User:Tohline/SSC/FreeEnergy/PowerPoint - Revision history

Tohline: /* Case M Equilibrium Conditions */ Add link back to discussion of Lane-Emden equation

2016-09-07T15:42:43Z

Case M Equilibrium Conditions: Add link back to discussion of Lane-Emden equation

← Older revision		Revision as of 15:42, 7 September 2016
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	</td>		</td>
	</tr>		</tr>
			</table>
			</div>
			which matches the expression derived in an [[User:Tohline/SSC/Structure/Polytropes#Lane-Emden_Equation\|ASIDE box found with our introduction of the Lane-Emden equation]], and
			<div align="center">
			<table border="0" cellpadding="3">

	<tr>		<tr>

Tohline: /* Five-One Bipolytropes */ Begin inquiring whether the system becomes dynamically unstable before reaching the nu_max turning point

2016-08-31T00:58:26Z

Five-One Bipolytropes: Begin inquiring whether the system becomes dynamically unstable before reaching the nu_max turning point

← Older revision		Revision as of 00:58, 31 August 2016
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	===Five-One Bipolytropes===		===Five-One Bipolytropes===

			====Basic Properties====
	For bipolytropes, in general, let <math>~n =n_c</math> and <math>~j = n_e</math>. The statement of virial equilibrium is, then,		For bipolytropes, in general, let <math>~n =n_c</math> and <math>~j = n_e</math>. The statement of virial equilibrium is, then,

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	[[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#Stability_Condition\|Table 1 of an accompanying chapter]] — and the red-dashed curve in the figure adjacent to that table — identifies some key properties of the model that marks the transition from stable to unstable configurations along equilibrium sequences that have various values of the mean-molecular weight ratio, <math>~\mu_e/\mu_c</math>.		[[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#Stability_Condition\|Table 1 of an accompanying chapter]] — and the red-dashed curve in the figure adjacent to that table — identifies some key properties of the model that marks the transition from stable to unstable configurations along equilibrium sequences that have various values of the mean-molecular weight ratio, <math>~\mu_e/\mu_c</math>.

			====Coincidence Between Points of Secular and Dynamical Instability====
			From the [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#Model_Sequences\|accompanying graphical display of equilibrium sequences]], it seems that a <math>~\nu_\mathrm{max}</math> turning point will only exist in five-one bipolytropes for <math>~\mu_e/\mu_c</math> less than some value — call it, <math>~(\mu_e/\mu_c)_\mathrm{begin}</math> — which is less than but approximately equal to <math>~\tfrac{1}{3}</math>. As we move along any sequence for which <math>~\mu_e/\mu_c < (\mu_e/\mu_c)_\mathrm{begin}</math>, in the direction of increasing <math>~\ell_i</math>, it is fair to ask whether the system becomes dynamically unstable (at <math>~[x_\mathrm{eq}]_\mathrm{crit}</math>) before or after it encounters the point of secular instability marked by <math>~\nu_\mathrm{max}</math>.


	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: /* Five-One Bipolytropes */

2016-08-30T19:20:54Z

Five-One Bipolytropes

← Older revision		Revision as of 19:20, 30 August 2016
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~(3\mathfrak{L}_i +12\mathfrak{K}_i)~~\chi_~~\mathrm{eq}		<math>~(3\mathfrak{L}_i +12\mathfrak{K}_i) x_\mathrm{eq}
	\biggl[ \biggl( \frac{3^6}{2^5\pi} \biggr)^{1/2} \frac{\ell_i^3}{(1+\ell_i^2)^3} \biggr] </math>		\biggl[ \biggl( \frac{3^6}{2^5\pi} \biggr)^{1/2} \frac{\ell_i^3}{(1+\ell_i^2)^3} \biggr] </math>
	</td>		</td>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~(5 \mathfrak{L}_i)~~\chi_~~\mathrm{eq}^{3/5} \biggl[ \biggl( \frac{3^6}{2^5\pi} \biggr)^{1/2} \frac{\ell_i^3}{(1+\ell_i^2)^3} \biggr] </math>		<math>~(5 \mathfrak{L}_i) x_\mathrm{eq}^{3/5} \biggl[ \biggl( \frac{3^6}{2^5\pi} \biggr)^{1/2} \frac{\ell_i^3}{(1+\ell_i^2)^3} \biggr] </math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~(4\mathfrak{K}_i)~~\chi_~~\mathrm{eq}^{3} \biggl[ \biggl( \frac{3^6}{2^5\pi} \biggr)^{1/2} \frac{\ell_i^3}{(1+\ell_i^2)^3} \biggr] </math>		<math>~(4\mathfrak{K}_i) x_\mathrm{eq}^{3} \biggl[ \biggl( \frac{3^6}{2^5\pi} \biggr)^{1/2} \frac{\ell_i^3}{(1+\ell_i^2)^3} \biggr] </math>
	</td>		</td>
	</tr>		</tr>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\~~chi_~~\mathrm{eq}</math>		<math>~x_\mathrm{eq} = \frac{R_\mathrm{eq}}{R_\mathrm{norm}}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</table>		</table>
	</div>		</div>


