<td align="center"> 2.4178 </td>
			<td align="center"> 1.8399 </td>
	</tr>		</tr>
	<tr>		<tr>

Tohline: /* Incompressible Roche Ellipsoids (λ = 0) */ Begin filling out Table 3

2016-10-19T20:32:15Z

Incompressible Roche Ellipsoids (λ = 0): Begin filling out Table 3

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			<table border="1" align="center" cellpadding="5">

			<tr>
			<th align="center" colspan="8"><font size="+1"><b>Table 3:</b></font>  Incompressible <math>~(n=0)</math> Roche Ellipsoids with <math>~\lambda = p = 0</math></th>
			</tr>

			<tr>
			<th align="center" colspan="4">
			Extracted from Table 1 of [http://adsabs.harvard.edu/abs/1963ApJ…138.1182C Chandrasekhar (1963)]<br />
			same as [<b>[[User:Tohline/Appendix/References#EFE\|<font color="red">EFE</font>]]</b>] Table XVI</th>
			<th align="center" colspan="5">EFE Check</th>
			</tr>
			<tr>
			<td align="center"> (1) </td>
			<td align="center"> (2) </td>
			<td align="center"> (3) </td>
			<td align="center"> (4) </td>
			<td align="center"> (5) </td>
			<td align="center"> (6) </td>
			<td align="center"> (7) </td>
			<td align="center"> (8) </td>
			</tr>
			<tr>
			<td align="center"><math>~\cos^{-1}(a_3/a_1)</math></td>
			<td align="center"><math>~a_2/a_1</math></td>
			<td align="center"><math>~a_3/a_1</math></td>
			<td align="center"><math>~\Omega^2</math></td>
			<td align="center"><math>~r/R</math></td>
			<td align="center"><math>~R/a_1</math></td>
			<td align="center"><math>~\mathfrak{J}</math></td>
			<td align="center"><math>~j/(GM^' R)^{1/2}</math></td>
			</tr>
			<tr>
			<td align="center"> 24° </td>
			<td align="center"> 0.93188 </td>
			<td align="center"> 0.91355 </td>
			<td align="center"> 0.022624 </td>
			<td align="center"> 3.8916 </td>
			<td align="center"> 0.9478 </td>
			<td align="center"> 1.0400 </td>
			<td align="center"> 2.0269 </td>
			</tr>
			<tr>
			<td align="center"> 36° </td>
			<td align="center"> 0.84112 </td>
			<td align="center"> 0.80902 </td>
			<td align="center"> 0.047871 </td>
			<td align="center"> 3.0312 </td>
			<td align="center"> 0.8796 </td>
			<td align="center"> 1.1035 </td>
			<td align="center"> 1.8247 </td>
			</tr>
			<tr>
			<td align="center"> 48° </td>
			<td align="center"> 0.70687 </td>
			<td align="center"> 0.66913 </td>
			<td align="center"> 0.074799 </td>
			<td align="center"> 2.6122 </td>
			<td align="center"> 0.7791 </td>
			<td align="center"> 1.2352 </td>
			<td align="center"> 1.7333 </td>
			</tr>
			<tr>
			<td align="center"> 57° </td>
			<td align="center"> 0.57787 </td>
			<td align="center"> 0.54464 </td>
			<td align="center"> 0.088267 </td>
			<td align="center"> 2.4720 </td>
			<td align="center"> 0.68022 </td>
			<td align="center"> 1.4415 </td>
			<td align="center"> 1.7206 </td>
			</tr>
			<tr>
			<td align="center" colspan="8">
			[[File:P0Diagram.png\|600px\|Roche, Darwin, and Chandrasekhar p=0]]
			</td>
			</tr>
			</table>

	===References===		===References===