User:Tohline/SSC/Stability BoundedCompositePolytropes - Revision history

Tohline: /* Free-Energy Function and Its Derivatives */ Show what R^4 P_i product is implied by virial expression

2015-03-30T00:57:53Z

Free-Energy Function and Its Derivatives: Show what R^4 P_i product is implied by virial expression

← Older revision		Revision as of 00:57, 30 March 2015
Line 1,515:		Line 1,515:
	<math>~		<math>~
	\biggl[ \biggl(\frac{\nu}{q^2 \ell_i} \biggr)^2		\biggl[ \biggl(\frac{\nu}{q^2 \ell_i} \biggr)^2
	- 8\pi \~~cdot x~~^4 \biggl( \frac{P_{ic} }{P_\mathrm{norm}} \biggr)_\mathrm{eq} \biggr] \biggl\{		- 8\pi \biggl(\frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^4 \biggl( \frac{P_{i} }{P_\mathrm{norm}} \biggr)_\mathrm{eq} \biggr] \biggl\{
	\frac{3}{2^4}\biggl( \frac{q}{\ell_i} \biggr)^3 (1 + \ell_i^2)^3 \cdot \mathfrak{L}_1		\frac{3}{2^4}\biggl( \frac{q}{\ell_i} \biggr)^3 (1 + \ell_i^2)^3 \cdot \mathfrak{L}_1
	+\frac{3A^2 }{8b_\eta^3} \biggl[ \mathfrak{L}_2\biggr]_q^1		+\frac{3A^2 }{8b_\eta^3} \biggl[ \mathfrak{L}_2\biggr]_q^1
	\biggr\} \, .		\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~~ \biggl(\frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^4 \biggl( \frac{P_{i} }{P_\mathrm{norm}} \biggr)_\mathrm{eq}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{1}{8\pi} \biggl(\frac{\nu}{q^2 \ell_i} \biggr)^2
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~~ \frac{R_\mathrm{eq}^4 P_{i} }{GM_\mathrm{tot}^2} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{3}{8\pi} \biggl(\frac{\nu}{q^2 \xi_i} \biggr)^2 \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Free-Energy Function and Its Derivatives */ Clean up presentation and, while doing so, identify & fix critical sign error

2015-03-29T23:54:34Z

Free-Energy Function and Its Derivatives: Clean up presentation and, while doing so, identify & fix critical sign error

Show changes

Tohline: /* Free-Energy Function and Its Derivatives */ Write out virial expression

