Apps/GoldreichWeber80 - Revision history

Jet53man at 17:45, 29 October 2021

2021-10-29T17:45:43Z

← Older revision		Revision as of 13:45, 29 October 2021
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	<td align="right">		<td align="right">
	<math>		<math>
	\frac{\partial v}{\partial t} - \frac{\dot{a}}{a} \mathbf{r}\cdot\~~bold~~{\nabla}v		\frac{\partial v}{\partial t} - \frac{\dot{a}}{a} \mathbf{r}\cdot\mathbf{\nabla}v
	+ \tfrac{1}{2} a^{-2} \|\mathbf{\nabla}v\|^2 + h + \phi		+ \tfrac{1}{2} a^{-2} \|\mathbf{\nabla}v\|^2 + h + \phi
	</math>		</math>

Jet53man at 17:44, 29 October 2021

2021-10-29T17:44:36Z

← Older revision		Revision as of 13:44, 29 October 2021
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	<math>		<math>
	\frac{1}{\rho} \frac{\partial\rho}{\partial t}		\frac{1}{\rho} \frac{\partial\rho}{\partial t}
	+ a^{-1}(a^{-1} \~~bold~~{\nabla} v - \dot{a}\~~bold~~{r} ) \cdot \~~bold~~{\nabla}\rho/\rho		+ a^{-1}(a^{-1} \mathbf{\nabla} v - \dot{a}\mathbf{r} ) \cdot \mathbf{\nabla}\rho/\rho
	+ a^{-2} \nabla^2 v		+ a^{-2} \nabla^2 v
	</math>		</math>
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	<td align="right">		<td align="right">
	<math>		<math>
	\frac{\partial v}{\partial t} - \frac{\dot{a}}{a} \~~bold~~{r}\cdot\bold{\nabla}v		\frac{\partial v}{\partial t} - \frac{\dot{a}}{a} \mathbf{r}\cdot\bold{\nabla}v
	+ \tfrac{1}{2} a^{-2} \|\~~bold~~{\nabla}v\|^2 + h + \phi		+ \tfrac{1}{2} a^{-2} \|\mathbf{\nabla}v\|^2 + h + \phi
	</math>		</math>
	</td>		</td>

Jet53man: /* Reconciling with Goldreich & Weber */

2021-10-29T17:15:36Z

Reconciling with Goldreich & Weber

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 The set of three principal governing equations, as just derived, are intended to match equations (7) - (9) of {{ GW80 }}.  The following is a framed image of equations (7) - (9) as they appear in the {{ GW80 }} publication:
-<div align="center">
+<table border="1" width="80%" cellpadding="8" align="center">
 <tr>
-   <th align="center">
+   <td align="center">
-Principal Governing Equations from {{ GW80 }}
+Principal Governing Equations extracted without modification from &hellip;<br />{{ GW80figure }}
-   </th>
+   </td>
 </tr>
 <tr>
 <td>
-[[File:GW80Equations.png|500px|center|Goldreich &amp; Weber (1980)]]
+<!-- [[File:GW80Equations.png|500px|center|Goldreich &amp; Weber (1980)]] -->
 </td>
 </tr>
 </table>
-</div>
 For discussion purposes, next we will retype this set of equations, altering only the variable names and notation to correspond with ours.  Assuming that we have interpreted their typeset expressions correctly, the governing equations, as derived by {{ GW80 }}, are,

Jet53man: /* Homologous Solution */

2021-10-29T16:50:37Z

Homologous Solution

← Older revision		Revision as of 12:50, 29 October 2021
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	===Homologous Solution===		===Homologous Solution===
	~~[http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich & Weber (1980)]~~ discovered that the governing equations admit to an homologous, self-similar solution if they adopted a stream function of the form,		{{ GW80 }} discovered that the governing equations admit to an homologous, self-similar solution if they adopted a stream function of the form,
	<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>
	This matches equation (12) of ~~[http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich & Weber (1980)]~~.		This matches equation (12) of {{ GW80 }}.

