Joel2: /* Emden's Numerical Solution */

2024-07-18T17:50:31Z

Emden's Numerical Solution

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Joel2: /* Emden's Numerical Solution */

2024-07-18T17:46:25Z

Emden's Numerical Solution

← Older revision		Revision as of 13:46, 18 July 2024
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	[[File:EmdenTable14.~~jpg~~\|600px\|center\|Emden's (1907) Table 14]]		[[File:EmdenTable14.png\|600px\|center\|Emden's (1907) Table 14]]
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Joel2: Created page with "FORCETOC =Isothermal Sphere= {| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 125px; width: 125px; background-color:#ffff99;" | Isothermal
Sphere |} Here we supplement the simplified set of principal governing equations with an isothermal equation..."

2023-12-16T23:25:57Z

Created page with "__FORCETOC__  =Isothermal Sphere= {| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 125px; width: 125px; background-color:#ffff99;" | <font size="-1"><b>Isothermal<br />Sphere</b></font> |} Here we supplement the simplified set of principal governing equations with an isothermal equation..."

New page

__FORCETOC__


=Isothermal Sphere=
{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:#ffff99;" |
<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures|<b>Isothermal<br />Sphere</b>]]</font>
|}
Here we supplement the [[SSCpt1/PGE|simplified set of principal governing equations]] with an isothermal equation of state, that is, {{Math/VAR_Pressure01}} is related to {{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.
 <br />
 <br />
 <br />

Comparing this {{Math/VAR_Pressure01}}-{{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,

{{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, {{Math/C_GasConstant}}, {{Math/C_BoltzmannConstant}}, {{Math/C_AtomicMassUnit}}, and {{Math/MP_MeanMolecularWeight}} are all defined in the accompanying [[Appendix/VariablesTemplates|variables appendix]]. It will be useful to note that, for an isothermal gas, the enthalpy, {{Math/VAR_Enthalpy01}}, is related to {{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 [[SSCpt2/SolutionStrategies#Technique_2|solution technique #2]], we need to solve the following second-order ODE relating the two unknown functions, {{Math/VAR_Density01}} and {{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 {{Math/VAR_Enthalpy01}}-{{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 the book by {{ Emden07full }}.

<div align="center">
<table border="1" cellpadding="4">
<tr>
<td colspan="2" align="center">
Derivation Appearing on p. 131 of {{ Emden07 }} (edited)
</td>
</tr>
<tr>
<td align="center" bgcolor="black">
[[File:EmdenBookCover1907.png|240px|center|Emden (1907)]]
</td>
<td align="left">

§2. Wir gehen wieder aus von der Gleichung (59)
<div align="center"><math>\frac{d}{dr}\biggl(\frac{r^2}{\rho} \frac{dp}{dr}\biggr) = -4\pi G\rho r^2 \, .</math></div>
Da wir haben <math>p = \rho H T, T = ~\mathrm{konst.}</math>, so ergibt sich
<div align="center"><math>\frac{dp}{\rho} = HT \frac{d\rho}{\rho} = HT d\log\rho \, ,</math></div>
und setzen wir
<div align="center"><math>\beta^2 = \frac{4\pi G}{HT}</math> (gramm<sup>-1</sup> cent)</div>
und führen die Differentiation aus, so ergibt sich die
<div align="center">
<font color="maroon"><b>''Differentialgleichung der isothermal Gaskugel''</b></font><br />
<math>\frac{d^2 ~\mathrm{lg}\rho}{dr^2} + \frac{2}{r} \frac{d~\mathrm{lg}\rho}{r} + \beta^2 \rho = 0 \, .</math>
</div>
</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 {{ Emden07 }}.
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 {{ CW49full }}. In this published work as well as in §22 of Chapter IV in [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>], Chandrasekhar has written the governing ODE in a form that we will refer to as the,
<div align="center" id="Chandrasekhar">
<font color="maroon"><b>Isothermal Lane-Emden Equation</b></font><br />
{{ Math/EQ_SSLaneEmden02 }}
</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>        and         <math>~v_1 \rightarrow -\psi </math>.
</div>
Across the astrophysics community, Chadrasekhar's notation has been widely — although not universally — adopted as the standard.