2015-03-29T16:37:31Z

Free-Energy Function and Its Derivatives: Write out virial expression

← Older revision		Revision as of 16:37, 29 March 2015
Line 1,317:		Line 1,317:
	\biggl[ - \frac{8}{3} \ell_i^3 + ( 1 + \ell_i^2 )^{3}\mathrm{term1} \biggr]		\biggl[ - \frac{8}{3} \ell_i^3 + ( 1 + \ell_i^2 )^{3}\mathrm{term1} \biggr]
	- 8\pi \cdot x^4 \biggl[ \frac{P_{ic} }{P_\mathrm{norm}} \biggr]_\mathrm{eq} \biggl[ (1 + \ell_i^2)^3 \mathrm{term1} \biggr]\biggr\}		- 8\pi \cdot x^4 \biggl[ \frac{P_{ic} }{P_\mathrm{norm}} \biggr]_\mathrm{eq} \biggl[ (1 + \ell_i^2)^3 \mathrm{term1} \biggr]\biggr\}
	~~\, ,~~
	</math>		</math>
	</td>		</td>
	</tr>		</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>
			\frac{3}{2^4}\biggl( \frac{q}{\ell_i} \biggr)^3 \biggl\{
			- \frac{2^3 \nu^2 \ell_i}{3 q^4 } +\biggl[ \biggl(\frac{\nu}{q^2 \ell_i} \biggr)^2
			- 8\pi \cdot x^4 \biggl( \frac{P_{ic} }{P_\mathrm{norm}} \biggr)_\mathrm{eq} \biggr] (1 + \ell_i^2)^3 \mathrm{term1}
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>
			- \frac{ \nu^2 }{2 q \ell_i^2} +\biggl[ \biggl(\frac{\nu}{q^2 \ell_i} \biggr)^2
			- 8\pi \cdot x^4 \biggl( \frac{P_{ic} }{P_\mathrm{norm}} \biggr)_\mathrm{eq} \biggr] \frac{3}{2^4}\biggl( \frac{q}{\ell_i} \biggr)^3 (1 + \ell_i^2)^3 \mathrm{term1} \, ,
			</math>
			</td>
			</tr>
	</table>		</table>
	</div>		</div>
Line 1,343:		Line 1,373:
	</tr>		</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>
			\frac{3A^2 }{8b_\eta^3} \biggl\{
			- \frac{2^2 \nu^2 b_\eta }{3 q^4 \ell_i^2} \biggl[ x \sin^2(b_\eta x - B) \biggr]_q^1
			+\biggl[ \biggl( \frac{\nu }{q^2 \ell_i} \biggr)^2
			- 8\pi \cdot x^4 \biggl( \frac{P_{ic} }{P_\mathrm{norm}} \biggr)_\mathrm{eq}\biggr] \biggl[ \mathrm{term2}\biggr]_q^1
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>
			- \frac{\nu^2 }{ 2q^4 \ell_i^2} \cdot \frac{A^2 }{b_\eta^2} \biggl[ x \sin^2(b_\eta x - B) \biggr]_q^1
			+\biggl[ \biggl( \frac{\nu }{q^2 \ell_i} \biggr)^2
			- 8\pi \cdot x^4 \biggl( \frac{P_{ic} }{P_\mathrm{norm}} \biggr)_\mathrm{eq}\biggr] \frac{3A^2 }{8b_\eta^3} \biggl[ \mathrm{term2}\biggr]_q^1
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>
			- \frac{\nu^2 }{ 2q \ell_i^2} +\biggl[ \biggl( \frac{\nu }{q^2 \ell_i} \biggr)^2
			- 8\pi \cdot x^4 \biggl( \frac{P_{ic} }{P_\mathrm{norm}} \biggr)_\mathrm{eq}\biggr] \frac{3A^2 }{8b_\eta^3} \biggl[ \mathrm{term2}\biggr]_q^1 \, .
			</math>
			</td>
			</tr>
			</table>
			</div>
			Adding these two expressions together and setting the result equal to zero (as prescribed by the virial theorem) gives,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{ \nu^2 }{q \ell_i^2} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl[ \biggl(\frac{\nu}{q^2 \ell_i} \biggr)^2
			- 8\pi \cdot x^4 \biggl( \frac{P_{ic} }{P_\mathrm{norm}} \biggr)_\mathrm{eq} \biggr] \biggl\{
			\frac{3}{2^4}\biggl( \frac{q}{\ell_i} \biggr)^3 (1 + \ell_i^2)^3 \cdot \mathrm{term1}
			+\frac{3A^2 }{8b_\eta^3} \biggl[ \mathrm{term2}\biggr]_q^1
			\biggr\} \, .
			</math>
			</td>
			</tr>
	</table>		</table>
	</div>		</div>

Tohline: /* Free-Energy Function and Its Derivatives */Fix coefficient error in a_e

2015-03-29T15:53:43Z

Free-Energy Function and Its Derivatives: Fix coefficient error in a_e

← Older revision		Revision as of 15:53, 29 March 2015
Line 1,193:		Line 1,193:
	<td align="left">		<td align="left">
	<math>		<math>
	~~\frac{1}{2}~~\biggl( \frac{\nu q}{\ell_i} \biggr)^2		\biggl( \frac{\nu }{q^2 \ell_i} \biggr)^2
	\frac{3A^2 }{~~4b_~~\eta^3} \biggl[2 b_\eta x - \sin[2(b_\eta x-B)] - \frac{4}{3}\cdot b_\eta x \sin^2(b_\eta x - B) \biggr]_q^1 \, ,		\frac{3A^2 }{8b_\eta^3} \biggl[2 b_\eta x - \sin[2(b_\eta x-B)] - \frac{4}{3}\cdot b_\eta x \sin^2(b_\eta x - B) \biggr]_q^1 \, ,
	</math>		</math>
	</td>		</td>
Line 1,324:		Line 1,324:
	</table>		</table>
	</div>		</div>