	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. {{ GW80 }} (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,
	<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 border="1" cellpadding="10" align="center" width="75%">		<table border="1" cellpadding="10" align="center" width="75%">
	<tr><td align="left">		<tr><td align="left">
	As ~~[http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich & Weber (1980)]~~ point out, this nonlinear differential equation can be integrated twice to produce an algebraic relationship between <math>~a</math> and time, <math>~t</math>. The required mathematical steps are identical to the steps used to analytically solve the [[ProjectsUnderway/CoreCollapseSupernovae#Nonrotating.2C_Spherically_Symmetric_Collapse\|classic, spherically symmetric free-fall collapse problem]]. First, rewrite the equation as,		As {{ GW80 }} point out, this nonlinear differential equation can be integrated twice to produce an algebraic relationship between <math>~a</math> and time, <math>~t</math>. The required mathematical steps are identical to the steps used to analytically solve the [[ProjectsUnderway/CoreCollapseSupernovae#Nonrotating.2C_Spherically_Symmetric_Collapse\|classic, spherically symmetric free-fall collapse problem]]. First, rewrite the equation as,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	</div>		</div>

	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 equation 15 in {{ GW80 }}),

	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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	As ~~[http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich & Weber (1980)]~~ point out, because all terms in this equation are inside the gradient operator, the sum of the terms inside the square brackets must equal a constant — that is, the sum must be independent of spatial position throughout the spherically symmetric configuration. If, following ~~Goldreich & Weber's~~ lead, we simply fold this integration constant into the potential, the Euler equation becomes (see their equation 8),		As {{ GW80 }} point out, because all terms in this equation are inside the gradient operator, the sum of the terms inside the square brackets must equal a constant — that is, the sum must be independent of spatial position throughout the spherically symmetric configuration. If, following the lead of {{ GW80 }}, we simply fold this integration constant into the potential, the Euler equation becomes (see their equation 8),
	<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 border="1" cellpadding="5" align="center" width="75%">		<table border="1" cellpadding="5" align="center" width="75%">
	<tr><td align="center" colspan="1">		<tr><td align="center" colspan="1">
	~~[http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich & Weber's (1980)]~~ Governing Equations After Initial ''Length'' Scaling (yet to be demonstrated)		Governing Equations from {{ GW80 }} After Initial ''Length'' Scaling (yet to be demonstrated)
	</td></tr>		</td></tr>

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	</div>		</div>

	Next, we note that ~~Goldreich & Weber~~ introduce a variable to track the dimensionless density,		Next, we note that {{ GW80 }} introduce a variable to track the dimensionless density,

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Jet53man: /* Mass-Density and Speed */

2021-10-29T16:46:30Z

Mass-Density and Speed

← Older revision		Revision as of 12:46, 29 October 2021
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	====Mass-Density and Speed====		====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,		Next, {{ GW80 }} (see their equation 10) choose to normalize the density by the central density, specifically defining a dimensionless function,
	<div align="center">		<div align="center">
	<math>f \equiv \biggl( \frac{\rho}{\rho_c} \biggr)^{1/3} \, ,</math>		<math>f \equiv \biggl( \frac{\rho}{\rho_c} \biggr)^{1/3} \, ,</math>
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	</div>		</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,		Also, {{ GW80 }} (see their equation 11) normalize the gravitational potential to the square of the central sound speed,
	<div align="center">		<div align="center">
	<math>c_s^2 = \frac{\gamma P_c}{\rho_c} = \frac{4}{3} \kappa \rho_c^{1/3}		<math>c_s^2 = \frac{\gamma P_c}{\rho_c} = \frac{4}{3} \kappa \rho_c^{1/3}

Jet53man: /* Reconciling with Goldreich & Weber */

2021-10-29T16:45:21Z

Reconciling with Goldreich & Weber

← Older revision		Revision as of 12:45, 29 October 2021
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	====Reconciling with Goldreich & Weber====		====Reconciling with Goldreich & Weber====
	The set of three principal governing equations, as just derived, are intended to match equations (7) - (9) of ~~[http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich & Weber (1980)]~~. The following is a framed image of equations (7) - (9) as they appear in the ~~Goldreich & Weber~~ publication:		The set of three principal governing equations, as just derived, are intended to match equations (7) - (9) of {{ GW80 }}. The following is a framed image of equations (7) - (9) as they appear in the {{ GW80 }} publication:

