Appendix/Ramblings/Radiation/InitialTemperatures - Revision history

Jet53man at 21:40, 11 April 2022

2022-04-11T21:40:50Z

← Older revision		Revision as of 17:40, 11 April 2022
Line 3:		Line 3:

	=Initial Temperature Distributions=		=Initial Temperature Distributions=
	In an [[~~User:Tohline/~~Appendix/Ramblings/Radiation/CodeUnits\|accompanying Wiki page]] we've discussed in detail (or see the [[~~User:Tohline/~~Appendix/Ramblings/Radiation/SummaryScalings#Summary_of_Scalings\|summary page]]) how to transform back and forth between cgs units and the dimensionless code units that have been adopted by Dominic Marcello in his radiation-hydro simulations of binary mass-transfer. Here we want to probe in more depth what temperature distributions are obtained from the initial polytropic structure once Dominic chooses particular values of the four scaling parameters: <math>\tilde{r}</math>, <math>\tilde{a}</math>, <math>\tilde{g}</math>, and <math>\tilde{c}</math>.		In an [[Appendix/Ramblings/Radiation/CodeUnits\|accompanying Wiki page]] we've discussed in detail (or see the [[Appendix/Ramblings/Radiation/SummaryScalings#Summary_of_Scalings\|summary page]]) how to transform back and forth between cgs units and the dimensionless code units that have been adopted by Dominic Marcello in his radiation-hydro simulations of binary mass-transfer. Here we want to probe in more depth what temperature distributions are obtained from the initial polytropic structure once Dominic chooses particular values of the four scaling parameters: <math>\tilde{r}</math>, <math>\tilde{a}</math>, <math>\tilde{g}</math>, and <math>\tilde{c}</math>.

	Our derivation of the temperature distribution will center around the following ideas. First, the initial binary model that Dominic obtains from Wes Even's self-consistent-field (SCF) code obeys a polytropic equation of state (EOS), namely,		Our derivation of the temperature distribution will center around the following ideas. First, the initial binary model that Dominic obtains from Wes Even's self-consistent-field (SCF) code obeys a polytropic equation of state (EOS), namely,
	<div align="center">		<div align="center">
	{{~~User:Tohline/~~Math/EQ_Polytrope01}}		{{Math/EQ_Polytrope01}}
	</div>		</div>
	with an adopted polytropic index {{~~User:Tohline/~~Math/MP_PolytropicIndex}} <math>= 3/2</math>. Hence, at any point inside either star, the pressure (in code units), <math>P_\mathrm{code}</math>, can be obtained from knowledge of the mass-density (in code units), <math>\rho_\mathrm{code}</math>, and the polytropic constant, <math>K_\mathrm{code}</math>, via the relation,		with an adopted polytropic index {{Math/MP_PolytropicIndex}} <math>= 3/2</math>. Hence, at any point inside either star, the pressure (in code units), <math>P_\mathrm{code}</math>, can be obtained from knowledge of the mass-density (in code units), <math>\rho_\mathrm{code}</math>, and the polytropic constant, <math>K_\mathrm{code}</math>, via the relation,
	<div align="center">		<div align="center">
	<math>		<math>
Line 20:		Line 20:
	P_\mathrm{total} = P_\mathrm{gas} + P_\mathrm{deg} + P_\mathrm{rad} ,		P_\mathrm{total} = P_\mathrm{gas} + P_\mathrm{deg} + P_\mathrm{rad} ,
	</math></div>		</math></div>
	where mathematical expressions for the ideal gas pressure, <math>P_\mathrm{gas}</math>, the electron degeneracy pressure, <math>P_\mathrm{deg}</math>, and the photon radiation pressure, <math>P_\mathrm{rad}</math>, are provided in an [[~~User:Tohline/~~SR#Equation_of_State\|accompanying discussion of analytically prescribed equations of state]]. (Actually, Dominic is presently ignoring the effects of <math>P_\mathrm{deg}</math>, but because it allows for a more general treatment at some later date, we will assume the more general expression for <math>P_\mathrm{total}</math> and set <math>P_\mathrm{deg} = 0</math> near the end of our discussion.)		where mathematical expressions for the ideal gas pressure, <math>P_\mathrm{gas}</math>, the electron degeneracy pressure, <math>P_\mathrm{deg}</math>, and the photon radiation pressure, <math>P_\mathrm{rad}</math>, are provided in an [[SR#Equation_of_State\|accompanying discussion of analytically prescribed equations of state]]. (Actually, Dominic is presently ignoring the effects of <math>P_\mathrm{deg}</math>, but because it allows for a more general treatment at some later date, we will assume the more general expression for <math>P_\mathrm{total}</math> and set <math>P_\mathrm{deg} = 0</math> near the end of our discussion.)

