User:Tohline/SSC/BipolytropeGeneralization Version2 - Revision history

Tohline at 23:18, 10 July 2016

2016-07-10T23:18:38Z

← Older revision		Revision as of 23:18, 10 July 2016
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	* [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|Analytic solution of Detailed-Force-Balance BiPolytrope]] with <math>~n_c = 5</math> and <math>~n_e=1</math>.		* [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|Analytic solution of Detailed-Force-Balance BiPolytrope]] with <math>~n_c = 5</math> and <math>~n_e=1</math>.
	* [[User:Tohline/SSC/BipolytropeGeneralization\|Old ''Bipolytrope Generalization'' derivations]].		* [[User:Tohline/SSC/BipolytropeGeneralization\|Old ''Bipolytrope Generalization'' derivations]].



			=See Also=
			<ul>
			<li>[[User:Tohline/SphericallySymmetricConfigurations/IndexFreeEnergy#Index_to_Free-Energy_Analyses\|Index to a Variety of Free-Energy and/or Virial Analyses]]</li>
			</ul>


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Tohline: /* More Utilitarian Form */ Change subscript typo from "core" to "env"

2015-03-26T19:30:36Z

More Utilitarian Form: Change subscript typo from "core" to "env"

← Older revision		Revision as of 19:30, 26 March 2015
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	</table>		</table>
	</div>		</div>
	Inserting the generic definitions of the coefficients <math>~\mathcal{B}_\mathrm{core}</math> and <math>~\mathcal{B}_\mathrm{~~core~~}</math> — expressed in [[User:Tohline/SSC/BipolytropeGeneralization_Version2#Generalized_Free-Energy_Expression\|shorthand notation as referenced above]] — and demanding that the interface pressures be identical, gives,		Inserting the generic definitions of the coefficients <math>~\mathcal{B}_\mathrm{core}</math> and <math>~\mathcal{B}_\mathrm{env}</math> — expressed in [[User:Tohline/SSC/BipolytropeGeneralization_Version2#Generalized_Free-Energy_Expression\|shorthand notation as referenced above]] — and demanding that the interface pressures be identical, gives,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Bipolytrope Generalization */ Delete examples of equilibrium using utilitarian form of equations because content has been moved to other H_Book locations

2014-08-29T20:11:40Z

Bipolytrope Generalization: Delete examples of equilibrium using utilitarian form of equations because content has been moved to other H_Book locations

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Tohline: /* Bipolytrope */ Delete large "text box" sections because the relevant text has been transferred to separate H_Book pages

2014-08-29T20:04:48Z

Bipolytrope: Delete large "text box" sections because the relevant text has been transferred to separate H_Book pages

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Tohline: /* Partitioning the Mass */ Delete large "examples" text box because its contents have been transferred to other H_Book pages

2014-08-29T19:57:39Z

Partitioning the Mass: Delete large "examples" text box because its contents have been transferred to other H_Book pages

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Tohline: /* Related Discussions */ Expand "related discussions" subsection

2014-08-29T19:08:32Z

Related Discussions: Expand "related discussions" subsection

← Older revision		Revision as of 19:08, 29 August 2014
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	=Related Discussions=		=Related Discussions=
			* [[User:Tohline/SSC/Structure/BiPolytropes/FreeEnergy0_0\|Free-energy determination of equilibrium configurations for BiPolytropes]] with <math>~n_c = 0</math> and <math>~n_e=0</math>.
			* [[User:Tohline/SSC/Structure/BiPolytropes/FreeEnergy5_1#Free_Energy_of_BiPolytrope_with\|Free-energy determination of equilibrium configurations for BiPolytropes]] with <math>~n_c = 5</math> and <math>~n_e=1</math>.
			* [[User:Tohline/SSC/Structure/BiPolytropes/Analytic0_0#BiPolytrope_with_nc_.3D_0_and_ne_.3D_0\|Analytic solution of Detailed-Force-Balance BiPolytrope]] with <math>~n_c = 0</math> and <math>~n_e=0</math>.
			* [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|Analytic solution of Detailed-Force-Balance BiPolytrope]] with <math>~n_c = 5</math> and <math>~n_e=1</math>.
	* [[User:Tohline/SSC/BipolytropeGeneralization\|Old ''Bipolytrope Generalization'' derivations]].		* [[User:Tohline/SSC/BipolytropeGeneralization\|Old ''Bipolytrope Generalization'' derivations]].
	* [[User:Tohline/SSC/Structure/BiPolytropes/Analytic0_0#BiPolytrope_with_nc_.3D_0_and_ne_.3D_0\|Analytic solution]] with <math>~n_c = 0</math> and <math>~n_e=0</math>.
	* [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|Analytic solution]] with <math>~n_c = 5</math> and <math>~n_e=1</math>.

