User:Tohline/SSC/BipolytropeGeneralization - Revision history

Tohline: /* Bipolytrope Generalization */Explain that this chapter contains old derivations and is labeled in entirety as Work in Progress

2014-08-26T18:24:37Z

Bipolytrope Generalization: Explain that this chapter contains old derivations and is labeled in entirety as Work in Progress

← Older revision		Revision as of 18:24, 26 August 2014
Line 2:		Line 2:
	=Bipolytrope Generalization=		=Bipolytrope Generalization=
	{{LSU_HBook_header}}		{{LSU_HBook_header}}

			On <font color="red">26 August 2014</font>, Tohline finished rewriting the chapter titled "Bipolytrope Generalization" in a very concise manner ([[User:Tohline/SSC/BipolytropeGeneralization_Version2#Bipolytrope_Generalization\|go here for this Version2 chapter]]) then set this chapter aside to provide a collection of older attempts at the derivations. While much of what follows is technically correct, it is overly detailed and cumbersome. Because it likely also contains some misguided steps, we label it in entirety as Work in Progress.


			{{LSU_WorkInProgress}}

	==Old Stuff==		==Old Stuff==

Tohline at 13:55, 26 August 2014

2014-08-26T13:55:17Z

Show changes

Tohline: /* Related Discussions */

2014-08-26T13:42:39Z

Related Discussions

← Older revision		Revision as of 13:42, 26 August 2014
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	=Related Discussions=		=Related Discussions=
			* [[User:Tohline/SSC/BipolytropeGeneralization_Old\|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/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>.		* [[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: /* Extrema and Virial Equilibrium */ Virial equilibrium statement

2014-08-25T21:29:35Z

Extrema and Virial Equilibrium: Virial equilibrium statement

← Older revision		Revision as of 21:29, 25 August 2014
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~- \chi_\mathrm{eq}^{-1}		<math>~- \frac{\chi_\mathrm{eq}^{-1} }{E_\mathrm{norm}}
	\biggl[ ~~\biggl~~( ~~\frac{~~W_\mathrm{grav}~~}{E_\mathrm{norm}} \biggr~~)_\mathrm{core} + ~~\biggl~~( ~~\frac{~~W_\mathrm{grav}~~}{E_\mathrm{norm}} \biggr~~)_\mathrm{env}		\biggl[ ( W_\mathrm{grav})_\mathrm{core} + ( W_\mathrm{grav} )_\mathrm{env}
	+ 3 (\gamma_c-1) ~~\biggl~~( ~~\frac{~~\mathfrak{S}_A~~}{E_\mathrm{norm}} \biggr~~)_\mathrm{core}		+ 3 (\gamma_c-1) ( \mathfrak{S}_A )_\mathrm{core}
	+ 3 (\gamma_e-1) ~~\biggl~~( ~~\frac{~~\mathfrak{S}_A~~}{E_\mathrm{norm}} \biggr~~)_\mathrm{env}		+ 3 (\gamma_e-1) ( \mathfrak{S}_A)_\mathrm{env}
	\biggr]_\mathrm{eq} \, .		\biggr]_\mathrm{eq} \, .
	</math>		</math>
Line 2,099:		Line 2,099:
	</table>		</table>
	</div>		</div>
	Drawing from our [[User:Tohline/VE#Relationship_to_the_System.27s_Internal_Energy\|introductory discussion of the reservoir of thermodynamic energy]], we note that, for adiabatic systems, <math>~\mathfrak{G}_A</math> is equivalent to the internal energy of the system and therefore its relationship to ~~the~~ the thermal energy, <math>~S_\mathrm{therm}</math>, is,		Drawing from our [[User:Tohline/VE#Relationship_to_the_System.27s_Internal_Energy\|introductory discussion of the reservoir of thermodynamic energy]], we note that, for adiabatic systems, <math>~\mathfrak{G}_A</math> is equivalent to the internal energy of the system and therefore its relationship to the thermal energy, <math>~S_\mathrm{therm}</math>, is,
	<div align="center">		<div align="center">
	<math>~\mathfrak{G}_A = \frac{2}{3(\gamma-1)} S_\mathrm{therm} \, .</math>		<math>~\mathfrak{G}_A = \frac{2}{3(\gamma-1)} S_\mathrm{therm} \, .</math>
	</div>		</div>
	(This applies separately for the core and the envelope.) We therefore recognize that our derived expression for equilibrium systems is none other than the virial theorem applied to bipolytropic configurations.		(This applies separately for the core and the envelope.) We therefore recognize that our derived expression for equilibrium systems is none other than the virial theorem applied to bipolytropic configurations, specifically, in equilibrium,
			<div align="center">
			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="right">
			<math>~
			( W_\mathrm{grav})_\mathrm{core} + ( W_\mathrm{grav} )_\mathrm{env} + 2 [
			( S_\mathrm{therm} )_\mathrm{core} + ( S_\mathrm{therm} )_\mathrm{env} ]
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~0 \, .</math>
			</td>
			</tr>
			</table>
			</div>

