View source for User:Tohline/Apps/GoldreichWeber80

=Homologously Collapsing Stellar Cores=
{{LSU_HBook_header}}
==Review of Goldreich and Weber (1980)==
This is principally a review of the dynamical model that Peter Goldreich &amp; 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 &amp; 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 &amp; Weber as equations (7) and (8), for example.  At first, I turned to [http://www.sciencedirect.com/science/article/pii/S0021999106002555 Poludnenko &amp; Khokhlov (2007, Journal of Computational Physics, 220, 678)] &#8212; hereafter, PK07 &#8212; 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 &sect;19.11 (pp. 187 - 190) of Kippenhahn & Weigert [ [[User:Tohline/Appendix/References#KW94|KW94]] ].

===Governing Equations===
Goldreich &amp; Weber begin with the identical set of [[User:Tohline/PGE#Principal_Governing_Equations|principal governing equations]] that serves as the foundation for all of the discussions throughout this H_Book.  In particular, as is documented by their equation (1), their adopted equation of state is adiabatic/polytropic,
<div align="center">
<math>~P = \kappa \rho^\gamma \, ,</math>
</div>
&#8212; where both <math>~\kappa</math> and <math>~\gamma</math> are constants &#8212; and therefore satisfies what [[User:Tohline/PGE#PGE:AdiabaticFirstLaw|we have referred to]] as the

<div align="center">
<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />
('''Specific Entropy Conservation''')

{{User:Tohline/Math/EQ_FirstLaw02}} .
</div>
 
their equation (3) is what [[User:Tohline/PGE/Euler#in_terms_of_the_vorticity:|we have referred to]] as the

<div align="center">
Euler Equation<br />
<span id="ConservingMomentum:Lagrangian"><font color="#770000">'''in terms of the Vorticity'''</font></span>,

{{User:Tohline/Math/EQ_Euler04}}
</div>
where, <math>~\vec\zeta \equiv \nabla\times \vec{v}</math> is the fluid vorticity; their equation (4) is the

<div align="center">
<font color="#770000">'''Poisson Equation'''</font><br />

{{User:Tohline/Math/EQ_Poisson01}}
</div>

and their equation (2) is what [[User:Tohline/PGE/ConservingMass#Eulerian_Representation|we have referred to]] as the

<div align="center">
<span id="ConservingMass:Eulerian"><font color="#770000">'''Eulerian Representation'''</font></span><br />
or<br />
<span id="ConservingMass:Conservative"><font color="#770000">'''Conservative Form'''</font></span><br />
of the Continuity Equation,

{{User:Tohline/Math/EQ_Continuity02}}
</div>
although, for the derivation, below, we prefer to start with what [[User:Tohline/PGE/ConservingMass#Lagrangian_Representation|we have referred to]] as the
<div align="center">
<span id="ConservingMass:Lagrangian"><font color="#770000">'''Standard Lagrangian Representation'''</font></span><br />
of the Continuity Equation,

{{User:Tohline/Math/EQ_Continuity01}}
</div>

Tweaking the set of principal governing equations, as we have written them, to even more precisely match equations (1) - (4) in Goldreich &amp; Weber (1980), we should replace the state variable <math>~P</math> (pressure) with <math>~H</math> (enthalpy), keeping in mind that, <math>~\gamma = 1 + 1/n</math>, 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>
and,
<div align="center">
<math>~\nabla H = \frac{\nabla P}{\rho} \, .</math>
</div>

===Imposed Constraints===
[http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich &amp; 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,
<div align="center">
<math>~H = 4 \kappa \rho^{1/3} \, .</math>
</div>
And because a strictly radial flow-field exhibits no vorticity (i.e., <math>\vec\zeta = 0</math>), the Euler equation can be rewritten as,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{\partial v_r}{\partial t} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~ \nabla_r \biggl[ H + \Phi + \frac{1}{2}v_r^2 \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
Goldreich &amp; 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,
<div align="center">
<math>~\vec{v} = \nabla\psi ~~~~~\Rightarrow~~~~~v_r = \nabla_r\psi </math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp;
<math>~\nabla\cdot \vec{v} = \nabla_r^2 \psi \, .</math>
</div>
Hence, the continuity equation becomes,
<div 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>~-~ \nabla_r^2 \psi \, ,</math>
  </td>
</tr>
</table>
</div>

and the Euler equation becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{\partial }{\partial t} \biggl[ \nabla_r \psi \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~ \nabla_r \biggl[ H + \Phi + \frac{1}{2}(\nabla_r \psi)^2 \biggr] \, .</math>
  </td>
</tr>
</table>
</div>

Since we are, up to this point in the discussion, still referencing the inertial-frame radial coordinate, the <math>~\nabla_r</math> operator can be moved outside of the partial time-derivative on the lefthand side of this equation to give,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\nabla_r \biggl[ \frac{\partial \psi}{\partial t} + H + \Phi + \frac{1}{2}(\nabla_r \psi)^2 \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>
</table>
</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 &amp; Weber, this "integration constant" will be incorporated into the potential, in which case we have,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{\partial \psi}{\partial t} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~ \biggl[ H + \Phi + \frac{1}{2} ( \nabla_r \psi )^2 \biggr] \, ,</math>
  </td>
</tr>
</table>
</div>
which matches equation (5) of [http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich &amp; Weber (1980)].