	==Various Scalings==		==Various Scalings==
Line 98:		Line 98:

	</table>		</table>
	where {{~~User:Tohline/~~Math/C_ElectronMass}}, {{~~User:Tohline/~~Math/C_AtomicMassUnit}} and {{~~User:Tohline/~~Math/C_FermiPressure}} (the characteristic Fermi pressure) are physical constants defined in our [[~~User:Tohline/~~Appendix/~~Variables_templates~~\|accompanying variables appendix]]. (Numerical values of these constants can be obtained by scrolling the cursor over the symbols for the constants in this last sentence.) This relation also means that, generally,		where {{Math/C_ElectronMass}}, {{Math/C_AtomicMassUnit}} and {{Math/C_FermiPressure}} (the characteristic Fermi pressure) are physical constants defined in our [[Appendix/VariablesTemplates\|accompanying variables appendix]]. (Numerical values of these constants can be obtained by scrolling the cursor over the symbols for the constants in this last sentence.) This relation also means that, generally,
	<div align="center">		<div align="center">
	<math>\frac{P_\mathrm{cgs}}{A_\mathrm{F}} = \biggl[ \frac{ 8\pi^4}{5} \biggl(\frac{m_u}{m_e} \biggr)^4 \biggl( \frac{\tilde{r}^4 {\bar\mu}^4 }{\tilde{a} \tilde{c}^8} \biggr) \biggr] P_\mathrm{code} ;		<math>\frac{P_\mathrm{cgs}}{A_\mathrm{F}} = \biggl[ \frac{ 8\pi^4}{5} \biggl(\frac{m_u}{m_e} \biggr)^4 \biggl( \frac{\tilde{r}^4 {\bar\mu}^4 }{\tilde{a} \tilde{c}^8} \biggr) \biggr] P_\mathrm{code} ;
Line 239:		Line 239:
	==Total Pressure Relation==		==Total Pressure Relation==

	In an [[~~User:Tohline/~~SR/PressureCombinations#Total_Pressure\|accompanying page]] of our Wiki-based H_Book, we show that, when normalized to {{~~User:Tohline/~~Math/C_FermiPressure}}, the analytic expression for the dimensionless total pressure takes the form,		In an [[SR/PressureCombinations#Total_Pressure\|accompanying page]] of our Wiki-based H_Book, we show that, when normalized to {{Math/C_FermiPressure}}, the analytic expression for the dimensionless total pressure takes the form,
	<div align="center">		<div align="center">
	{{~~User:Tohline/~~Math/EQ_PressureTotal01}}		{{Math/EQ_PressureTotal01}}
	</div>		</div>

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	</table>		</table>

	that is, it is a quartic equation describing the relationship between <math>T_\mathrm{code}</math> and <math>\rho_\mathrm{code}</math>. Following [[~~User:Tohline/~~SR/Ptot_QuarticSolution\|our accompanying H_Book Wiki discussion]], the solution to this quartic equation is,		that is, it is a quartic equation describing the relationship between <math>T_\mathrm{code}</math> and <math>\rho_\mathrm{code}</math>. Following [[SR/Ptot_QuarticSolution\|our accompanying H_Book Wiki discussion]], the solution to this quartic equation is,
	<div align="center">		<div align="center">
	<math>		<math>
Line 504:		Line 504:
	</div>		</div>

	As the [[~~User:Tohline/~~SR/Ptot_QuarticSolution#Limiting_Regimes\|accompanying discussion]] points out, the limiting behavior of the quartic solution is as follows:		As the [[SR/Ptot_QuarticSolution#Limiting_Regimes\|accompanying discussion]] points out, the limiting behavior of the quartic solution is as follows:

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

	Now let's plug in numerical values for the two stars in Dominic's ''Case A,'' <math>q_0 = 0.7</math> model evolution, as drawn from the [[~~User:Tohline/~~Appendix/Ramblings/Radiation/SummaryScalings#TableProperties\|accompanying ''properties'' table]].		Now let's plug in numerical values for the two stars in Dominic's ''Case A,'' <math>q_0 = 0.7</math> model evolution, as drawn from the [[Appendix/Ramblings/Radiation/SummaryScalings#TableProperties\|accompanying ''properties'' table]].

