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

Tohline: /* Summary */ Finished proving that all four expressions are correct for case of n=5 as well!

2016-09-23T20:43:52Z

Summary: Finished proving that all four expressions are correct for case of n=5 as well!

← Older revision		Revision as of 20:43, 23 September 2016
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	<tr>		<tr>
			<td align="left">
			For <math>~n \ne 5</math>:
			</td>
	<td align="right">		<td align="right">
	<math>~		<math>~
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	<td align="left">		<td align="left">
	<math>~		<math>~
	3(n-3) \biggl[ (-{\tilde\theta}^')^2 - \biggl( -\frac{\tilde\theta {\tilde\theta}^'}{\tilde\xi} \biggr)\biggr]		3(n-3) \biggl[ (-{\tilde\theta}^')^2 - \biggl( -\frac{\tilde\theta {\tilde\theta}^'}{\tilde\xi} \biggr)\biggr] \, ;
			</math>
			</td>
			</tr>

			<tr>
			<td align="left">
			For <math>~n = 5</math>:
			</td>
			<td align="right">
			<math>~
			\biggl[\frac{2^4\cdot 5}{3}\biggr] \ell^3
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			(1+\ell^2)^3 \tan^{-1}\ell + \ell(\ell^4-1) \, .
	</math>		</math>
	</td>		</td>
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	If we adopt the shorthand notation,		If (for <math>n\ne 5</math>) we adopt the shorthand notation,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Summary */

2016-09-22T22:41:35Z

Summary

← Older revision		Revision as of 22:41, 22 September 2016
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	<td align="right">		<td align="right">
	<math>~\biggl[ \frac{P_\mathrm{e}}{P_\mathrm{norm}}\biggr]_\mathrm{crit}^{n-3}</math>		<math>~\biggl[ \frac{P_\mathrm{e}}{P_\mathrm{norm}}\biggr]_\mathrm{crit}^{n-3}</math>
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	~~\biggl( \frac{3}{4\pi}\biggr)^4 \biggl[\frac{ (n-3)}{4\pi (n+1)}\biggr]^{n-3} \biggl[\frac{3(n+1) }{4n} \biggr]^{4n}~~
	~~\biggl[ \frac{ 5 }{ \tilde{\mathfrak{f}}_W } \biggr]^{3(n+1)} \tilde{\mathfrak{f}}_M^{2(n+1)} \tilde{\mathfrak{f}}_A^{4n}~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~
	~~<tr>~~
	~~<td align="right">~~
	~~ ~~
	</td>		</td>
	<td align="center">		<td align="center">

Tohline: /* Summary */ Finished proving that all four expressions are correct

2016-09-22T22:40:48Z

Summary: Finished proving that all four expressions are correct

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Tohline: /* Summary */ Finished showing that first table relation is correct

2016-09-22T20:28:37Z

Summary: Finished showing that first table relation is correct

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Tohline: /* Free-Energy of Truncated Polytropes */ Finished constructing table with all expressions for dynamical instability criteria

2016-09-22T19:30:54Z

Free-Energy of Truncated Polytropes: Finished constructing table with all expressions for dynamical instability criteria

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Tohline: /* Case P */ Finalized in a form suitable for PowerPoint presentation

2016-09-21T21:38:54Z

Case P: Finalized in a form suitable for PowerPoint presentation

← Older revision		Revision as of 21:38, 21 September 2016
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\biggl[\frac{ (n-3)}{4\pi n}\biggr]^{n-3} \biggl(\frac{3 }{4} \biggr)^{4n} \mathcal{A}^{-3(n+1)} \mathcal{B}^{4n} \, .		\biggl[\frac{ (n-3)}{4\pi n}\biggr]^{n-3} \biggl(\frac{3 }{4} \biggr)^{4n} \mathcal{A}^{-3(n+1)} \mathcal{B}^{4n} \, ,
	</math>		</math>
	</td>		</td>
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	</table>		</table>
	</div>		</div>

			where the constants,
			<div align="center">
			<table border="0" cellpadding="5">
			<tr>
			<td align="right">
			<math>~\mathcal{A} \equiv \frac{1}{5} \cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2}</math>
			</td>
			<td align="center">
			and
			</td>
			<td align="left">
			<math>\mathcal{B} \equiv \biggl(\frac{4\pi}{3} \biggr)^{-1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{f}}_M^{(n+1)/n}} \, .</math>
			</td>
			</tr>
			</table>
			</div>

