User:Tohline/Apps/HomologousCollapse - Revision history

Tohline: /* Homologous Solution */ Correct continuity equation

2014-11-25T23:26:56Z

Homologous Solution: Correct continuity equation

← Older revision		Revision as of 23:26, 25 November 2014
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\dot{a}\mathfrak{x} \, ;		<math>~\dot{a}\mathfrak{x} \, .
	</math>		</math>
	</td>		</td>
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	</table>		</table>
	</div>		</div>
	~~But~~ in our more general case, the continuity equation gives,		Also, in our more general case, the continuity equation gives,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\frac{d\ln ~~f^n~~}{dt} + \frac{d\ln a^3}{dt} </math>		<math>~\frac{d\ln \rho_c}{dt} + \frac{d\ln a^3}{dt} </math>
	</td>		</td>
	<td align="center">		<td align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>\Rightarrow~~~~~~f^n~~ a^3</math>		<math>\Rightarrow~~~ a^3\rho_c</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</table>		</table>
	</div>		</div>
	independent of time~~; and~~ the Euler equation becomes,		independent of time. But the Euler equation becomes,

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

Tohline: /* Governing Equations */

2014-11-25T23:23:52Z

Governing Equations

← Older revision		Revision as of 23:23, 25 November 2014
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\frac{d\ln f^n}{dt} </math>		<math>~\cancelto{0}{\frac{d\ln f^n}{dt}} + \frac{d\ln \rho_c}{dt} </math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~-~ a^{-2} \nabla_\mathfrak{x}^2 \psi \, ,</math>		<math>~-~ a^{-2} \nabla_\mathfrak{x}^2 \psi \, ;</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Homologous Solution */ Showed that Poisson equation is very similar to Lane-Emden equation for arbitrary index, n

2014-11-10T03:00:04Z

Homologous Solution: Showed that Poisson equation is very similar to Lane-Emden equation for arbitrary index, n

← Older revision		Revision as of 03:00, 10 November 2014
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	</table>		</table>
	</div>		</div>
			Inserting this expression into the Poisson equation gives,
			<div align="center">
			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="right">
			<math>~\nabla^2_\mathfrak{x} \biggl( \frac{n}{6} \lambda ~\mathfrak{x}^2 - nf \biggr)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~nf^n </math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~\frac{1}{\mathfrak{x}^2} \frac{d}{d\mathfrak{x}} \biggl[ \mathfrak{x}^2 \frac{d}{d\mathfrak{x}}
			\biggl(f - \frac{\lambda}{6} ~\mathfrak{x}^2 \biggr)\biggr]</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~-f^n </math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~\frac{1}{\mathfrak{x}^2} \frac{d}{d\mathfrak{x}} \biggl[ \mathfrak{x}^2 \frac{df}{d\mathfrak{x}}
			\biggr] </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\lambda - f^n \, ,</math>
			</td>
			</tr>
			</table>
			</div>

			which becomes the familiar Lane-Emden equation ''for arbitrary n'' when <math>~\lambda = 0 \, .</math>

	From the terms on the righthand side we conclude, furthermore, that the nonlinear differential equation governing the time-dependent variation of the scale length, <math>~a</math>, is,		From the terms on the righthand side we conclude, furthermore, that the nonlinear differential equation governing the time-dependent variation of the scale length, <math>~a</math>, is,
	<div align="center">		<div align="center">