	<table align="center" border="1" cellpadding="5" width="60%">		<table align="center" border="1" cellpadding="5" width="60%">
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	</div>		</div>

	In order to avoid establishing stellar structures that are convectively unstable, Dominic also needs to choose an ''evolutionary'' ratio of specific heats, <math>\gamma</math>, such that its value is everywhere greater than a critical value, <math>\gamma_c</math>, established at the center of the accretor. From Equation (131) in Chapter II of [[~~User:Tohline/~~Appendix/References#C67\|Chandrasekhar (1967)]] we see that <math>\gamma_c</math> depends on each star's central value of <math>\beta</math> — that is, it depends on <math>\beta_c</math> — and on each star's ''structural'' <math>\Gamma_1 \equiv d\ln P/d\ln \rho</math> (which is <math>5/3</math> for our two {{~~User:Tohline/~~Math/MP_PolytropicIndex}} <math>=3/2</math> polytropic stars) in the following way:		In order to avoid establishing stellar structures that are convectively unstable, Dominic also needs to choose an ''evolutionary'' ratio of specific heats, <math>\gamma</math>, such that its value is everywhere greater than a critical value, <math>\gamma_c</math>, established at the center of the accretor. From Equation (131) in Chapter II of [[Appendix/References#C67\|Chandrasekhar (1967)]] we see that <math>\gamma_c</math> depends on each star's central value of <math>\beta</math> — that is, it depends on <math>\beta_c</math> — and on each star's ''structural'' <math>\Gamma_1 \equiv d\ln P/d\ln \rho</math> (which is <math>5/3</math> for our two {{Math/MP_PolytropicIndex}} <math>=3/2</math> polytropic stars) in the following way:
	<div align="center">		<div align="center">
	<math>		<math>

Jet53man at 16:40, 23 July 2021

2021-07-23T16:40:15Z

← Older revision		Revision as of 12:40, 23 July 2021
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	__FORCETOC__		__FORCETOC__
	<!-- __NOTOC__ will force TOC off -->		<!-- __NOTOC__ will force TOC off -->
	~~{{LSU_HBook_header}}~~

	=Initial Temperature Distributions=		=Initial Temperature Distributions=

Jet53man: Created page with "FORCETOC {{LSU_HBook_header}} =Initial Temperature Distributions= In an User:Tohline/Appendix/Ramblings/Radiation/CodeUnits|accom..."

2021-07-23T16:39:37Z

Created page with "__FORCETOC__  {{LSU_HBook_header}} =Initial Temperature Distributions= In an User:Tohline/Appendix/Ramblings/Radiation/CodeUnits|accom..."

New page

__FORCETOC__

{{LSU_HBook_header}}

=Initial Temperature Distributions=
In an [[User:Tohline/Appendix/Ramblings/Radiation/CodeUnits|accompanying Wiki page]] we've discussed in detail (or see the [[User:Tohline/Appendix/Ramblings/Radiation/SummaryScalings#Summary_of_Scalings|summary page]]) how to transform back and forth between cgs units and the dimensionless code units that have been adopted by Dominic Marcello in his radiation-hydro simulations of binary mass-transfer. Here we want to probe in more depth what temperature distributions are obtained from the initial polytropic structure once Dominic chooses particular values of the four scaling parameters: <math>\tilde{r}</math>, <math>\tilde{a}</math>, <math>\tilde{g}</math>, and <math>\tilde{c}</math>.

Our derivation of the temperature distribution will center around the following ideas. First, the initial binary model that Dominic obtains from Wes Even's self-consistent-field (SCF) code obeys a polytropic equation of state (EOS), namely,
<div align="center">
{{User:Tohline/Math/EQ_Polytrope01}}
</div>
with an adopted polytropic index {{User:Tohline/Math/MP_PolytropicIndex}} <math>= 3/2</math>. Hence, at any point inside either star, the pressure (in code units), <math>P_\mathrm{code}</math>, can be obtained from knowledge of the mass-density (in code units), <math>\rho_\mathrm{code}</math>, and the polytropic constant, <math>K_\mathrm{code}</math>, via the relation,
<div align="center">
<math>
[P_\mathrm{total}]_\mathrm{code} = K_\mathrm{code} \rho_\mathrm{code}^{5/3} .
</math>
</div>
Second, Dominic's models are ''evolved'' assuming a more realistic EOS. Specifically, he assumes that the total pressure is given by the expression,
<div align="center">
<math>
P_\mathrm{total} = P_\mathrm{gas} + P_\mathrm{deg} + P_\mathrm{rad} ,
</math></div>
where mathematical expressions for the ideal gas pressure, <math>P_\mathrm{gas}</math>, the electron degeneracy pressure, <math>P_\mathrm{deg}</math>, and the photon radiation pressure, <math>P_\mathrm{rad}</math>, are provided in an [[User:Tohline/SR#Equation_of_State|accompanying discussion of analytically prescribed equations of state]]. (Actually, Dominic is presently ignoring the effects of <math>P_\mathrm{deg}</math>, but because it allows for a more general treatment at some later date, we will assume the more general expression for <math>P_\mathrm{total}</math> and set <math>P_\mathrm{deg} = 0</math> near the end of our discussion.)

