View source for User:Tohline/SSC/Structure/IsothermalSphere

__FORCETOC__

=Isothermal Sphere=
{{LSU_HBook_header}}
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, 
<div align="center">
<math>P = c_s^2 \rho \, ,</math>
</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

<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">
<math>c_s^2 = \frac{\Re T}{\bar{\mu}} = \frac{k T}{m_u \bar{\mu}} \, ,</math>
</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,
<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>
<span id="keyExpression">or, equivalently,</span>
<div align="center">
<math>
\frac{d^2\ln\rho}{dr^2} +\frac{2}{r} \frac{d\ln\rho}{dr} + \beta^2 \rho = 0 \, ,
</math> 
</div>
where,
<div align="center">
<math>
\beta^2 \equiv \frac{4\pi G}{c_s^2} \, .
</math>
</div>
This matches the governing ODE whose derivation was published on p. 131 of [http://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Robert Emden's (1907) book titled, ''Gaskugeln'']. 

<div align="center">
<table border="1" cellpadding="4">
<tr>
  <td colspan="2" align="center">
Derivation by [http://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Emden] (edited)
  </td>
</tr>
<tr>
  <td align="center" bgcolor="black">
[[File:EmdenBookCover1907.jpg|240px|center|Emden (1907)]]
  </td>
  <td align="center">
[[File:EmdenIsothermalDerivation.jpg|500px|center|Emden (1907)]]
  </td>
</tr>
<tr>
  <td align="center" colspan="2">
<font size="-1">
Note that, in Emden's derivation, <math>H</math> is not enthalpy but, rather, the effective gas constant, <math>H = c_s^2/T</math>.
</font>
  </td>
</tr>
</table>
</div>
By adopting the following dimensionless variables,
<div align="center">
<math>
\mathfrak{r}_1 \equiv \rho_c^{1/2} \beta r \, , ~~~~\mathrm{and}~~~~v_1 \equiv \ln(\rho/\rho_c) \, ,
</math>
</div>
where <math>\rho_c</math> is the configuration's central density, the governing ODE can be rewritten in dimensionless form as,
<div align="center">
<math>
\frac{d^2v_1}{d\mathfrak{r}_1^2} +\frac{2}{\mathfrak{r}_1} \frac{dv_1}{d\mathfrak{r}_1} + e^{v_1} = 0 \, ,
</math> 
</div>
which is exactly the equation numbered (II"a) that can be found on p. 133 of [http://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Emden (1907)].
Emden numerically determined the behavior of the function <math>~v_1(\mathfrak{r}_1)</math>, its first derivative with respect to <math>~\mathfrak{r}_1</math>, <math>~v_1'</math>, along with <math>~e^{v_1}</math> and several other useful products, and published his results as Table 14, on p. 135 of his book.  This table has been reproduced [[#Emden.27s_Numerical_Solution|immediately below]], primarily for historical purposes.

Note that a somewhat more extensive tabulation of the structural properties of isothermal spheres is provided by [http://adsabs.harvard.edu/abs/1949ApJ...109..551C Chandrasekhar &amp; Wares (1949, ApJ, 109, 551)].  In this published work as well as in &sect;22 of Chapter IV in C67, Chandrasekhar has written the governing ODE as,
<div align="center" id="Chandrasekhar">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{\xi^2}\frac{d}{d\xi}\biggl( \xi^2 \frac{d\psi}{d\xi}\biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~e^{-\psi} \, .</math>
  </td>
</tr>
</table>
</div>
It is straightforward to show that this is identical to Emden's governing expression after making the variable substitutions:
<div align="center">
<math>~\mathfrak{r}_1 \rightarrow \xi</math>&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; <math>~v_1 \rightarrow -\psi </math>.
</div>
Across the astrophysics community, Chadrasekhar's notation has been widely &#8212; although not universally &#8212; adopted as the standard.