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 [[User:Tohline/PGE/Euler#Eulerian_Representation|general discussion of the Euler equation]], namely,
<div align="center" id="TimeDerivativeTransformation">
<math>~\frac{\partial\psi}{\partial t} ~~ \rightarrow ~~ \frac{d\psi}{dt} - \vec{v}\cdot \nabla\psi \, .</math>
</div>
In the context of Goldreich &amp; Weber's model, we are dealing with a one-dimension (spherically symmetric), radial flow, so,
<div align="center">
<math>\vec{v}\cdot \nabla\psi = v_r \nabla_r \psi \, .</math>
</div>
But, given that we have adopted a stream-function representation of the flow in which <math>~v_r = \nabla_r\psi</math>, we appreciate that this term can either be written as <math>~v_r^2</math> or <math>~(\nabla_r\psi)^2</math>.  We choose the latter representation, so the Euler equation becomes,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{d\psi}{dt} - (\nabla_r\psi)^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~ \biggl[ H + \Phi + \frac{1}{2} ( \nabla_r \psi )^2 \biggr] \, ,</math>
  </td>
</tr>
</table>
</div>
or, combining like terms on the left and right,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{d\psi}{dt} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2} ( \nabla_r \psi )^2 - H - \Phi  \, .</math>
  </td>
</tr>
</table>
</div>

===Dimensionless and ''Time-Dependent'' Normalization===
====Length====

In their investigation, [http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich &amp; Weber (1980)] chose 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.  In this case <math>~(n = 3)</math>, substitution of the equation of state expression for <math>~H_c</math> leads to,
<div align="center">
<math>
a = \rho_c^{-1/3} \biggl(\frac{\kappa}{\pi G}\biggr)^{1/2} \, .
</math>
</div>
''Most significantly'', Goldreich &amp; 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 &amp; 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>&nbsp; &nbsp; &nbsp; &nbsp;
and
&nbsp; &nbsp; &nbsp; &nbsp;<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" id="GoverningWithStreamFunction">
<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>

====Reconciling with Goldreich &amp; 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 &amp; Weber (1980)].  The following is a framed image of equations (7) - (9) as they appear in the Goldreich &amp; Weber publication:

<div align="center">
<table border="2">
<tr>
  <th align="center">
Principal Governing Equations from [http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich &amp; Weber (1980)]
  </th>
</tr>
<tr>
<td>
[[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 Goldreich &amp; Weber, are,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{1}{\rho} \frac{\partial\rho}{\partial t} 
~+  a^{-1}(a^{-1}\nabla_\mathfrak{x}\psi - \dot{a}\mathfrak{x})\cdot \frac{\nabla_\mathfrak{x}\rho}{\rho}+~ a^{-2} \nabla_\mathfrak{x}^2 \psi  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>0 \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial\psi}{\partial t} - \frac{\dot{a} \mathfrak{x}}{a} \cdot \nabla_\mathfrak{x} \psi~+ \frac{1}{2} a^{-2}( \nabla_\mathfrak{x} \psi )^2 
+ H + \Phi </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>0 \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~a^{-2}\nabla_\mathfrak{x}^2 \Phi - 4\pi G  \rho</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>
</table>
</div>

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

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{1}{\rho} \biggl[ \frac{\partial\rho}{\partial t} 
+ (a^{-1}\nabla_\mathfrak{x}\psi - \dot{a}\mathfrak{x})\cdot a^{-1}\nabla_\mathfrak{x}\rho \biggr]
</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{\partial\psi}{\partial t} +(a^{-1}\nabla_\mathfrak{x}\psi - \dot{a}\mathfrak{x})\cdot a^{-1}\nabla_\mathfrak{x}\psi</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>
</table>
</div>

Written in this way, the righthand-sides of Goldreich &amp; 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 &amp; Weber express the time-variation of the principal physical variable (either <math>~\rho</math> or <math>~\psi</math>) as a ''partial'' derivative &#8212; traditionally denoting an Eulerian perspective of the flow &#8212; while we have chosen to express the time-variation of both variables as a ''total'' derivative &#8212; to denote a Lagrangian perspective of the flow;
* Goldreich &amp; 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,
<table border="0" cellpadding="10" align="center">
<tr><td align="center">
<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 &amp; 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 &amp; 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 &amp; 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 &amp; Weber.

