Tohline: /* Thermodynamic Energy Reservoir */ Add ID tag to expressions for the internal energy of core and envelope

2016-07-25T15:19:36Z

Thermodynamic Energy Reservoir: Add ID tag to expressions for the internal energy of core and envelope

← Older revision		Revision as of 15:19, 25 July 2016
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	Finally, then, we have,		<span id="InternalEnergies">Finally, then, we have,</span>
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 734:		Line 734:
	</table>		</table>
	</div>		</div>



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

Tohline at 23:19, 10 July 2016

2016-07-10T23:19:31Z

← Older revision		Revision as of 23:19, 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: /* Related Discussions */ Expand "related discussions" subsection

2014-08-29T19:09:11Z

Related Discussions: Expand "related discussions" subsection

← Older revision		Revision as of 19:09, 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>.

	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: /* Virial Theorem */

2014-08-29T18:28:48Z

Virial Theorem

← Older revision		Revision as of 18:28, 29 August 2014
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	</div>		</div>

	Plugging these expressions into the equilibrium condition ~~derived~~ above, and setting the interface pressures equal to one another, gives,		Plugging these expressions into the equilibrium condition shown above, and setting the interface pressures equal to one another, gives,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Free Energy of BiPolytrope with ~(n_c, n_e) = (0, 0) */ Finished moving derivations from earlier overview chapter into this page

2014-08-29T16:29:15Z

Free Energy of BiPolytrope with ~(n_c, n_e) = (0, 0): Finished moving derivations from earlier overview chapter into this page

Show changes

Tohline: Begin building page to present bipolytrope with (n_c, n_e) = (0, 0)

2014-08-29T16:14:56Z

Begin building page to present bipolytrope with (n_c, n_e) = (0, 0)

New page

__FORCETOC__
=Free Energy of BiPolytrope with <math>~(n_c, n_e) = (0, 0)</math>=
{{LSU_HBook_header}}

Here we present a specific example of the equilibrium structure of a bipolytrope as determined from a free-energy analysis. The example is a bipolytrope whose core has a polytropic index, <math>~n_c = 0</math>, and whose envelope has a polytropic index, <math>~n_e = 0</math>. The details presented here build upon an [[User:Tohline/SSC/BipolytropeGeneralization_Version2|overview of the free energy of bipolytropes that has been presented elsewhere]].

==Preliminaries==

==Mass Profile==

In this case, <math>~\rho_\mathrm{core}(x) = \rho_c = </math> constant — hence, also, <math>~[\rho(x)/\bar\rho]_\mathrm{core} = 1</math> — and <math>~\rho_\mathrm{env}(x) = \rho_e = </math> constant — hence, also, <math>~[\rho(x)/\bar\rho]_\mathrm{env} = 1</math> — but in general <math>~\rho_e \ne \rho_c</math>. Performing the separate integrals to obtain expressions for <math>~M_r(r)</math> inside the core and the envelope, [[User:Tohline/SSC/BipolytropeGeneralization_Version2#Partitioning_the_Mass|as established in our accompanying overview]], we obtain:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>(\mathrm{For}~0 \leq x \leq q)</math>      
<math>~M_r </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
M_\mathrm{tot} \biggl( \frac{\nu}{q^3} \biggr) \int_0^{x} 3x^2 dx = \nu M_\mathrm{tot} \biggl( \frac{x}{q} \biggr)^3 \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>(\mathrm{For}~q \leq x \leq 1)</math>      
<math>~M_r </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
M_\mathrm{tot} \biggl\{\nu + \biggl( \frac{1-\nu}{1-q^3} \biggr) \int_{q}^{x} 3
x^2 dx \biggr\}
= M_\mathrm{core} + (1-\nu) M_\mathrm{tot}\biggl( \frac{x^3 - q^3}{1-q^3} \biggr)\, .
</math>
</td>
</tr>
</table>
</div>
When <math>~x = q</math>, both expressions give,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~M_r</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~M_\mathrm{core} = \nu M_\mathrm{tot} \, ,</math>
</td>
</tr>
</table>
</div>
as they should. We deduce, as well, that the mass contained in the envelope is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~M_\mathrm{env}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~M_\mathrm{tot} - M_\mathrm{core} = (1-\nu) M_\mathrm{tot} \, ,</math>
</td>
</tr>
</table>
</div>
and that the volumes occupied by the core and envelope are, respectively,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~V_\mathrm{core}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~q^3 R_\mathrm{edge}^3 \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~V_\mathrm{env}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(1- q^3) R_\mathrm{edge}^3 \, .</math>
</td>
</tr>

