User:Tohline/SSC/IsothermalCollapse - Revision history

Tohline: /* See Especially */

2017-07-21T20:06:21Z

See Especially

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	* [http://adsabs.harvard.edu/abs/1993ApJ...416..303F Prudence N. Foster & Roger A. Chevalier (1993, ApJ, 416, 303)]:   ''Gravitational Collapse of an Isothermal Sphere''		* [http://adsabs.harvard.edu/abs/1993ApJ...416..303F Prudence N. Foster & Roger A. Chevalier (1993, ApJ, 416, 303)]:   ''Gravitational Collapse of an Isothermal Sphere''
	* [http://adsabs.harvard.edu/abs/2013RMxAA..49..127R A. C. Raga, J. C. Rodríguez-Ramírez, A. Rodríguez-González, V. Lora, & A. Esquivel (2013, Revista Mexicana de Astronomía y Astrofísica, 49, 127-135)]:   ''Analytic and Numerical Calculations of the Radial Stability of the Isothermal Spheres''		* [http://adsabs.harvard.edu/abs/2013RMxAA..49..127R A. C. Raga, J. C. Rodríguez-Ramírez, A. Rodríguez-González, V. Lora, & A. Esquivel (2013, Revista Mexicana de Astronomía y Astrofísica, 49, 127-135)]:   ''Analytic and Numerical Calculations of the Radial Stability of the Isothermal Spheres''
			* [http://adsabs.harvard.edu/abs/2017ApJ...835…40C E. Coughlin (2017, ApJ, 835, article id. 40)]:   ''Spherically Symmetric Cold Collapse: The Exact Solutions and a Comparison with Self-Similar Solutions''


	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: /* See Especially */

2017-07-19T23:08:47Z

See Especially

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	* [http://adsabs.harvard.edu/abs/1967LIACo..14..305P M. V. Penston (1967, …)]:   ''A Numerical Integration of the Nonlinear Equations of Spherical Collapse''		* [http://adsabs.harvard.edu/abs/1967LIACo..14..305P M. V. Penston (1967, …)]:   ''A Numerical Integration of the Nonlinear Equations of Spherical Collapse''
	* [http://adsabs.harvard.edu/abs/1966RGOB..117..299P M. V. Penston (1966, Royal Observatory Bulletins. Series E, No. 117)]:   ''Dynamics of Self-Gravitating Gaseous Spheres I''[Not available digitally]		* [http://adsabs.harvard.edu/abs/1966RGOB..117..299P M. V. Penston (1966, Royal Observatory Bulletins. Series E, No. 117)]:   ''Dynamics of Self-Gravitating Gaseous Spheres (Part I)'' [Not available digitally]
	* [http://adsabs.harvard.edu/abs/1968ApJ...152..515B P. Bodenheimer & A. Sweigart (1968, ApJ, 152, 515)]:   ''Dynamic Collapse of the Isothermal Sphere''		* [http://adsabs.harvard.edu/abs/1968ApJ...152..515B P. Bodenheimer & A. Sweigart (1968, ApJ, 152, 515)]:   ''Dynamic Collapse of the Isothermal Sphere''
			* [http://adsabs.harvard.edu/abs/1969MNRAS.145..457P M. V. Penston (1969, MNRAS, 145, 457)]:   ''Dynamics of Self-Gravitating Gaseous Sphers - II. Collapses of Gas Spheres With Cooling and the Behavior of Polytropic Gas Spheres''
	* [http://adsabs.harvard.edu/abs/1969MNRAS.144..425P M. V. Penston (1969, MNRAS, 144, 425)]:   ''Dynamics of Self-Gravitating Gaseous Sphers - III. Analytic Results in the Free-Fall of Isothermal Cases''		* [http://adsabs.harvard.edu/abs/1969MNRAS.144..425P M. V. Penston (1969, MNRAS, 144, 425)]:   ''Dynamics of Self-Gravitating Gaseous Sphers - III. Analytic Results in the Free-Fall of Isothermal Cases''
	* [http://adsabs.harvard.edu/abs/1969MNRAS.145..271L Richard B. Larson (1969, MNRAS, 145, 271)]:   ''Numerical Calculations of the Dynamics of Collapsing Proto-Star''		* [http://adsabs.harvard.edu/abs/1969MNRAS.145..271L Richard B. Larson (1969, MNRAS, 145, 271)]:   ''Numerical Calculations of the Dynamics of Collapsing Proto-Star''

Tohline: /* See Especially */

2017-07-19T16:51:54Z

See Especially

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	=See Especially=		=See Especially=