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

Next, [http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich &amp; 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>
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, 
<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 &amp; 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">
&nbsp;
  </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">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~3f \, .</math>
  </td>
</tr>

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

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\cancelto{0}{\frac{d\ln f^3}{dt}}  +  \frac{d\ln \rho_c}{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>

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,

<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>

===Homologous Solution===
[http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich &amp; Weber (1980)] discovered that the governing equations admit to an homologous, self-similar solution if they adopted a stream function of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\psi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2}a \dot{a} \mathfrak{x}^2 \, ,</math>
  </td>
</tr>
</table>
</div>
which, when acted upon by the various relevant operators, gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\nabla_\mathfrak{x}\psi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a \dot{a} \mathfrak{x} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\nabla^2_\mathfrak{x}\psi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{1}{2}a \dot{a} \biggr) \frac{1}{\mathfrak{x}^2} \frac{d}{d\mathfrak{x}} \biggl[\mathfrak{x}^2 \frac{d}{d\mathfrak{x}}  \mathfrak{x}^2 \biggr]
= 3 a \dot{a} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{d\psi}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathfrak{x}^2 \biggl[ \frac{1}{2}\dot{a}^2 + \frac{1}{2}a\ddot{a} \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
Hence, the radial velocity profile is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~v_r = a^{-1}\nabla_\mathfrak{x} \psi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\dot{a}\mathfrak{x} \, ,
</math>
  </td>
</tr>
</table>
</div>
which, as foreshadowed above, exactly matches the radial velocity of the collapsing coordinate frame; the continuity equation gives,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{d\ln \rho_c}{dt}  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>-~ \frac{3\dot{a}}{a}  </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~~\frac{d\ln \rho_c}{dt} + \frac{d\ln a^3}{dt} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, ,</math>
  </td>
</tr>
</table>
</div>
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,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~ - 3 f - \sigma </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]  
\biggl\{ \mathfrak{x}^2 \biggl[ \frac{1}{2}\dot{a}^2 + \frac{1}{2}a\ddot{a} \biggr]  - 
\frac{1}{2} ( \dot{a} \mathfrak{x} )^2 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3}{8} \biggl( \frac{\pi G}{\kappa^3} \biggr)^{1/2}   (a \mathfrak{x})^2 \ddot{a} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~ \frac{(f + \sigma/3)}{\mathfrak{x}^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{1}{8} \biggl( \frac{\pi G}{\kappa^3} \biggr)^{1/2}   a^2 \ddot{a} \, .
</math>
  </td>
</tr>

</table>
</div>
This matches equation (12) of [http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich &amp; 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 &#8212; via the parameter, <math>~a(t)</math> &#8212; both expressions must equal the same (dimensionless) constant.  [http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich &amp; 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,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\sigma</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2} \lambda ~\mathfrak{x}^2 - 3f \, .</math>
  </td>
</tr>
</table>
</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,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
a^2 \ddot{a} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~\frac{4\lambda}{3} \biggl( \frac{\kappa^3}{\pi G} \biggr)^{1/2} \, .</math>
  </td>
</tr>
</table>
</div>



{{LSU_WorkInProgress}}

<table border="1" cellpadding="10" align="center" width="75%">
<tr><td align="left">
As [http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich &amp; 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 [[User:Tohline/ProjectsUnderway/Core_Collapse_Supernovae#Nonrotating.2C_Spherically_Symmetric_Collapse|classic, spherically symmetric free-fall collapse problem]].  First, rewrite the equation as,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\frac{d \dot{a} }{dt}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\frac{B}{2a^2} \, , </math>
  </td>
</tr>

</table>
where,
<div align="center">
<math>
~B \equiv \frac{8\lambda}{3} \biggl( \frac{\kappa^3}{\pi G} \biggr)^{1/2}  \, ,
</math>
</div>
has the same dimensions as the product, <math>~GM</math> (see the  [[User:Tohline/ProjectsUnderway/Core_Collapse_Supernovae#Nonrotating.2C_Spherically_Symmetric_Collapse|free-fall collapse problem]]), that is, the dimensions of "length-cubed per unit time-squared."   Then, multiply both sides by <math>~2\dot{a} = 2(da/dt)</math> to obtain,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
2\dot{a} \frac{d\dot{a}}{dt}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-B \biggl( a^{-2} \frac{da}{dt} \biggr) </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~
\frac{d\dot{a}^2}{dt}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~B \frac{d}{dt} \biggl( \frac{1}{a} \biggr) \, ,</math>
  </td>
</tr>

</table>
which integrates once to give,
<div align="center">
<math>
~\dot{a}^2 = \frac{B}{a} + C \, ,
</math>
</div>
or,
<div align="center">
<math>
~dt = \biggl( \frac{B}{a} + C \biggr)^{-1/2} da  \, .
</math>
</div>