	==Old Stuff==		==Old Stuff==

Tohline: /* Generalized Free-Energy Expression */ STart discussion virial theorem

2014-08-25T19:54:25Z

Generalized Free-Energy Expression: STart discussion virial theorem

← Older revision		Revision as of 19:54, 25 August 2014
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	</div>		</div>

			==Extrema and Virial Equilibrium==
			Extrema arise in the free-energy function wherever,
			<div align="center">
			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="right">
			<math>~\frac{\partial \mathfrak{G}^*}{\partial \chi}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~0 \, ,</math>
			</td>
			</tr>
			</table>
			</div>
			that is, when,
			<div align="center">
			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="right">
			<math>~3\mathcal{A}\chi^{-2} - 3\mathcal{B}_\mathrm{core} \chi^{2-3\gamma_c} - 3\mathcal{B}_\mathrm{env} \chi^{2-3\gamma_e}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~0 \, .</math>
			</td>
			</tr>
			</table>
			</div>
			Values of the dimensionless variable, <math>~\chi</math>, that provide solutions to this algebraic equation identify the size of equilibrium configurations and will henceforth be labeled with the "eq" subscript, that is,
			<div align="center">
			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="right">
			<math>~\chi ~~~ \rightarrow ~~~ \chi_\mathrm{eq}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{R_\mathrm{eq}}{R_\mathrm{norm}} \, .</math>
			</td>
			</tr>
			</table>
			</div>
			We can rewrite the equilibrium condition as,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~0 </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~- \chi_\mathrm{eq}^{-1}
			\biggl[ -3\mathcal{A}\chi_\mathrm{eq}^{-1} -
			3\mathcal{B}_\mathrm{core} \chi_\mathrm{eq}^{3-3\gamma_c} - 3\mathcal{B}_\mathrm{env} \chi_\mathrm{eq}^{3-3\gamma_e} \biggr]_\mathrm{eq}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~- \chi_\mathrm{eq}^{-1}
			\biggl[ \biggl( \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr)_\mathrm{core} + \biggl( \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr)_\mathrm{env}
			+ 3 (\gamma_c-1) \biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{core}
			+ 3 (\gamma_e-1) \biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{env}
			\biggr]_\mathrm{eq} \, .
			</math>
			</td>
			</tr>
			</table>
			</div>
			Drawing from our [[User:Tohline/VE#Relationship_to_the_System.27s_Internal_Energy\|introductory discussion of the reservoir of thermodynamic energy]], we note that, for adiabatic systems, <math>~\mathfrak{G}_A</math> is equivalent to the internal energy of the system and therefore its relationship to the the thermal energy, <math>~S_\mathrm{therm}</math>, is,
			<div align="center">
			<math>~\mathfrak{G}_A = \frac{2}{3(\gamma-1)} S_\mathrm{therm} \, .</math>
			</div>
			(This applies separately for the core and the envelope.) We therefore recognize that our derived expression for equilibrium systems is none other than the virial theorem applied to bipolytropic configurations.



