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

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(→‎Summary of Analytic Results: Insert analytic coefficients for virial n = 6/5; and delete second table)
(Begin discussion of Bonnor-Ebert Sphere)
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=Mass Upper Limits=
=Isothermal Sphere (structure)=
==Virial Equilibrium==
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,  
By examining the [[User:Tohline/SphericallySymmetricConfigurations/Virial#P-V_Diagram|virial equilibrium of nonrotating, spherically symmetric configurations that are embedded in an external medium]] of pressure <math>P_e</math>, one can begin to appreciate that there is a mass above which no equilibrium exists if the effective adiabatic exponent of the gas is <math>\gamma_g < 4/3</math>.  Assuming uniform density and uniform specific entropy configurations for simplicity, there is an analytic expression for this limiting mass and for the equilibrium radius of that limiting configuration.  Specifically,
<div align="center">
<div align="center">
<math>
<math>P = c_s^2 \rho \, ,</math>
M_\mathrm{max}^2 = \frac{3\cdot 5}{2^2 \pi} \biggl(\frac{4}{3\gamma_g} - 1 \biggr) \biggl[ \frac{3 \gamma_g}{4} \biggr] ^{4/(4-3\gamma_g)} \biggl[ \frac{\bar{c_s}^8}{G^3 P_e} \biggr] \, ,
</math>
</div>
</div>
and,
where, <math>c_s</math> is the isothermal sound speed.  Comparing this {{User:Tohline/Math/VAR_Pressure01}}-{{User:Tohline/Math/VAR_Density01}} relationship to
<div align="center">
<math>
R_\mathrm{eq}^2 = \frac{1}{5^2} \biggl[ \frac{4}{3\gamma_g} \biggr]^{2/(4-3\gamma_g)} \biggl[ \frac{GM_\mathrm{max}}{\bar{c_s}^2 } \biggr]^2
= \frac{3}{2^2 \cdot 5\pi} \biggl[  \frac{3\gamma_g}{4} \biggr]^{2/(4-3\gamma_g)} \biggl( \frac{4}{3\gamma_g} - 1 \biggr) \biggl[ \frac{\bar{c_s}^4 }{GP_e} \biggr] \, .
</math>
</div>
 
==Polytropes Embedded in an External Medium==
[[User:Tohline/SSC/Structure/PolytropesEmbedded|Polytropes embedded in an external medium]]
 
==Summary of Analytic Results==


<div align="center">
<div align="center">
<table border="1" width="90%" cellpadding="4">
<span id="IdealGas:FormA"><font color="#770000">'''Form A'''</font></span><br />
  <tr>
of the Ideal Gas Equation of State,
  <td colspan="8" align="center">'''Example:''' &nbsp; Specify <math>c_s</math> and <math>P_e</math></td>
  </tr>
<tr>
  <td align="center" rowspan="2">
{{User:Tohline/Math/MP_PolytropicIndex}}
  </td>
  <td align="center" rowspan="2">
<math>\eta = 1+1/n</math>
  </td>
  <td align="center" colspan="3">
<math>M_\mathrm{max}</math>
  </td>
  <td align="center" colspan="3">
<math>R_\mathrm{eq}</math>
  </td>
</tr>
 
<tr>
  <td align="center" colspan="1">
<math>\alpha_\mathrm{virial}</math>
  </td>
  <td align="center">
<math>\alpha_\mathrm{DHB}</math>
  </td>
  <td align="center">
<math>\mathrm{scale}</math>
  </td>
  <td align="center" colspan="1">
<math>\alpha_\mathrm{virial}</math>
  </td>
  <td align="center">
<math>\alpha_\mathrm{DHB}</math>
  </td>
  <td align="center">
<math>\mathrm{scale}</math>
  </td>
</tr>
 
<tr>
  <td align="center">
<math>\infty</math>
  </td>
  <td align="center">
1
  </td>
  <td align="center">
<math>\biggr(\frac{3^4 \cdot 5^3}{2^{10} \pi}\biggr)^{1/2} </math><br>&nbsp;<br><math> \approx 1.77408</math>
  </td>
  <td align="center">
--
  </td>
  <td align="center">
<math>\biggl[ \frac{c_s^8}{G^3 P_e}\biggr]^{1/2}</math>
  </td>
  <td align="center">
<math>\biggl(\frac{3^2 \cdot 5}{2^6\pi} \bigg)^{1/2}</math><br>&nbsp;<br><math> \approx 0.47309</math>
  </td>
  <td align="center">
--
  </td>
  <td align="center">
<math>\biggl( \frac{c_s^4}{GP_e} \biggr)^{1/2}</math>
  </td>
</tr>


<tr>
{{User:Tohline/Math/EQ_EOSideal0A}}
  <td align="center">
5
  </td>
  <td align="center">
6/5
  </td>
  <td align="center">
<math>\biggl( \frac{3^{19}}{2^{12}\cdot 5^9 \pi} \biggr)^{1/2} </math><br>&nbsp;<br><math> \approx 0.21505</math>
  </td>
  <td align="center">
<math>\biggl( \frac{1}{2} \biggr)^{3/10} \biggl( \frac{3^7}{2^8 \pi} \biggr)^{1/2} </math><br>&nbsp;<br><math> \approx 1.33943</math>
  </td>
  <td align="center">
<math>\biggl[ \frac{\bar{c_s}^8}{G^3 P_e}\biggr]^{1/2}</math>
  </td>
  <td align="center">
<math>\frac{3^{9}}{2^{7}\cdot 5^{6} \pi} </math><br>&nbsp;<br><math> \approx 0.0031326</math>
  </td>
  <td align="center">
--
  </td>
  <td align="center">
<math>\biggl( \frac{\bar{c_s}^4}{GP_e} \biggr)^{1/2}</math>
  </td>
</tr>
 
