User:Tohline/SSC/Structure/BiPolytropes/FreeEnergy5 1 - Revision history

Tohline at 23:20, 10 July 2016

2016-07-10T23:20:18Z

← Older revision		Revision as of 23:20, 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>


	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: /* The Core */

2015-03-28T22:16:40Z

The Core

← Older revision		Revision as of 22:16, 28 March 2015
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	\biggl[\biggl(\frac{3}{2^4}\biggr) \biggl( \frac{q}{\ell_i}\biggr)^{5}\biggl(\frac{\nu}{q^3} \biggr)^2 \biggl( 1 + \ell_i^2 \biggr)^{3} \biggr]_\mathrm{eq}		\biggl[\biggl(\frac{3}{2^4}\biggr) \biggl( \frac{q}{\ell_i}\biggr)^{5}\biggl(\frac{\nu}{q^3} \biggr)^2 \biggl( 1 + \ell_i^2 \biggr)^{3} \biggr]_\mathrm{eq}
	\biggl[		\biggl[
			\ell_i (\ell_i^4 - \frac{8}{3}\ell_i^2 - 1) (\ell_i^2 +1)^{-3} + \tan^{-1}\ell_i
			\biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>\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 (\ell_i^4 - \frac{8}{3}\ell_i^2 - 1) (\ell_i^2 +1)^{-3} + \tan^{-1}\ell_i		\ell_i (\ell_i^4 - \frac{8}{3}\ell_i^2 - 1) (\ell_i^2 +1)^{-3} + \tan^{-1}\ell_i
	\biggr] \, .		\biggr] \, .

Tohline: /* The Core */ Rewrite (W_grav)_core in more useful form

2015-03-28T22:00:20Z

The Core: Rewrite (W_grav)_core in more useful form

← Older revision		Revision as of 22:00, 28 March 2015
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	</table>		</table>
	</div>		</div>

			<table border="1" cellpadding="5" align="center" width="90%">
			<tr><td align="left">

			MORE USEFUL:

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

			<tr>
			<td align="right">
			<math>~- \chi \biggl[ \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]_\mathrm{core}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>
			\biggl[\biggl(\frac{3}{2^4}\biggr) \biggl( \frac{q}{\ell_i}\biggr)^{5}\biggl(\frac{\nu}{q^3} \biggr)^2 \biggl( 1 + \ell_i^2 \biggr)^{3} \biggr]_\mathrm{eq}
			\biggl[
			\ell_i (\ell_i^4 - \frac{8}{3}\ell_i^2 - 1) (\ell_i^2 +1)^{-3} + \tan^{-1}\ell_i
			\biggr] \, .
			</math>
			</td>
			</tr>
			</table>
			</div>

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

	But, also from our above discussion of the mass profile, we can write,		But, also from our above discussion of the mass profile, we can write,
	<div align="center">		<div align="center">

Tohline: /* The Envelope */

2015-03-28T20:15:42Z

The Envelope

← Older revision		Revision as of 20:15, 28 March 2015
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	</table>		</table>
	</div>		</div>

			<table border="1" cellpadding="5" align="center" width="90%">
			<tr><td align="left">

			MORE USEFUL:

			Letting, <math>\ell \equiv \xi/\sqrt{3}</math>,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~b_\eta = \eta_s</math>
			</td>
			<td align="center">

			and

			</td>
			<td align="left">
			<math>~b_\eta q = \eta_i = 3\biggl( \frac{\mu_e}{\mu_c} \biggr) \ell_i (1 + \ell_i^2)^{-1} \, ,</math>
			</td>
			</tr>
			</table>
			</div>

			<div align="center">
			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="right">
			<math>~\tilde\mathfrak{f}_\mathrm{Menv} \equiv \biggl( \frac{\bar\rho}{\rho_c}\biggr)_\mathrm{env}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ \frac{q^3(1-\nu)}{\nu(1-q^3)} \cdot \tilde\mathfrak{f}_\mathrm{Mcore}
			</math>
			</td>
			</tr>
			</table>
			</div>