For the case, <math>~C = 0</math>, this differential equation can be integrated straightforwardly to give (see Goldreich &amp; 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,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~t</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{a}{C} \biggl( \frac{B}{a} + C \biggr)^{1/2}
- \frac{B}{2C^{3/2}} \ln \biggl[2aC^{1/2}  \biggl( \frac{B}{a} + C \biggr)^{1/2}  + B + 2aC \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>


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


As [http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich &amp; 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 &#8212; that is, the sum must be independent of spatial position throughout the spherically symmetric configuration.  If, following Goldreich &amp; Weber's lead, we simply fold this integration constant into the potential, the Euler equation becomes (see their equation 8),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial \psi}{\partial t}  - \biggl( \frac{\dot{a}}{a} \biggr)\psi +
H + \Phi + \frac{1}{2}\biggl(\frac{1}{a} \nabla_x\psi  \biggr)^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>~\frac{\partial \rho}{\partial t} + \rho \nabla_r \cdot \vec{v} + \vec{v}\cdot \nabla_r \rho</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left" width="25%">
<math>~0</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \frac{1}{\rho} \frac{\partial \rho}{\partial t} + \nabla_r \cdot \vec{v} + \vec{v}\cdot \frac{\nabla_r \rho}{\rho}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left" width="25%">
<math>~0</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \frac{1}{\rho} \frac{\partial \rho}{\partial t} + a^{-1} \nabla_x \cdot \biggl[  a^{-1} \nabla_x \psi \biggr] 
+ a^{-1} \nabla_x \psi  \cdot \frac{a^{-1}\nabla_x \rho}{\rho}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left" width="25%">
<math>~0</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{1}{\rho} \frac{\partial \rho}{\partial t} + a^{-1}(a^{-1} \nabla_x\psi - \dot{a} \vec{x}) \cdot \frac{\nabla_x\rho}{\rho}
 + a^{-2} \nabla_x^2\psi </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left" width="25%">
<math>~0</math>
  </td>
</tr>
</table>
</div>

<table border="1" cellpadding="5" align="center" width="75%">
<tr><td align="center" colspan="1">
[http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich &amp; Weber's (1980)] Governing Equations After Initial ''Length'' Scaling (yet to be demonstrated)
</td></tr>

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

<tr>
  <td align="right">
<math>~\frac{1}{\rho} \frac{\partial \rho}{\partial t} + a^{-1}(a^{-1} \nabla_x\psi - \dot{a} \vec{x}) \cdot \frac{\nabla_x\rho}{\rho}
 + a^{-2} \nabla_x^2\psi </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left" width="25%">
<math>~0</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial \psi}{\partial t} - \frac{\dot{a}}{a} \vec{x}\cdot \nabla_x\psi + \frac{1}{2} a^{-2} | \nabla_x\psi|^2 
+ H + \Phi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~
a^{-2} \nabla_x^2\Phi - 4\pi G \rho
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0</math>
  </td>
</tr>

<tr><td align="left" colspan="3">
where, 
<div align="center">
<math>~\vec{x} \equiv \frac{\vec{r}}{a} \, ,</math>
</div>
and it is understood that derivatives in the <math>~\nabla_x</math> and <math>~\nabla_x^2</math> operators are taken with respect to the dimensionless radial coordinate, <math>~x</math>.
</td></tr>

</table>

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

<!-- BEGIN PK07 ASIDE 


<div align="center">
<table border="1" width="90%" cellpadding="8">
<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 &amp; Weber as equations (7) and (8), for example.  I turned to [http://www.sciencedirect.com/science/article/pii/S0021999106002555 Poludnenko &amp; Khokhlov (2007, Journal of Computational Physics, 220, 678)] &#8212; hereafter, PK07 &#8212; 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.