	<div align="center">		<div align="center">
Line 1,339:		Line 1,336:
	<td align="left">		<td align="left">
	<math>		<math>
	\frac{3A^2 }{8b_\eta^3} \biggl\{ \biggl( \frac{\nu q}{\ell_i} \biggr)^2		\frac{3A^2 }{8b_\eta^3} \biggl\{ \biggl( \frac{\nu }{q^2 \ell_i} \biggr)^2
	\biggl[\mathrm{term2}- \frac{4}{3}\cdot b_\eta x \sin^2(b_\eta x - B) \biggr]_q^1		\biggl[\mathrm{term2}- \frac{4}{3}\cdot b_\eta x \sin^2(b_\eta x - B) \biggr]_q^1
	- 8\pi \cdot x^4 \biggl[ \frac{P_{ic} }{P_\mathrm{norm}} \biggr]_\mathrm{eq} \biggl[ \mathrm{term2}\biggr]_q^1 \biggr\}		- 8\pi \cdot x^4 \biggl[ \frac{P_{ic} }{P_\mathrm{norm}} \biggr]_\mathrm{eq} \biggl[ \mathrm{term2}\biggr]_q^1 \biggr\}

Tohline: /* Free-Energy Function and Its Derivatives */ First stab at detailing both "core" and "env" terms that form the virial theorem

2015-03-29T02:29:57Z

Free-Energy Function and Its Derivatives: First stab at detailing both "core" and "env" terms that form the virial theorem