Tohline: /* Homologous Solution */ Finished detailing homology solution

2014-11-10T02:31:16Z

Homologous Solution: Finished detailing homology solution

← Older revision		Revision as of 02:31, 10 November 2014
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	Because everything on the lefthand side of ~~Goldreich & Weber's~~ scaled Euler equation depends only on the dimensionless spatial coordinate, <math>~\mathfrak{x}</math>, while everything on the righthand side depends only on time — via the parameter, <math>~a(t)</math> — both expressions must equal the same (dimensionless) constant. ~~[http://adsabs.harvard.edu/abs/1980ApJ...238..991G~~ Goldreich & Weber (1980)~~] (see their equation 12)~~ call this constant, <math>~\lambda/6</math>. From the terms on the lefthand side, ~~they~~ conclude ~~(see their equation 13)~~ that the dimensionless gravitational potential is,		Because everything on the lefthand side of this scaled Euler equation depends only on the dimensionless spatial coordinate, <math>~\mathfrak{x}</math>, while everything on the righthand side depends only on time — via the parameter, <math>~a(t)</math> — both expressions must equal the same (dimensionless) constant. We will follow the lead of Goldreich & Weber (1980) and call this constant, <math>~\lambda/6</math>. From the terms on the lefthand side, we conclude that the dimensionless gravitational potential 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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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{1}{2} \lambda ~\mathfrak{x}^2 - 3f \, .</math>		<math>~\frac{n}{6} \lambda ~\mathfrak{x}^2 - nf \, .</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	</div>		</div>
	From the terms on the righthand side ~~they~~ conclude, furthermore, that the nonlinear differential equation governing the time-dependent variation of the scale length, <math>~a</math>, is,		From the terms on the righthand side we conclude, furthermore, that the nonlinear differential equation governing the time-dependent variation of the scale length, <math>~a</math>, 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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	<td align="right">		<td align="right">
	<math>~		<math>~
	a^2 \ddot{a}		a^{(n+1)/(n-1)} \ddot{a}
	</math>		</math>
	</td>		</td>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~-~\frac{4\lambda}{3} \biggl( \frac{\kappa^3}{\pi G} \biggr)^{1/2} \, .</math>		<math>~-~\frac{\lambda}{3} \biggl[ \frac{(n+1)^n \kappa^n}{4\pi G} \biggr]^{1/(n-1)} \, .</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Governing Equations */

2014-11-10T02:21:43Z

Governing Equations

← Older revision		Revision as of 02:21, 10 November 2014
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	==Governing Equations==		==Governing Equations==
	We begin with the set of [[User:Tohline/Apps/GoldreichWeber80#GoverningWithStreamFunction\|principal governing equations already written in terms of a stream function]], as developed in the accompanying discussion of Goldreich & Weber's (1980) work. ~~Specifically, the~~ continuity equation, the Euler equation, and the Poisson equation ~~become~~, respectively,		We begin with the set of [[User:Tohline/Apps/GoldreichWeber80#GoverningWithStreamFunction\|principal governing equations]] that are already written in terms of a stream function and a time-varying radial coordinate, <math>~\vec\mathfrak{x} \equiv \vec{r}/a(t)</math>, as developed in the accompanying discussion of Goldreich & Weber's (1980) work. The continuity equation, the Euler equation, and the Poisson equation are, respectively,

	<div align="center" id="GoverningWithStreamFunction">		<div align="center" id="GoverningWithStreamFunction">
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	</math>		</math>
	</div>		</div>
	~~Again, following~~ Goldreich & Weber, we allow the normalizing scale length to vary with time an adopt an ''accelerating'' coordinate system with a time-dependent dimensionless radial coordinate,		Following Goldreich & Weber, we allow the normalizing scale length to vary with time and adopt an ''accelerating'' coordinate system with a time-dependent dimensionless radial coordinate,
	<div align="center">		<div align="center">
	<math>~\vec\mathfrak{x} \equiv \frac{1}{a(t)} \vec{r} \, .</math>		<math>~\vec\mathfrak{x} \equiv \frac{1}{a(t)} \vec{r} \, .</math>
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	And we normalize both the gravitational potential and the enthalpy to the square of the central sound speed,		And we normalize both the gravitational potential and the enthalpy to the square of the central sound speed,
	~~<div align="center">~~
	~~<math>c_s^2 = \frac{\gamma P_c}{\rho_c} = \frac{H_c}{n} = \biggl( \frac{n+1}{n} \biggr) \kappa \rho_c^{1/n}~~
	~~= \biggl( \frac{n+1}{n} \biggr) \kappa~~
	~~\biggl\{ \biggl[\frac{(n+1)\kappa}{4\pi G}\biggr]^{1/2} \frac{1}{a(t)} \biggr\}^{2/(n-1)}~~
	~~= \, .</math>~~
	~~</div>~~