			<!-- MORE EXPRESSIONS FOR POWERPOINT PRESENTATION
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~x_\mathrm{eq}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl(\frac{\pi}{2^3}\biggr)^{1/2} \frac{(1+\ell_i^2)^3}{3^3\ell_i^3} \biggl[ \frac{1 }{(1+\Lambda_i^2)} \biggr]
			\biggl\{ 1 + \frac{1}{\eta_i}\biggl[\frac{\pi}{2} + \tan^{-1}\Lambda_i \biggr] \biggr\}^{-1}</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Lambda_i</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~\frac{1}{\eta_i} - \ell_i </math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\eta_i</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \biggl[\frac{\ell_i }{(1+\ell_i^2)}\biggr] </math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~R_0 = R_\mathrm{norm}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl( \frac{G}{K_c} \biggr)^{5/2} M_\mathrm{tot}^2
			</math>
			</td>
			</tr>
			</table>
			</div>
			-->

	In rewriting this last expression, we have made use of the two relations,		In rewriting this last expression, we have made use of the two relations,
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	Consistent with our [[User:Tohline/SSC/BipolytropeGeneralization#Free_Energy_and_Its_Derivatives\|generic discussion of the stability of bipolytropes]] and the ''specific'' discussion of [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#Stability_Condition\|the stability of bipolytropes having]] <math>~(n_c, n_e) = (5, 1)</math>, it can straightforwardly be shown that <math>~\partial \mathfrak{G}/\partial \chi = 0</math> is satisfied by setting <math>~\Chi = 1</math>; that is, the equilibrium condition is,		Consistent with our [[User:Tohline/SSC/BipolytropeGeneralization#Free_Energy_and_Its_Derivatives\|generic discussion of the stability of bipolytropes]] and the ''specific'' discussion of [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#Stability_Condition\|the stability of bipolytropes having]] <math>~(n_c, n_e) = (5, 1)</math>, it can straightforwardly be shown that <math>~\partial \mathfrak{G}/\partial \chi = 0</math> is satisfied by setting <math>~\Chi = 1</math>; that is, the equilibrium condition is,

	Furthermore, the equilibrium configuration is unstable whenever <math>~\partial^2 \mathfrak{G}/\partial \chi^2 < 0</math>, that is, ~~it is~~ unstable whenever,		Furthermore, the equilibrium configuration is unstable whenever <math>~\partial^2 \mathfrak{G}/\partial \chi^2 < 0</math>, that is, the transition from stable to unstable configurations whenever,

	<div align="center">		<div align="center">
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	</td>		</td>
	<td align="center">		<td align="center">
	<math>~></math>		<math>~=</math>
	</td>		</td>
	<td align="left">		<td align="left">

Tohline: /* Five-One Bipolytropes */ Typed in some hidden equations for use in PowerPoint presentation

2016-08-30T00:57:58Z

Five-One Bipolytropes: Typed in some hidden equations for use in PowerPoint presentation

← Older revision		Revision as of 00:57, 30 August 2016
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	</table>		</table>
	</div>		</div>

			<!-- EQUATIONS USEFUL in POWERPOINT PRESENTATION
			<div align="center">
			<math>~\theta_5(\xi) ~~~~\theta_1(\eta)</math>