==Various Scalings==

===Pressure===

Now, realizing that pressure has units of energy per unit volume, we conclude that in order to transform between cgs units and code units, Dominic must adopt the relation,

<table align="center" border="0" cellpadding="5">
<tr>
<td align="right">
<math>
\frac{P_\mathrm{cgs}}{P_\mathrm{code}}
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\biggl( \frac{m_\mathrm{cgs}}{m_\mathrm{code}} \biggr) \biggl( \frac{\ell_\mathrm{cgs}}{\ell_\mathrm{code}} \biggr)^{-1} \biggl( \frac{t_\mathrm{cgs}}{t_\mathrm{code}} \biggr)^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\frac{\Lambda}{ G \bar{\mu}^2} \biggl( \frac{\tilde{g}^3 \tilde{a} }{\tilde{r}^4} \biggr)^{1/2}
\biggl[ \frac{\Lambda}{ c^2 \bar{\mu}^2} \biggl( \frac{\tilde{c}^4 \tilde{g} \tilde{a} }{\tilde{r}^4} \biggr)^{1/2} \biggr]^{-1}
\biggl[ \frac{\Lambda}{ c^3 \bar{\mu}^2} \biggl( \frac{\tilde{c}^6 \tilde{g} \tilde{a} }{\tilde{r}^4} \biggr)^{1/2} \biggr]^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\frac{ c^8 a_\mathrm{rad}}{(\Re / \bar{\mu})^4} \biggl( \frac{\tilde{r}^4 }{\tilde{a} \tilde{c}^8} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\frac{ 8\pi^4}{5} \biggl(\frac{m_u}{m_e} \biggr)^4 A_\mathrm{F} \biggl( \frac{\tilde{r}^4 {\bar\mu}^4 }{\tilde{a} \tilde{c}^8} \biggr)
</math>
</td>
</tr>

</table>
where {{User:Tohline/Math/C_ElectronMass}}, {{User:Tohline/Math/C_AtomicMassUnit}} and {{User:Tohline/Math/C_FermiPressure}} (the characteristic Fermi pressure) are physical constants defined in our [[User:Tohline/Appendix/Variables_templates|accompanying variables appendix]]. (Numerical values of these constants can be obtained by scrolling the cursor over the symbols for the constants in this last sentence.) This relation also means that, generally,
<div align="center">
<math>\frac{P_\mathrm{cgs}}{A_\mathrm{F}} = \biggl[ \frac{ 8\pi^4}{5} \biggl(\frac{m_u}{m_e} \biggr)^4 \biggl( \frac{\tilde{r}^4 {\bar\mu}^4 }{\tilde{a} \tilde{c}^8} \biggr) \biggr] P_\mathrm{code} ;
</math>
</div>
and, ''specifically'' when <math>P_\mathrm{cgs} = P_\mathrm{total}</math>, we have,
<div align="center">
<math>
p_\mathrm{total} \equiv \frac{P_\mathrm{total}}{A_\mathrm{F}} = \biggl[ \frac{ 8\pi^4}{5} \biggl(\frac{m_u}{m_e} \biggr)^4 \biggl( \frac{\tilde{r}^4 {\bar\mu}^4 }{\tilde{a} \tilde{c}^8} \biggr) \biggr] K_\mathrm{code} \rho_\mathrm{code}^{5/3} .
</math>
</div>

===Density===

In a similar manner we recognize that the density transformation must be governed by the relation ... (express this in terms of <math>\chi^3</math> so that it is obvious how to introduce <math>\rho_\mathrm{code}</math> into the quartic equation, below).

<table align="center" border="0" cellpadding="5">
<tr>
<td align="right">
<math>
\frac{\rho_\mathrm{cgs}}{\rho_\mathrm{code}}
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\biggl( \frac{m_\mathrm{cgs}}{m_\mathrm{code}} \biggr) \biggl( \frac{\ell_\mathrm{cgs}}{\ell_\mathrm{code}} \biggr)^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\frac{\Lambda}{ G \bar{\mu}^2} \biggl( \frac{\tilde{g}^3 \tilde{a} }{\tilde{r}^4} \biggr)^{1/2}
\biggl[ \frac{\Lambda}{ c^2 \bar{\mu}^2} \biggl( \frac{\tilde{c}^4 \tilde{g} \tilde{a} }{\tilde{r}^4} \biggr)^{1/2} \biggr]^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\frac{ c^6 a_\mathrm{rad}}{(\Re / \bar{\mu})^4} \biggl( \frac{\tilde{r}^4 }{\tilde{a} \tilde{c}^6} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\frac{ \pi^4}{5} \biggl(\frac{m_u}{m_e} \biggr)^4 \biggl( \frac{m_e}{m_p} \biggr) \frac{B_\mathrm{F}}{\mu_e} \biggl( \frac{\tilde{r}^4 {\bar\mu}^4 }{\tilde{a} \tilde{c}^6} \biggr)
</math>
</td>
</tr>