==Emden's Numerical Solution==
<div align="center">
<table border="1" cellpadding="8" align="center" width="80%">
<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:
* {{ CW49full }}:   ''The Isothermal Function''
* {{ Horedt86full }}:  ''Seven-Digit Tables of Lane-Emden Functions'' — See, in particular, pp. 405-406 (Sphere of polytropic index <math>~n = \pm \infty</math>).

An analytic — but ''approximate'' — solution to the isothermal Lane-Emden equation can be found:
* [http://adsabs.harvard.edu/abs/1996MNRAS.281.1197L F. K. Liu (1996, MNRAS, 281, 1197-1205)]:   ''Polytropic Gas Spheres: An Approximate Analytic Solution of the Lane-Emden Equation''
* [http://adsabs.harvard.edu/abs/1997MNRAS.286..268N Priyamvada Natarajan & Donald Lynden-Bell (1997, MNRAS, 286, 268-270)]:   '' An Analytic Approximation to the Isothermal Sphere''
* [http://adsabs.harvard.edu/abs/2013RMxAA..49...63R A. C. Raga, J. C. Rodríguez-Ramírez, M. Villasante, A. Rodríguez-González, & V. Lora (2013, Revista Mexicana de Astronomía y Astrofísica, 49, 63-69)]:   ''A New Analytic Approximation to the Isothermal, Self-Gravitating Sphere''

</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 — specifically, <math>~ \rho \propto r^{-2}</math>. This is consistent with our general discussion, [[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.png|350px|center|Plotted from Emden's (1907) tabulated data]]
</td>
<td align="center" valign="top">
[[File:EmdenMassProfile.png|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 — 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 [[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 — 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 />

{{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 — 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 />

{{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 — specifically, <math> ~M_r \propto r^{+1}</math>.

As was realized independently by {{ Ebert55full }} and {{ Bonnor56full }}, 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 [[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 & 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 — and, hence, also the pressure profile — 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 & 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.png|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=
{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:white;" |
<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures|<b>via<br />Direct<br />Numerical<br />Integration</b>]]</font>
|}
The [[#keyExpression|above governing relation]] — see especially [[#Chandrasekhar|Chandrasekhar's notation]] — may be rewritten as (see also, for example, §19.8, eq. 19.35 of [<b>[[User:Tohline/Appendix/References#KW94|<font color="red">KW94</font>]]</b>]),
<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 — where, <math>~w_i = w_0 = 0</math> and <math>~r_i = r_0 = 0</math> — we will use the following [[Appendix/Ramblings/PowerSeriesExpressions#IsothermalLaneEmden|power-series expansion]] (see, for example, eq. 377 from §22 in Chapter IV of [<b>[[User:Tohline/Appendix/References#C67|<font color="red">C67</font>]]</b>]) 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>

=Related Discussions=
==Journal Articles==
* [http://adsabs.harvard.edu/abs/1870AmJS...50...57L J. H. Lane (1870)], '''American Journal of Science''', ''On the Theoretical Temperature of the Sun''
* [https://archive.org/details/mobot31753002152772/page/56 J. H. Lane (1870)], '''The American Journal of Science and Arts''', Vol. 50, pp. 57 - 74: ''On the Theoretical Temperature of the Sun under the Hypothesis of a Gaseous Mass Maintaining Its Volume by Its Internal Heat and Depending on the Laws of Gases Known to Terrestrial Experiment''

==Wikipedia==
* [https://en.wikipedia.org/wiki/Robert_Emden Robert Emden]
* [https://en.wikipedia.org/wiki/Emden–Chandrasekhar_equation Emden-Chandrasekhar equation]
* [https://en.wikipedia.org/wiki/Jonathan_Homer_Lane Jonathan Home Lane]
* [https://en.wikipedia.org/wiki/Lane–Emden_equation Lane-Emden equation]

{{ SGFfooter }}

SSC/Structure/IsothermalSphere - Revision history

Joel2: /* Emden's Numerical Solution */

Joel2: /* Emden's Numerical Solution */