==Emden's Numerical Solution==
<div align="center">
<table border="1" cellpadding="1">
<tr>
  <td align="center" valign="top" rowspan="1">
[[File:EmdenTable14.jpg|600px|center|Emden's (1907) Table 14]]
  </td>
</tr>
<tr>
   <td align="center">
<font size="-1">
Note:  The entry highlighted in blue in the <math>3^\mathrm{rd}</math> column must be a typesetting error.
</font>
  </td>
</tr>
<tr>
   <td align="left">
<font size="-1">
A more recent and more extensive tabulation of the structural properties of isothermal spheres is provided by [http://adsabs.harvard.edu/abs/1949ApJ...109..551C Chandrasekhar &amp; Wares (1949, ApJ, 109, 551)].
</font>
  </td>
</tr>
</table>
</div>


A plot of <math>v_1</math> versus <math>\ln\mathfrak{r}_1</math>, as shown below in Figure 1a, translates into a log-log plot of the equilibrium configuration's <math>\rho(r)</math> density profile.  Notice that this isolated isothermal configuration extends to infinity and that, at large radii, the density profile displays a simple power-law behavior &#8212; specifically, <math> \rho \propto r^{-2}</math>.  This is consistent with our general discussion, [[User:Tohline/SSC/Structure/PowerLawDensity#Isothermal_Equation_of_State|presented elsewhere]], of power-law density distributions as solutions of the Lane-Emden equation.  


<div align="center">
<table border="0" cellpadding="5" width="85%">

<tr>
  <td align="center" colspan="2">
'''Figure 1:  Emden's Numerical Solution'''
  </td>
</tr>
<tr>
  <td align="center" valign="top">
[[File:IsothermalDensityPlot.jpg|350px|center|Plotted from Emden's (1907) tabulated data]]
  </td>
  <td align="center" valign="top">
[[File:EmdenMassProfile.jpg|350px|center|Plotted from Emden's (1907) tabulated data]]
  </td>
</tr>
<tr>
  <td valign="top">
(a) The <math>(x,y)</math> locations of the data points plotted in blue are drawn directly from column 1 and column 3 of Emden's Table 14 &#8212; specifically, <math>x = \ln(\mathfrak{r}_1)</math> and <math>y = v_1</math>.  The dashed red line has a slope of <math>-2</math> and serves to illustrate that, at large radii, the [[User:Tohline/SSC/Structure/PowerLawDensity#Isothermal_Equation_of_State|isothermal density profile tends toward a <math>\rho \propto r^{-2}</math> distribution]].
  </td>
  <td valign="top">
(b) The <math>(x,y)</math> locations of the data points plotted in purple are drawn directly from column 1 and column 7 of Emden's Table 14 &#8212; specifically, <math>x = \ln(\mathfrak{r}_1)</math> and <math>y = \mathfrak{r}_1^2 v_1'</math>.  The dashed green line has a slope of <math>+1</math> and serves to illustrate that, at large radii, the isothermal <math>M(r)</math> distribution tends toward a <math>M_r \propto r</math> distribution.
  </td>
</tr>
</table>
</div>

==Mass Profile==
The mass enclosed within a given radius, <math>M_r</math>, can be determined by performing an appropriate volume-weighted integral over the density distribution.  Specifically, based on the key expression for,

<div align="center">
<span id="HydrostaticBalance"><font color="#770000">'''Mass Conservation'''</font></span><br />

{{User:Tohline/Math/EQ_SSmassConservation01}}
</div>
in spherically symmetric configurations, the relevant integral is,
<div align="center">
<math>
M_r = \int_0^r 4\pi r^2 \rho(r) dr \, .
</math>
</div>
But <math>M_r</math> also can be determined from the information provided in column 7 of Emden's Table 14 &#8212; that is, from knowledge of the first derivative of <math>v_1</math>.  The appropriate expression can be obtained from the mathematical prescription for

<div align="center">
<span id="HydrostaticBalance"><font color="#770000">'''Hydrostatic Balance'''</font></span><br />