</table>
</div>
</div>
 
we see that,
=Bonnor-Ebert Limiting Mass=
Here we will refer to the original works of [http://http://adsabs.harvard.edu/abs/1955ZA.....37..217E Ebert] (1955, ZA, 37, 217) and [http://http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor] (1956, MNRAS, 116, 351) .
 
In his study of the "global gravitational stability [of] one-dimensional polytropes,"  [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth] (1981, MNRAS, 195, 967) normalizes (or "references") various derived mathematical expressions for configuration radii, <math>R</math>, and for the pressure exerted by an external bounding medium, <math>P_\mathrm{ex}</math>, to quantities he refers to as, respectively, <math>R_\mathrm{rf}</math> and <math>P_\mathrm{rf}</math>.  The paragraph from his paper in which these two reference quantities are defined is shown here:
<div align="center">
<div align="center">
<table border="2">
<math>c_s^2 = \frac{\Re T}{\bar{\mu}} = \frac{k T}{m_u \bar{\mu}} \, ,</math>
<tr><td>
[[File:WhitworthScalingText.jpg|600px|center|Whitworth (1981, MNRAS, 195, 967)]]
</td></tr>
</table>
</div>
</div>
In order to map Whitworth's terminology to ours, we note, first, that he uses <math>M_0</math> to represent the spherical configuration's total mass, which we refer to simply as <math>M</math>; and his parameter <math>\eta</math> is related to our {{User:Tohline/Math/MP_PolytropicIndex}} via the relation,
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,
<div align="center">
<math>\eta = 1 + \frac{1}{n} \, .</math>
</div>
Hence, Whitworth writes the polytropic equation of state as,
<div align="center">
<math>P = K_\eta \rho^\eta \, ,</math>
</div>
whereas, using our standard notation, this same key relation is written as,
<div align="center">
{{User:Tohline/Math/EQ_Polytrope01}} ;
</div>
and his parameter <math>K_\eta</math> is identical to our {{User:Tohline/Math/MP_PolytropicConstant}}. 
 
According to the second (bottom) expression identified by the red outlined box drawn above,
<div align="center">
<div align="center">
<math>
<math>
P_\mathrm{rf} = \frac{3^4 5^3}{2^{10} \pi} \biggl( \frac{K_1^4}{G^3 M^2} \biggr) \, ,
dH = \frac{dP}{\rho} = c_s^2 d\ln\rho \, .
</math>
</math>
</div>
</div>
and inverting the expression inside the green outlined box gives,
 
==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">
<div align="center">
<math>
<math>\frac{1}{r^2} \frac{d}{dr}\biggl( r^2 \frac{dH}{dr} \biggr) =-  4\pi G \rho</math> .
K_1 = \biggl[ K_n (4 P_\mathrm{rf})^{\eta - 1} \biggr]^{1/\eta} \, .
</math>
</div>
</div>
Hence,
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">
<div align="center">
<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> .
P_\mathrm{rf} = \frac{3^4 5^3}{2^{10} \pi} \biggl( \frac{1}{G^3 M^2} \biggr)\biggl[ K_n (4 P_\mathrm{rf})^{\eta - 1} \biggr]^{4/\eta} \, ,
</math>
</div>
</div>
or, gathering all factors of <math>P_\mathrm{rf}</math> to the left-hand side,
In their studies of pressure-bounded isothermal spheres, [http://http://adsabs.harvard.edu/abs/1955ZA.....37..217E Ebert] (1955, ZA, 37, 217) and [http://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>
P_\mathrm{rf}^{(4-3\eta)} = 2^{-2(4+\eta)} \biggl( \frac{3^4 5^3}{\pi} \biggr)^\eta \biggl[ \frac{K_n^4}{G^{3\eta} M^{2\eta}} \biggr] \, .
</math>
</div>
Analogously, according to the first (top) expression identified inside the red outlined box,
<div align="center">
<math>
R_\mathrm{rf} = \frac{2^2 GM}{3\cdot 5 K_1} = 2^{2/\eta} \biggl( \frac{GM}{3\cdot 5}\biggr) K_n^{-1/\eta}  P_\mathrm{rf}^{(1-\eta)/\eta}
~~~~\Rightarrow~~~~ R_\mathrm{rf}^\eta = \frac{2^{2}}{K_n} \biggl( \frac{GM}{3\cdot 5}\biggr)^\eta P_\mathrm{rf}^{(1-\eta)} \, .
</math>
</div>
 




Revision as of 01:23, 28 October 2012

Whitworth's (1981) Isothermal Free-Energy Surface
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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,

LSU Key.png

<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.


Related Wikipedia Discussions


Whitworth's (1981) Isothermal Free-Energy Surface

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