			<math>~\mu_e/\mu_c</math>


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

			<tr>
			<td align="right">
			<math>~\ell_i</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~\frac{\xi_i}{\sqrt{3}}</math>
			</td>
			</tr>
			</table>
			</div>

			-->

	Comparing the general free-energy expression at the beginning of this chapter with the free-energy expression provided via our [[User:Tohline/SSC/FreeEnergy/PolytropesEmbedded#BiPolytropeFreeEnergy\|accompanying summary discussion of five-one bipolytropes]], we find that,		Comparing the general free-energy expression at the beginning of this chapter with the free-energy expression provided via our [[User:Tohline/SSC/FreeEnergy/PolytropesEmbedded#BiPolytropeFreeEnergy\|accompanying summary discussion of five-one bipolytropes]], we find that,

Tohline: /* Five-One Bipolytropes */ Simplify coefficient expressions somewhat

2016-08-29T16:40:59Z

Five-One Bipolytropes: Simplify coefficient expressions somewhat

← Older revision		Revision as of 16:40, 29 August 2016
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~(3\mathfrak{L}_i +12\mathfrak{K}_i)\chi_\mathrm{eq} \biggl\{ \biggl( \frac{2^5\pi~~}{3^6~~} \biggr)^{1/2} \~~biggl[~~ \~~frac~~{(1+\ell_i^2)~~}{\ell_i~~} \biggr]~~^{3}~~ ~~\biggr\}^{-1}~~</math>		<math>~(3\mathfrak{L}_i +12\mathfrak{K}_i)\chi_\mathrm{eq}
			\biggl[ \biggl( \frac{3^6}{2^5\pi} \biggr)^{1/2} \frac{\ell_i^3}{(1+\ell_i^2)^3} \biggr] </math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~(5 \mathfrak{L}_i)\chi_\mathrm{eq}^{3/5} \biggl\{ \biggl( \frac{2^5\pi~~}{3^6~~} \biggr)^{1/2} \~~biggl[~~ \~~frac~~{(1+\ell_i^2)~~}{\ell_i} \biggr]~~^{3} \biggr~~\}^{-1}~~</math>		<math>~(5 \mathfrak{L}_i)\chi_\mathrm{eq}^{3/5} \biggl[ \biggl( \frac{3^6}{2^5\pi} \biggr)^{1/2} \frac{\ell_i^3}{(1+\ell_i^2)^3} \biggr] </math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~(4\mathfrak{K}_i)\chi_\mathrm{eq}^{3} \biggl\{ \biggl( \frac{2^5\pi~~}{3^6~~} \biggr)^{1/2} \~~biggl[~~ \~~frac~~{(1+\ell_i^2)~~}{\ell_i} \biggr]~~^{3} \biggr~~\}^{-1}~~</math>		<math>~(4\mathfrak{K}_i)\chi_\mathrm{eq}^{3} \biggl[ \biggl( \frac{3^6}{2^5\pi} \biggr)^{1/2} \frac{\ell_i^3}{(1+\ell_i^2)^3} \biggr] </math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Five-One Bipolytropes */ Enter all of the coefficient expressions related to free-energy of five-one bipolytropes

2016-08-29T16:01:38Z

Five-One Bipolytropes: Enter all of the coefficient expressions related to free-energy of five-one bipolytropes

Show changes

Tohline: /* General Free-Energy Expression */ Begin five-one bipolytrope synopsis

2016-08-29T15:11:39Z

General Free-Energy Expression: Begin five-one bipolytrope synopsis

← Older revision		Revision as of 15:11, 29 August 2016
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	<math>~		<math>~
	\sqrt{3} \, .		\sqrt{3} \, .
			</math>
			</td>
			</tr>
			</table>
			</div>

			===Five-One Bipolytropes===
			For bipolytropes, in general, let <math>~n =n_c</math> and <math>~j = n_e</math>. The statement of virial equilibrium is, then,