← Older revision		Revision as of 02:29, 29 March 2015
Line 1,179:		Line 1,179:
	<math>		<math>
	\biggl( \frac{\mu_e}{\mu_c} \biggr)^2 \biggl[ \frac{3^2 A^2 }{b_\eta^4 } \biggr] \biggl( \frac{\nu^2 \theta_i^4}{q^6} \biggr)		\biggl( \frac{\mu_e}{\mu_c} \biggr)^2 \biggl[ \frac{3^2 A^2 }{b_\eta^4 } \biggr] \biggl( \frac{\nu^2 \theta_i^4}{q^6} \biggr)
	\frac{1}{8b_\eta} \biggl[6 b_\eta x - 3\sin[2(b_\eta x-B)] - 4b_\eta x \sin^2(b_\eta x - B) \biggr]_q^1 \, .		\frac{1}{8b_\eta} \biggl[6 b_\eta x - 3\sin[2(b_\eta x-B)] - 4b_\eta x \sin^2(b_\eta x - B) \biggr]_q^1
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>
			\frac{1}{2}\biggl( \frac{\nu q}{\ell_i} \biggr)^2
			\frac{3A^2 }{4b_\eta^3} \biggl[2 b_\eta x - \sin[2(b_\eta x-B)] - \frac{4}{3}\cdot b_\eta x \sin^2(b_\eta x - B) \biggr]_q^1 \, ,
	</math>		</math>
	</td>		</td>
Line 1,193:		Line 1,208:
	<td align="left">		<td align="left">
	<math>		<math>
	~\frac{3}{2^3}\biggl( \frac{\ell_i~~}{q~~} \biggr)^3 (1 + \ell_i^2)^3		~\frac{3}{2^3}\biggl( \frac{q}{\ell_i} \biggr)^3 (1 + \ell_i^2)^3
	\biggl[ \ell_i (\ell_i^4 - 1 )(1+\ell_i^2)^{-3} + \tan^{-1}(\ell_i) \biggr]		\biggl[ \ell_i (\ell_i^4 - 1 )(1+\ell_i^2)^{-3} + \tan^{-1}(\ell_i) \biggr]
	\, ,		\, ,
Line 1,209:		Line 1,224:
	<td align="left">		<td align="left">
	<math>		<math>
	~~~\frac{3}{4}~~ \biggl( \frac{P_{ic}}{P_{ie}} \biggr) ~~\biggl(~~ \frac{A^2}{b_\eta^3} ~~\biggr)~~ \biggl[ 2b_\eta x -\sin[2(b_\eta x - B)] \biggr]_q^1 \ ,		~\biggl( \frac{P_{ic}}{P_{ie}} \biggr) \frac{3A^2}{4b_\eta^3} \biggl[ 2b_\eta x -\sin[2(b_\eta x - B)] \biggr]_q^1 \ ,
	</math>		</math>
	</td>		</td>
Line 1,284:		Line 1,299:
	</td>		</td>
	</tr>		</tr>
			</table>
			</div>

			Let's try putting these terms all together in the context of the virial theorem.
			<div align="center">
			<table border="0" cellpadding="5">
			<tr>
			<td align="right">
			<math>a_c - \frac{3b_c}{n_c}\cdot x^{(n_c-3)/n_c} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>
			\frac{3}{2^4}\biggl( \frac{q}{\ell_i} \biggr)^3 \biggl\{ \biggl(\frac{\nu}{q^2 \ell_i} \biggr)^2
			\biggl[ - \frac{8}{3} \ell_i^3 + ( 1 + \ell_i^2 )^{3}\mathrm{term1} \biggr]
			- 8\pi \cdot x^4 \biggl[ \frac{P_{ic} }{P_\mathrm{norm}} \biggr]_\mathrm{eq} \biggl[ (1 + \ell_i^2)^3 \mathrm{term1} \biggr]\biggr\}
			\, ,
			</math>
			</td>
			</tr>

			</table>
			</div>




			<div align="center">
			<table border="0" cellpadding="5">
			<tr>
			<td align="right">
			<math>a_e - \frac{3b_e}{n_e}\cdot x^{(n_e-3)/n_e} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>
			\frac{3A^2 }{8b_\eta^3} \biggl\{ \biggl( \frac{\nu q}{\ell_i} \biggr)^2
			\biggl[\mathrm{term2}- \frac{4}{3}\cdot b_\eta x \sin^2(b_\eta x - B) \biggr]_q^1
			- 8\pi \cdot x^4 \biggl[ \frac{P_{ic} }{P_\mathrm{norm}} \biggr]_\mathrm{eq} \biggl[ \mathrm{term2}\biggr]_q^1 \biggr\}
			</math>
			</td>
			</tr>

	</table>		</table>
	</div>		</div>

Tohline: /* Free-Energy Function and Its Derivatives */ Inserted more useful expressions for free-energy coefficients

2015-03-28T22:55:43Z

Free-Energy Function and Its Derivatives: Inserted more useful expressions for free-energy coefficients

← Older revision		Revision as of 22:55, 28 March 2015
Line 1,160:		Line 1,160:
	<td align="left">		<td align="left">
	<math>		<math>