			* [http://adsabs.harvard.edu/abs/1967LIACo..14..305P M. V. Penston (1967, …)]:   ''A Numerical Integration of the Nonlinear Equations of Spherical Collapse''
			* [http://adsabs.harvard.edu/abs/1966RGOB..117..299P M. V. Penston (1966, Royal Observatory Bulletins. Series E, No. 117)]:   ''Dynamics of Self-Gravitating Gaseous Spheres I''[Not available digitally]
	* [http://adsabs.harvard.edu/abs/1968ApJ...152..515B P. Bodenheimer & A. Sweigart (1968, ApJ, 152, 515)]:   ''Dynamic Collapse of the Isothermal Sphere''		* [http://adsabs.harvard.edu/abs/1968ApJ...152..515B P. Bodenheimer & A. Sweigart (1968, ApJ, 152, 515)]:   ''Dynamic Collapse of the Isothermal Sphere''
	* [http://adsabs.harvard.edu/abs/1969MNRAS.144..425P M. V. Penston (1969, MNRAS, 144, 425)]:   ''Dynamics of Self-Gravitating Gaseous Sphers - III. Analytic Results in the Free-Fall of Isothermal Cases''		* [http://adsabs.harvard.edu/abs/1969MNRAS.144..425P M. V. Penston (1969, MNRAS, 144, 425)]:   ''Dynamics of Self-Gravitating Gaseous Sphers - III. Analytic Results in the Free-Fall of Isothermal Cases''

Tohline: /* Similarity Solution */

2017-07-09T17:45:04Z

Similarity Solution

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	====Larson (1969)====		====Larson (1969)====
	In Appendix C of ~~his~~ seminal paper, [http://adsabs.harvard.edu/abs/1969MNRAS.145..271L Richard B. Larson (1969, MNRAS, 145, 271)] presents an ''asymptotic similarity solution for the isothermal collapse of a sphere.'' He begins by stating that, <font color="darkgreen">in Eulerian form, the equations governing the isothermal collapse may be written</font> (see his equation ''set'' C1),		In Appendix C of a seminal paper, [http://adsabs.harvard.edu/abs/1969MNRAS.145..271L Richard B. Larson (1969, MNRAS, 145, 271)] presents an ''asymptotic similarity solution for the isothermal collapse of a sphere.'' He begins by stating that, <font color="darkgreen">in Eulerian form, the equations governing the isothermal collapse may be written</font> (see his equation ''set'' C1),
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</table>		</table>
	</div>		</div>
	Apart from the different adopted variable ''names'', it is clear that the first and third of these equations have exact counterparts in the set of [[#IsothermalEulerianFrame\|"Eulerian Frame" governing equations]] that we have identified, above. The second ~~of his equations can be shown to be equivalent to~~ the continuity equation, ~~as follows~~. ~~From an Eulerian perspective, the radial coordinate~~ is ~~independent of time, so it can be moved inside or outside of~~ the ~~(partial) time derivative without consequence. Hence~~, ~~multiplying our version of the continuity equation through by~~ <math>~~~4\pi r^2 dr~~</math> ~~— while~~ switching to ~~Larson's variable notation — gives~~,		Apart from the different adopted variable ''names'', it is clear that the first and third of these equations have exact counterparts in the set of [[#IsothermalEulerianFrame\|"Eulerian Frame" governing equations]] that we have identified, above. The second equation replaces the continuity equation, providing a different but equally valid statement of mass conservation. It is most straightforwardly derived by recognizing that the quantity, <math>~m</math>, can be used as a Lagrangian tracer whose (Lagrangian) time-derivative is zero throughout an evolution, then switching to an Eulerian frame of reference. That is,

	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\frac{~~\partial~~}{~~\partial t~~} ~~\biggl[ 4\pi r^2 \rho dr \biggr]~~</math>		<math>~\frac{dM_r}{dt}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~~~-d(r^2 \rho u) \, .~~</math>		<math>~0 </math>
	</td>		</td>
	</tr>		</tr>
	~~</table>~~
	~~</div>~~
	~~Integrating both sides from the center of the configuration to any radial coordinate location within the configuration then gives,~~
	~~<div align="center">~~
	~~<table border="0" cellpadding="5" align="center">~~