	<div align="center">		<div align="center">
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	<tr>		<tr>
	<th align="center">		<th align="center">
	Principal Governing Equations from ~~[http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich & Weber (1980)]~~		Principal Governing Equations from {{ GW80 }}
	</th>		</th>
	</tr>		</tr>
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	</table>		</table>
	</div>		</div>
	For discussion purposes, next we will retype this set of equations, altering only the variable names and notation to correspond with ours. Assuming that we have interpreted their typeset expressions correctly, the governing equations, as derived by ~~Goldreich & Weber~~, are,		For discussion purposes, next we will retype this set of equations, altering only the variable names and notation to correspond with ours. Assuming that we have interpreted their typeset expressions correctly, the governing equations, as derived by {{ GW80 }}, are,

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

	Notice that our expression for the Poisson equation matches the expression presented by ~~Goldreich & Weber~~, but it isn't immediately obvious whether or not the other two pairs of equations match. Let's rearrange the terms in ~~Goldreich & Weber's~~ continuity equation and in their Euler equation to emphasize overlap with ours:		Notice that our expression for the Poisson equation matches the expression presented by {{ GW80 }}, but it isn't immediately obvious whether or not the other two pairs of equations match. Let's rearrange the terms in the {{ GW80 }} continuity equation and in their Euler equation to emphasize overlap with ours:

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

	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:		Written in this way, the righthand-sides of the {{ GW80 }} 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;		* {{ GW80 }} 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,		* {{ GW80 }} include a term in which the principal physical variable (either <math>~\rho</math> or <math>~\psi</math>) is being acted upon by the operator,
	<table border="0" cellpadding="10" align="center">		<table border="0" cellpadding="10" align="center">
	<tr><td align="center">		<tr><td align="center">
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	<math>v_r = \nabla_r\psi = a^{-1} \nabla_\mathfrak{x}\psi \, ;</math>		<math>v_r = \nabla_r\psi = a^{-1} \nabla_\mathfrak{x}\psi \, ;</math>
	</div>		</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,		while the radial velocity of the coordinate frame that has been adopted by {{ GW80 }} 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">		<div align="center">
	<math>\|\vec{v}_T\| = (a^{-1}\nabla_\mathfrak{x}\psi - \dot{a}\mathfrak{x}) \, .</math>		<math>\|\vec{v}_T\| = (a^{-1}\nabla_\mathfrak{x}\psi - \dot{a}\mathfrak{x}) \, .</math>
	</div>		</div>
	It is therefore clear that the lefthand-sides of the continuity and Euler equations, as presented by ~~Goldreich & Weber~~, are simply the operator,		It is therefore clear that the lefthand-sides of the continuity and Euler equations, as presented by {{ GW80 }}, are simply the operator,
	<div align="center">		<div align="center">
	<math>~ \frac{\partial}{\partial t} + \|\vec{v}_T\| a^{-1} \nabla_\mathfrak{x} </math>		<math>~ \frac{\partial}{\partial t} + \|\vec{v}_T\| a^{-1} \nabla_\mathfrak{x} </math>
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	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~~.		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 {{ GW80 }}.

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

Jet53man: /* Length */

2021-10-29T16:41:47Z

Length

← Older revision		Revision as of 12:41, 29 October 2021
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	====Length====		====Length====

	In their investigation, ~~[http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich & Weber (1980)]~~ chose the same length scale for normalization that is used in deriving the [[SSC/Structure/Polytropes#Lane-Emden_equation\|Lane-Emden equation]], which governs the hydrostatic structure of a polytrope of index {{ Math/MP_PolytropicIndex }}, that is,		In their investigation, {{ GW80 }} chose the same length scale for normalization that is used in deriving the [[SSC/Structure/Polytropes#Lane-Emden_equation\|Lane-Emden equation]], which governs the hydrostatic structure of a polytrope of index {{ Math/MP_PolytropicIndex }}, that is,
	<div align="center">		<div align="center">
	<math>		<math>
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	</math>		</math>
	</div>		</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,		''Most significantly'', {{ GW80 }} (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">		<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>
	</div>		</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,		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, {{ GW80 }} 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">		<div align="center">
	<math>		<math>