	x_\mathrm{eq} \biggl( \frac{~~3^8~~}{2^5\pi} \biggr)^{~~1/2~~}		\frac{3}{5}
			\biggl[\biggl(\frac{\nu}{q^3} \biggr)^2 \biggl( 1 + \ell_i^2 \biggr)^{3} \biggr]_\mathrm{eq}
			\biggl(\frac{5}{2^4}\biggr) \biggl( \frac{q}{\ell_i}\biggr)^{5}
	\biggl[ \ell_i \biggl(\ell_i^4 - \frac{8}{3} \ell_i^2 -1 \biggr) (1 + \ell_i^2)^{-3} + \tan^{-1}(\ell_i) \biggr]		\biggl[ \ell_i \biggl(\ell_i^4 - \frac{8}{3} \ell_i^2 -1 \biggr) (1 + \ell_i^2)^{-3} + \tan^{-1}(\ell_i) \biggr]
	\, ,		\, ,
Line 1,176:		Line 1,178:
	<td align="left">		<td align="left">
	<math>		<math>
	~~x_\mathrm{eq}~~\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \biggl( \frac{1}{2^~~3\pi~~} \biggr)^{1/2} ~~A^2~~		\biggl( \frac{\mu_e}{\mu_c} \biggr)^2 \biggl[ \frac{3^2 A^2 }{b_\eta^4 } \biggr] \biggl( \frac{\nu^2 \theta_i^4}{q^6} \biggr)
	\biggl[6 b_\eta x - 3\sin[2(b_\eta x-B)] - 4b_\eta x \sin^2(b_\eta x - B) \biggr]_q^1 \, ,		\frac{1}{8b_\eta} \biggl[6 b_\eta x - 3\sin[2(b_\eta x-B)] - 4b_\eta x \sin^2(b_\eta x - B) \biggr]_q^1 \, .
	</math>		</math>
	</td>		</td>
Line 1,184:		Line 1,186:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~~~b_c~~</math>		<math>~q^3 s_\mathrm{core}</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 1,191:		Line 1,193:
	<td align="left">		<td align="left">
	<math>		<math>
	~~\biggl(~~ \frac~~{2n_c}~~{3} ~~\biggr) x_\mathrm~~{~~eq}~~^{3~~/n_c~~}		~\frac{3}{2^3}\biggl( \frac{\ell_i}{q} \biggr)^3 (1 + \ell_i^2)^3
	~~\biggl\{~~\biggl( \frac{~~3^8~~}{~~2^7\pi~~} \biggr)^{1/2} \biggl[ \ell_i (\ell_i^4 - 1 )(1+\ell_i^2)^{-3} + \tan^{-1}(\ell_i) \biggr]		\biggl[ \ell_i (\ell_i^4 - 1 )(1+\ell_i^2)^{-3} + \tan^{-1}(\ell_i) \biggr]
	~~\biggr\}~~ \, ,		\, ,
	</math>		</math>
	</td>		</td>
Line 1,200:		Line 1,202:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~~~b_e~~</math>		<math>~(1-q^3)s_\mathrm{env}</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 1,207:		Line 1,209:
	<td align="left">		<td align="left">
	<math>		<math>
	~~\biggl(~~ \frac~~{2n_e}~~{3} ~~\biggr) x_\mathrm{eq}^~~{~~3/n_e~~} ~~\biggl\{~~		~\frac{3}{4} \biggl( \frac{P_{ic}}{P_{ie}} \biggr) \biggl( \frac{A^2}{b_\eta^3} \biggr) \biggl[ 2b_\eta x -\sin[2(b_\eta x - B)] \biggr]_q^1 \ ,
	\biggl( \frac{~~\mu_e~~}{~~\mu_c~~} \biggr)~~^{-3} A^2~~ \biggl( \frac{3^2}{2^~~5\pi~~} \biggr)~~^{1/2}~~ \biggl[ 2b_\eta x -\sin[2(b_\eta x - B)] \biggr]_q^1 ~~\biggr\}~~ \ ,
	</math>		</math>
	</td>		</td>

Tohline: /* Free-Energy Function and Its Derivatives */

2015-03-27T02:10:52Z

Free-Energy Function and Its Derivatives

← Older revision		Revision as of 02:10, 27 March 2015
Line 1,161:		Line 1,161:
	<math>		<math>
	x_\mathrm{eq} \biggl( \frac{3^8}{2^5\pi} \biggr)^{1/2}		x_\mathrm{eq} \biggl( \frac{3^8}{2^5\pi} \biggr)^{1/2}
	\biggl[ a_\~~xi^{1/2} q~~(a_\~~xi^2 q~~^4 - \frac{8}{3}a_\~~xi q~~^2 - 1) (a_\~~xi q~~^2 +1)^{-3} + \tan^{-1}(a_\~~xi^{1/2} q~~) \biggr]		\biggl[ \ell_i \biggl(\ell_i^4 - \frac{8}{3} \ell_i^2 -1 \biggr) (1 + \ell_i^2)^{-3} + \tan^{-1}(\ell_i) \biggr]
	\, ,		\, ,
	</math>		</math>
Line 1,192:		Line 1,192:
	<math>		<math>
	\biggl( \frac{2n_c}{3} \biggr) x_\mathrm{eq}^{3/n_c}		\biggl( \frac{2n_c}{3} \biggr) x_\mathrm{eq}^{3/n_c}
	\biggl\{\biggl( \frac{3^8}{2^7\pi} \biggr)^{1/2} \biggl[		\biggl\{\biggl( \frac{3^8}{2^7\pi} \biggr)^{1/2} \biggl[ \ell_i (\ell_i^4 - 1 )(1+\ell_i^2)^{-3} + \tan^{-1}(\ell_i) \biggr]