We note, first, that PK07 (see their equation 4) adopt an accelerated radial coordinate of the same form as Goldreich &amp; Weber,
<div align="center">
<math>~\tilde{r} \equiv \biggl[ \frac{1}{a(t)} \biggr] \vec{r} \, ,</math>
</div>
but the PK07 time-dependent scale factor is dimensionless, whereas the scale factor adopted by Goldreich &amp; Weber &#8212; denoted here as <math>~a_{GW}(t)</math> &#8212; has units of length.  To transform from the KP07 notation, we ultimately will set,
<div align="center">
<math>~\mathfrak{x} = \frac{1}{a_0} \tilde{r}  ~~~~~\Rightarrow ~~~~~ a_{GW}(t) = a_0 a(t) \, ,</math>
</div>
where, <math>~a_0</math> is understood to be the Goldreich &amp; 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>)
<div align="center">
<math>~\tau \equiv \int_0^t \frac{dt}{a(t)} \, .</math>
</div>
According to equation (7) of PK07 &#8212; again, setting their exponent <math>~\beta=0</math> &#8212; 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">
<math>~\vec{v} = \tilde{v} + \biggl[ \frac{d\ln a}{d\tau} \biggr] \tilde{r} \, .</math>
</div>
We note that, according to equation (8) of PK07, the first derivative of <math>~a(t)</math> with respect to ''physical'' time is,
<div align="center">
<math>~\dot{a} = \frac{d\ln a}{d\tau} \, ,</math>
</div>
so the transformation between velocities may equally well be written as,
<div align="center">
<math>~\vec{v} = \tilde{v} + \dot{a} \tilde{r} \, ;</math>
</div>
and we note that (see equation 9 of PK07),
<div align="center">
<math>~\ddot{a} = \frac{1}{a} \biggl[ \frac{d^2\ln a}{d\tau^2} \biggr] \, .</math>
</div>

Next, we note that Goldreich &amp; Weber introduce a variable to track the dimensionless density,

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

<tr>
  <td align="right">
<math>~f^3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\rho}{\rho_c} \biggr) = \biggl( \frac{\pi G}{\kappa} \biggr)^{3/2} [a_{GW}(t)]^3 \rho \, .</math>
  </td>
</tr>
</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,
<div align="center">
<math>~\tilde\rho = [a(t)]^\alpha \rho \, ,</math>
</div>
we see that, by setting the exponent <math>~\alpha = 3</math>, the Goldreich &amp; Weber dimensionless density can be retrieved from the PK07 work by setting,
<div align="center">
<math>~f^3= \frac{\tilde\rho}{\rho_0} \, ,</math>
</div>
where,
<div align="center">
<math>~\rho_0 \equiv \biggl( \frac{\kappa}{\pi G a_0^2} \biggr)^{3/2} \, .</math>
</div>
PK07 then claim that, in the accelerating reference frame, the continuity equation and Euler equation become, respectively,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial \tilde\rho}{\partial \tau} + \tilde{\nabla}\cdot(\tilde\rho \tilde{v})</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(3-\nu)\biggl[ \frac{d\ln a}{d\tau} \biggr] \tilde\rho \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial \tilde\rho \tilde{v} }{\partial \tau} + \tilde{\nabla} \cdot(\tilde\rho \tilde{v} \tilde{v})  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(2-\nu)\biggl[ \frac{d\ln a}{d\tau} \biggr] \tilde\rho \tilde{v} - 
\biggl[ \frac{d^2\ln a}{d\tau^2} \biggr] \tilde\rho \tilde{r} - \tilde{\nabla}\tilde{P} \, ,</math>
  </td>
</tr>
</table>
</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 &amp; Weber's Euler equation, we note that,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\nabla\cdot [(\tilde\rho \tilde{v}) \tilde{v}]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tilde\rho(\tilde{v}\cdot \tilde\nabla) \tilde{v} + \tilde{v}[\tilde\nabla \cdot (\tilde\rho \tilde{v})] \, ,</math>
  </td>
</tr>
</table>
</div>
and, with the help of the PK07 continuity equation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial (\tilde\rho \tilde{v})}{\partial\tau}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tilde\rho \frac{\partial \tilde{v}}{\partial\tau} + \tilde{v} \frac{\partial \tilde\rho}{\partial\tau} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tilde\rho \frac{\partial \tilde{v}}{\partial\tau} + \tilde{v} \biggl[ 
(3-\nu)\biggl( \frac{d\ln a}{d\tau} \biggr) \tilde\rho
- \tilde{\nabla}\cdot(\tilde\rho \tilde{v})
\biggr] \, .</math>
  </td>
</tr>
</table>
</div>
Hence, the Euler equation becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ \tilde\rho \frac{\partial \tilde{v}}{\partial\tau}  
+ \tilde\rho(\tilde{v}\cdot \tilde\nabla) \tilde{v} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~\biggl[ \frac{d\ln a}{d\tau} \biggr] \tilde\rho \tilde{v} - 
\biggl[ \frac{d^2\ln a}{d\tau^2} \biggr] \tilde\rho \tilde{r} - \tilde{\nabla}\tilde{P} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~ \Rightarrow ~~~ \frac{\partial \tilde{v}}{\partial\tau}  + (\tilde{v}\cdot \tilde\nabla) \tilde{v} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~\dot{a} \tilde{v} - 
a \ddot{a} \tilde{r} - \tilde{\nabla}\tilde{H} \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~ \Rightarrow ~~~ \frac{\partial \tilde{v}}{\partial\tau}  + \frac{1}{2} \tilde\nabla({\tilde{v}} \cdot \tilde{v} ) + \tilde{\zeta}\times \tilde{v}  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~\dot{a} \tilde{v} - 
a \ddot{a} \tilde{r} - \tilde{\nabla}\tilde{H} \, ,</math>
  </td>
</tr>
</table>
</div>
where the vector identity that has been used to obtain this last expression has been drawn from our [[User:Tohline/PGE/Euler#in_terms_of_the_vorticity:|separate presentation of the Euler equation written in terms of the fluid vorticity]], <math>~\tilde\zeta \equiv \tilde\nabla \times \tilde{v}</math>.  

