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.)
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> |
EOS Quartic Solution
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© 2014 - 2021 by Joel E. Tohline |