Editing SSC/Dynamics/IsothermalSimilaritySolution (section)

==Limiting Behavior==

It can be shown by analytic manipulation of the pair of coupled ODEs that the dimensionless density, <math>~\Rho</math>, and the dimensionless radial velocity, <math>~U</math>, have the following behaviors in various limits:

* As, <math>\zeta \rightarrow - \infty</math>:
<p>&nbsp;</p>
{| class="Chap1A" style="margin-right: auto; margin-left: 50px; vertical-align:top; text-align:left;"
|- 
|<math>~U \approx \frac{2}{3} \biggl( - \frac{1}{\zeta}\biggr) + \frac{1}{45} \biggl[ \frac{2}{3} - e^{Q_0}\biggr] \biggl( - \frac{1}{\zeta}\biggr)^3 \, ,</math><p>&nbsp;</p>
|-
|<math>~Q \equiv \ln(\zeta^2 \Rho) \approx Q_0 + \frac{1}{6}\biggr[ \frac{2}{3} - e^{Q_0}\biggr] \biggl( - \frac{1}{\zeta}\biggr)^2 \, ,</math><p>&nbsp;</p>
|-
|where, <math>~Q_0</math> is a positive constant.
|}
<p>&nbsp;</p>

* [[File:CommentButton02.png|right|100px|Comment by J. E. Tohline on 13 July 2017:  In this expression for U, a "plus" sign has been inserted between the &zeta; term and the &zeta;-squared term, correcting a typographical error in equation 4.12a of Tohline (1982). And the expression for &Rho; has been expanded to include a &zeta;-cubed term.]]For, <math>\zeta \approx 0</math>:
<p>&nbsp;</p>
{| class="Chap1B" style="margin-right: auto; margin-left: 50px; vertical-align:top; text-align:left;"
|- 
|<math>~U \approx U_0 + \zeta(\Rho_0 - 2)+ \zeta^2 U_0 + \zeta^3\biggl[(\Rho_0-2)(1-\Rho_0/6) - \frac{2}{3} U_0^2 \biggr] \, ,</math><p>&nbsp;</p>
|-
|<math>~\Rho \approx \Rho_0 - \zeta^2\biggl[\frac{1}{2} \Rho_0(\Rho_0 -2)\biggr] + \frac{1}{3} \zeta^3 U_0 \Rho_0(\Rho_0-4) \, ,</math><p>&nbsp;</p>
|-
|where, <math>~U_0</math> and  <math>~\Rho_0</math> are positive constants.
|}
<p>&nbsp;</p>

* As, <math>\zeta \rightarrow + \infty</math>:
<p>&nbsp;</p>
{| class="Chap1C" style="margin-right: auto; margin-left: 50px; vertical-align:top; text-align:left;"
|- 
|<math>~U \approx (2m_0 \zeta)^{1 / 2} \, ,</math><p>&nbsp;</p>
|-
|<math>~\Rho \approx \biggl( \frac{m_0}{2\zeta} \biggr)^{1 / 2} \, ,</math><p>&nbsp;</p>
|-
|where, <math>~m_0</math> is a positive constant.
|}
<p>&nbsp;</p>
The values of the three constants, <math>~U_0</math>, <math>~\Rho_0</math>, and <math>~m_0</math> depend on the chosen value of <math>~Q_0</math>, as demonstrated by [http://adsabs.harvard.edu/abs/1977ApJ...218..834H Hunter (1977)].  In terms of the physical quantities, <math>~v_r(r,t)</math> and  <math>~\rho(r,t)</math>, these asymptotic behaviors translate into the following.

* For, <math>~t < 0</math> and <math>~r \ll c_s|t|</math>:
<p>&nbsp;</p>
{| class="Chap1D" style="margin-right: auto; margin-left: 50px; vertical-align:top; text-align:left;"
|- 
|<math>~v_r(r,t) \approx - \frac{2r}{3(-t)} \, ,</math><p>&nbsp;</p>
|-
|<math>~\rho(r,t) \approx \biggl[\frac{e^{Q_0}}{4\pi G}\biggr] \frac{1}{t^2} \, .</math><p>&nbsp;</p>
|}
<p>&nbsp;</p>

* For, <math>~r \gg c_s|t|</math> at any time:
<p>&nbsp;</p>
{| class="Chap1E" style="margin-right: auto; margin-left: 50px; vertical-align:top; text-align:left;"
|- 
|<math>~v_r(r,t) \approx - c_s U_0\, ,</math><p>&nbsp;</p>
|-
|<math>~\rho(r,t) \approx \biggl[\frac{c_s^2 \Rho_0}{4\pi G}\biggr] \frac{1}{r^2} \, .</math><p>&nbsp;</p>
|}
<p>&nbsp;</p>

* For, <math>~t > 0</math> and <math>~r \ll c_s|t|</math>:
<p>&nbsp;</p>
{| class="Chap1F" style="margin-right: auto; margin-left: 50px; vertical-align:top; text-align:left;"
|- 
|<math>~v_r(r,t) \approx - \biggl( \frac{2m_0}{c_s} \biggr)^{1 / 2} \biggl( \frac{t}{r}\biggr)^{1 / 2} \, ,</math><p>&nbsp;</p>
|-
|<math>~\rho(r,t) \approx \frac{1}{4\pi G} \biggl[\frac{m_0 c_s^2}{2}\biggr]^{1 / 2} \biggl( \frac{1}{t r^3}\biggr)^{1 / 2} \, .</math><p>&nbsp;</p>
|}
<p>&nbsp;</p>