User:Tohline/Appendix/Ramblings/Radiation/InitialTemperatures
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Initial Temperature Distributions
In an accompanying Wiki page we've discussed in detail (or see the 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,
<math>~P = K_\mathrm{n} \rho^{1+1/n}</math>
with an adopted polytropic index <math>~n</math> <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,
<math> [P_\mathrm{total}]_\mathrm{code} = K_\mathrm{code} \rho_\mathrm{code}^{5/3} . </math>
Second, Dominic's models are evolved assuming a more realistic EOS. Specifically, he assumes that the total pressure is given by the expression,
<math> P_\mathrm{total} = P_\mathrm{gas} + P_\mathrm{deg} + P_\mathrm{rad} ,
</math>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 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,
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<math> \frac{P_\mathrm{cgs}}{P_\mathrm{code}} </math> |
<math> = </math> |
<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> |
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<math> = </math> |
<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> |
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<math> = </math> |
<math> \frac{ c^8 a_\mathrm{rad}}{(\Re / \bar{\mu})^4} \biggl( \frac{\tilde{r}^4 }{\tilde{a} \tilde{c}^8} \biggr) </math> |
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<math> = </math> |
<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> |
where <math>~m_e</math>, <math>~m_u</math> and <math>~A_\mathrm{F}</math> (the characteristic Fermi pressure) are physical constants defined in our 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,
<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>
and, specifically when <math>P_\mathrm{cgs} = P_\mathrm{total}</math>, we have,
<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>
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).
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<math> \frac{\rho_\mathrm{cgs}}{\rho_\mathrm{code}} </math> |
<math> = </math> |
<math> \biggl( \frac{m_\mathrm{cgs}}{m_\mathrm{code}} \biggr) \biggl( \frac{\ell_\mathrm{cgs}}{\ell_\mathrm{code}} \biggr)^{-3} </math> |
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<math> = </math> |
<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> |
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<math> = </math> |
<math> \frac{ c^6 a_\mathrm{rad}}{(\Re / \bar{\mu})^4} \biggl( \frac{\tilde{r}^4 }{\tilde{a} \tilde{c}^6} \biggr) </math> |
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<math> = </math> |
<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> |
Hence,
<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>Temperature
Also, the temperature scaling can be rewritten as follows.
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<math> \frac{T_\mathrm{cgs}}{T_\mathrm{code}} </math> |
<math> = </math> |
<math> \frac{c^2}{ (\Re/ \bar{\mu})} \biggl(\frac{\tilde{r} }{\tilde{c}^2}\biggr) </math> |
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<math> = </math> |
<math> \biggl(\frac{m_u}{m_e} \biggr) T_e \biggl(\frac{\tilde{r} \bar\mu }{\tilde{c}^2}\biggr) </math> |
Hence,
<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>
Total Pressure Relation
In an accompanying page of our Wiki-based H_Book, we show that, when normalized to <math>~A_\mathrm{F}</math>, the analytic expression for the dimensionless total pressure takes the form,
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<math>~p_\mathrm{total} = \biggl(\frac{\mu_e m_p}{\bar{\mu} m_u} \biggr) 8 \chi^3 \frac{T}{T_e} + F(\chi) + \frac{8\pi^4}{15} \biggl( \frac{T}{T_e} \biggr)^4</math> |
Based on the above-derived relations, this can now be rewritten in terms of the variables used in Dominic's simulations as follows:
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<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> |
= |
<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> |
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<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> |
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<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> |
= |
<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> |
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<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> |
= |
<math> \tilde{r} \rho_\mathrm{code} T_\mathrm{code} + \frac{\tilde{a}}{3} T_\mathrm{code}^4 . </math> |
EOS Quartic Solution
We can view this last expression as having the form,
<math> a_4 T_\mathrm{code}^4 + a_1 T_\mathrm{code} - a_0 = 0 , </math>
where,
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<math> a_4 </math> |
<math> \equiv </math> |
<math> \frac{\tilde{a}}{3} , </math> |
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<math> a_1 </math> |
<math> \equiv </math> |
<math> \tilde{r}\rho_\mathrm{code} , </math> |
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<math> a_0 </math> |
<math> \equiv </math> |
<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> |
that is, it is a quartic equation describing the relationship between <math>T_\mathrm{code}</math> and <math>\rho_\mathrm{code}</math>. Following our accompanying H_Book Wiki discussion, the solution to this quartic equation is,
<math> T_\mathrm{code} = \theta \mathcal{K}(\lambda) , </math>
where,
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<math> \theta </math> |
<math> \equiv </math> |
<math> \biggl[ \frac{a_1}{4 a_4} \biggr]^{1/3} </math> |
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<math> \lambda </math> |
<math> \equiv </math> |
<math> \biggl[ \frac{256~ a_0^3 a_4}{27 a_1^4} \biggr]^{1/3} , </math> |
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<math> \mathcal{K}(\phi(\lambda)) </math> |
<math> \equiv </math> |
<math> \phi^{-1/3} \biggl[ (\phi - 1)^{1/2} - 1 \biggr] , </math> |
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<math> \phi </math> |
<math> \equiv </math> |
<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> |
Application to Dominic Marcello's Rad-Hydro Models
Currently Dominic is ignoring the effects of electron degeneracy pressure, so in applying this <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,
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<math> \theta </math> |
<math> = </math> |
<math> \biggl[ \frac{3 \tilde{r}\rho_\mathrm{code}}{4 \tilde{a}} \biggr]^{1/3} , </math> |
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<math> \lambda </math> |
<math> = </math> |
<math> \biggl[ \biggl(\frac{256~ \tilde{a} }{81~ \tilde{r}^4} \biggr)K_\mathrm{code}^3 \rho_\mathrm{code} \biggr]^{1/3} . </math> |
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,
<math>\rho_1 \equiv \frac{81~ \tilde{r}^4}{256~ \tilde{a} K_\mathrm{code}^3} . </math>
As the accompanying discussion points out, the limiting behavior of the quartic solution is as follows:
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For <math>\rho_\mathrm{code} \ll \rho_1 </math> |
<math>\cdots </math> |
<math>T_\mathrm{code} \approx \frac{a_0}{a_1}</math> |
= |
<math>\biggl( \frac{K_\mathrm{code}}{\tilde{r}} \biggr) \rho_\mathrm{code}^{2/3}</math> |
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For <math>\rho_\mathrm{code} \gg \rho_1 </math> |
<math>\cdots </math> |
<math>T_\mathrm{code} \approx \biggl( \frac{a_0}{a_4} \biggr)^{1/4}</math> |
= |
<math> \biggl[ \biggl( \frac{3K_\mathrm{code}}{\tilde{a}} \biggr)\rho_\mathrm{code}^{5/3} \biggr]^{1/4} </math> |
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© 2014 - 2021 by Joel E. Tohline |