</table>
Hence,
<div align="center">
<math>\chi^3 \equiv \frac{\rho_\mathrm{cgs}}{B_\mathrm{F}} = \frac{ \pi^4}{5} \biggl(\frac{m_u}{m_e} \biggr)^4 \biggl( \frac{m_e}{m_p} \biggr) \frac{1}{\mu_e} \biggl( \frac{\tilde{r}^4 {\bar\mu}^4 }{\tilde{a} \tilde{c}^6} \biggr) \rho_\mathrm{code}
</math></div>

===Temperature===

Also, the temperature scaling can be rewritten as follows.

<table align="center" border="0" cellpadding="5">
<tr>
<td align="right">
<math>
\frac{T_\mathrm{cgs}}{T_\mathrm{code}}
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\frac{c^2}{ (\Re/ \bar{\mu})} \biggl(\frac{\tilde{r} }{\tilde{c}^2}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\biggl(\frac{m_u}{m_e} \biggr) T_e \biggl(\frac{\tilde{r} \bar\mu }{\tilde{c}^2}\biggr)
</math>
</td>
</tr>

</table>

Hence,
<div align="center">
<math>
\frac{T_\mathrm{cgs}}{T_e} = \biggl(\frac{m_u}{m_e} \biggr) \biggl(\frac{\tilde{r} \bar\mu }{\tilde{c}^2}\biggr) T_\mathrm{code} .

</math>
</div>

==Total Pressure Relation==

In an [[User:Tohline/SR/PressureCombinations#Total_Pressure|accompanying page]] of our Wiki-based H_Book, we show that, when normalized to {{User:Tohline/Math/C_FermiPressure}}, the analytic expression for the dimensionless total pressure takes the form,
<div align="center">
{{User:Tohline/Math/EQ_PressureTotal01}}
</div>

Based on the above-derived relations, this can now be rewritten in terms of the variables used in Dominic's simulations as follows:
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right">
<math>\biggl[ \frac{ 8\pi^4}{5} \biggl(\frac{m_u}{m_e} \biggr)^4 \biggl( \frac{\tilde{r}^4 {\bar\mu}^4 }{\tilde{a} \tilde{c}^8} \biggr) \biggr] K_\mathrm{code} \rho_\mathrm{code}^{5/3}
</math>
</td>
<td align="center">
=
</td>
<td align="left">
<math>\biggl(\frac{\mu_e m_p}{\bar\mu m_u}\biggr) 8 \biggl[\frac{ \pi^4}{5} \biggl(\frac{m_u}{m_e} \biggr)^4 \biggl( \frac{m_e}{m_p} \biggr) \frac{1}{\mu_e} \biggl( \frac{\tilde{r}^4 {\bar\mu}^4 }{\tilde{a} \tilde{c}^6} \biggr) \rho_\mathrm{code} \biggr] \biggl(\frac{m_u}{m_e} \biggr) \biggl(\frac{\tilde{r} \bar\mu }{\tilde{c}^2}\biggr) T_\mathrm{code}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+ F(\chi) + \frac{8\pi^4}{15} \biggl[ \biggl(\frac{m_u}{m_e} \biggr) \biggl(\frac{\tilde{r} \bar\mu }{\tilde{c}^2}\biggr) T_\mathrm{code}\biggr]^4
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~~~\biggl( \frac{\tilde{r}^4 {\bar\mu}^4 }{\tilde{a} \tilde{c}^8} \biggr) K_\mathrm{code} \rho_\mathrm{code}^{5/3} - \biggl[ \frac{5}{ 8\pi^4} \biggl(\frac{m_e}{m_u} \biggr)^4 \biggr]F(\chi)
</math>
</td>
<td align="center">
=
</td>
<td align="left">
<math>\biggl(\frac{\mu_e}{\bar\mu}\biggr) \biggl[ \frac{1}{\mu_e} \biggl( \frac{\tilde{r}^4 {\bar\mu}^4 }{\tilde{a} \tilde{c}^6} \biggr) \rho_\mathrm{code} \biggr] \biggl(\frac{\tilde{r} \bar\mu }{\tilde{c}^2}\biggr) T_\mathrm{code} + \frac{1}{3} \biggr[ \biggl(\frac{\tilde{r} \bar\mu }{\tilde{c}^2}\biggr) T_\mathrm{code}\biggr]^4
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~~~ K_\mathrm{code} \rho_\mathrm{code}^{5/3} - \biggl( \frac{\tilde{a} \tilde{c}^8}{\tilde{r}^4 {\bar\mu}^4 } \biggr)\biggl[ \frac{5}{ 8\pi^4} \biggl(\frac{m_e}{m_u} \biggr)^4 \biggr]F(\chi)
</math>
</td>
<td align="center">
=
</td>
<td align="left">
<math> \tilde{r} \rho_\mathrm{code} T_\mathrm{code} + \frac{\tilde{a}}{3} T_\mathrm{code}^4 .
</math>
</td>
</tr>