	===Old Stuff===


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

Tohline: /* Generalized Free-Energy Expression */

2014-08-25T19:00:21Z

Generalized Free-Energy Expression

← Older revision		Revision as of 19:00, 25 August 2014
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	<td align="left">		<td align="left">
	<math>		<math>
	4\pi \biggl[ \frac{P_{ic} \chi^{3\gamma_c}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \int_0^q \biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr] x^2 dx \, ,		\frac{4\pi}{3} \biggl[ \frac{P_{ic} \chi^{3\gamma_c}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \int_0^q 3\biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr] x^2 dx \, ,
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>		<math>
	4\pi\biggl[ \frac{P_{ie} \chi^{3\gamma_e}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \int_q^1 \biggl[1 - p_e(x) \biggr] x^2 dx \, .		\frac{4\pi}{3} \biggl[ \frac{P_{ie} \chi^{3\gamma_e}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \int_q^1 3\biggl[1 - p_e(x) \biggr] x^2 dx \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Bipolytrope Generalization */ Gathered terms to write generalized free-energy expression

2014-08-25T18:55:02Z

Bipolytrope Generalization: Gathered terms to write generalized free-energy expression

← Older revision		Revision as of 18:55, 25 August 2014
Line 1,925:		Line 1,925:
	</table>		</table>
	</div>		</div>

			==Generalized Free-Energy Expression==
			Bringing all of these expressions together, the normalized free-energy function for bipolytropes is,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\mathfrak{G}^* \equiv \frac{\mathfrak{G}}{E_\mathrm{norm}}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl( \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr)_\mathrm{core}
			+ \biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{core}
			+ \biggl( \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr)_\mathrm{env}
			+ \biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{env}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			-3\mathcal{A} \chi^{-1} - \frac{\mathcal{B}_\mathrm{core}}{(1-\gamma_c)} \chi^{3-3\gamma_c} - \frac{\mathcal{B}_\mathrm{env}}{(1-\gamma_e)} \chi^{3-3\gamma_e} \, ,
			</math>
			</td>
			</tr>
			</table>
			</div>
			where,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\mathcal{A}</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>
			\biggl( \frac{\nu}{q^3} \biggr) \int_0^{q} \biggl[\frac{M_r(x)}{M_\mathrm{tot}} \biggr]_\mathrm{core} \biggl[ \frac{\rho(x)}{\bar\rho} \biggr]_\mathrm{core} x dx
			~+
			\biggl( \frac{1-\nu}{1-q^3} \biggr) \int_{q}^{1} \biggl[\frac{M_r(x)}{M_\mathrm{tot}} \biggr]_\mathrm{env} \biggl[ \frac{\rho(x)}{\bar\rho} \biggr]_\mathrm{env} x dx \, ,
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\mathcal{B}_\mathrm{core}</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>
			4\pi \biggl[ \frac{P_{ic} \chi^{3\gamma_c}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \int_0^q \biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr] x^2 dx \, ,
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\mathcal{B}_\mathrm{env}</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>
			4\pi\biggl[ \frac{P_{ie} \chi^{3\gamma_e}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \int_q^1 \biggl[1 - p_e(x) \biggr] x^2 dx \, .
			</math>
			</td>
			</tr>
			</table>
			</div>





	===Old Stuff===		===Old Stuff===

Tohline: /* Sanity Check */ Success!

2014-08-25T17:13:56Z

Sanity Check: Success!

← Older revision		Revision as of 17:13, 25 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>
			</table>
			</div>
			Now, a bit of algebra shows that,
			<div align="center">
			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="right">
			<math>~\frac{2}{5}q^5 (f - 1 - \mathfrak{F} )</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\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] \, .</math>
			</td>
			</tr>
			</table>
			</div>

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

			<tr>
			<td align="right">
			<math>~\frac{1}{b_\xi} =
			\biggl( \frac{2^3 \pi}{3} \biggr) \frac{P_0 R_\mathrm{edge}^4}{G M_\mathrm{tot}^2 } \biggl( \frac{q^3}{\nu}\biggr)^2
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~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] </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>

Tohline: /* Sanity Check */ Improve sanity Check discussion

2014-08-25T16:46:30Z

Sanity Check: Improve sanity Check discussion

Show changes

Tohline: /* Separate Thermodynamic Energy Reservoirs */ Rework the definition of coefficient b_\xi

