User:Tohline/SSC/Structure/BonnorEbert
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Isothermal Sphere (structure)
Here we supplement the simplified set of principal governing equations with an isothermal equation of state, that is, <math>~P</math> is related to <math>~\rho</math> through the relation,
<math>P = c_s^2 \rho \, ,</math>
where, <math>c_s</math> is the isothermal sound speed. Comparing this <math>~P</math>-<math>~\rho</math> relationship to
Form A
of the Ideal Gas Equation of State,
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<math>~P_\mathrm{gas} = \frac{\Re}{\bar{\mu}} \rho T</math> |
we see that,
<math>c_s^2 = \frac{\Re T}{\bar{\mu}} = \frac{k T}{m_u \bar{\mu}} \, ,</math>
where, <math>~\Re</math>, <math>~k</math>, <math>~m_u</math>, and <math>~\bar{\mu}</math> are all defined in the accompanying variables appendix. It will be useful to note that, for an isothermal gas, <math>~H</math> is related to <math>~\rho</math> via the expression,
<math> dH = \frac{dP}{\rho} = c_s^2 d\ln\rho \, . </math>
Governing Relations
Adopting solution technique #2, we need to solve the following second-order ODE relating the two unknown functions, <math>~\rho</math> and <math>~H</math>:
<math>\frac{1}{r^2} \frac{d}{dr}\biggl( r^2 \frac{dH}{dr} \biggr) =- 4\pi G \rho</math> .
Using the <math>~H</math>-<math>~\rho</math> relationship for an isothermal gas presented above, this can be rewritten entirely in terms of the density as,
<math>\frac{1}{r^2} \frac{d}{dr}\biggl( r^2 \frac{d\ln\rho}{dr} \biggr) =- \frac{4\pi G}{c_s^2} \rho</math> .
In their studies of pressure-bounded isothermal spheres, Ebert (1955, ZA, 37, 217) and Bonnor (1956, MNRAS, 116, 351) both started with this governing ODE, but developed its solution in different ways. Here we present both developments while highlighting transformations between the two.
| Derivation by Bonnor (edited) | translation | Derivation by Ebert (edited) |
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<math>G \Leftrightarrow \gamma</math> |
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| <math>\rho_c \Leftrightarrow \rho_0</math> | ||
| <math>\frac{kT}{m} \Leftarrow c_s^2 \Rightarrow \frac{\Re T_0}{\mu}</math> | ||
| <math>\beta^{1/2}\lambda^{-1/2} \Leftrightarrow l_0</math> | ||
| <math>e^{-\psi} \Leftrightarrow \eta</math> |
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