Jet53man: /* Imposed Constraints */

2021-10-29T16:40:09Z

Imposed Constraints

← Older revision		Revision as of 12:40, 29 October 2021
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	===Imposed Constraints===		===Imposed Constraints===
	~~[http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich & Weber (1980)]~~ specifically choose to examine the spherically symmetric collapse of a <math>~\gamma = 4/3</math> fluid. With this choice of adiabatic index, the equation of state becomes,		{{ GW80 }} specifically choose to examine the spherically symmetric collapse of a <math>~\gamma = 4/3</math> fluid. With this choice of adiabatic index, the equation of state becomes,
	<div align="center">		<div align="center">
	<math>~H = 4 \kappa \rho^{1/3} \, .</math>		<math>~H = 4 \kappa \rho^{1/3} \, .</math>
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	</table>		</table>
	</div>		</div>
	~~Goldreich & Weber~~ also realize that, because the flow is vorticity free, the velocity can be obtained from a stream function, <math>~\psi</math>, via the relation,		{{ GW80 }} also realize that, because the flow is vorticity free, the velocity can be obtained from a stream function, <math>~\psi</math>, via the relation,
	<div align="center">		<div align="center">
	<math>~\vec{v} = \nabla\psi ~~~~~\Rightarrow~~~~~v_r = \nabla_r\psi </math>		<math>~\vec{v} = \nabla\psi ~~~~~\Rightarrow~~~~~v_r = \nabla_r\psi </math>
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	</table>		</table>
	</div>		</div>
	This means that the terms inside the square brackets must sum to a constant that is independent of spatial position. Following the lead of ~~Goldreich & Weber~~, this "integration constant" will be incorporated into the potential, in which case we have,		This means that the terms inside the square brackets must sum to a constant that is independent of spatial position. Following the lead of {{ GW80 }}, this "integration constant" will be incorporated into the potential, in which case we have,

	<div align="center">		<div align="center">
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	</table>		</table>
	</div>		</div>
	which matches equation (5) of ~~[http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich & Weber (1980)]~~.		which matches equation (5) of {{ GW80 }}.

	Now, because it is more readily integrable, we ultimately would like to work with a differential equation that contains the total, rather than partial, time derivative of <math>~\psi</math>. So we will take this opportunity to shift from an Eulerian representation of the Euler equation to a Lagrangian representation, invoking the same (familiar to fluid dynamicists) operator transformation as we have used in our [[PGE/Euler#Eulerian_Representation\|general discussion of the Euler equation]], namely,		Now, because it is more readily integrable, we ultimately would like to work with a differential equation that contains the total, rather than partial, time derivative of <math>~\psi</math>. So we will take this opportunity to shift from an Eulerian representation of the Euler equation to a Lagrangian representation, invoking the same (familiar to fluid dynamicists) operator transformation as we have used in our [[PGE/Euler#Eulerian_Representation\|general discussion of the Euler equation]], namely,
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	<math>~\frac{\partial\psi}{\partial t} ~~ \rightarrow ~~ \frac{d\psi}{dt} - \vec{v}\cdot \nabla\psi \, .</math>		<math>~\frac{\partial\psi}{\partial t} ~~ \rightarrow ~~ \frac{d\psi}{dt} - \vec{v}\cdot \nabla\psi \, .</math>
	</div>		</div>
	In the context of ~~Goldreich & Weber's~~ model, we are dealing with a one-dimension (spherically symmetric), radial flow, so,		In the context of the {{ GW80 }} model, we are dealing with a one-dimension (spherically symmetric), radial flow, so,
	<div align="center">		<div align="center">
	<math>\vec{v}\cdot \nabla\psi = v_r \nabla_r \psi \, .</math>		<math>\vec{v}\cdot \nabla\psi = v_r \nabla_r \psi \, .</math>

Jet53man: /* PK2007 ASIDE */

2021-10-29T16:37:08Z

PK2007 ASIDE

← Older revision		Revision as of 12:37, 29 October 2021
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	</td></tr>		</td></tr>
	</table>		</table>

			<!-- BEGIN PK2007 ASIDE

	=PK2007 ASIDE=		=PK2007 ASIDE=
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	</table>		</table>
	</div>		</div>