	\~~tan~~^{-1~~}[a_~~\xi^~~{1/~~2~~}q] - a_\xi~~^{~~1/2~~}~~q ~~~\~~frac~~{(1 ~~- a_\xi q^2)~~}{(~~1 + a_~~\~~xi q^2~~)~~^2}~~ \biggr]
	\biggr\} \, ,		\biggr\} \, ,
	</math>		</math>
Line 1,209:		Line 1,208:
	<math>		<math>
	\biggl( \frac{2n_e}{3} \biggr) x_\mathrm{eq}^{3/n_e} \biggl\{		\biggl( \frac{2n_e}{3} \biggr) x_\mathrm{eq}^{3/n_e} \biggl\{
	\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} A^2 \biggl( \frac{3^2}{2^5\pi} \biggr)^{1/2} \biggl[ 2b_\eta x -\sin[2(b_\eta x - B)] \biggr]_q^1 \biggr\} \ .		\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} A^2 \biggl( \frac{3^2}{2^5\pi} \biggr)^{1/2} \biggl[ 2b_\eta x -\sin[2(b_\eta x - B)] \biggr]_q^1 \biggr\} \ ,
	</math>		</math>
			</td>
			</tr>
			</table>
			</div>
			where,
			<div align="center">
			<math>~\ell_i \equiv \frac{\xi_i}{3^{1/2}} \, .</math>
			</div>
			<div align="center">
			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="right">
			<math>~b_\eta</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~3^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr) \frac{\theta_i^2 \xi_i}{q}
			=
			3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \frac{\ell_i}{q} (1+\ell_i^2 )^{-1}\, ,</math>
			</td>
			</tr>
			</table>
			</div>

			<div align="center">
			<table border="0" cellpadding="3">

			<tr>
			<td align="right">
			<math>~A</math>
			</td>
			<td align="center">
			<math>~=~</math>
			</td>
			<td align="left">
			<math>~\frac{\eta_i}{\sin(\eta_i - B)} \, ,</math>
			</td>
			<td align="left">
			[because <math>~\phi_i = 1</math>]
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~(\eta_s - B)</math>
			</td>
			<td align="center">
			<math>~=~</math>
			</td>
			<td align="left">
			<math>~\pi \, ,</math>
			</td>
			<td align="left">
			[hence, <math>~\sin^2(\eta_s - B) = 0</math>]
			</td>
			</tr>

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

	</td>		</td>
	</tr>		</tr>

Tohline: /* Composite Polytropes (Bipolytropes) */ Type in specific expressions for four coefficients relevant to (n_c,n_e) = (5,1)

2015-03-27T01:00:10Z

Composite Polytropes (Bipolytropes): Type in specific expressions for four coefficients relevant to (n_c,n_e) = (5,1)

Show changes

Tohline: Begin discussing composite polytropes

2015-03-26T18:33:40Z

Begin discussing composite polytropes

← Older revision		Revision as of 18:33, 26 March 2015
Line 954:		Line 954:
	</table>		</table>
	</div>		</div>


			==Composite Polytropes (Bipolytropes)==

			Throughout this section we will rely on the following definitions of two normalization constants:
			<div align="center">
			<math>M_\mathrm{SWS} \equiv
			\biggl( \frac{n+1}{nG} \biggr)^{3/2} K_n^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} \, ,</math>
			</div>
			<div align="center">
			<math>
			R_\mathrm{SWS} \equiv \biggl( \frac{n+1}{nG} \biggr)^{1/2} K_n^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]} \, .
			</math>
			</div>

			===Free-Energy Function and Its Derivatives===
			The free-energy function that is relevant to a discussion of the structure and stability of a composite polytropic configurations of index, <math>~n_c</math> for the core and <math>~n_e</math> for the envelope, has the form,