{{User:Tohline/Math/EQ_SShydrostaticBalance01}}
</div>
in a spherically symmetric configuration.  Since, for an isothermal equation of state (see above),
<div align="center">
<math>
\frac{dP}{\rho} = c_s^2 {d\ln\rho} \, ,
</math>
</div>
the statement of hydrostatic balance can be rewritten as,
<div align="center">
<math>
M_r = \frac{c_s^2}{G} \biggl[ - r^2 \frac{d\ln\rho}{dr} \biggr] = \frac{c_s^2}{G \rho_c^{1/2} \beta} \biggl[ - \mathfrak{r}_1^2 \frac{dv_1}{d\mathfrak{r}_1} \biggr]
= \biggl( \frac{c_s^6}{4\pi G^3 \rho_c} \biggr)^{1/2}  \biggl[ - \mathfrak{r}_1^2 v_1' \biggr] \, .
</math>
</div>
The quantity tabulated in column 7 of Emden's Table 14 is precisely the dimensionless term inside the square brackets of this last expression; having units of mass, the coefficient out front sets the mass scale of the equilibrium configuration and depends only on the choice of central density and isothermal sound speed.  Hence, a plot of <math>\ln(\mathfrak{r}_1^2 v_1')</math> versus <math>\ln\mathfrak{r}_1</math>, as shown above in Figure 1b, translates into a log-log plot of the equilibrium configuration's <math>M_r</math> mass profile.  Notice that, along with the radius, the mass of this isolated isothermal configuration extends to infinity and that, at large radii, the mass profile displays a simple power-law behavior &#8212; specifically, <math> M_r \propto r^{+1}</math>. 

As was realized independently by [http://adsabs.harvard.edu/abs/1955ZA.....37..217E Ebert (1955)] and [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor (1956)], a spherically symmetric isothermal equilibrium configuration of finite radius and finite mass can be constructed if the system is embedded in a pressure-confining external medium.  We discuss their findings [[User:Tohline/SSC/Structure/BonnorEbert|elsewhere]].

<font color="darkblue">
==Summary==
</font>

Based on the above derivations, the internal structural properties of an equilibrium isothermal sphere can be described in terms of the tabulated quantities provided in Emden's Table 14 as follows:
* <font color="red">Radial Coordinate Position</font>: 
: Given the isothermal sound speed, <math>c_s</math>, and the central density, <math>\rho_c</math>, the radial coordinate is,
<div align="center">
<math>r = ( \rho_c \beta^2 )^{-1/2} \mathfrak{r}_1 = \biggl( \frac{c_s^2}{4\pi G \rho_c} \biggr)^{1/2} \mathfrak{r}_1 </math> .
</div>

* <font color="red">Density &amp; Pressure</font>: 
: As a function of the radial coordinate, <math>r(\mathfrak{r}_1)</math>, the density profile is,
<div align="center">
<math>\rho(r(\mathfrak{r}_1))= \rho_c e^{v_1(\mathfrak{r}_1)}</math>;
</div>
: and the pressure profile is,
<div align="center">
<math>P(r(\mathfrak{r}_1))= (c_s^2 \rho_c) e^{v_1(\mathfrak{r}_1)}</math>.
</div>
: As has been explicitly pointed out in the above discussion associated with Figure 1a, the density profile &#8212; and, hence, also the pressure profile &#8212; extends to infinity and, at large radii, behaves as a power law; specifically, <math>\rho \propto r^{-2}</math>. 

* <font color="red">Mass</font>:  
: Given <math>c_s</math> and <math>\rho_c</math>, the natural mass scale is,
<div align="center">
<math>M_0 \equiv \biggl( \frac{c_s^6}{4\pi G^3 \rho_c} \biggr)^{1/2}  </math> ;
</div>
: and, expressed in terms of <math>M_0</math>, the mass that lies interior to radius <math>r</math> is,
<div align="center">
<math>
M_r = M_0 [ - \mathfrak{r}_1^2 v_1' ] \, .
</math>
</div>
: As discussed above in the context of Figure 1b, at large radii, the mass increases linearly with <math>r</math>.  Because the density and pressure profiles extend to infinity, this means that the mass of an isolated isothermal sphere is infinite.

