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Similarity Solution

Much of the material in this chapter has been drawn from §4.1 of a review article by Tohline (1982) titled, Hydrodynamic Collapse.

Whitworth's (1981) Isothermal Free-Energy Surface
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Several authors (references given, below) have shown that when isothermal pressure gradients are important during a gas cloud's collapse, the equations governing the collapse admit a set of similarity solutions. Certain properties of these solutions can be described analytically and are instructive models for comparison with more detailed, numerical collapse calculations.

Establishing Set of Governing Equations

Drawing from an accompanying chapter's introductory discussion, we begin with the set of governing equations that describe the collapse of isothermal spheres from an Eulerian frame of reference.

Eulerian Frame

<math>~\frac{\partial M_r}{\partial r} </math>


<math>~4\pi r^2 \rho \, , </math>

<math>~\frac{\partial M_r}{\partial t} </math>


<math>~- 4\pi r^2 \rho v_r \, ,</math>

<math>~\frac{\partial v_r}{\partial t} + v_r \frac{\partial v_r}{\partial r}</math>


<math>~- c_s^2 \biggl( \frac{\partial \ln \rho}{\partial r}\biggr) - \frac{GM_r}{r^2} \, .</math>

Notice that, following Larson's (1969) lead, we have replaced the standard continuity equation with the following equivalent statement of mass conservation:



<math>~0 </math>

<math>~\Rightarrow ~~~ 0</math>


<math>~\frac{\partial M_r}{\partial t} + v_r ~\frac{\partial M_r}{\partial r} </math>



<math>~\frac{\partial M_r}{\partial t} +4\pi r^2 \rho v_r \, .</math>

Mathematical Solution


A similarity solution becomes possible for these equations when the single independent variable,

<math>~\zeta = \frac{c_s t}{r} \, ,</math>

is used to replace both <math>~r</math> and <math>~t</math>. Then, if <math>~M_r</math>, <math>~\rho</math>, and <math>~v_r</math> assume the following forms,



<math>~\biggl(\frac{c_s^3 t}{G}\biggr) m(\zeta) \, ,</math>



<math>~\biggl(\frac{c_s^2 }{4\pi G r^2}\biggr) \Rho (\zeta) \, ,</math>



<math>~- c_s U(\zeta) \, ,</math>

the three coupled partial differential equations reduce to two coupled ordinary differential equations for the functions, <math>~\Rho (\zeta)</math> and <math>~U(\zeta)</math>, namely,



<math>~ \frac{(\zeta U +1) [\Rho (\zeta U +1) -2)]}{[ (\zeta U +1)^2 - \zeta^2]} \, , </math>



<math>~\frac{\zeta \Rho [2-\Rho (\zeta U +1)]}{[ (\zeta U +1)^2 - \zeta^2]} \, ,</math>

and a single equation defining <math>~m(\zeta)</math>,



<math>~\Rho \biggl[ U + \frac{1}{\zeta} \biggr] \, .</math>

The parameters <math>~\zeta, m, \Rho</math>, and <math>~U</math>, and this summary set of equations are exactly those used by Hunter (1977) in his analysis of this problem. But they differ in form from the relations used by Larson (1969), Penston (1969), and Shu (1977) primarily because these authors chose to use a similarity variable,

<math>~x = \pm \frac{1}{\zeta} \, ,</math>

instead of <math>~\zeta</math>. Hunter's analysis is the most complete and his relations will be used here, but a transformation between his presentation and those of the other authors can be easily obtained from Table 1 of Hunter (1977) which, for convenience, is reproduced here.

Analogous to Table 1 from Hunter (1977)
Relations Between the Variables Used by Different Authors

Herein Larson (1969) Penston (1969) Shu (1977)
<math>~\frac{c_s t}{r}</math> <math>~\zeta</math> <math>~- \frac{1}{x}</math> <math>~- \frac{1}{x}</math> <math>~+ \frac{1}{x}</math>
<math>~- \frac{v_r}{c_s}</math> <math>~U</math> <math>~\xi</math> <math>~- V</math> <math>~-v</math>
<math>~\frac{4\pi G\rho r^2}{c_s^2}</math> <math>~\Rho</math> <math>~x^2\eta</math> <math>~x^2 e^Q</math> <math>~x^2\alpha</math>
<math>~\frac{GM_r}{c_s^3 t}</math> <math>~m</math> <math>~-N</math> <math>~m</math>
<math>~\ln(4\pi G\rho t^2)</math> <math>~Q</math> <math>~\ln\eta</math> <math>~Q</math> <math>~\ln\alpha</math>
<math>~\frac{r}{(- c_s t)}</math> <math>~y</math> <math>~x</math> <math>~x</math> <math>~-x</math>

Adopting Hunter's notation, this dimensionless variable name, <math>~\Rho</math> (the capital Greek letter, <math>~\rho</math>), should not be confused with the variable name, <math>~P</math>, that represents herein the ideal gas pressure.