----
Now, let's shift to ''physical'' parameters &#8212; or example,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\tilde{v}</math>
  </td>
  <td align="center">
<math>~~~\rightarrow~~~</math>
  </td>
  <td align="left">
<math>~\vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} = \vec{v} - \dot{a} \tilde{r} ~~ \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial}{\partial\tau}</math>
  </td>
  <td align="center">
<math>~~~\rightarrow~~~</math>
  </td>
  <td align="left">
<math>~\frac{\partial t}{\partial\tau} \frac{\partial}{\partial t} = a \frac{\partial}{\partial t} </math>
  </td>
</tr>
</table>
</div>
&#8212; and, following Goldreich & Weber, set the vorticity to zero.  The Euler equation becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial }{\partial t}  \biggl[ \vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr]
+ \frac{1}{2}a^{-1} \tilde\nabla \biggl[ \biggl( \vec{v} - \dot{a} \tilde{r} \biggr) \cdot \biggl( \vec{v} - \dot{a} \tilde{r} \biggr) \biggr]  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~\frac{\dot{a}}{a} \biggl( \vec{v} - \dot{a} \tilde{r} \biggr)  - 
\ddot{a} \tilde{r} - a^{-1}\tilde{\nabla}\tilde{H} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~
\frac{\partial \vec{v} }{\partial t}  - \biggl[ \biggl(\frac{\dot{a}}{a} \biggr)\frac{\partial\vec{r}}{\partial t} + \frac{\ddot{a}}{a} \vec{r} - \biggl( \frac{\dot{a}}{a}\biggr)^2 \vec{r} \biggr]
+ \frac{1}{2}a^{-1} \tilde\nabla \biggl[ \vec{v} \cdot \vec{v} -2\dot{a} \vec{v} \tilde{r} + (\dot{a} \tilde{r} )^2 \biggr]  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~\frac{\dot{a}}{a} \biggl( \vec{v} - \frac{\dot{a}}{a} \vec{r} \biggr)  - 
\frac{\ddot{a}}{a} \vec{r} - a^{-1}\tilde{\nabla}\tilde{H} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~
\frac{\partial \vec{v} }{\partial t}  
+ \frac{1}{2}a^{-1} \tilde\nabla \biggl[ \vec{v} \cdot \vec{v} -2\dot{a} \vec{v} \tilde{r} + (\dot{a} \tilde{r} )^2 \biggr]  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\frac{\dot{a}}{a} \biggl(\frac{\partial\vec{r}}{\partial t} - \vec{v} \biggr)  - a^{-1}\tilde{\nabla}\tilde{H}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~
\frac{\partial \vec{v} }{\partial t}  
+ a^{-1} \tilde\nabla \biggl[ \frac{1}{2}(\vec{v} \cdot \vec{v}) - \dot{a} \vec{v} \tilde{r} + \tilde{H} \biggr]  + \biggl( \frac{\dot{a}}{a} \biggr)^2 \vec{r} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\frac{\dot{a}}{a} \biggl(\frac{\partial\vec{r}}{\partial t} - \vec{v} \biggr)  \, .</math>
  </td>
</tr>
</table>
</div>

Now, let's tackle the continuity equation:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial \tilde\rho}{\partial \tau} + \tilde\rho \tilde{\nabla}\cdot \tilde{v} + \tilde{v} \cdot \tilde\nabla \tilde\rho  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(3-\nu)\biggl[ \frac{d\ln a}{d\tau} \biggr] \tilde\rho </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~\frac{a}{\tilde\rho}\frac{\partial \tilde\rho}{\partial t} + \tilde{\nabla}\cdot \tilde{v} + \tilde{v} \cdot \frac{\tilde\nabla \tilde\rho}{\tilde\rho}  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(3-\nu) \dot{a} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~
\frac{a}{\tilde\rho}\frac{\partial \tilde\rho}{\partial t} 
+ (\vec{v} - \dot{a}\tilde{r})\cdot \frac{\tilde\nabla \tilde\rho}{\tilde\rho}  
+ \tilde{\nabla}\cdot (\vec{v} - \dot{a}\tilde{r}) 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(3-\nu) \dot{a} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~
\frac{1}{a^3\rho}\frac{\partial (a^3\rho)}{\partial t} 
+ a^{-1}(\vec{v} - \dot{a}\tilde{r})\cdot \frac{\tilde\nabla \rho}{\rho}  
+ a^{-1}\tilde{\nabla}\cdot \vec{v} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(3-\nu) \frac{\dot{a}}{a}  + a^{-1}\tilde{\nabla}\cdot (\dot{a}\tilde{r}) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~
\frac{1}{\rho}\frac{\partial \rho}{\partial t} 
+ a^{-1}(\vec{v} - \dot{a}\tilde{r})\cdot \frac{\tilde\nabla \rho}{\rho}  
+ a^{-1}\tilde{\nabla}\cdot \vec{v} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(3-\nu) \frac{\dot{a}}{a}  + a^{-1}\tilde{\nabla}\cdot (\dot{a}\tilde{r}) -3 \frac{\dot{a}}{a}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\dot{a}}{a} (\tilde{\nabla}\cdot \tilde{r}-\nu)  
</math>
  </td>
</tr>