* <font color="red">Enthalpy &amp; Gravitational Potential</font>: 
: To within an additive constant, the enthalpy distribution is,
<div align="center">
<math>H(r(\mathfrak{r}_1))= c_s^2  [- v_1(\mathfrak{r}_1)]</math>;
</div>
: and the gravitational potential is,
<div align="center">
<math>\Phi(r(\mathfrak{r}_1)) = - H(r(\mathfrak{r}_1))= c_s^2  v_1(\mathfrak{r}_1)</math>.
</div>

* <font color="red">Mean-to-Local Density Ratio</font>:
: The ratio of the configuration's mean density, inside a given radius, to its local density at that radius is,
<div align="center">
<math>\frac{\bar{\rho}}{\rho} = \frac{3M_r}{4\pi r^3 \rho} = 3\biggl[- \frac{v_1'}{\mathfrak{r}_1 e^{v_1}} \biggr] </math> .
</div>
: As Figure 2 shows, at large <math>r</math> this density ratio goes to the value of 3, which means that the term inside the square brackets goes to unity at large <math>r</math>.  This behavior is consistent with the limiting power-law behavior of both <math>M_r</math> and <math>\rho</math>, discussed above.

<div align="center">
<table border="0" cellpadding="5" width="360">
<tr>
  <td align="center">
'''Figure 2:  From Emden's Tabulated Data'''
  </td>
</tr>
<tr>
  <td align="center">
[[File:PlotMeanToLocalDensity.jpg|350px|center|Plot based on data from Emden's (1907) Table 14]]
  </td>
</tr>
<tr>
  <td align="left">
The blue curve displays an evaluation of the density ratio, <math>[- 3v_1'/(\mathfrak{r}_1 e^{v_1}) ]</math>, as a function of <math>\ln (\mathfrak{r}_1)</math>, as determined from the data presented in Emden's Table 14, shown above.
  </td>
</tr>
</table>
</div>

=Our Numerical Integration=
The [[#keyExpression|above governing relation]] &#8212; see especially [[#Chandrasekhar|Chandrasekhar's notation]] &#8212; may be rewritten as (see also, for example, &sect;19.8, eq. 19.35 of [[User:Tohline/Appendix/References#KW94|KW94]]),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d^2w}{dr^2} +\frac{2}{r} \frac{d w}{dr} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~e^{-w} \, ,</math>
  </td>
</tr>
</table>
</div>
where we appreciate that,
<div align="center">
<math>~w \equiv \ln\biggl(\frac{\rho}{\rho_c}\biggr) \, .</math>
</div>

We'll adopt the following finite-difference approximations for the first and second derivatives on a grid of radial spacing, <math>~\Delta_r</math>:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~w_i'</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\frac{w_+ - w_-}{2\Delta_r}</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~w_i''</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\frac{w_+ - 2w_i +w_-}{\Delta_r^2} \, .</math>
  </td>
</tr>
</table>
</div>
Our finite-difference approximation of the governing equation is, then,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~r_i \biggl[ \frac{w_+ - 2w_i +w_-}{\Delta_r^2} \biggr] + 2\biggl[  \frac{w_+ - w_-}{2\Delta_r} \biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r_i e^{-w_i} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ r_i [ w_+ - 2w_i +w_- ] + \Delta_r [ w_+ - w_- ]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\Delta_r^2 r_i e^{-w_i} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~w_+ 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\Delta_r^2 r_i e^{-w_i} +  2r_i w_i  + w_- (\Delta_r - r_i)}{( \Delta_r + r_i)} \, .</math>
  </td>
</tr>
</table>
</div>

Now, for the first two steps away from the center &#8212; where, <math>~w_i = w_0 = 0</math> and <math>~r_i = r_0 = 0</math> &#8212;  we will use the following [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#IsothermalLaneEmden|power-series expansion]] (see, for example, eq. 377 from &sect;22 in Chapter IV of [[User:Tohline/Appendix/References#C67|C67]]) to determine the value of <math>~w_i</math>:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~w_1 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\Delta_r^2}{6} - \frac{\Delta_r^4}{120} + \frac{\Delta_r^6}{1890} \, ,</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~w_2 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(2\Delta_r)^2}{6} - \frac{(2\Delta_r)^4}{120} + \frac{(2\Delta_r)^6}{1890} \, .</math>
  </td>
</tr>
</table>
</div>

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