SSC/Structure/IsothermalSphere - Revision history

Jet53man: /* Mass Profile */

2022-10-17T21:56:02Z

Mass Profile

← Older revision		Revision as of 17:56, 17 October 2022
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	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>.		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 ~~[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 [[SSC/Structure/BonnorEbert\|elsewhere]].		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">		<font color="darkblue">

Jet53man: /* Emden's Numerical Solution */

2022-10-17T21:52:12Z

Emden's Numerical Solution

← Older revision		Revision as of 17:52, 17 October 2022
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	<font size="-1">		<font size="-1">
	A more recent and more extensive tabulation of the structural properties of isothermal spheres is provided by:		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 & Wares (1949, ApJ, 109, 551)]~~:   ''The Isothermal Function''		* {{ CW49full }}:   ''The Isothermal Function''
	* ~~[http://adsabs.harvard.edu/abs/1986Ap%26SS.126..357H G. P. Horedt (1986, Astrophys. & Space Science, 126, 357-408)]~~:  ''Seven-Digit Tables of Lane-Emden Functions'' — See, in particular, pp. 405-406 (Sphere of polytropic index <math>~n = \pm \infty</math>).		* {{ 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:		An analytic — but ''approximate'' — solution to the isothermal Lane-Emden equation can be found:

Jet53man: /* Governing Relations */

2022-10-17T21:38:01Z

Governing Relations

← Older revision		Revision as of 17:38, 17 October 2022
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	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.		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 & Wares (1949, ApJ, 109, 551)]~~. 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,		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">		<div align="center" id="Chandrasekhar">
	<font color="maroon"><b>Isothermal Lane-Emden Equation</b></font><br />		<font color="maroon"><b>Isothermal Lane-Emden Equation</b></font><br />

Jet53man: /* Governing Relations */

2022-10-17T17:46:52Z

Governing Relations

← Older revision		Revision as of 13:46, 17 October 2022
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	</div>		</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'']~~.		This matches the governing ODE whose derivation was published on p. 131 of the book by {{ Emden07full }}.

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	<tr>		<tr>
	<td colspan="2" align="center">		<td colspan="2" align="center">
	Derivation Appearing on p. 131 of ~~[http://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Emden]~~ (edited)		Derivation Appearing on p. 131 of {{ Emden07 }} (edited)
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	</div>		</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)]~~.		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.		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.

Jet53man: /* Summary */

2021-09-20T16:52:09Z

Summary

← Older revision		Revision as of 12:52, 20 September 2021
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	<tr>		<tr>
	<td align="center">		<td align="center">
	[[File:PlotMeanToLocalDensity.~~jpg~~\|350px\|center\|Plot based on data from Emden's (1907) Table 14]]		[[File:PlotMeanToLocalDensity.png\|350px\|center\|Plot based on data from Emden's (1907) Table 14]]
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Emden's Numerical Solution */

2021-09-20T16:49:57Z

Emden's Numerical Solution

← Older revision		Revision as of 12:49, 20 September 2021
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	<tr>		<tr>
	<td align="center" valign="top">		<td align="center" valign="top">
	[[File:IsothermalDensityPlot.~~jpg~~\|350px\|center\|Plotted from Emden's (1907) tabulated data]]		[[File:IsothermalDensityPlot.png\|350px\|center\|Plotted from Emden's (1907) tabulated data]]
	</td>		</td>
	<td align="center" valign="top">		<td align="center" valign="top">
	[[File:EmdenMassProfile.~~jpg~~\|350px\|center\|Plotted from Emden's (1907) tabulated data]]		[[File:EmdenMassProfile.png\|350px\|center\|Plotted from Emden's (1907) tabulated data]]
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Our Numerical Integration */

2021-07-29T03:45:51Z

Our Numerical Integration

← Older revision		Revision as of 23:45, 28 July 2021
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	{\| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"		{\| 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~~;" \|		! 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>		<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures\|<b>via<br />Direct<br />Numerical<br />Integration</b>]]</font>
	\|}		\|}

Jet53man: /* Our Numerical Integration */

2021-07-29T03:25:53Z

Our Numerical Integration

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	=Our Numerical Integration=		=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:#ffff99;" \|
			<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>]),		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">		<div align="center">

Jet53man: /* Isothermal Sphere */

2021-07-28T20:32:24Z

Isothermal Sphere

← Older revision		Revision as of 16:32, 28 July 2021
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	<math>P = c_s^2 \rho \, ,</math>		<math>P = c_s^2 \rho \, ,</math>
	</div>		</div>
	where, <math>~c_s</math> is the isothermal sound speed. ~~Comparing this {{Math/VAR_Pressure01}}-{{Math/VAR_Density01}} relationship to~~		where, <math>~c_s</math> is the isothermal sound speed.
	~~ <br />~~
	<br />		<br />
	<br />		<br />
	<br />		<br />

			Comparing this {{Math/VAR_Pressure01}}-{{Math/VAR_Density01}} relationship to

	<div align="center">		<div align="center">

Jet53man: /* Governing Relations */

2021-07-28T17:48:34Z

Governing Relations

← Older revision		Revision as of 13:48, 28 July 2021
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	<tr>		<tr>
	<td colspan="2" align="center">		<td colspan="2" align="center">
	Derivation by [http://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Emden] (edited)		Derivation Appearing on p. 131 of [http://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Emden] (edited)
	</td>		</td>
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	[[File:EmdenBookCover1907.png\|240px\|center\|Emden (1907)]]		[[File:EmdenBookCover1907.png\|240px\|center\|Emden (1907)]]
	</td>		</td>
	<td align="~~center~~">		<td align="left">
	[[File:EmdenIsothermalDerivation.jpg\|500px\|center\|Emden (1907)]]		<!-- [[File:EmdenIsothermalDerivation.jpg\|500px\|center\|Emden (1907)]] -->
			§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>		</td>
	</tr>		</tr>