2014-08-24T21:59:28Z

Separate Thermodynamic Energy Reservoirs: Rework the definition of coefficient b_\xi

← Older revision		Revision as of 21:59, 24 August 2014
Line 1,382:		Line 1,382:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~R_\mathrm{~~norm~~} ~~\biggl[~~ \biggl( \frac{G \~~rho_0^~~{2- {\~~gamma_c~~} ~~}}{K_c~~} \biggr)^{1/2} ~~\chi \biggr]~~_\mathrm{eq} x</math>		<math>~\biggl[
			\frac{G R_\mathrm{edge}^2}{P_0} \biggl( \frac{3 \nu M_\mathrm{tot}}{4\pi q^3 R_\mathrm{edge}^3} \biggr)^2
			\biggr]^{1/2}_\mathrm{eq} x</math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\~~chi_~~\~~mathrm~~{eq} \~~biggl[~~ \frac{G R_\mathrm{~~norm~~}^2 ~~\rho_~~\mathrm{~~norm~~}^~~{2-~~ {\~~gamma_c} }~~}{~~K_c~~} \biggr]^~~{1/~~2~~} (~~\~~rho_0^*)~~^{~~(2- {\gamma_c} )~~/2}_\mathrm{eq} ~~\cdot~~ x</math>		<math>~\biggl[ \biggl( \frac{3^2}{2^4 \pi^2} \biggr)
			\frac{G M_\mathrm{tot}^2 }{P_0 R_\mathrm{edge}^4} \biggl( \frac{\nu}{q^3}\biggr)^2
			\biggr]^{1/2}_\mathrm{eq} x</math>
	</td>		</td>
	</tr>		</tr>

	~~<tr>~~
	~~<td align="right">~~
	~~ ~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~\chi_\mathrm{eq} \biggl( \frac{3}{4\pi} \biggr)^{1/2}~~
	~~\biggl[ \biggl( \frac{\rho_0}{\bar\rho} \biggr)_\mathrm{core} \frac{M_\mathrm{core}/M_\mathrm{tot}}{R_\mathrm{core}^3/R_\mathrm{norm}^3}~~
	~~\biggr]^{(2- {\gamma_c} )/2}_\mathrm{eq} x</math>~~
	~~</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right">~~
	~~ ~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~\chi_\mathrm{eq} \biggl( \frac{3}{4\pi} \biggr)^{1/2}~~
	~~\biggl( \frac{\rho_0}{\bar\rho} \biggr)_\mathrm{core}^{(2- {\gamma_c} )/2}~~
	~~\biggl[ \frac{\nu}{q^3} \cdot \chi^{-3}~~
	~~\biggr]^{(2- {\gamma_c} )/2}_\mathrm{eq} x</math>~~
	~~</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right">~~
	~~ ~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~\chi_\mathrm{eq}^{(3 {\gamma_c} -4)/2} \biggl( \frac{3}{4\pi} \biggr)^{1/2}~~
	~~\biggl[ \frac{\nu}{q^3} \biggr]^{(2- {\gamma_c} )/2}_\mathrm{eq} x \, .</math>~~
	~~</td>~~
	~~</tr>~~
	</table>		</table>
	</div>		</div>
	~~[In this last step, we set <math>~(\rho_0/\bar\rho)_\mathrm{core} \rightarrow 1</math> because we are examining the case of a uniform density core.]~~ So, defining the coefficient,		So, defining the coefficient,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 1,452:		Line 1,415:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{1}{2}~ \~~chi_~~\mathrm{eq}^{3 {~~\gamma_c~~} -4} \biggl( \frac{\nu}{q^3} \biggr)^{2~~- {\gamma_c} }~~ \, ,</math>		<math>~\biggl( \frac{3}{2^3 \pi} \biggr)
			\frac{G M_\mathrm{tot}^2 }{P_0 R_\mathrm{edge}^4} \biggl( \frac{\nu}{q^3}\biggr)^2\, ,</math>
	</td>		</td>
	</tr>		</tr>