Difference between revisions of "User:Tohline/SSC/Structure/BonnorEbert"

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(→‎Governing Relations: Insert separate derivations from Bonnor and Ebert)
(Blend Bonnor-Ebert discussion in with earlier discussion of Emden's isothermal sphere)
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=Isothermal Sphere (structure)=
=Pressure-Bounded Isothermal Sphere (structure)=
Here we supplement the [[User:Tohline/SphericallySymmetricConfigurations/PGE|simplified set of principal governing equations]] with an isothermal equation of state, that is, {{User:Tohline/Math/VAR_Pressure01}} is related to {{User:Tohline/Math/VAR_Density01}} through the 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_(Structure)|discussed elsewhere]].  From this separate discussion we appreciate that the governing ODE is,
<div align="center">
<div align="center">
<math>P = 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>
</div>
</div>
where, <math>c_s</math> is the isothermal sound speed.  Comparing this {{User:Tohline/Math/VAR_Pressure01}}-{{User:Tohline/Math/VAR_Density01}} relationship to
where,
 
<div align="center">
<span id="IdealGas:FormA"><font color="#770000">'''Form A'''</font></span><br />
of the Ideal Gas Equation of State,
 
{{User:Tohline/Math/EQ_EOSideal0A}}
</div>
we see that,
<div align="center">
<div align="center">
<math>c_s^2 = \frac{\Re T}{\bar{\mu}} = \frac{k T}{m_u \bar{\mu}} \, ,</math>
<math>c_s^2 = \frac{\Re T}{\bar{\mu}} = \frac{k T}{m_u \bar{\mu}} \, ,</math>
</div>
</div>
where, {{User:Tohline/Math/C_GasConstant}}, {{User:Tohline/Math/C_BoltzmannConstant}}, {{User:Tohline/Math/C_AtomicMassUnit}}, and {{User:Tohline/Math/MP_MeanMolecularWeight}} are all defined in the accompanying [[User:Tohline/Appendix/Variables_templates|variables appendix]].  It will be useful to note that, for an isothermal gas, {{User:Tohline/Math/VAR_Enthalpy01}} is related to {{User:Tohline/Math/VAR_Density01}} via the expression,
is the square of the isothermal sound speed.  In their studies of ''pressure-bounded'' isothermal spheres, [http://adsabs.harvard.edu/abs/1955ZA.....37..217E Ebert] (1955, ZA, 37, 217) and [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B 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.
<div align="center">
<math>
dH = \frac{dP}{\rho} = c_s^2 d\ln\rho \, .
</math>
</div>
 
==Governing Relations==
 
Adopting [[User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies#Technique_2|solution technique #2]], we need to solve the following second-order ODE relating the two unknown functions, {{User:Tohline/Math/VAR_Density01}} and {{User:Tohline/Math/VAR_Enthalpy01}}:
<div align="center">
<math>\frac{1}{r^2} \frac{d}{dr}\biggl( r^2 \frac{dH}{dr} \biggr) =-  4\pi G \rho</math> .
</div>
Using the {{User:Tohline/Math/VAR_Enthalpy01}}-{{User:Tohline/Math/VAR_Density01}} relationship for an isothermal gas presented above, this can be rewritten entirely in terms of the density as,
<div align="center">
<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> .
</div>
In their studies of pressure-bounded isothermal spheres, [http://adsabs.harvard.edu/abs/1955ZA.....37..217E Ebert] (1955, ZA, 37, 217) and [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B 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.


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<div align="center">
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</tr>
</tr>
</table>
</table>
</div>
Both of these dimensionless governing ODEs &#8212; Bonnor's Eq. (2.8) and Ebert's Eq. (17) &#8212; are identical to the one derived by Emden (see the [[User:Tohline/SSC/Structure/IsothermalSphere#Governing_Relations|presentation elsewhere]]), namely,
<div align="center">
<math>
\frac{d^2v_1}{d\mathfrak{r}_1^2} +\frac{2}{\mathfrak{r}_1} \frac{dv_1}{dr} + e^{v_1} = 0 \, .
</math>
</div>
The translation from Emden-to-Bonnor-to-Ebert is straightforward:
<div align="center">
<math>
\mathfrak{r}_1 = \xi|_\mathrm{Bonner} = \xi|_\mathrm{Ebert}~~~~\mathrm{and}~~~~e^{v_1} = e^{-\psi} = \eta \, .
</math>
</div>
</div>


Revision as of 20:57, 31 October 2012

Whitworth's (1981) Isothermal Free-Energy Surface
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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)
Bonnor (1956, MNRAS, 116, 351)
 <math>G \Leftrightarrow \gamma</math>
Ebert (1955, ZA, 37, 217)
<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 one 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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Whitworth's (1981) Isothermal Free-Energy Surface

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