</table>
</div>

Hence, the ratio of envelope density to core density is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\frac{\rho_e}{\rho_c} = \frac{\bar\rho_\mathrm{env}}{\bar\rho_\mathrm{core}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{M_\mathrm{env}/V_\mathrm{env}}{M_\mathrm{core}/V_\mathrm{core}} = \frac{q^3(1-\nu)}{\nu(1-q^3)} \, .
</math>
</td>
</tr>
</table>
</div>

These relations should be compared to — and ultimately must match — the prescriptions for <math>~M_r</math> that have been presented elsewhere in connection with [[User:Tohline/SSC/Structure/BiPolytropes/Analytic0_0#BiPolytrope_with_nc_.3D_0_and_ne_.3D_0|detailed force-balance models of <math>~(n_c, n_e) = (0, 0)</math> bipolytropes]] and in our introductory discussion of [[User:Tohline/SSC/VirialStability#Expressions_for_Mass|the virial stability of bipolytropes]].

==Gravitational Potential Energy==
Here we follow the steps that have been [[User:Tohline/SSC/BipolytropeGeneralization_Version2#Separate_Contributions_to_Gravitational_Potential_Energy|outlined in an accompanying overview]] to determine the separate contributions to the gravitational potential energy. Let's do the core first. In this case, <math>~\rho_\mathrm{core}(x) = \rho_c = </math> constant — hence, also, <math>~[\rho(x)/\bar\rho]_\mathrm{core} = 1</math>. As has been demonstrated above, the corresponding <math>~M_r</math> function is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl[ \frac{M_r(x)}{M_\mathrm{tot}}\biggr]_\mathrm{core} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\nu}{q^3} \biggr) x^3 \, .
</math>
</td>
</tr>

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

<tr>
<td align="right">
<math>~W_\mathrm{grav}\biggr|_\mathrm{core}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- E_\mathrm{norm} \cdot \chi^{-1} \biggl( \frac{\nu}{q^3} \biggr)
\int_0^{q} 3\biggl( \frac{\nu}{q^3} \biggr)x^4 dx
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- E_\mathrm{norm} \cdot \chi^{-1} \biggl( \frac{\nu}{q^3} \biggr)^2 \biggl( \frac{3}{5} q^5 \biggr)
</math>
</td>
</tr>

</table>
</div>

Now, let's do the envelope. In this case, <math>~\rho_\mathrm{env}(x) = \rho_e = </math> constant; hence, also, <math>~[\rho(x)/\bar\rho]_\mathrm{env} = 1</math>. As [[User:Tohline/SSC/BipolytropeGeneralization#ExampleMass|shown elsewhere]], the corresponding <math>~M_r</math> function is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl[ \frac{M_r(x)}{M_\mathrm{tot}} \biggr]_\mathrm{env} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
\nu + \biggl(\frac{1-\nu}{1-q^3} \biggr) (x^3 - q^3) \, .
</math>
</td>
</tr>
</table>
</div>
Hence,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~W_\mathrm{grav}\biggr|_\mathrm{env}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- E_\mathrm{norm} \cdot \chi^{-1} \biggl( \frac{1-\nu}{1-q^3} \biggr)
\biggl\{ \int_{q}^{1}
\biggl[ \nu -q^3 \biggl(\frac{1-\nu}{1-q^3} \biggr) \biggr]3x dx + \int_{q}^{1} \biggl[ \biggl(\frac{1-\nu}{1-q^3} \biggr) \biggr] 3x^4 dx
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- E_\mathrm{norm} \cdot \chi^{-1} \biggl( \frac{1-\nu}{1-q^3} \biggr)
\biggl\{ \frac{3}{2}
\biggl[ \nu -q^3 \biggl(\frac{1-\nu}{1-q^3} \biggr) \biggr] (1-q^2) + \frac{3}{5} \biggl[ \biggl(\frac{1-\nu}{1-q^3} \biggr) \biggr] (1-q^5)
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- \frac{3}{5} \biggl(\frac{\nu^2}{q} \biggr) E_\mathrm{norm} \cdot \chi^{-1}
\biggl[ \frac{1}{\nu} \biggl( \frac{1-\nu}{1-q^3} \biggr)\biggr] \biggl\{ \frac{5}{2}
\biggl[ 1 - \frac{q^3}{\nu} \biggl(\frac{1-\nu}{1-q^3} \biggr) \biggr] (q-q^3) + \biggl[ \frac{q}{\nu} \biggl(\frac{1-\nu}{1-q^3} \biggr) \biggr] (1-q^5)
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- \frac{3}{5} \biggl(\frac{\nu^2}{q} \biggr) E_\mathrm{norm} \cdot \chi^{-1}
\biggl[ \frac{q^3}{\nu} \biggl( \frac{1-\nu}{1-q^3} \biggr)\biggr] \biggl\{ \frac{5}{2}
\biggl[ 1 - \frac{q^3}{\nu} \biggl(\frac{1-\nu}{1-q^3} \biggr) \biggr] \biggl(\frac{1}{q^2}-1 \biggr) + \biggl[ \frac{q^3}{\nu} \biggl(\frac{1-\nu}{1-q^3} \biggr)
\biggr] \biggl( \frac{1}{q^5}-1\biggr) \biggr\} \, .
</math>
</td>
</tr>