</table>

==EOS Quartic Solution==

We can view this last expression as having the form,
<div align="center">
<math>
a_4 T_\mathrm{code}^4 + a_1 T_\mathrm{code} - a_0 = 0 ,
</math>
</div>
where,

<table border="0" align="center" cellpadding="5">
<tr>
<td align="right">
<math>
a_4
</math>
</td>
<td align="center">
<math>
\equiv
</math>
</td>
<td align="left">
<math>
\frac{\tilde{a}}{3} ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
a_1
</math>
</td>
<td align="center">
<math>
\equiv
</math>
</td>
<td align="left">
<math>
\tilde{r}\rho_\mathrm{code} ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
a_0
</math>
</td>
<td align="center">
<math>
\equiv
</math>
</td>
<td align="left">
<math>
K_\mathrm{code} \rho_\mathrm{code}^{5/3} - \biggl( \frac{\tilde{a} \tilde{c}^8}{\tilde{r}^4 {\bar\mu}^4 } \biggr)\biggl[ \frac{5}{ 8\pi^4} \biggl(\frac{m_e}{m_u} \biggr)^4 \biggr]F(\chi) ,
</math>
</td>
</tr>

</table>

that is, it is a quartic equation describing the relationship between <math>T_\mathrm{code}</math> and <math>\rho_\mathrm{code}</math>. Following [[User:Tohline/SR/Ptot_QuarticSolution|our accompanying H_Book Wiki discussion]], the solution to this quartic equation is,
<div align="center">
<math>
T_\mathrm{code} = \theta \mathcal{K}(\lambda) ,
</math>
</div>
where,
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right">
<math>
\theta
</math>
</td>
<td align="center">
<math>
\equiv
</math>
</td>
<td align="left">
<math>
\biggl[ \frac{a_1}{4 a_4} \biggr]^{1/3}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\lambda
</math>
</td>
<td align="center">
<math>
\equiv
</math>
</td>
<td align="left">
<math>
\biggl[ \frac{256~ a_0^3 a_4}{27 a_1^4} \biggr]^{1/3} ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\mathcal{K}(\phi(\lambda))
</math>
</td>
<td align="center">
<math>
\equiv
</math>
</td>
<td align="left">
<math>
\phi^{-1/3} \biggl[ (\phi - 1)^{1/2} - 1 \biggr] ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\phi
</math>
</td>
<td align="center">
<math>
\equiv
</math>
</td>
<td align="left">
<math>
2^{3/2} \biggl[ 1 + (1 + \lambda^3)^{1/2} \biggr]^{1/2}
\biggl\{ \biggl[ 1 + (1 + \lambda^3)^{1/2} \biggr]^{2/3} - \lambda \biggr\}^{-3/2} .
</math>
</td>
</tr>

</table>

==Application to Dominic Marcello's Rad-Hydro Models==
===Temperature===
Currently Dominic is ignoring the effects of electron degeneracy pressure, so in applying the above <math>T(\rho)</math> solution to his models we can set <math>F(\chi) = 0</math> in the definition of <math>a_0</math>. Doing this, we find that,

<table border="0" align="center" cellpadding="5">
<tr>
<td align="right">
<math>
\theta
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\biggl[ \frac{3 \tilde{r}\rho_\mathrm{code}}{4 \tilde{a}} \biggr]^{1/3} ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\lambda
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\biggl[ \biggl(\frac{256~ \tilde{a} }{81~ \tilde{r}^4} \biggr)K_\mathrm{code}^3 \rho_\mathrm{code} \biggr]^{1/3} .
</math>
</td>
</tr>