			END PK2007 ASIDE -->

	=See Also=		=See Also=

Jet53man: /* PK2007 ASIDE */

2021-10-29T16:36:14Z

PK2007 ASIDE

← Older revision		Revision as of 12:36, 29 October 2021
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	=PK2007 ASIDE=		=PK2007 ASIDE=


	<div align="center">		<div align="center">
	<table border="1" width="90%" cellpadding="8">		<table border="1" width="90%" cellpadding="8">
	<tr><td align="left">		<tr><td align="left">
	<font color="red">'''ASIDE:'''</font> It wasn't immediately obvious to me 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. 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 set of governing equations that allows for coordinate rotation as well as expansion or contraction; here we will ignore any modifications due to rotation.		<font color="red">'''ASIDE:'''</font> It wasn't immediately obvious to me 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 {{ GW80 }} as equations (7) and (8), for example. I turned to {{ PK2007full }} — hereafter, {{ PK2007hereafter }} — for guidance. {{ PK2007hereafter }} develop a set of governing equations that allows for coordinate rotation as well as expansion or contraction; here we will ignore any modifications due to rotation.

	We note, first, that ~~PK07~~ (see their equation 4) adopt an accelerated radial coordinate of the same form as ~~Goldreich & Weber~~,		We note, first, that {{ PK2007hereafter }} (see their equation 4) adopt an accelerated radial coordinate of the same form as {{ GW80 }},
	<div align="center">		<div align="center">
	<math>~\tilde{r} \equiv \biggl[ \frac{1}{a(t)} \biggr] \vec{r} \, ,</math>		<math>~\tilde{r} \equiv \biggl[ \frac{1}{a(t)} \biggr] \vec{r} \, ,</math>
	</div>		</div>
	but the ~~PK07~~ time-dependent scale factor is dimensionless, whereas the scale factor adopted by ~~Goldreich & Weber~~ — denoted here as <math>~a_{GW}(t)</math> — has units of length. To transform from the KP07 notation, we ultimately will set,		but the {{ PK2007hereafter }} time-dependent scale factor is dimensionless, whereas the scale factor adopted by {{ GW80 }} — denoted here as <math>~a_{GW}(t)</math> — has units of length. To transform from the KP07 notation, we ultimately will set,
	<div align="center">		<div align="center">
	<math>~\mathfrak{x} = \frac{1}{a_0} \tilde{r} ~~~~~\Rightarrow ~~~~~ a_{GW}(t) = a_0 a(t) \, ,</math>		<math>~\mathfrak{x} = \frac{1}{a_0} \tilde{r} ~~~~~\Rightarrow ~~~~~ a_{GW}(t) = a_0 a(t) \, ,</math>
	</div>		</div>
	where, <math>~a_0</math> is understood to be the ~~Goldreich & Weber~~ scale length at the onset of collapse, that is, at <math>~t = 0</math>. According to ~~PK07~~, this leads to a new "accelerated" time (see, again, their equation 4 with the exponent, <math>~\beta = 0</math>)		where, <math>~a_0</math> is understood to be the {{ GW80 }} scale length at the onset of collapse, that is, at <math>~t = 0</math>. According to {{ PK2007hereafter }}, this leads to a new "accelerated" time (see, again, their equation 4 with the exponent, <math>~\beta = 0</math>)
	<div align="center">		<div align="center">
	<math>~\tau \equiv \int_0^t \frac{dt}{a(t)} \, .</math>		<math>~\tau \equiv \int_0^t \frac{dt}{a(t)} \, .</math>
	</div>		</div>
	According to equation (7) of ~~PK07~~ — again, setting their exponent <math>~\beta=0</math> — the relationship between the fluid velocity in the inertial frame, <math>~\vec{v}</math>, to the fluid velocity measured in the accelerated frame, <math>~\tilde{v}</math>, is		According to equation (7) of {{ PK2007hereafter }} — again, setting their exponent <math>~\beta=0</math> — the relationship between the fluid velocity in the inertial frame, <math>~\vec{v}</math>, to the fluid velocity measured in the accelerated frame, <math>~\tilde{v}</math>, is
	<div align="center">		<div align="center">