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

			<tr>
			<td align="right">
			<math>~\frac{ b}{n_c c}\cdot x^{(n_c-3)/n_c }_\mathrm{eq} - \frac{a}{3c} + \frac{1}{n_e}\cdot x^{(n_e-3)/n_e}_\mathrm{eq} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ 0 \, .</math>
			</td>
			</tr>
			</table>
			</div>
			And the critical equilibrium configuration has,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~[x_\mathrm{eq}^{(n_e-3)/n_e}]_\mathrm{crit} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{a}{3^2c}\biggl[ \frac{n_e^2(n_c-3)}{n_c-n_e} \biggr]
			\, .
			</math>
			</td>
			</tr>
			</table>
			</div>

			Here we choose to set <math>~n_c = 5</math> and <math>~n_e = 1</math>. Hence, these two conditions become, respectively,

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

			<tr>
			<td align="right">
			<math>~\frac{ b}{5 c}\cdot x^{2/5 }_\mathrm{eq} + x^{-2}_\mathrm{eq} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ \frac{a}{3c} \, ,</math>
			</td>
			</tr>
			</table>
			</div>
			and,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~[x_\mathrm{eq}]_\mathrm{crit} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl[ \frac{2\cdot 3^2c}{a} \biggr]^{1/2}
			\, .
	</math>		</math>
	</td>		</td>

Tohline: /* Case M Equilibrium Conditions */ Insert 50% size movie; the animated gif appears to work

2016-08-29T00:48:39Z

Case M Equilibrium Conditions: Insert 50% size movie; the animated gif appears to work

← Older revision		Revision as of 00:48, 29 August 2016
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	<tr>		<tr>
	<td align="center">		<td align="center">
	[[File:~~CaseMfreeEnergy~~.gif\|~~600px~~\|Case M movie of free-energy structures]]		[[File:CaseMfreeEnergy50.gif\|750px\|Case M movie of free-energy structures]]
	</td>		</td>
	</tr>		</tr>

Tohline: /* Case M Equilibrium Conditions */ Insert movie

2016-08-29T00:39:30Z

Case M Equilibrium Conditions: Insert movie

← Older revision		Revision as of 00:39, 29 August 2016
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	[[File:CaseMcompareTogether.png\|700px\|center\|Case M pressure-truncated n=5 polytrope]]		[[File:CaseMcompareTogether.png\|700px\|center\|Case M pressure-truncated n=5 polytrope]]
	</td></tr>		</td></tr>
			</table>
			</div>


			Now for a movie!
			<div align="center">
			<table border="1" align="center">
			<tr>
			<td align="center">
			[[File:CaseMfreeEnergy.gif\|600px\|Case M movie of free-energy structures]]
			</td>
			</tr>
	</table>		</table>
	</div>		</div>

Tohline: /* Case M Equilibrium Conditions */ Add ID tag to Figure 3

2016-08-25T01:55:23Z

Case M Equilibrium Conditions: Add ID tag to Figure 3

← Older revision		Revision as of 01:55, 25 August 2016
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	This agrees with the [[User:Tohline/SSC/FreeEnergy/PolytropesEmbedded#Case_M\|expression derived in a separate <font color="red">ASIDE</font>]]; as was pointed out in that context, the root of this equation is: <math>~\ell \approx 2.223175</math>, that is, <math>~\tilde\xi \approx 3.85065</math>. This point along the equilibrium sequence is identified by the red circular dot in Figure 3, below. It is almost — but definitely not — coincident with the configuration along the sequence (marked by the yellow circular dot) that is associated with the minimum-radius turning point.		This agrees with the [[User:Tohline/SSC/FreeEnergy/PolytropesEmbedded#Case_M\|expression derived in a separate <font color="red">ASIDE</font>]]; as was pointed out in that context, the root of this equation is: <math>~\ell \approx 2.223175</math>, that is, <math>~\tilde\xi \approx 3.85065</math>. This point along the equilibrium sequence is identified by the red circular dot in Figure 3, below. It is almost — but definitely not — coincident with the configuration along the sequence (marked by the yellow circular dot) that is associated with the minimum-radius turning point.

	<div align="center">		<div align="center" id="Figure3">
	<table border="0" align="center"><tr><th align="center">		<table border="0" align="center"><tr><th align="center">
	Figure 3:   Case M Equilibrium Sequence		Figure 3:   Case M Equilibrium Sequence
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	</table>		</table>
	</div>		</div>


	=====Case P Equilibrium Conditions=====		=====Case P Equilibrium Conditions=====