	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\~~frac{\partial}{\partial t} \biggl[ 4\pi \int_0^r r^2 \rho dr \biggr]~~</math>		<math>~\Rightarrow ~~~ 0</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~~~- 4~~\pi \~~int_0^~~r ~~d(r^2 \rho u)~~ </math>		<math>~\frac{\partial M_r}{\partial t} + v_r ~\frac{\partial M_r}{\partial r} </math>
	</td>		</td>
	</tr>		</tr>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	~~<math>~\frac{\partial m}{\partial t} </math>~~
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~- 4\pi r^2 \rho u \, . </math>		<math>~\frac{\partial M_r}{\partial t} +4\pi r^2 \rho v_r \, .</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	</div>		</div>

	~~Q.E.D.~~

	=See Especially=		=See Especially=

Tohline: /* Establishing Set of Governing Equations */

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Establishing Set of Governing Equations

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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\frac{~~dM_r~~}{dr} </math>		<math>~\frac{\partial M_r}{\partial r} </math>
	</td>		</td>
	<td align="center">		<td align="center">

Tohline: /* See Especially */

2017-07-08T23:21:15Z

See Especially

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	* [http://adsabs.harvard.edu/abs/1977ApJ...214..488S F. H. Shu (1977, ApJ, 214, 488-497)]:   ''Self-Similar Collapse of Isothermal Spheres and Star Formation''		* [http://adsabs.harvard.edu/abs/1977ApJ...214..488S F. H. Shu (1977, ApJ, 214, 488-497)]:   ''Self-Similar Collapse of Isothermal Spheres and Star Formation''
	* [http://adsabs.harvard.edu/abs/1977ApJ...218..834H C. Hunter (1977, ApJ, 218, 834-845)]:   ''The Collapse of Unstable Isothermal Spheres''		* [http://adsabs.harvard.edu/abs/1977ApJ...218..834H C. Hunter (1977, ApJ, 218, 834-845)]:   ''The Collapse of Unstable Isothermal Spheres''
			* [http://adsabs.harvard.edu/abs/1979ApJ...230..839B J. Buff, H. Gerola, & R. F. Stellingwerf (1979, ApJ, 230, 839-846)]:   ''A Predictability Limit for Collapsing Isothermal Spheres'' — they numerically study properties of eigenvectors near the Bonnor-Ebert limit.
	* [http://adsabs.harvard.edu/abs/1985MNRAS.214....1W A. Whitworth & D. Summers (1985, MNRAS, 214, 1 - 25)]:  ''Self-Similar Condensation of Spherically Symmetric Self-Gravitting Isothermal Gas Clouds''		* [http://adsabs.harvard.edu/abs/1985MNRAS.214....1W A. Whitworth & D. Summers (1985, MNRAS, 214, 1 - 25)]:  ''Self-Similar Condensation of Spherically Symmetric Self-Gravitting Isothermal Gas Clouds''
	* [http://adsabs.harvard.edu/abs/1993ApJ...416..303F Prudence N. Foster & Roger A. Chevalier (1993, ApJ, 416, 303)]:   ''Gravitational Collapse of an Isothermal Sphere''		* [http://adsabs.harvard.edu/abs/1993ApJ...416..303F Prudence N. Foster & Roger A. Chevalier (1993, ApJ, 416, 303)]:   ''Gravitational Collapse of an Isothermal Sphere''

Tohline: /* Larson (1969) */

2017-07-08T21:09:33Z

Larson (1969)

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	====Larson (1969)====		====Larson (1969)====
	In Appendix C of his seminal paper, [http://adsabs.harvard.edu/abs/1969MNRAS.145..271L Richard B. Larson (1969, MNRAS, 145, 271)] presents an ''asymptotic similarity solution for the isothermal collapse of a sphere.''		In Appendix C of his seminal paper, [http://adsabs.harvard.edu/abs/1969MNRAS.145..271L Richard B. Larson (1969, MNRAS, 145, 271)] presents an ''asymptotic similarity solution for the isothermal collapse of a sphere.'' He begins by stating that, <font color="darkgreen">in Eulerian form, the equations governing the isothermal collapse may be written</font> (see his equation ''set'' C1),
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial r} + \frac{Gm}{r^2} + \mathfrak{R} T ~\frac{d\ln\rho}{d r}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~0 \, ,</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\frac{\partial m}{\partial t} + 4\pi r^2\rho u</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~0 \, ,</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\frac{\partial m}{\partial r} - 4\pi r^2 \rho</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ 0 \, . </math>
			</td>
			</tr>
			</table>
			</div>
			Apart from the different adopted variable ''names'', it is clear that the first and third of these equations have exact counterparts in the set of [[#IsothermalEulerianFrame\|"Eulerian Frame" governing equations]] that we have identified, above. The second of his equations can be shown to be equivalent to the continuity equation, as follows. From an Eulerian perspective, the radial coordinate is independent of time, so it can be moved inside or outside of the (partial) time derivative without consequence. Hence, multiplying our version of the continuity equation through by <math>~4\pi r^2 dr</math> — while switching to Larson's variable notation — gives,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{\partial}{\partial t} \biggl[ 4\pi r^2 \rho dr \biggr]</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~-d(r^2 \rho u) \, .</math>
			</td>
			</tr>
			</table>
			</div>
			Integrating both sides from the center of the configuration to any radial coordinate location within the configuration then gives,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{\partial}{\partial t} \biggl[ 4\pi \int_0^r r^2 \rho dr \biggr]</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~- 4\pi \int_0^r d(r^2 \rho u) </math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\frac{\partial m}{\partial t} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~- 4\pi r^2 \rho u \, . </math>
			</td>
			</tr>
			</table>
			</div>

