User:Tohline/Apps/GoldreichWeber80 - Revision history

Tohline: /* Related Discussions */

2017-03-08T01:48:06Z

Related Discussions

← Older revision		Revision as of 01:48, 8 March 2017
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	=Related Discussions=		=Related Discussions=
			* Homologous collapse deduced from an analysis of the [[User:Tohline/SSC/Stability/n3PolytropeLAWE#Specific_case_of_n_.3D_3_Polytropes\|LAWE that governs low-amplitude, homentropic radial oscillations in n = 3 polytropes]].


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Tohline: Shift location of "work in progress" road sign

2014-11-25T23:41:33Z

Shift location of "work in progress" road sign

← Older revision		Revision as of 23:41, 25 November 2014
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	This matches equation (12) of [http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich & Weber (1980)].		This matches equation (12) of [http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich & Weber (1980)].


	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. [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. [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,
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	For the case, <math>~C = 0</math>, this differential equation can be integrated straightforwardly to give (see Goldreich & Weber's equation 15),		For the case, <math>~C = 0</math>, this differential equation can be integrated straightforwardly to give (see Goldreich & Weber's equation 15),



	For the cases when <math>~C \ne 0</math>, [http://integrals.wolfram.com/index.jsp Wolfram Mathematica's online integrator] can be called upon to integrate this equation and provide the following closed-form solution,		For the cases when <math>~C \ne 0</math>, [http://integrals.wolfram.com/index.jsp Wolfram Mathematica's online integrator] can be called upon to integrate this equation and provide the following closed-form solution,
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	~~=Related Discussions=~~



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			=Related Discussions=



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Tohline: /* Homologous Solution */

2014-11-25T23:36:52Z

Homologous Solution

← Older revision		Revision as of 23:36, 25 November 2014
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	<math>~\dot{a}\mathfrak{x} \, ;		<math>~\dot{a}\mathfrak{x} \, ,
	</math>		</math>
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	the continuity equation gives,		which, as foreshadowed above, exactly matches the radial velocity of the collapsing coordinate frame; the continuity equation gives,
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	<math>~		<math>~
	- \frac{1}{8} \biggl( \frac{\pi G}{\kappa^3} \biggr)^{1/2} a^2 \ddot{a} \, ,		- \frac{1}{8} \biggl( \frac{\pi G}{\kappa^3} \biggr)^{1/2} a^2 \ddot{a} \, .
	</math>		</math>
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	~~which~~ matches equation (12) of [http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich & Weber (1980)].		This matches equation (12) of [http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich & Weber (1980)].


	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. [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,
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Tohline: /* Mass-Density and Speed */

2014-11-25T23:32:48Z

Mass-Density and Speed

← Older revision		Revision as of 23:32, 25 November 2014
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	With these additional scalings, ~~the~~ continuity equation becomes,		With these additional scalings, our continuity equation becomes,

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	where the first term on the lefthand side has been set to zero because, as stated above, <math>~f</math> may be a function of space but not of time; ~~the~~ Euler equation becomes,		where the first term on the lefthand side has been set to zero because, as stated above, <math>~f</math> may be a function of space but not of time; our Euler equation becomes,

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Tohline: /* Reconciling with Goldreich & Weber */

2014-11-25T23:31:23Z

Reconciling with Goldreich & Weber

← Older revision		Revision as of 23:31, 25 November 2014
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	<math>\frac{1}{2} a^{-2}( \nabla_\mathfrak{x} \psi )^2 - H - \Phi		<math>\frac{1}{2} a^{-2}( \nabla_\mathfrak{x} \psi )^2 - H - \Phi \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Homologous Solution */ Correct continuity equation

2014-11-25T23:20:34Z

Homologous Solution: Correct continuity equation

← Older revision		Revision as of 23:20, 25 November 2014
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	<math>~3 ~\frac{d\ln f}{dt} </math>		<math>~\frac{d\ln \rho_c}{dt} </math>
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	<math>\Rightarrow~~~~\frac{d\ln f}{dt} + \frac{d\ln a}{dt} </math>		<math>\Rightarrow~~~~\frac{d\ln \rho_c}{dt} + \frac{d\ln a^3}{dt} </math>
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	which means that the product, <math>~af</math>, is independent of time; and the Euler equation becomes,		which means that, consistent with the expected relationship between the central density and the time-varying length scale [[User:Tohline/Apps/GoldreichWeber80#Length\|established above]], the product, <math>~a^3 \rho_c</math>, is independent of time; and the Euler equation becomes,

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Tohline: /* Mass-Density and Speed */ Use "cancelto" latex operation in order to improve clarity of continuity equation

2014-11-25T23:07:10Z

Mass-Density and Speed: Use "cancelto" latex operation in order to improve clarity of continuity equation

