Difference between revisions of "User:Tohline/SSC/Structure/BonnorEbert"
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=Pressure-Bounded Isothermal Sphere (structure)= | =Pressure-Bounded Isothermal Sphere (structure)= | ||
==Governing Relation== | ==Governing Relation== | ||
The equilibrium structure of an ''isolated'' isothermal sphere, as derived by [http://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Emden] (1907), has been [[User:Tohline/SSC/Structure/IsothermalSphere#Isothermal_Sphere_( | The equilibrium structure of an ''isolated'' isothermal sphere, as derived by [http://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Emden] (1907), has been [[User:Tohline/SSC/Structure/IsothermalSphere#Isothermal_Sphere_(structure)|discussed elsewhere]]. From this separate discussion we appreciate that the governing ODE is, | ||
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<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> | <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> | ||
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Both of these dimensionless governing ODEs — Bonnor's Eq. (2.8) and Ebert's Eq. (17) — are identical to the | Both of these dimensionless governing ODEs — Bonnor's Eq. (2.8) and Ebert's Eq. (17) — are identical to the dimensionless expression derived by Emden (see the [[User:Tohline/SSC/Structure/IsothermalSphere#Governing_Relations|presentation elsewhere]]), namely, | ||
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<math> | <math> | ||
Revision as of 21:00, 31 October 2012
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Pressure-Bounded Isothermal Sphere (structure)
Governing Relation
The equilibrium structure of an isolated isothermal sphere, as derived by Emden (1907), has been discussed elsewhere. From this separate discussion we appreciate that the governing ODE is,
<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>
where,
<math>c_s^2 = \frac{\Re T}{\bar{\mu}} = \frac{k T}{m_u \bar{\mu}} \, ,</math>
is the square of the isothermal sound speed. 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) |
 |
<math>G \Leftrightarrow \gamma</math> |
 |
| <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> |
Both of these dimensionless governing ODEs — Bonnor's Eq. (2.8) and Ebert's Eq. (17) — are identical to the dimensionless expression derived by Emden (see the presentation elsewhere), namely,
<math> \frac{d^2v_1}{d\mathfrak{r}_1^2} +\frac{2}{\mathfrak{r}_1} \frac{dv_1}{dr} + e^{v_1} = 0 \, . </math>
The translation from Emden-to-Bonnor-to-Ebert is straightforward:
<math> \mathfrak{r}_1 = \xi|_\mathrm{Bonner} = \xi|_\mathrm{Ebert}~~~~\mathrm{and}~~~~e^{v_1} = e^{-\psi} = \eta \, . </math>
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