			Q.E.D.

	=See Especially=		=See Especially=

Tohline: /* Establishing Set of Governing Equations */

2017-07-08T20:16:36Z

Establishing Set of Governing Equations

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	where, <math>~c_s</math>, is the isothermal sound speed. This is equivalent to adopting an [[User:Tohline/SR#Equation_of_State\|ideal-gas equation of state]] and setting,		where, <math>~c_s</math>, is the isothermal sound speed. This is equivalent to adopting an [[User:Tohline/SR#Equation_of_State\|ideal-gas equation of state]] and setting,
	<div align="center">		<div align="center">
	<math>~c_s^2 = \frac{\mathfrak{R}T}{\mu} \, ,</math>		<math>~c_s^2 = \frac{\mathfrak{R}T}{\bar\mu} \, ,</math>
	</div>		</div>
	then implementing the isothermality condition by holding the gas temperature constant. Recognizing that the differential element of mass that is contained within a spherical shell of width, <math>~dr</math>, at radial location, <math>~r</math>, is,		then implementing the isothermality condition by holding the gas temperature constant. Recognizing that the differential element of mass that is contained within a spherical shell of width, <math>~dr</math>, at radial location, <math>~r</math>, is,

Tohline: /* Establishing Set of Governing Equations */

2017-07-08T20:11:30Z

Establishing Set of Governing Equations

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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. Recognizing that the differential element of mass that is contained within a spherical shell of width, <math>~dr</math>, at radial location, <math>~r</math>, is,		where, <math>~c_s</math>, is the isothermal sound speed. This is equivalent to adopting an [[User:Tohline/SR#Equation_of_State\|ideal-gas equation of state]] and setting,
			<div align="center">
			<math>~c_s^2 = \frac{\mathfrak{R}T}{\mu} \, ,</math>
			</div>
			then implementing the isothermality condition by holding the gas temperature constant. Recognizing that the differential element of mass that is contained within a spherical shell of width, <math>~dr</math>, at radial location, <math>~r</math>, is,
	<div align="center">		<div align="center">
	<math>~dm = 4\pi r^2 \rho dr \, ,</math>		<math>~dm = 4\pi r^2 \rho dr \, ,</math>
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	</td></tr></table>		</td></tr></table>
			</div>

			In either case, we can reduce the number of dependent variables by one — and, in conjunction, reduce the set of equations by one — by explicitly replacing <math>~P</math> by the term on the righthand side of the isothermal equation of state, in which case the first term on the righthand side of the Euler + Poisson equation becomes,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{1}{\rho} \frac{dP}{dr}</math>
			</td>
			<td align="center">
			<math>~~\rightarrow~~</math>
			</td>
			<td align="left">
			<math>~\biggl( \frac{c_s^2}{\rho} \biggr) \frac{d\rho}{dr} = c_s^2 \biggl( \frac{d\ln\rho}{dr} \biggr) \, .</math>
			</td>
			</tr>
			</table>
	</div>		</div>

Tohline: /* Bodenheimer & Sweigart (1968) */

2017-07-08T19:50:58Z

Bodenheimer & Sweigart (1968)

← Older revision		Revision as of 19:50, 8 July 2017
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	</div>		</div>

	to track how the mass density varies as the coordinate location of each mass shell, <math>~r</math>, varies. ~~Finally, they adopt an ideal-gas equation of state (see their equation 3) and implement the isothermality condition by holding the gas temperature constant, that is to say, they effectively set,~~		to track how the mass density varies as the coordinate location of each mass shell, <math>~r</math>, varies.
	~~<div align="center">~~
	~~<math>~c_s^2 = \frac{\mathfrak{R}T}{\mu} \,.</math>~~
	~~</div>~~

	===Similarity Solution===		===Similarity Solution===