	<math>~\vec{v} = \tilde{v} + \biggl[ \frac{d\ln a}{d\tau} \biggr] \tilde{r} \, .</math>		<math>~\vec{v} = \tilde{v} + \biggl[ \frac{d\ln a}{d\tau} \biggr] \tilde{r} \, .</math>
	</div>		</div>
	We note that, according to equation (8) of ~~PK07~~, the first derivative of <math>~a(t)</math> with respect to ''physical'' time is,		We note that, according to equation (8) of {{ PK2007hereafter }}, the first derivative of <math>~a(t)</math> with respect to ''physical'' time is,
	<div align="center">		<div align="center">
	<math>~\dot{a} = \frac{d\ln a}{d\tau} \, ,</math>		<math>~\dot{a} = \frac{d\ln a}{d\tau} \, ,</math>
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	<math>~\vec{v} = \tilde{v} + \dot{a} \tilde{r} \, ;</math>		<math>~\vec{v} = \tilde{v} + \dot{a} \tilde{r} \, ;</math>
	</div>		</div>
	and we note that (see equation 9 of ~~PK07~~),		and we note that (see equation 9 of {{ PK2007hereafter }}),
	<div align="center">		<div align="center">
	<math>~\ddot{a} = \frac{1}{a} \biggl[ \frac{d^2\ln a}{d\tau^2} \biggr] \, .</math>		<math>~\ddot{a} = \frac{1}{a} \biggl[ \frac{d^2\ln a}{d\tau^2} \biggr] \, .</math>
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	</tr>		</tr>
	</table>		</table>
	Comparing this to equation (10) of ~~PK07~~, which introduces a density field, <math>~\tilde\rho</math>, as viewed in the accelerated frame of reference of the form,		Comparing this to equation (10) of {{ PK2007hereafter }}, which introduces a density field, <math>~\tilde\rho</math>, as viewed in the accelerated frame of reference of the form,
	<div align="center">		<div align="center">
	<math>~\tilde\rho = [a(t)]^\alpha \rho \, ,</math>		<math>~\tilde\rho = [a(t)]^\alpha \rho \, ,</math>
	</div>		</div>
	we see that, by setting the exponent <math>~\alpha = 3</math>, the ~~Goldreich & Weber~~ dimensionless density can be retrieved from the ~~PK07~~ work by setting,		we see that, by setting the exponent <math>~\alpha = 3</math>, the {{ GW80 }} dimensionless density can be retrieved from the {{ PK2007hereafter }} work by setting,
	<div align="center">		<div align="center">
	<math>~f^3= \frac{\tilde\rho}{\rho_0} \, ,</math>		<math>~f^3= \frac{\tilde\rho}{\rho_0} \, ,</math>
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	<math>~\rho_0 \equiv \biggl( \frac{\kappa}{\pi G a_0^2} \biggr)^{3/2} \, .</math>		<math>~\rho_0 \equiv \biggl( \frac{\kappa}{\pi G a_0^2} \biggr)^{3/2} \, .</math>
	</div>		</div>
	~~PK07~~ then claim that, in the accelerating reference frame, the continuity equation and Euler equation become, respectively,		{{ PK2007hereafter }} then claim that, in the accelerating reference frame, the continuity equation and Euler equation become, respectively,
	<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>
	where ~~PK07~~ have introduced <math>~\nu</math> as a "dimensionality parameter of the problem." In an effort to rewrite the left-hand-side of ~~PK07's~~ Euler equation in a form that matches ~~Goldreich & Weber's~~ Euler equation, we note that,		where {{ PK2007hereafter }} have introduced <math>~\nu</math> as a "dimensionality parameter of the problem." In an effort to rewrite the left-hand-side of their Euler equation in a form that matches the {{ GW80 }} Euler equation, we note that,
	<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>
	and, with the help of the ~~PK07~~ continuity equation,		and, with the help of the {{ PK2007hereafter }} continuity equation,
	<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>
	— and, following ~~Goldreich & Weber~~, set the vorticity to zero. The Euler equation becomes,		— and, following {{ GW80 }}, set the vorticity to zero. The Euler equation becomes,
	<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>
	If we set <math>~\nu = 3</math>, this last expression appears to match equation (7) of ~~Goldreich & Weber~~.		If we set <math>~\nu = 3</math>, this last expression appears to match equation (7) of {{ GW80 }}.