Joel2: /* The Envelope */

2024-01-18T19:05:32Z

The Envelope

← Older revision		Revision as of 15:05, 18 January 2024
Line 279:		Line 279:
	</table>		</table>
	</div>		</div>
	But we also know, [[SSC/Structure/~~BiPolytropes~~/FreeEnergy51#~~The_Envelope~~\|from above]], that,		But we also know, [[SSC/Structure/BiiPolytropes/FreeEnergy51#EquilibriumRadius\|from above]], that,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Joel2: Created page with "FORCETOC =Free Energy of BiPolytrope with (n_c, n_e) = (5, 1)=

Part I: Mass Profile
Part II: Gravitational Potential Energy
2024-01-18T18:45:56Z

Created page with "FORCETOC =Free Energy of BiPolytrope with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1)= <table border="1" align="center" width="100%" colspan="8"> <td align="center" bgcolor="lightblue" width="33%"><br />Part I: Mass Profile </td> <td align="center" bgcolor="lightblue"3 width="33%"><br />Part II: Gravitational Potential Energy </td> <td align="c..."

New page
FORCETOC
=Free Energy of BiPolytrope with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1)=
<table border="1" align="center" width="100%" colspan="8">
<td align="center" bgcolor="lightblue" width="33%"><br />[[SSC/Structure/BiiPolytropes/FreeEnergy51|Part I:  Mass Profile]]
 
</td>
<td align="center" bgcolor="lightblue"3 width="33%"><br />[[SSC/Structure/BiPolytropes/FreeEnergy51/Pt2|Part II:  Gravitational Potential Energy]]
 
</td>
<td align="center" bgcolor="lightblue"><br />[[SSC/Structure/BiPolytropes/FreeEnergy51/Pt3|Part III:  Thermal Energy Reservoir]]
 
</td>
</tr>
</table>

==Gravitational Potential Energy==

===The Core===

Borrowing from our derivation, above, of the mass distribution in this type of bipolytrope, the expression for the gravitational potential energy in the core that has been [[SSC/BipolytropeGeneralizationVersion2#Separate_Contributions_to_Gravitational_Potential_Energy|outlined in our accompanying overview]] may be written as,
<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} \biggl(\frac{\rho_0}{\bar\rho} \biggr)_\mathrm{core} \biggr]_\mathrm{eq}
\int_0^{q} 3x \biggl[\frac{M_r(x)}{M_\mathrm{tot}} \biggr]_\mathrm{core} \biggl[ \frac{\rho(x)}{\rho_0} \biggr]_\mathrm{core} 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} \biggl( 1 + a_\xi q^2 \biggr)^{3/2} \biggr]_\mathrm{eq}
\int_0^{q} 3x \biggl\{
\nu \biggl( \frac{x^3}{q^3} \biggr) \biggl[ \frac{ 1 + a_\xi x^2 }{ 1 + a_\xi q^2 } \biggr]^{-3/2}
\biggr\}
\biggl( 1 + a_\xi x^2 \biggr)^{-5/2} 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[ 3\biggl(\frac{\nu}{q^3} \biggr)^2 \biggl( 1 + a_\xi q^2 \biggr)^{3} \biggr]_\mathrm{eq}
\int_0^{q} x^4 \biggl( 1 + a_\xi x^2 \biggr)^{-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[ 3\biggl(\frac{\nu}{q^3} \biggr)^2 \biggl( 1 + a_\xi q^2 \biggr)^{3} \biggr]_\mathrm{eq}
\biggl\{
\frac{a_\xi^{1/2} q(3a_\xi^2 q^4 - 8a_\xi q^2 - 3) + 3(a_\xi q^2 +1)^3 \tan^{-1}(a_\xi^{1/2} q)}{48 a_\xi^{5/2}(a_\xi q^2 + 1)^3}
\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[\biggl(\frac{3}{2^4}\biggr) a_\xi^{-5/2}\biggl(\frac{\nu}{q^3} \biggr)^2 \biggl( 1 + a_\xi q^2 \biggr)^{3} \biggr]_\mathrm{eq}
\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] \, .
</math>
</td>
</tr>

</table>
</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>

<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
\biggr] \, .
</math>
</td>
</tr>
</table>
</div>

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

But, also from our above discussion of the mass profile, we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~a_\xi^{-5/2} \biggl( \frac{\nu}{q^3} \biggr)^2 (1 + a_\xi q^2)^3</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\chi_\mathrm{eq} \biggl( \frac{2^3 \cdot 3^6}{\pi} \biggr)^{1/2} \, .</math>
</td>
</tr>
</table>
</div>
Hence,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl( \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr)_\mathrm{core} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- \frac{\chi_\mathrm{eq}}{\chi} \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] \, .
</math>
</td>
</tr>
</table>
</div>
After making the substitution, <math>~(a_\xi^{1/2} q) \rightarrow x_i</math>, this expression agrees with a result for the dimensionless energy, <math>~W^*_\mathrm{core}</math>, [[SSC/Structure/BiPolytropes/Analytic51#Expression_for_Free_Energy|derived by Tohline in the context of detailed force-balanced bipolytropes]].