	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<math>\nabla_\mathfrak{x}^2 \sigma = n f^n \, .</math>		<math>\nabla_\mathfrak{x}^2 \sigma = n f^n \, .</math>
	</div>		</div>


	==Homologous Solution==		==Homologous Solution==

Tohline: /* Review of Goldreich and Weber (1980) */

2014-11-10T02:10:38Z

Review of Goldreich and Weber (1980)

← Older revision		Revision as of 02:10, 10 November 2014
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	{{LSU_HBook_header}}		{{LSU_HBook_header}}
	==Review of Goldreich and Weber (1980)==		==Review of Goldreich and Weber (1980)==
	In an [[User:Tohline/Apps/GoldreichWeber80#Homologously_Collapsing_Stellar_Cores\|accompanying discussion]], we ~~review~~ the dynamical model that Peter Goldreich & Stephen Weber [http://adsabs.harvard.edu/abs/1980ApJ...238..991G (1980, ApJ, 238, 991)] developed to describe the near-homologous collapse of stellar cores that obey an equation of state having an adiabatic index, <math>~\gamma=4/3</math> — and polytropic index, <math>~n=3</math>. Here we examine whether or not this self-similar collapse model can be extended to configurations with arbitrary polytropic index.		In an [[User:Tohline/Apps/GoldreichWeber80#Homologously_Collapsing_Stellar_Cores\|accompanying discussion]], we have reviewed the self-similar, dynamical model that Peter Goldreich & Stephen Weber [http://adsabs.harvard.edu/abs/1980ApJ...238..991G (1980, ApJ, 238, 991)] developed to describe the near-homologous collapse of stellar cores that obey an equation of state having an adiabatic index, <math>~\gamma=4/3</math> — and polytropic index, <math>~n=3</math>. Here we examine whether or not this self-similar collapse model can be extended to configurations with arbitrary polytropic index.

	==Governing Equations==		==Governing Equations==

Tohline: /* Homologously Collapsing Polytropic Spheres */ More derivation, but with a few lines still from the old derivation

2014-11-09T23:50:25Z

Homologously Collapsing Polytropic Spheres: More derivation, but with a few lines still from the old derivation

Show changes

Tohline: /* Homologously Collapsing Polytropic Spheres */Work through some details of generalized homologous collapse

2014-11-09T23:28:00Z

Homologously Collapsing Polytropic Spheres: Work through some details of generalized homologous collapse

Show changes

Tohline: /* Governing Equations */ Working on raw generalization to arbitrary polytropic index

2014-11-08T22:19:28Z

Governing Equations: Working on raw generalization to arbitrary polytropic index

← Older revision		Revision as of 22:19, 8 November 2014
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	where the subscript, "c", denotes central values and, [[User:Tohline/SR#Barotropic_Structure\|as presented in our introductory discussion of barotropic supplemental relations]],		where the subscript, "c", denotes central values and, [[User:Tohline/SR#Barotropic_Structure\|as presented in our introductory discussion of barotropic supplemental relations]],
	<div align="center">		<div align="center">
	<math>~H = ~~\biggl~~( ~~\frac{\gamma}{\gamma-~~1~~} \biggr~~) \kappa \rho^{~~\gamma-~~1} \, .</math>		<math>~H = (n+1) \kappa \rho^{1/n} \, .</math>
	</div>		</div>
	Substitution of this equation of state expression leads to,		Substitution of this equation of state expression leads to,
	<div align="center">		<div align="center">
	<math>		<math>
	a = ~~\rho_c^{-1/3}~~ \biggl(\frac{\kappa}{\pi G}\biggr)^{1/2} \, .		a = \biggl[\frac{(n+1)\kappa}{4\pi G}\biggr]^{1/2} \rho_c^{-(n-1)/(2n)} \, .
	</math>		</math>
	</div>		</div>
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	<div align="center">		<div align="center">
	<math>		<math>
	\rho_c = \biggl(\frac{\kappa}{\pi G}\biggr)^{3/2} [a(t)]^{-3} \, .		\rho_c = \biggl\{ \biggl[\frac{(n+1)\kappa}{4\pi G}\biggr]^{1/2} \frac{1}{a(t)} \biggr\}^{2n/(n-1)} \, .
	</math>		</math>
	</div>		</div>

