Editing Apps/GoldreichWeber80 (section)

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

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">
<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'', {{ 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">
<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, {{ 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">
<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 {{ GW80 }}.  The following is a framed image of equations (7) - (9) as they appear in the {{ GW80 }} publication:

<table border="1" width="80%" cellpadding="8" align="center">
<tr>
  <td align="center">
Principal Governing Equations extracted without modification from &hellip;<br />{{ GW80figure }}
  </td>
</tr>
<tr>
<td>
<!-- [[File:GW80Equations.png|500px|center|Goldreich &amp; Weber (1980)]] -->

<table border="0" align="center" cellpadding="8" width="100%">
<tr>
  <td align="right">
<math>
\frac{1}{\rho} \frac{\partial\rho}{\partial t} 
+ a^{-1}(a^{-1} \mathbf{\nabla} v - \dot{a}\mathbf{r} ) \cdot \mathbf{\nabla}\rho/\rho
+ a^{-2} \nabla^2 v
</math>
  </td>
  <td align="center" width="5%"><math>=</math></td>
  <td align="left" width="25%"><math>0 \, ,</math></td>
  <td align="right" width="8%">(7)</td>
</tr>

<tr>
  <td align="right">
<math>
\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
</math>
  </td>
  <td align="center" width="5%"><math>=</math></td>
  <td align="left" width="25%"><math>0 \, ,</math></td>
  <td align="right" width="8%">(8)</td>
</tr>

<tr>
  <td align="right">
<math>
a^{-2} \nabla^2\phi - 4\pi G\rho
</math>
  </td>
  <td align="center" width="5%"><math>=</math></td>
  <td align="left" width="25%"><math>0 \, .</math></td>
  <td align="right" width="8%">(9)</td>
</tr>
</table>

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

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">
<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 {{ 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">
<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 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:
* {{ GW80 }} 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;
* {{ 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">
<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 [[#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 {{ 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">
<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 {{ GW80 }}, 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 {{ GW80 }}.

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

Next, {{ GW80 }} (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 [[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, {{ GW80 }} (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>