===The Envelope===

Again, borrowing from our derivation, above, of the mass distribution in this type of bipolytrope, the expression for the gravitational potential energy in the envelope that has been [[SSC/BipolytropeGeneralizationVersion2#Separate_Contributions_to_Gravitational_Potential_Energy|outlined in our accompanying overview]] may be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl( \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr)_\mathrm{env}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- \chi^{-1} \biggl[ \biggl( \frac{1-\nu}{1-q^3} \biggr) \biggl( \frac{\rho_0}{\bar\rho}\biggr)_\mathrm{env} \biggr]_\mathrm{eq}
\int_{q}^{1} 3x \biggl[\frac{M_r(x)}{M_\mathrm{tot}} \biggr]_\mathrm{env} \biggl[ \frac{\rho(x)}{\rho_0} \biggr]_\mathrm{env} dx
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- \chi^{-1} \biggl[ \frac{\nu}{q^3 \theta_i^3} \biggr]
\int_{q}^{1} 3x \biggl\{ \frac{\nu}{C_1} \biggl[ \sin(B-b_\eta x) + xb_\eta \cos(B - b_\eta x) \biggr]\biggr\}
\biggl\{ A \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta_i^5 \biggl[ \frac{\sin(b_\eta x - B)}{b_\eta x} \biggr] \biggr\} dx
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- \chi^{-1} \biggl( \frac{\mu_e}{\mu_c} \biggr) \biggl[ \frac{3\nu^2 A \theta_i^2}{b_\eta q^3 } \biggr] \frac{1}{C_1}
\int_{q}^{1} [ xb_\eta \cos(b_\eta x-B) - \sin(b_\eta x- B) ]\sin(b_\eta x - B) dx
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- \chi^{-1} \biggl( \frac{\mu_e}{\mu_c} \biggr) \biggl[ \frac{3\nu^2 A \theta_i^2}{b_\eta q^3 } \biggr]
\biggl[ - 3A\theta_i^2 \biggl( \frac{\mu_e}{\mu_c} \biggr) (b_\eta q)^{-3} \biggr]
\int_{q}^{1} [ xb_\eta \cos(b_\eta x-B) - \sin(b_\eta x- B) ]\sin(b_\eta x - B) dx
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- \chi^{-1} \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)
\int_{q}^{1} [\sin(b_\eta x- B) - xb_\eta \cos(b_\eta x-B) ]\sin(b_\eta x - B) dx \, .
</math>
</td>
</tr>

</table>
</div>
But we also know, [[SSC/Structure/BiPolytropes/FreeEnergy51#The_Envelope|from above]], that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\chi_\mathrm{eq} = b_\eta^{-5} \biggl( \frac{3^4 \pi}{2^3} \biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^5 \frac{\nu^2 \theta_i^4}{q^6}</math>
</td>
<td align="center">
    <math>\Rightarrow </math>    
</td>
<td align="left">
<math>
~\frac{\nu^2 \theta_i^4}{q^6} = \chi_\mathrm{eq} b_\eta^{5} \biggl( \frac{2^3}{3^4 \pi} \biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5} \, .
</math>
</td>
</tr>
</table>
</div>
So we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl( \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr)_\mathrm{env}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- \frac{\chi_\mathrm{eq}}{\chi} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \biggl( \frac{2^3}{\pi} \biggr)^{1/2} A^2 b_\eta
\int_{q}^{1} [\sin(b_\eta x- B) - xb_\eta \cos(b_\eta x-B) ]\sin(b_\eta x - B) dx \, .
</math>
</td>
</tr>

</table>
</div>

The integral can be broken into two separate parts:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\int_{q}^{1} \sin^2(b_\eta x - B) dx
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{4b_\eta} \biggl\{2 b_\eta x - \sin[2(b_\eta x-B)] \biggr\}_q^1 \, ,
</math>
</td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
- \int_{q}^{1} xb_\eta \cos(b_\eta x-B)\sin(b_\eta x - B) dx
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{8b_\eta} \biggl\{2b_\eta x \cos[2(b_\eta x - B)] - \sin[2(b_\eta x-B)] \biggr\}_q^1 \, .</math>
</td>
</tr>
</table>
</div>
(Note: We have dropped integration constants that might result from carrying out an indefinite integral because such constants would disappear upon application of our specified limits of integration.) When added together, they give,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\int_q^1 \ldots ~dx</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
\frac{1}{8b_\eta} \biggl\{2b_\eta x \cos[2(b_\eta x - B)] - 3\sin[2(b_\eta x-B)] + 4 b_\eta x \biggr\}_q^1
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
\frac{1}{8b_\eta} \biggl\{2b_\eta x \biggl[ 1 - 2\sin^2(b_\eta x - B) \biggr] - 3\sin[2(b_\eta x-B)] + 4 b_\eta x \biggr\}_q^1
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
\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>

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

<tr>
<td align="right">
<math>~\biggl( \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr)_\mathrm{env}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- \frac{\chi_\mathrm{eq}}{\chi} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \biggl( \frac{1}{2^3\pi} \biggr)^{1/2} A^2
\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>

</table>
</div>

This expression matches in detail the expression for the gravitational potential energy of the envelope derived in the context of our [[SSC/Structure/BiPolytropes/Analytic51#Expression_for_Free_Energy|derivation of detailed force-balanced models of this bipolytrope]].

=See Also=

{{ SGFfooter }}

SSC/Structure/BiPolytropes/FreeEnergy51/Pt2 - Revision history

Joel2: /* The Envelope */