</table>

For purposes of discussion, we will define <math>\rho_1</math> as the value of <math>\rho_\mathrm{code}</math> when <math>\lambda = 1</math>, that is,
<div align="center">
<math>\rho_1 \equiv \frac{81~ \tilde{r}^4}{256~ \tilde{a} K_\mathrm{code}^3} .
</math>
</div>

As the [[User:Tohline/SR/Ptot_QuarticSolution#Limiting_Regimes|accompanying discussion]] points out, the limiting behavior of the quartic solution is as follows:

<table align="center" border="0" cellpadding="5">
<tr>
<td align="right">
For <math>\rho_\mathrm{code} \ll \rho_1 </math>
</td>
<td align="right">
    <math>\cdots </math>    
</td>
<td align="left">
<math>T_\mathrm{code} \approx \frac{a_0}{a_1}</math>
</td>
<td align="center">
=
</td>
<td align="left">
<math>\biggl( \frac{K_\mathrm{code}}{\tilde{r}} \biggr) \rho_\mathrm{code}^{2/3}</math>
</td>
</tr>

<tr>
<td align="right">
For <math>\rho_\mathrm{code} \gg \rho_1 </math>
</td>
<td align="right">
    <math>\cdots </math>    
</td>
<td align="left">
<math>T_\mathrm{code} \approx \biggl( \frac{a_0}{a_4} \biggr)^{1/4}</math>
</td>
<td align="center">
=
</td>
<td align="left">
<math>
\biggl[ \biggl( \frac{3K_\mathrm{code}}{\tilde{a}} \biggr)\rho_\mathrm{code}^{5/3} \biggr]^{1/4}
</math>
</td>
</tr>
</table>

Now let's plug in numerical values for the two stars in Dominic's ''Case A,'' <math>q_0 = 0.7</math> model evolution, as drawn from the [[User:Tohline/Appendix/Ramblings/Radiation/SummaryScalings#TableProperties|accompanying ''properties'' table]].

<table align="center" border="1" cellpadding="5" width="60%">
<tr>
<td align="center" colspan="7">
<font color="red">Case A:</font>   <math>\tilde{g} = 1</math>; <math>\tilde{c} = 198</math>; <math>\tilde{r} = 0.44</math>; <math>\tilde{a} = 0.044</math>; <math>\bar\mu = 4/3</math>;
</td>
</tr>

<tr>
<td align="center">
'''Star'''
</td>
<td align="center">
<math>K_\mathrm{code}</math>
</td>
<td align="center">
<math>\theta</math>
</td>

<td align="center">
<math>\lambda</math>
</td>
<td align="center">
<math>\rho_1</math>
</td>
<td align="center">
<math>T_\mathrm{code}</math> (for <math>\rho_\mathrm{code} \ll \rho_1</math>)
</td>
<td align="center">
<math>P_\mathrm{rad}/P_\mathrm{gas}</math> (for <math>\rho_\mathrm{code} \ll \rho_1</math>)
</td>
</tr>

<tr>
<td align="center">
Accretor
</td>
<td align="center">
<math>0.2571</math>
</td>
<td align="center">
<math>1.957 \rho_\mathrm{code}^{1/3}</math>
</td>

<td align="center">
<math>0.3980\rho_\mathrm{code}^{1/3}</math>
</td>
<td align="center">
<math>15.86</math>
</td>
<td align="center">
<math>0.5843~\rho_\mathrm{code}^{2/3}</math>
</td>
<td align="center">
<math>6.65\times 10^{-3}~\rho_\mathrm{code}</math>
</td>
</tr>

<tr>
<td align="center">
Donor
</td>
<td align="center">
<math>0.2364</math>
</td>
<td align="center">
<math>1.957 \rho_\mathrm{code}^{1/3}</math>
</td>

<td align="center">
<math>0.3660\rho_\mathrm{code}^{1/3}</math>
</td>
<td align="center">
<math>20.4</math>
</td>
<td align="center">
<math>0.5373~\rho_\mathrm{code}^{2/3}</math>
</td>
<td align="center">
<math>5.17\times 10^{-3}~\rho_\mathrm{code}</math>
</td>
</tr>

</table>

===Other Physical Variables===

The ratio of radiation pressure to gas pressure (see the last column of the above table) is calculated via the relation,
<div align="center">
<math>
\frac{1}{\Gamma} = \frac{P_\mathrm{rad}}{P_\mathrm{gas}} = \biggl( \frac{\tilde{a}}{3\tilde{r}} \biggr) \frac{T_\mathrm{code}^3}{\rho_\mathrm{code}} .
</math>
</div>
Also note that,
<div align="center">
<math>
\beta \equiv \frac{P_\mathrm{gas}}{P_\mathrm{total}} = \frac{1}{1+P_\mathrm{rad}/P_\mathrm{gas}} .
</math>
</div>