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Tohline: /* Separate Thermodynamic Energy Reservoirs */ Finished deriving q^2s_core for n_c = 5 core

2014-08-28T18:51:41Z

Separate Thermodynamic Energy Reservoirs: Finished deriving q^2s_core for n_c = 5 core

← Older revision		Revision as of 18:51, 28 August 2014
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\biggl( 1 + \xi^2 \biggr)^{-3} \, ,</math>		<math>~\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3} \, ,</math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\biggl( 1 + ~~3a_~~\xi x^2 \biggr)^{-3} \, .</math>		<math>~\biggl( 1 + a_\xi x^2 \biggr)^{-3} \, ,</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	</div>		</div>
			where, <math>~a_\xi</math> is defined above in connection with our illustration of [[User:Tohline/SSC/BipolytropeGeneralization_Version2#ExampleMass\|example mass profiles]].
	The desired integral over this pressure distribution therefore gives,		The desired integral over this pressure distribution therefore gives,
	<div align="center">		<div align="center">
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	<td align="left">		<td align="left">
	<math>~		<math>~
	3\biggl( 1 + ~~3a_~~\xi q^2 \biggr)^{3} \int_0^q \frac{x^2 dx}{(1+~~3a_~~\xi x^2)^3}		3\biggl( 1 + a_\xi q^2 \biggr)^{3} \int_0^q \frac{x^2 dx}{(1+a_\xi x^2)^3}
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>~		<math>~
	3\biggl( 1 + ~~3a_~~\xi q^2 \biggr)^{3} \biggl\{		3\biggl( 1 + a_\xi q^2 \biggr)^{3} \biggl\{
	\frac{\tan^{-1}[~~(3a_~~\xi)^{1/2}q]}{2^~~3 \cdot (~~3 a_\xi)^{3/2}}		\frac{\tan^{-1}[a_\xi^{1/2}q]}{2^3 a_\xi^{3/2}}
	+ \frac{q}{2^~~3\cdot (~~3 a_\xi) (~~3a_~~\xi q^2 +1)}		+ \frac{q}{2^3 a_\xi (a_\xi q^2 +1)}
	- \frac{q}{2^2~~\cdot (3~~ a_\xi)(~~3a_~~\xi q^2 +1)^2}		- \frac{q}{2^2 a_\xi (a_\xi q^2 +1)^2}
	\biggr\}		\biggr\}
	</math>		</math>
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\frac{3}{2^~~3 \cdot (~~3 a_\xi)^{3/2}} \biggl( 1 + ~~3a_~~\xi q^2 \biggr)^{3} \biggl\{		\frac{3}{2^3 a_\xi^{3/2}} \biggl( 1 + a_\xi q^2 \biggr)^{3} \biggl\{
	\tan^{-1}[~~(3a_~~\xi)^{1/2}q]		\tan^{-1}[a_\xi^{1/2}q]
	~+~ \frac{~~(3a_~~\xi)^{1/2}q}{(~~3a_~~\xi q^2 +1)}		+ \frac{a_\xi^{1/2}q}{(a_\xi q^2 +1)}
	~-~ \frac{~~2(3a_~~\xi)^{1/2}q}{(~~3a_~~\xi q^2 +1)^2}		- \frac{2a_\xi^{1/2}q}{(a_\xi q^2 +1)^2}
	\biggr\}		\biggr\}
	</math>		</math>
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\frac{3}{2^~~3 \cdot (~~3 a_\xi)^{3/2}} \biggl( 1 + ~~3a_~~\xi q^2 \biggr)^{3} \biggl\{		\frac{3}{2^3 a_\xi^{3/2}} \biggl( 1 + a_\xi q^2 \biggr)^{3} \biggl[
	\tan^{-1}[~~(3a_~~\xi)^{1/2}q]		\tan^{-1}[a_\xi^{1/2}q] - a_\xi^{1/2}q ~\frac{(1 - a_\xi q^2)}{(1 + a_\xi q^2)^2}
	~~~+~\frac{(3a_~~\xi)^{1/2}q (~~3a_~~\xi q^2 -1) }{(~~3a_~~\xi q^2 +1)^2}		\biggr] \, .
	\biggr\}
	</math>		</math>
	</td>		</td>

Tohline: /* Separate Thermodynamic Energy Reservoirs */ Begin derivation of thermal content of n_c=5 core

2014-08-27T21:33:36Z

Separate Thermodynamic Energy Reservoirs: Begin derivation of thermal content of n_c=5 core

← Older revision		Revision as of 21:33, 27 August 2014
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	</div>		</div>

			----


			Core of <math>~(n_c, n_e) = (5, 1)</math> bipolytrope:


			According to our [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#Profile\|derivation of the properties of detailed force-balance <math>~(n_c, n_e) = (5, 1)</math> bipolytropes]] — see also the relevant derivations above — in this case the pressure throughout the core is defined by the dimensionless function,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~P^* \equiv \frac{P_\mathrm{core}(\xi)}{P_0} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl( 1 + \xi^2 \biggr)^{-3} \, ,</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>\Rightarrow ~~~~ 1-p_c(x) = \frac{P_\mathrm{core}(x)}{P_0} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl( 1 + 3a_\xi x^2 \biggr)^{-3} \, .</math>
			</td>
			</tr>
			</table>
			</div>
			The desired integral over this pressure distribution therefore gives,
			<div align="center">
			<table border="0" cellpadding="8" align="center">

			<tr>
			<td align="right">
			<math>~q^3 s_\mathrm{core} </math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~
			\int_0^q 3\biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr] x^2 dx
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			3\biggl( 1 + 3a_\xi q^2 \biggr)^{3} \int_0^q \frac{x^2 dx}{(1+3a_\xi x^2)^3}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			3\biggl( 1 + 3a_\xi q^2 \biggr)^{3} \biggl\{
			\frac{\tan^{-1}[(3a_\xi)^{1/2}q]}{2^3 \cdot (3 a_\xi)^{3/2}}
			+ \frac{q}{2^3\cdot (3 a_\xi) (3a_\xi q^2 +1)}
			- \frac{q}{2^2\cdot (3 a_\xi)(3a_\xi q^2 +1)^2}
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{3}{2^3 \cdot (3 a_\xi)^{3/2}} \biggl( 1 + 3a_\xi q^2 \biggr)^{3} \biggl\{
			\tan^{-1}[(3a_\xi)^{1/2}q]
			~+~ \frac{(3a_\xi)^{1/2}q}{(3a_\xi q^2 +1)}
			~-~ \frac{2(3a_\xi)^{1/2}q}{(3a_\xi q^2 +1)^2}
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{3}{2^3 \cdot (3 a_\xi)^{3/2}} \biggl( 1 + 3a_\xi q^2 \biggr)^{3} \biggl\{
			\tan^{-1}[(3a_\xi)^{1/2}q]
			~+~\frac{(3a_\xi)^{1/2}q (3a_\xi q^2 -1) }{(3a_\xi q^2 +1)^2}
			\biggr\}
			</math>
			</td>
			</tr>
			</table>
			</div>

Tohline: /* More Utilitarian Form */ Final step of derivation, to match earlier solutions

2014-08-27T16:48:17Z

More Utilitarian Form: Final step of derivation, to match earlier solutions

← Older revision		Revision as of 16:48, 27 August 2014
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	q^2 + \frac{2}{5} q^5( f - 1-\mathfrak{F} )		q^2 + \frac{2}{5} q^5( f - 1-\mathfrak{F} )
	</math>		</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>\Rightarrow ~~~~
			\frac{P_0 R_\mathrm{edge}^4}{G M_\mathrm{tot}^2 }
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl( \frac{3}{2^3 \pi} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^2 \biggl\{ q^2 + \biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl[
			2q^2(1-q) + \biggl( \frac{\rho_e}{\rho_c} \biggr) (1-3q^2 + 2q^3) \biggr] \biggr\} \, .</math>
	</td>		</td>
	</tr>		</tr>
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	</div>		</div>

			This exactly matches the equilibrium relation that was derived from our [[User:Tohline/SSC/Structure/BiPolytropes/Analytic0_0#CentralPressure\|detailed force-balance analysis of]] <math>~(n_c, n_e) = (0, 0)</math> bipolytropes.

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

Tohline: /* More Utilitarian Form */ Derive solution from this more utilitarian form

2014-08-27T16:45:10Z

More Utilitarian Form: Derive solution from this more utilitarian form

Show changes