			<div align="center">
			<table border="0" cellpadding="5">
			<tr>
			<td align="right">
			<math>~\mathcal{G}(x)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>
			-ax^{-1} +b_c x^{-3/n_c} +b_e x^{-3/n_e} + \mathcal{G}_0
			\, ,
			</math>
			</td>
			</tr>

			</table>
			</div>
			where <math>~x \equiv R/R_\mathrm{SWS}</math> identifies the radius of the configuration and <math>\mathcal{G}_0</math> is an arbitrary constant. If the coefficients, <math>~a, b_c</math>, and <math>~b_e</math>, are held constant while varying the configuration's size, we see that,
			<div align="center">
			<table border="0" cellpadding="5">
			<tr>
			<td align="right">
			<math>~\frac{d\mathcal{G}}{dx}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>
			ax^{-2} - \frac{3b_c}{n_c}\cdot x^{-(3+n_c)/n_c} - \frac{3b_e}{n_e}\cdot x^{-(3+n_e)/n_e}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>
			x^{-2} \biggl[ a - \frac{3b_c}{n_c}\cdot x^{(n_c-3)/n_c} - \frac{3b_e}{n_e}\cdot x^{(n_e-3)/n_e} \biggr]
			\, ,
			</math>
			</td>
			</tr>

			</table>
			</div>
			and,
			<div align="center">
			<table border="0" cellpadding="5">
			<tr>
			<td align="right">
			<math>~\frac{d^2\mathcal{G}}{dx^2}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>
			x^{-3} \biggl[ -2a + \frac{3(3+n_c)b_c}{n_c^2}\cdot x^{(n_c-3)/n_c} + \frac{3(3+n_e)b_e}{n_e^2}\cdot x^{(n_e-3)/n_e} \biggr]
			\, .
			</math>
			</td>
			</tr>

			</table>
			</div>
			The three relevant coefficients are,

			<div align="center">
			<table border="0" cellpadding="5">
			<tr>
			<td align="right">
			<math>~a</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~3\mathcal{A}
			=
			\frac{3}{5}\biggl(\frac{n+1}{n}\biggr) \biggl[ \biggl( \frac{M}{M_\mathrm{SWS}}\biggr) \cdot \frac{1}{\tilde\mathfrak{f}_M} \biggr]^2
			\tilde\mathfrak{f}_W \, ,</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~b_c</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~n_c\mathcal{B}_\mathrm{core}
			=
			n \biggl( \frac{3}{4\pi}\biggr)^{1/n} \biggl[ \biggl( \frac{M}{M_\mathrm{SWS}}\biggr) \cdot \frac{1}{\tilde\mathfrak{f}_M} \biggr]^{(n+1)/n}
			\tilde\mathfrak{f}_A \, ,
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~b_e</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~n_e\mathcal{B}_\mathrm{core} \, .
			</math>
			</td>
			</tr>
			</table>
			</div>


	=Related Discussions=		=Related Discussions=

Tohline: /* Bounded (Pressure-Truncated) Polytropes */ Derive stability condition on-the-fly

2015-03-25T22:23:31Z

Bounded (Pressure-Truncated) Polytropes: Derive stability condition on-the-fly

← Older revision		Revision as of 22:23, 25 March 2015
Line 808:		Line 808:
	= \frac{4\pi}{(n+1)^n} \cdot \tilde\xi^{-(n+1)} (-\tilde\Theta_H^')^{1-n} \, .		= \frac{4\pi}{(n+1)^n} \cdot \tilde\xi^{-(n+1)} (-\tilde\Theta_H^')^{1-n} \, .
	</math>		</math>
			</div>

			===Stability===

			As before, if the first derivative of the free-energy function is zero — that is, if you have identified an equilibrium configuration — and the second derivative of the free-energy function for that configuration is ''positive'', then the equilibrium system is dynamically stable. If, however, the second derivative is ''negative'', then the equilibrium system is dynamically unstable.