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

	==Gravitational Potential Energy==		==Gravitational Potential Energy==

Tohline: /* The Core */ Begin adding "boxed in more useful" relations

2015-03-28T19:56:12Z

The Core: Begin adding "boxed in more useful" relations

← Older revision		Revision as of 19:56, 28 March 2015
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	</div>		</div>

	<table border="1" cellpadding="8" align="center" width="90%">		<table border="1" cellpadding="5" align="center" width="90%">
	<tr><td align="left">		<tr><td align="left">

	MORE USEFUL		MORE USEFUL:

			Letting, <math>\ell \equiv \xi/\sqrt{3}</math>,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~a_\xi</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl( \frac{\ell_i}{q} \biggr)^2 \, ,</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\tilde\mathfrak{f}_\mathrm{Mcore}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl( \frac{\bar\rho}{\rho_c} \biggr)_\mathrm{core}
			=
			(1 + \ell_i^2)^{-3/2} \, .
			</math>
			</td>
			</tr>
			</table>
			</div>

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

Tohline: /* The Core */

2015-03-28T19:33:14Z

The Core

← Older revision		Revision as of 19:33, 28 March 2015
Line 360:		Line 360:
	</table>		</table>
	</div>		</div>

			<table border="1" cellpadding="8" align="center" width="90%">
			<tr><td align="left">

			MORE USEFUL

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

	===The Envelope===		===The Envelope===

Tohline: /* The Envelope */ Finished deriving the thermal energy content of the envelope

2014-09-16T20:15:35Z

The Envelope: Finished deriving the thermal energy content of the envelope

Show changes

Tohline: /* Thermodynamic Energy Reservoir */ Begin deriving envelope's thermal energy

2014-09-16T18:13:32Z

Thermodynamic Energy Reservoir: Begin deriving envelope's thermal energy

← Older revision		Revision as of 18:13, 16 September 2014
Line 1,421:		Line 1,421:
	</div>		</div>
	As it should, the term inside the curly brackets precisely matches the analytic expression for the dimensionless thermal energy of the core, <math>~S^*_\mathrm{core}</math>, that has been derived elsewhere in conjunction with our [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#Expression_for_Free_Energy\|discussion of the detailed force-balanced structure of this bipolytrope]].		As it should, the term inside the curly brackets precisely matches the analytic expression for the dimensionless thermal energy of the core, <math>~S^*_\mathrm{core}</math>, that has been derived elsewhere in conjunction with our [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#Expression_for_Free_Energy\|discussion of the detailed force-balanced structure of this bipolytrope]].


			===The Envelope===

			Similarly, the energy contained in the envelope's thermodynamic reservoir may be written as,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{env}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>
			\frac{2}{3({\gamma_e}-1)} \biggl( \frac{\chi}{\chi_\mathrm{eq}} \biggr)^{3-3\gamma_e} \biggl( \frac{P_{ie}}{P_{ic}} \biggr)
			\biggl[ \frac{2\pi P_{ic} \chi^3}{P_\mathrm{norm}} \biggr]_\mathrm{eq}
			\biggl[ (1-q^3) s_\mathrm{env} \biggr] \, ,
			</math>
			</td>
			</tr>
			</table>
			</div>
			where,
			<div align="center">
			<table border="0" cellpadding="8" align="center">

			<tr>
			<td align="right">
			<math>~(1-q^3) s_\mathrm{env} </math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~
			\int_q^1 3\biggl[ 1 - p_e(x) \biggr] x^2 dx \, ,
			</math>
			</td>
			</tr>
			</table>
			</div>
			defines the relevant integral over the core's pressure distribution. 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 [[User:Tohline/SSC/BipolytropeGeneralization_Version2#Bipolytrope_Generalization\|in our accompanying overview]] — 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 + \frac{1}{3}\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 + a_\xi x^2 \biggr)^{-3} \, ,</math>
			</td>
			</tr>
			</table>
			</div>
			where, <math>~a_\xi</math> is defined above in connection with our [[User:Tohline/SSC/Structure/BiPolytropes/FreeEnergy5_1#Mass_Profile\|derivation of the mass profile]].