			So we can furthermore conclude that,

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

			<tr>
			<td align="right">
			<math>~\biggl[\frac{M}{M_\mathrm{SWS}} \biggr]_\mathrm{crit}^{2(n+1)}
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl[\frac{ (n-3)}{4\pi n}\biggr]^{n-3} \biggl(\frac{3 }{4} \biggr)^{4n} \biggl\{ \frac{1}{5} \cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggr\}^{-3(n+1)}
			\biggl\{ \biggl(\frac{4\pi}{3} \biggr)^{-1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{f}}_M^{(n+1)/n}} \biggr\}^{4n}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl(\frac{3}{4\pi} \biggr)^{4}
			\biggl[\frac{ (n-3)}{4\pi n}\biggr]^{n-3} \biggl(\frac{3\tilde{\mathfrak{f}}_A }{4} \biggr)^{4n} \biggl[ \frac{5^3\tilde{\mathfrak{f}}_M^2}{\tilde{\mathfrak{f}}_W^3} \biggr]^{(n+1)} \, .
			</math>
			</td>
			</tr>
			</table>
			</div>


	Our expression for <math>~[M]_\mathrm{crit}^{2(n+1)}</math> can also be combined with the "Case 2 stability criterion" to eliminate the mass entirely, giving,		Our expression for <math>~[M]_\mathrm{crit}^{2(n+1)}</math> can also be combined with the "Case 2 stability criterion" to eliminate the mass entirely, giving,
	<div align="center">		<div align="center">

Tohline: /* Case M */ Correct factor of 4pi/3 in Case M

2016-09-21T19:06:05Z

Case M: Correct factor of 4pi/3 in Case M

← Older revision		Revision as of 19:06, 21 September 2016
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\frac{4n}{15(n+1) } \biggl(\frac~~{3}~~{4\pi}\biggr)^{1/n}\cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_A \tilde{\mathfrak{f}}_M^{(n-1)/n}}		\frac{4n}{15(n+1) } \biggl(\frac{4\pi}{3}\biggr)^{1/n}\cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_A \tilde{\mathfrak{f}}_M^{(n-1)/n}}
	</math>		</math>
	</td>		</td>
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	</table>		</table>
	</div>		</div>


	===Case P===		===Case P===

Tohline: /* Free-Energy of Truncated Polytropes */ Elaborate on expression for "Case P" critical radius

2016-09-21T17:09:13Z

Free-Energy of Truncated Polytropes: Elaborate on expression for "Case P" critical radius

← Older revision		Revision as of 17:09, 21 September 2016
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	<math>~		<math>~
	\biggl[ \frac{ (n-3)}{4\pi n} \biggr]^{(n-1)} \biggl(\frac{3 }{4} \biggr)^{2n}		\biggl[ \frac{ (n-3)}{4\pi n} \biggr]^{(n-1)} \biggl(\frac{3 }{4} \biggr)^{2n}
	\mathcal{A}^{-(n+1)} \mathcal{B}^{2n} \, .		\mathcal{A}^{-(n+1)} \mathcal{B}^{2n}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl[ \frac{n}{n-3} \biggr]^{(1-n)} (4\pi)^{1-n}\biggl(\frac{3 }{4} \biggr)^{2n}
			\biggl[ \frac{1}{5} \cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggr]^{-(n+1)}
			\biggl[ \biggl(\frac{4\pi}{3} \biggr)^{-1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{f}}_M^{(n+1)/n}} \biggr]^{2n}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl[ \frac{n}{n-3} \biggr]^{(1-n)} (4\pi)^{1-n -2} 3^{2n+2} 4^{-2n}
			\biggl[ \frac{5\tilde{\mathfrak{f}}_M^2}{\tilde{\mathfrak{f}}_W}\biggr]^{(n+1)}
			\biggl[ \frac{\tilde{\mathfrak{f}}_A^{2n}}{\tilde{\mathfrak{f}}_M^{2(n+1)}} \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl[ \frac{n}{n-3} \biggr]^{(1-n)}
			\biggl[ \frac{3^2\cdot 5}{4\pi \tilde{\mathfrak{f}}_W}\biggr]^{(n+1)}
			\biggl[\frac{ \tilde{\mathfrak{f}}_A}{4} \biggr]^{2n} \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Virial Equilibrium and Dynamical Stability */ Finished relating expressions to "Case P"

2016-09-21T16:49:20Z

Virial Equilibrium and Dynamical Stability: Finished relating expressions to "Case P"

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Tohline: /* Virial Equilibrium and Dynamical Stability */ Finished "Case M"

2016-09-21T00:52:59Z

Virial Equilibrium and Dynamical Stability: Finished "Case M"