			Using the expression for the second derivative of the free-energy function [[User:Tohline/SSC/Stability_BoundedCompositePolytropes#Free-Energy_Function_and_Its_Derivatives_2\|derived above]], we deduce that equilibrium configurations are ''dynamically unstable'' when,
			<div align="center">
			<table border="0" cellpadding="5">
			<tr>
			<td align="right">
			<math>~\biggl[\frac{3(3+n)b}{n^2}\cdot x^{(n-3)/n} - 2a + 6cx^4\biggr]_{x_\mathrm{eq}} </math>
			</td>
			<td align="center">
			<math>~<</math>
			</td>
			<td align="left">
			<math>
			~0
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~~~ x_\mathrm{eq}^{(n-3)/n} </math>
			</td>
			<td align="center">
			<math>~<</math>
			</td>
			<td align="left">
			<math>
			~\frac{n^2}{3(3+n)b}\biggl[ 2a - 6cx^4_\mathrm{eq} \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~~~ \frac{n}{b} \biggl[\frac{a}{3} + c\cdot x^4_\mathrm{eq} \biggr] </math>
			</td>
			<td align="center">
			<math>~<</math>
			</td>
			<td align="left">
			<math>
			~\frac{n^2}{3(3+n)b}\biggl[ 2a - 6cx^4_\mathrm{eq} \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~~~ (3+n) [a + 3c x^4_\mathrm{eq} ] </math>
			</td>
			<td align="center">
			<math>~<</math>
			</td>
			<td align="left">
			<math>
			~n[ 2a - 6cx^4_\mathrm{eq} ]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~~~ [3c(3+n) + 6nc] x^4_\mathrm{eq} </math>
			</td>
			<td align="center">
			<math>~<</math>
			</td>
			<td align="left">
			<math>
			~[2n - (3+n)]a
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~~~ x^4_\mathrm{eq} </math>
			</td>
			<td align="center">
			<math>~<</math>
			</td>
			<td align="left">
			<math>
			~\frac{(n-3)a}{9c(n+1)} \, .
			</math>
			</td>
			</tr>
			</table>
			</div>

			Plugging in the expressions for the free-energy coefficients, <math>~a</math> and <math>~c</math>, this in turn implies that an equilibrium system will be dynamically unstable if,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\biggl(\frac{R_\mathrm{eq}}{R_\mathrm{SWS}}\biggr)^4 </math>
			</td>
			<td align="center">
			<math>~<</math>
			</td>
			<td align="left">
			<math>
			~\frac{(n-3)}{20\pi n}
			\biggl[ \biggl( \frac{M}{M_\mathrm{SWS}}\biggr) \cdot \frac{1}{\tilde\mathfrak{f}_M} \biggr]^2
			\tilde\mathfrak{f}_W
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~~~ \biggl(\frac{R_\mathrm{eq}}{R_\mathrm{SWS}}\biggr)^4 \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^{-2}</math>
			</td>
			<td align="center">
			<math>~<</math>
			</td>
			<td align="left">
			<math>
			~\biggl[ \frac{(n-3)}{12\pi n (5-n)} \biggr] \biggl[\frac{1}{\tilde\xi(-\tilde\Theta_H^')^2} \biggr]
			\biggl[\tilde\xi \tilde\Theta_H^{n+1} + 3\tilde\xi (\tilde\Theta_H^')^2 - 3(-\tilde\Theta_H^')\tilde\Theta_H \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~~~ \frac{P_e R_\mathrm{eq}^4}{GM^2}</math>
			</td>
			<td align="center">
			<math>~<</math>
			</td>
			<td align="left">
			<math>
			~\biggl[ \frac{(n-3)}{12\pi (5-n)(n+1)} \biggr] \biggl[\frac{1}{\tilde\xi(-\tilde\Theta_H^')^2} \biggr]
			\biggl[\tilde\xi \tilde\Theta_H^{n+1} + 3\tilde\xi (\tilde\Theta_H^')^2 - 3(-\tilde\Theta_H^')\tilde\Theta_H \biggr] \, .
			</math>
			</td>
			</tr>
			</table>
	</div>		</div>