Tohline: Save after cut-and-pasting several subsections from GW80 chapter

2014-11-08T22:04:55Z

Save after cut-and-pasting several subsections from GW80 chapter

New page

=Homologously Collapsing Polytropic Spheres=
{{LSU_HBook_header}}
==Review of Goldreich and Weber (1980)==
This is principally a review of the dynamical model that Peter Goldreich & Stephen Weber [http://adsabs.harvard.edu/abs/1980ApJ...238..991G (1980, ApJ, 238, 991)] developed to describe the near-homologous collapse of stellar cores. As we began to study the Goldreich & Weber paper, it wasn't immediately obvious how the set of differential governing equations should be modified in order to accommodate a radially contracting (accelerating) coordinate system. I did not understand the transformed set of equations presented by Goldreich & Weber as equations (7) and (8), for example. At first, I turned to [http://www.sciencedirect.com/science/article/pii/S0021999106002555 Poludnenko & Khokhlov (2007, Journal of Computational Physics, 220, 678)] — hereafter, PK07 — for guidance. PK07 develop a very general set of governing equations that allows for coordinate rotation as well as expansion or contraction. Ultimately, the most helpful additional reference proved to be §19.11 (pp. 187 - 190) of Kippenhahn & Weigert [ [[User:Tohline/Appendix/References#KW94|KW94]] ].

==Governing Equations==

===Length===

Following [http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich & Weber (1980)], we choose the same length scale for normalization that is used in deriving the [[User:Tohline/SSC/Structure/Polytropes#Lane-Emden_equation|Lane-Emden equation]], which governs the hydrostatic structure of a polytrope of index {{ User:Tohline/Math/MP_PolytropicIndex }}, that is,
<div align="center">
<math>
a_\mathrm{n} \equiv \biggl[\frac{1}{4\pi G}~ \biggl( \frac{H_c}{\rho_c} \biggr)\biggr]^{1/2} \, ,
</math>
</div>
where the subscript, "c", denotes central values and, [[User:Tohline/SR#Barotropic_Structure|as presented in our introductory discussion of barotropic supplemental relations]],
<div align="center">
<math>~H = \biggl( \frac{\gamma}{\gamma-1} \biggr) \kappa \rho^{\gamma-1} \, .</math>
</div>
Substitution of this equation of state expression leads to,
<div align="center">
<math>
a = \rho_c^{-1/3} \biggl(\frac{\kappa}{\pi G}\biggr)^{1/2} \, .
</math>
</div>
''Most significantly'', Goldreich & Weber (see their equation 6) allow the normalizing scale length to vary with time in order for the governing equations to accommodate a self-similar dynamical solution. In doing this, they effectively adopted an ''accelerating'' coordinate system with a time-dependent dimensionless radial coordinate,
<div align="center">
<math>~\vec\mathfrak{x} \equiv \frac{1}{a(t)} \vec{r} \, .</math>
</div>

This, in turn, will mean that either the central density varies with time, or the specific entropy of all fluid elements (captured by the value of <math>~\kappa</math>) varies with time, or both. In practice, Goldreich & Weber assume that <math>~\kappa</math> is held fixed, so the time-variation in the scale length, <math>~a</math>, reflects a time-varying central density; specifically,
<div align="center">
<math>
\rho_c = \biggl(\frac{\kappa}{\pi G}\biggr)^{3/2} [a(t)]^{-3} \, .
</math>
</div>