← Older revision		Revision as of 00:52, 21 September 2016
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	<math>~		<math>~
	\biggl[\frac{4n\mathcal{A}}{3(n+1) \mathcal{B}} \biggr]^n \biggl(\frac{G}{K}\biggr)^n M^{n-1}		\biggl[\frac{4n\mathcal{A}}{3(n+1) \mathcal{B}} \biggr]^n \biggl(\frac{G}{K}\biggr)^n M^{n-1}
			</math>
			</td>
			</tr>
			</table>
			</div>

			===Case M===
			Now, in our discussion of "Case M" sequence analyses, the configuration's radius is normalized to,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~R_\mathrm{norm}</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~[ G^n K^{-n} M^{n-1} ]^{1/(n-3)} \, .</math>
			</td>
			</tr>
			</table>
			</div>
			Our "Case 3" stability criterion directly relates. We conclude that the transition from stability to dynamical instability occurs when,

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

			<tr>
			<td align="right">
			<math>~
			\biggl[\frac{R_\mathrm{eq}}{R_\mathrm{norm}}\biggr]_\mathrm{crit}^{n-3}
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl[\frac{4n\mathcal{A}}{3(n+1) \mathcal{B}} \biggr]^n
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~
			\biggl[\frac{R_\mathrm{eq}}{R_\mathrm{norm}}\biggr]_\mathrm{crit}^{(n-3)/n}
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{4n}{15(n+1) } \biggl(\frac{3}{4\pi}\biggr)^{1/n}\cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_A \tilde{\mathfrak{f}}_M^{(n-1)/n}}
			</math>
			</td>
			</tr>
			</table>
			</div>
			Also in the "Case M" discussions, the external pressure is normalized to,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~P_\mathrm{norm}</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~[ G^{-3(n+1)} K^{4n} M^{-2(n+1)} ]^{1/(n-3)} \, .</math>
			</td>
			</tr>
			</table>
			</div>

			If we raise the "Case 1" stability criterion expression to the <math>~(n-3)</math> power, then divide it by the "Case 3" stability criterion expression raised to the fourth power, we find,

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

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~
			[P_e]_\mathrm{crit}^{n-3}
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl[\frac{\mathcal{A} GM^2 (n-3)}{4\pi (n+1)}\biggr]^{n-3}\biggl\{ \biggl[\frac{4n\mathcal{A}}{3(n+1) \mathcal{B}} \biggr]^n \biggl(\frac{G}{K}\biggr)^n M^{n-1} \biggr\}^{-4}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl[\frac{\mathcal{A} (n-3)}{4\pi (n+1)}\biggr]^{n-3} G^{n-3} M^{2(n-3)} \biggl[\frac{3(n+1) \mathcal{B}}{4n\mathcal{A}} \biggr]^{4n} \biggl(\frac{K}{G}\biggr)^{4n} M^{4(1-n)}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl[\frac{ (n-3)}{4\pi (n+1)}\biggr]^{n-3} \biggl[\frac{3(n+1) }{4n} \biggr]^{4n} \mathcal{A}^{-3(n+1)} \mathcal{B}^{4n} K^{4n} M^{-2(n+1)} G^{-3(n+1)} M^{2(n-3)}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~
			\biggl[\frac{P_e}{P_\mathrm{norm}} \biggr]_\mathrm{crit}^{n-3}
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl[\frac{ (n-3)}{4\pi (n+1)}\biggr]^{n-3} \biggl[\frac{3(n+1) }{4n} \biggr]^{4n} \mathcal{A}^{-3(n+1)} \mathcal{B}^{4n}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl[\frac{ (n-3)}{4\pi (n+1)}\biggr]^{n-3} \biggl[\frac{3(n+1) }{4n} \biggr]^{4n} \biggl[ \frac{ 5\tilde{\mathfrak{f}}_M^2 }{ \tilde{\mathfrak{f}}_W } \biggr]^{3(n+1)}
			\biggl( \frac{3}{4\pi}\biggr)^4 \biggl[ \frac{ \tilde{\mathfrak{f}}_A }{ \tilde{\mathfrak{f}}_M^{(n+1)/n} } \biggr]^{4n}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl( \frac{3}{4\pi}\biggr)^4 \biggl[\frac{ (n-3)}{4\pi (n+1)}\biggr]^{n-3} \biggl[\frac{3(n+1) }{4n} \biggr]^{4n}
			\biggl[ \frac{ 5 }{ \tilde{\mathfrak{f}}_W } \biggr]^{3(n+1)} \tilde{\mathfrak{f}}_M^{2(n+1)} \tilde{\mathfrak{f}}_A^{4n}
	</math>		</math>
	</td>		</td>