</table>
</div>
Realizing from the above mass segregation derivation that,
<div align="center">
<math>~\frac{q^3}{\nu} \biggl( \frac{1-\nu}{1-q^3} \biggr) = \frac{\rho_e}{\rho_c} \, ,</math>
</div>
this last expression can be rewritten as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~W_\mathrm{grav}\biggr|_\mathrm{env}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- \frac{3}{5} \biggl(\frac{\nu^2}{q} \biggr) E_\mathrm{norm} \cdot \chi^{-1}
\biggl[ \frac{\rho_e}{\rho_c} \biggr] \biggl\{ \frac{5}{2}
\biggl[ 1 - \frac{\rho_e}{\rho_c} \biggr] \biggl(\frac{1}{q^2}-1 \biggr) + \biggl[ \frac{\rho_e}{\rho_c}
\biggr] \biggl( \frac{1}{q^5}-1\biggr) \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- \frac{3}{5} \biggl(\frac{\nu^2}{q} \biggr) E_\mathrm{norm} \cdot \chi^{-1}
\biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl\{ \frac{5}{2}\biggl(\frac{1}{q^2}-1 \biggr) +\biggl( \frac{\rho_e}{\rho_c}
\biggr) \biggl[ \frac{1}{q^5}-1 + \frac{5}{2} \biggl( 1-\frac{1}{q^2}\biggr)\biggr] \biggr\} \, .
</math>
</td>
</tr>

</table>
</div>
So, when put together to obtain the total gravitational potential energy, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~W_\mathrm{grav} = W_\mathrm{grav}\biggr|_\mathrm{core} + W_\mathrm{grav}\biggr|_\mathrm{env}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- \frac{3}{5} E_\mathrm{norm} \cdot \chi^{-1} \biggl(\frac{\nu^2}{q} \biggr) f(\nu,q) \, ,
</math>
</td>
</tr>

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

<tr>
<td align="right">
<math>~f(\nu,q)</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>
1+
\frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl(\frac{1}{q^2}-1 \biggr) +\biggl( \frac{\rho_e}{\rho_c}
\biggr)^2 \biggl[ \frac{1}{q^5}-1 + \frac{5}{2} \biggl( 1-\frac{1}{q^2}\biggr)\biggr] \, .
</math>
</td>
</tr>

</table>
</div>
(This result agrees with Tohline's earlier derivations in other sections of this H_Book, which may now be erased to avoid repetition.)

==Thermodynamic Energy Reservoir==

==Virial Theorem==

=Related Discussions=
* [[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>.

{{LSU_HBook_footer}}

User:Tohline/SSC/Structure/BiPolytropes/FreeEnergy0 0 - Revision history

Tohline: /* Thermodynamic Energy Reservoir */ Add ID tag to expressions for the internal energy of core and envelope

Tohline at 23:19, 10 July 2016

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

Tohline: /* Virial Theorem */

Tohline: /* Free Energy of BiPolytrope with ~(n_c, n_e) = (0, 0) */ Finished moving derivations from earlier overview chapter into this page

Tohline: Begin building page to present bipolytrope with (n_c, n_e) = (0, 0)