	==Virial Theorem==		==Virial Theorem==

Tohline: /* The Core */

2014-09-16T18:03:07Z

The Core

← Older revision		Revision as of 18:03, 16 September 2014
Line 1,420:		Line 1,420:
	</table>		</table>
	</div>		</div>
	As it should, the term inside the curly brackets precisely matches the analytic expression for the dimensionless thermal energy of the core, <math>~S^*_\mathrm{core}</math>, derived in conjunction with our [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#Expression_for_Free_Energy\|discussion of the detailed force-balanced structure of this bipolytrope]].		As it should, the term inside the curly brackets precisely matches the analytic expression for the dimensionless thermal energy of the core, <math>~S^*_\mathrm{core}</math>, that has been derived elsewhere in conjunction with our [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#Expression_for_Free_Energy\|discussion of the detailed force-balanced structure of this bipolytrope]].

	==Virial Theorem==		==Virial Theorem==

Tohline: /* The Core */ Finished derivation of thermodynamic energy reservoir for the core

2014-09-16T16:41:47Z

The Core: Finished derivation of thermodynamic energy reservoir for the core

← Older revision		Revision as of 16:41, 16 September 2014
Line 1,335:		Line 1,335:
	</table>		</table>
	</div>		</div>

			Next, let's examine the factor in square brackets with an "eq" subscript. From 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]], we know that,
			<div align="center">
			<math>P_{ic} = K_c \rho_0^{6/5} \biggl(1 + \frac{1}{3}\xi_i^2 \biggr)^{-3} \, ,</math>
			</div>
			and,
			<div align="center">
			<math>
			\chi_\mathrm{eq} = \biggl(\frac{R_\mathrm{edge}}{R_\mathrm{norm}} \biggr)_\mathrm{eq}
			= \frac{1}{q}\biggl(\frac{r_i}{R_\mathrm{norm}}\biggr)_\mathrm{eq}
			= \frac{1}{q}\biggl[ \frac{K_c^{1/2} G^{-1/2} \rho_0^{-2/5}}{R_\mathrm{norm}}\biggr] \biggl( \frac{3}{2\pi} \biggr)^{1/2} \xi_i \, .
			</math>
			</div>

			Hence, the relevant factor may be rewritten as,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>\frac{2\pi P_{ic} \chi_\mathrm{eq}^3}{P_\mathrm{norm}} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>
			2\pi \biggl[ \frac{K_c \rho_0^{6/5}}{P_\mathrm{norm}}\biggr] \biggl(1 + \frac{1}{3}\xi_i^2 \biggr)^{-3}
			\biggl\{ \frac{1}{q}\biggl[ \frac{K_c^{1/2} G^{-1/2} \rho_0^{-2/5}}{R_\mathrm{norm}}\biggr] \biggl( \frac{3}{2\pi} \biggr)^{1/2} \xi_i \biggr\}^3
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>
			\biggl( \frac{3^3}{2\pi} \biggr)^{1/2} \biggl[ \frac{K_c^{5/2} G^{-3/2}}{P_\mathrm{norm} R_\mathrm{norm}^3 }\biggr] \biggl(1 + \frac{1}{3}\xi_i^2 \biggr)^{-3}
			\biggl( \frac{\xi_i }{q}\biggr)^3
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>
			\biggl( \frac{3^6}{2\pi} \biggr)^{1/2} \biggl(1 + a_\xi q^2 \biggr)^{-3} a_\xi^{3/2} \, ,
			</math>
			</td>
			</tr>
			</table>
			</div>
			where, the last expression has been obtained by employing the substitution, defined above, <math>~\xi_i = (3a_\xi)^{1/2}q</math>. Finally, then, we have,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{core}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>
			\frac{2}{3({\gamma_c}-1)} \biggl( \frac{\chi}{\chi_\mathrm{eq}} \biggr)^{3-3\gamma_c}
			\biggl\{\biggl( \frac{3^8}{2^7\pi} \biggr)^{1/2} \biggl[
			\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\} \, .
			</math>
			</td>
			</tr>
			</table>
			</div>
			As it should, the term inside the curly brackets precisely matches the analytic expression for the dimensionless thermal energy of the core, <math>~S^*_\mathrm{core}</math>, derived in conjunction with our [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#Expression_for_Free_Energy\|discussion of the detailed force-balanced structure of this bipolytrope]].

	==Virial Theorem==		==Virial Theorem==