In order to avoid establishing stellar structures that are convectively unstable, Dominic also needs to choose an ''evolutionary'' ratio of specific heats, <math>\gamma</math>, such that its value is everywhere greater than a critical value, <math>\gamma_c</math>, established at the center of the accretor. From Equation (131) in Chapter II of [[User:Tohline/Appendix/References#C67|Chandrasekhar (1967)]] we see that <math>\gamma_c</math> depends on each star's central value of <math>\beta</math> — that is, it depends on <math>\beta_c</math> — and on each star's ''structural'' <math>\Gamma_1 \equiv d\ln P/d\ln \rho</math> (which is <math>5/3</math> for our two {{User:Tohline/Math/MP_PolytropicIndex}} <math>=3/2</math> polytropic stars) in the following way:
<div align="center">
<math>
\gamma_c = \biggl[ \frac{12(1-\beta_c)(\Gamma_1 - \beta_c) -\beta_c(\Gamma_1 - \beta_c) - (4-3\beta_c)^2}{12(1-\beta_c)(\Gamma_1 - \beta_c)- (4-3\beta_c)^2} \biggr].
</math>
</div>

Plugging <math>\rho_\mathrm{code}^\mathrm{max}</math> into these expressions lets us tabulate various properties at the center of both stars.

<table align="center" border="1" cellpadding="5" width="75%">
<tr>
<td align="center" colspan="10">
<b><font color="darkblue">Central Stellar Values</font></b>
</td>
</tr>
<tr>
<td colspan="2">
 
</td>
<td align="center" colspan="3">
Approximations
</td>
<td align="center" colspan="5">
From Quartic Solution
</td>
</tr>

<tr>
<td align="center">
'''Star'''
</td>
<td align="center">
<math>\rho^\mathrm{max}_\mathrm{code}</math>
</td>
<td align="center">
<math>T_\mathrm{code}^\mathrm{max}</math>
</td>
<td align="center">
<math>\frac{P_\mathrm{rad}}{P_\mathrm{gas}}\biggr|_c</math>
</td>
<td align="center">
<math>\beta_c</math>
</td>

<td align="center">
<math>T_\mathrm{code}^\mathrm{max}</math>
</td>
<td align="center">
<math>\frac{P_\mathrm{rad}}{P_\mathrm{gas}}\biggr|_c</math>
</td>
<td align="center">
<math>\beta_c</math>
</td>
<td align="center">
<math>\gamma_c</math>
</td>
<td align="center">
<math>T_\mathrm{cgs}^\mathrm{max}</math>
</td>
</tr>

<tr>
<td align="center">
Accretor
</td>
<td align="center">
<math>1.0000</math>
</td>
<td align="center">
<math>0.5843</math>
</td>
<td align="center">
<math>6.65 \times 10^{-3}</math>
</td>
<td align="center">
<math>0.99339</math>
</td>

<td align="center" colspan="1">
0.5805
</td>
<td align="center" colspan="1">
<math>6.522\times 10^{-3}</math>
</td>
<td align="center" colspan="1">
0.99352
</td>
<td align="center" colspan="1">
1.67765
</td>
<td align="center" colspan="1">
<math>9.39\times 10^{7}~\mathrm{K}</math>
</td>
</tr>

<tr>
<td align="center">
Donor
</td>
<td align="center">
<math>0.6077</math>
</td>
<td align="center">
<math>0.3854</math>
</td>
<td align="center">
<math>3.14\times 10^{-3}</math>
</td>
<td align="center">
<math>0.99687</math>
</td>

<td align="center" colspan="1">
0.3842
</td>
<td align="center" colspan="1">
<math>3.111\times 10^{-3}</math>
</td>
<td align="center" colspan="1">
0.99690
</td>
<td align="center" colspan="1">
1.67188
</td>
<td align="center" colspan="1">
<math>6.22\times 10^{7}~\mathrm{K}</math>
</td>
</tr>

</table>

The central values of <math>P_\mathrm{rad}/P_\mathrm{gas}</math> obtained via the quartic solution match exactly the values that Dominic read straight from the rad-hydrocode, namely, <math>6.5\times 10^{-3}</math> (for the accretor) and <math>3.1\times 10^{-3}</math> (for the donor). (See email from Dominic to Joel dated 8/4/2010.) In this email, Dominic also stated that he chose <math>\gamma = 1.67114094</math>; I'm not quite sure how he derived this value.

{{ SGFfooter }}

Appendix/Ramblings/Radiation/InitialTemperatures - Revision history

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Jet53man at 16:40, 23 July 2021

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