</table>
</div>
If we set <math>~\nu = 3</math>, this last expression appears to match equation (7) of Goldreich &amp; Weber.




----
With the aid of the continuity equation, the left-hand-side of the Euler equation can be rewritten as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial \tilde\rho \tilde{v} }{\partial \tau} + \tilde{\nabla} \cdot(\tilde\rho \tilde{v} \tilde{v})  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl[ \tilde\rho \frac{\partial \tilde{v} }{\partial \tau} + \tilde{v} \frac{\partial \tilde\rho }{\partial \tau} \biggr] + 
\biggl[ (\tilde{v} \cdot \tilde{\nabla} ) \tilde\rho \tilde{v} + (\tilde\rho \tilde{v} \cdot \tilde{\nabla})\tilde{v} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\tilde\rho \frac{\partial \tilde{v} }{\partial \tau} + 
\biggl[(3-\nu)\biggl( \frac{d\ln a}{d\tau} \biggr) \tilde\rho \tilde{v}  - (\tilde{v} \cdot \tilde{\nabla} ) \tilde\rho \tilde{v} \biggr] + 
\biggl[ (\tilde{v} \cdot \tilde{\nabla} ) \tilde\rho \tilde{v} + (\tilde\rho \tilde{v} \cdot \tilde{\nabla})\tilde{v} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\tilde\rho \biggl[ \frac{\partial \tilde{v} }{\partial \tau} + 
(3-\nu)\biggl( \frac{d\ln a}{d\tau} \biggr) \tilde{v}  + 
(\tilde{v} \cdot \tilde{\nabla})\tilde{v} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, the Euler equation becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\frac{\partial \tilde{v} }{\partial \tau} 
+ (\tilde{v} \cdot \tilde{\nabla})\tilde{v} 
+ \biggl( \frac{d\ln a}{d\tau} \biggr) \tilde{v}  
+ \biggl( \frac{d^2\ln a}{d\tau^2} \biggr) \tilde{r}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - \frac{\tilde{\nabla}\tilde{P}}{\tilde\rho} \, .</math>
  </td>
</tr>
</table>
</div>

----
Now, let's shift to ''physical'' parameters.  For example,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\tilde{v}</math>
  </td>
  <td align="center">
<math>~~~\rightarrow~~~</math>
  </td>
  <td align="left">
<math>~\vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial}{\partial\tau}</math>
  </td>
  <td align="center">
<math>~~~\rightarrow~~~</math>
  </td>
  <td align="left">
<math>~\frac{\partial t}{\partial\tau} \frac{\partial}{\partial t} = a \frac{\partial}{\partial t} \, .</math>
  </td>
</tr>
</table>
</div>