← Older revision		Revision as of 23:07, 25 November 2014
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	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,		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,
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	<math>f \equiv \biggl( \frac{\rho}{\rho_c} \biggr)^{1/3} \, .</math>		<math>f \equiv \biggl( \frac{\rho}{\rho_c} \biggr)^{1/3} \, ,</math>
	</div>		</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,		which, in order to successfully identify a similarity solution, may be a function of space but not of time. 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,
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	<math>\Theta_H \equiv \biggl( \frac{\rho}{\rho_c} \biggr)^{1/n} \, .</math>		<math>\Theta_H \equiv \biggl( \frac{\rho}{\rho_c} \biggr)^{1/n} \, .</math>
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	<math>~\frac{d\ln f^3}{dt} + \frac{d\ln \rho_c}{dt}</math>		<math>~\cancelto{0}{\frac{d\ln f^3}{dt}} + \frac{d\ln \rho_c}{dt}</math>
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	the Euler equation becomes,		where the first term on the lefthand side has been set to zero because, as stated above, <math>~f</math> may be a function of space but not of time; the Euler equation becomes,

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Tohline: /* Mass-Density and Speed */ Correct continuity equation

2014-11-25T22:52:35Z

Mass-Density and Speed: Correct continuity equation

← Older revision		Revision as of 22:52, 25 November 2014
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	<math>~\frac{d\ln f^3}{dt} </math>		<math>~\frac{d\ln f^3}{dt} + \frac{d\ln \rho_c}{dt}</math>
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Tohline: /* Length */

2014-11-09T21:56:06Z

Length

← Older revision		Revision as of 21:56, 9 November 2014
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	Specifically, the continuity equation, the Euler equation, and the Poisson equation become, respectively,		Specifically, the continuity equation, the Euler equation, and the Poisson equation become, respectively,

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Tohline: /* Reconciling with Goldreich & Weber */ Finish developing subsection that reconciles our derivations with the GW80 derivations

2014-11-07T21:40:04Z

Reconciling with Goldreich & Weber: Finish developing subsection that reconciles our derivations with the GW80 derivations

← Older revision		Revision as of 21:40, 7 November 2014
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			Written in this way, the righthand-sides of Goldreich & Weber's continuity equation and Euler equation match the righthand-sides of our derived versions of these two equations. But, in both cases, the lefthand-sides do not match for two reasons:
			* Goldreich & Weber express the time-variation of the principal physical variable (either <math>~\rho</math> or <math>~\psi</math>) as a ''partial'' derivative — traditionally denoting an Eulerian perspective of the flow — while we have chosen to express the time-variation of both variables as a ''total'' derivative — to denote a Lagrangian perspective of the flow;
			* Goldreich & Weber include a term in which the principal physical variable (either <math>~\rho</math> or <math>~\psi</math>) is being acted upon by the operator,
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			<math>(a^{-1}\nabla_\mathfrak{x}\psi - \dot{a}\mathfrak{x})\cdot a^{-1}\nabla_\mathfrak{x} </math>
			</td></tr>
			</table>

			In order to reconcile these differences, we remember, first, the [[User:Tohline/Apps/GoldreichWeber80#TimeDerivativeTransformation\|operator transformation (familiar to fluid dynamicists) used previously]],
			<div align="center">
			<math>~\frac{d}{dt} ~~ \rightarrow ~~ \frac{\partial}{\partial t} + \vec{v}_T\cdot \nabla </math>
			</div>
			where we have added a subscript <math>~T</math> to the velocity in order to emphasize that, in this context, <math>~\vec{v}</math> is a "transport" velocity measuring the fluid velocity ''relative to'' the adopted coordinate frame. Now, the radial velocity of the fluid (as measured in the inertial frame) is derivable from the stream function via the expression,
			<div align="center">
			<math>v_r = \nabla_r\psi = a^{-1} \nabla_\mathfrak{x}\psi \, ;</math>
			</div>
			while the radial velocity of the coordinate frame that has been adopted by Goldreich & Weber is <math>~\dot{a}\mathfrak{x}</math>. Hence, as measured in the radially collapsing coordinate frame, the magnitude of the (radially directed) transport velocity is,
			<div align="center">
			<math>\|\vec{v}_T\| = (a^{-1}\nabla_\mathfrak{x}\psi - \dot{a}\mathfrak{x}) \, .</math>
			</div>
			It is therefore clear that the lefthand-sides of the continuity and Euler equations, as presented by Goldreich & Weber, are simply the operator,
			<div align="center">
			<math>~ \frac{\partial}{\partial t} + \|\vec{v}_T\| a^{-1} \nabla_\mathfrak{x} </math>
			</div>
			acting on <math>~\rho</math> and <math>~\psi</math>, respectively. The lefthand sides of these equations ''do'', therefore, represent exactly the same physics as the lefthand sides of the equations we have derived.


			Finally, it should be appreciated that, if the evolutionary flow throughout the collapsing configuration is simple enough that a single scalar function, <math>a(t)</math>, suffices to track the location of all fluid elements simultaneously, then <math>~\|\vec{v}_T\|</math> will be zero everywhere and at all times. And the time-variation of the primary variables as deduced from Goldriech & Weber's Eulerian perspective will be identical to the time-variation of the primary variables as deduced from our Lagrangian perspective. This is precisely the outcome achieved via the similarity solution discovered by Goldreich & Weber.

	====Mass-Density and Speed====		====Mass-Density and Speed====