Given the newly adopted dimensionless radial coordinate, the following replacements for the spatial operators should be made, as appropriate, throughout the set of governing equations:
<div align="center">
<math>~\nabla_r ~\rightarrow~ a^{-1} \nabla_\mathfrak{x}</math>       
and
       <math>~\nabla_r^2 ~\rightarrow~ a^{-2} \nabla_\mathfrak{x}^2 \, .</math>
</div>

Specifically, the continuity equation, the Euler equation, and the Poisson equation become, respectively,

<div align="center">
<table border="1" align="center" cellpadding="10" width="55%">
<tr><td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\frac{1}{\rho} \frac{d\rho}{dt} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-~ a^{-2} \nabla_\mathfrak{x}^2 \psi \, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{d\psi}{dt} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2} a^{-2} ( \nabla_\mathfrak{x} \psi )^2 - H - \Phi \, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>~a^{-2}\nabla_\mathfrak{x}^2 \Phi </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~4\pi G \rho \, .</math>
</td>
</tr>
</table>

</td></tr>
</table>
</div>

===Mass-Density and Speed===

Next, [http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich & Weber (1980)] (see their equation 10) choose to normalize the density by the central density, specifically defining a dimensionless function,
<div align="center">
<math>f \equiv \biggl( \frac{\rho}{\rho_c} \biggr)^{1/3} \, .</math>
</div>
Keeping in mind that <math>~n = 3</math>, this is also in line with the formulation and evaluation of the [[User:Tohline/SSC/Structure/Polytropes#Lane-Emden_equation|Lane-Emden equation]], where the primary ''dependent'' structural variable is the dimensionless polytropic enthalpy,
<div align="center">
<math>\Theta_H \equiv \biggl( \frac{\rho}{\rho_c} \biggr)^{1/n} \, .</math>
</div>

Also, [http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich & Weber (1980)] (see their equation 11) normalize the gravitational potential to the square of the central sound speed,
<div align="center">
<math>c_s^2 = \frac{\gamma P_c}{\rho_c} = \frac{4}{3} \kappa \rho_c^{1/3}
= \frac{4}{3}\biggl(\frac{\kappa^3}{\pi G}\biggr)^{1/2} [a(t)]^{-1} \, .</math>
</div>
Specifically, their dimensionless gravitational potential is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\sigma</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\frac{\Phi}{c_s^2} = \biggl[ \frac{3}{4} \biggl( \frac{\pi G}{\kappa^3} \biggr)^{1/2} a(t) \biggr] \Phi \, ,</math>
</td>
</tr>
</table>
</div>
and the similarly normalized enthalpy may be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\frac{H}{c_s^2} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{3}{4} \biggl( \frac{\pi G}{\kappa^3} \biggr)^{1/2} a(t) \biggr] 4\kappa \rho^{1/3} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~3 \biggl( \frac{\rho}{\rho_c} \biggr)^{1/3} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~3f \, .</math>
</td>
</tr>

</table>
</div>
With these additional scalings, the continuity equation becomes,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\frac{d\ln f^3}{dt} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-~ a^{-2} \nabla_\mathfrak{x}^2 \psi \, ,</math>
</td>
</tr>
</table>
</div>

the Euler equation becomes,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~
\biggl[ \frac{3}{4} \biggl( \frac{\pi G}{\kappa^3} \biggr)^{1/2} a(t) \biggr]
\biggl[ \frac{d\psi}{dt} - \frac{1}{2a^2} ( \nabla_\mathfrak{x} \psi )^2 \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ - 3 f - \sigma \, ;</math>
</td>
</tr>
</table>
</div>

and the Poisson equation becomes,
<div align="center">
<math>\nabla_\mathfrak{x}^2 \sigma = 3f^3 \, .</math>
</div>

{{LSU_HBook_footer}}