Hence, the Euler equation becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ - \tilde{\nabla}\tilde{H} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
a\frac{\partial}{\partial t} \biggl[ \vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr]
+ (\tilde{v} \cdot \tilde{\nabla})\tilde{v} 
+ \dot{a} \biggl[ \vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr]  
+ \ddot{a} \vec{r}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>a\frac{\partial \vec{v} }{\partial t} - 
a \biggl[ \biggl(\frac{\ddot{a}}{a} \biggr) \vec{r} - \biggl(\frac{\dot{a}}{a} \biggr)^2 \vec{r} + \biggl(\frac{\dot{a}}{a} \biggr) \frac{\partial \vec{r} }{\partial t} \biggr]
+ (\tilde{v} \cdot \tilde{\nabla})\tilde{v} 
+ \dot{a} \biggl[ \vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr]  
+ \ddot{a} \vec{r}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>a\frac{\partial \vec{v} }{\partial t} + (\tilde{v} \cdot \tilde{\nabla})\tilde{v} 
+ \dot{a} \biggl[ \vec{v} - \frac{\partial \vec{r} }{\partial t} \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>a\frac{\partial \vec{v} }{\partial t} + \dot{a} \biggl[ \vec{v} - \frac{\partial \vec{r} }{\partial t} \biggr]  
+ \biggl\{\biggl[ \vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr] \cdot \tilde{\nabla} \biggr\} \biggl[ \vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>a\frac{\partial \vec{v} }{\partial t} + \dot{a} \biggl[ \vec{v} - \frac{\partial \vec{r} }{\partial t} \biggr]  
+ (\vec{v} \cdot \tilde\nabla)\biggl[ \vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr]
- \biggl(\frac{\dot{a}}{a} \biggr) \vec{r}  \cdot \tilde{\nabla} 
\biggl[ \vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl\{ a\frac{\partial \vec{v} }{\partial t} + \dot{a} \biggl[ \vec{v} - \frac{\partial \vec{r} }{\partial t} \biggr]  
+ (\vec{v} \cdot \tilde\nabla)\vec{v} 
- \biggl(\frac{\dot{a}}{a} \biggr) \vec{r}  \cdot \tilde{\nabla} \vec{v} \biggr\}
- (\vec{v} \cdot \tilde\nabla)\biggl[ \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr]
+ \biggl(\frac{\dot{a}}{a} \biggr) \vec{r}  \cdot \tilde{\nabla} \biggl[\biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl\{ a\frac{\partial \vec{v} }{\partial t} + \dot{a} \biggl[ \vec{v} - \frac{\partial \vec{r} }{\partial t} \biggr]  
+ (\vec{v} \cdot \tilde\nabla)\vec{v} 
- \biggl(\frac{\dot{a}}{a} \biggr) \vec{r}  \cdot \tilde{\nabla} \vec{v} \biggr\}
- \biggl[ \vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr] \cdot \tilde{\nabla} \biggl[\dot{a} \tilde{r}  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl\{ a\frac{\partial \vec{v} }{\partial t}  + (\vec{v} \cdot \tilde\nabla)\vec{v} 
- \biggl(\frac{\dot{a}}{a} \biggr) \vec{r}  \cdot \tilde{\nabla} \vec{v} \biggr\}
+ \dot{a} \biggl\{ \biggl[ \vec{v} - \frac{\partial \vec{r} }{\partial t} \biggr] 
- \biggl[ \vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr] \cdot \tilde{\nabla} \biggl[\tilde{r}  \biggr] \biggr\}
</math>
  </td>
</tr>
</table>
</div>

And the continuity equation becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(3-\nu) \dot{a} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a}{\tilde\rho} \frac{\partial \tilde\rho}{\partial t} 
+ \tilde{\nabla}\cdot \biggl[  \vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr]
+ \biggl[ \vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr] \cdot \frac{\tilde{\nabla} \tilde\rho}{\tilde\rho} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a}{\tilde\rho} \frac{\partial \tilde\rho}{\partial t} 
+ \tilde{\nabla}\cdot \vec{v} 
- \tilde{\nabla}\cdot \biggl[  \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr]
+ \biggl[ \vec{v} 
- \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr] \cdot \frac{\tilde{\nabla} \tilde\rho}{\tilde\rho} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ (3-\nu) \dot{a} 
+ \tilde{\nabla}\cdot \biggl[  \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{a^2 \rho} \frac{\partial (a^3\rho)}{\partial t} 
+ \tilde{\nabla}\cdot \vec{v} 
+ \biggl[ \vec{v} 
- \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr] \cdot \frac{\tilde{\nabla} \rho}{\rho} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a}{\rho} \frac{\partial \rho}{\partial t} + 3\dot{a} 
+ \tilde{\nabla}\cdot \vec{v} 
+ \biggl[ \vec{v} 
- \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr] \cdot \frac{\tilde{\nabla} \rho}{\rho} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{1}{\rho} \frac{\partial \rho}{\partial t} 
+ a^{-1}\tilde{\nabla}\cdot \vec{v} 
+ \biggl[ \vec{v} 
- \dot{a} \biggl( \frac{\vec{r}}{a}\biggr) \biggr] \cdot \frac{a^{-1}\tilde{\nabla} \rho}{\rho} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math> 
\frac{\dot{a}}{a} \biggl[\tilde{\nabla}\cdot \biggl( \frac{\vec{r}}{a} \biggr) -\nu \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{1}{\rho} \frac{\partial \rho}{\partial t} 
+ a^{-1}\tilde{\nabla}\cdot \vec{v} 
+ a^{-1} \biggl[ \vec{v} 
- \dot{a} \vec{\mathfrak{x}} \biggr] \cdot \frac{\tilde{\nabla} \rho}{\rho} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math> 
\frac{\dot{a}}{a} \biggl[\tilde{\nabla}\cdot \vec{\mathfrak{x}} -\nu \biggr]
</math>
  </td>
</tr>
</table>
</div>

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


-->

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