The following pair of images are reproductions of (left) Figure 1 and (right) Figure 3 from Hunter (1977). The solid curves show how (left) the dimensionless velocity, <math>~U</math>, and (right) the dimensionless density, <math>~\Rho</math>, behave as a function of the similarity variable, <math>~\zeta</math>, for models having several different prescribed values of Hunter's parameter, <math>~Q_0</math>. For each value of <math>~Q_0</math>, the table of numbers immediately below the pair of images provides corresponding values of several other numerical constants.

Figures extracted from Hunter (1977)

"The Collapse of Unstable Isothermal Spheres"

ApJ, vol. 218, pp. 834 - 845 © American Astronomical Society

Figure 1 from Hunter (1977, ApJ, 218, 836
Figure 3 from Hunter (1977, ApJ, 218, 836
Model <math>~Q_0</math> <math>~U_0</math> <math>~P_0</math> <math>~m_0</math>
LP 0.5139 3.278 8.854 46.915
H(b) 11.236 0.295 2.378 2.577
H(d) 20.975 -- -- --


Plugging the similarity solution expressions for <math>~M_r</math> and <math>~\rho</math> into the first of the three governing equations gives,

<math>~\frac{\partial}{\partial r} \biggl[ \biggl(\frac{c_s^3 t}{G}\biggr) m(\zeta) \biggr]</math>


<math>~4\pi r^2 \biggl[ \biggl(\frac{c_s^2 }{4\pi G r^2}\biggr) \Rho (\zeta) \biggr] </math>

<math>~\Rightarrow ~~~ (c_s t ) \frac{\partial}{\partial r} \biggl[ m(\zeta) \biggr]</math>


<math>~\Rho (\zeta) \, .</math>

Plugging the similarity solution expressions for <math>~M_r</math>, <math>~\rho</math>, and <math>~v_r</math> into the second of the three governing equations gives,

<math>~\frac{\partial}{\partial t} \biggl[ \biggl(\frac{c_s^3 t}{G}\biggr) m(\zeta) \biggr]</math>


<math>~- 4\pi r^2 \biggl[ \biggl(\frac{c_s^2 }{4\pi G r^2}\biggr) \Rho (\zeta) \biggr] \biggl[ -c_s U(\zeta)\biggr]</math>

<math>~\Rightarrow ~~~ \frac{\partial}{\partial t} \biggl[ t m(\zeta) \biggr]</math>


<math>~\Rho (\zeta) U(\zeta) </math>

<math>~\Rightarrow ~~~ m(\zeta) + t \biggl[ \frac{\partial m(\zeta)}{\partial t} \biggr]</math>


<math>~\Rho (\zeta) U(\zeta) \, .</math>

And, plugging the similarity solution expressions for <math>~M_r</math>, <math>~\rho</math>, and <math>~v_r</math> into the third of the three governing equations gives,

<math>~\frac{\partial }{\partial t} \biggl[ - c_s U(\zeta) \biggr] + \biggl[ - c_s U(\zeta) \biggr] \frac{\partial }{\partial r} \biggl[ - c_s U(\zeta) \biggr]</math>


<math>~- c_s^2 \biggl[\biggl(\frac{c_s^2 }{4\pi G r^2}\biggr) \Rho (\zeta) \biggr]^{-1} \frac{\partial }{\partial r}\biggl[ \biggl(\frac{c_s^2 }{4\pi G r^2}\biggr) \Rho (\zeta) \biggr] - \frac{G}{r^2}\biggl[ \biggl(\frac{c_s^3 t}{G}\biggr) m(\zeta) \biggr] </math>

<math>~\Rightarrow ~~~ \frac{\partial }{\partial t} \biggl[ U(\zeta) \biggr] - c_s U(\zeta) \frac{\partial }{\partial r} \biggl[ U(\zeta) \biggr]</math>


<math>~\biggl[ \frac{c_s r^2}{\Rho (\zeta)} \biggr]\frac{\partial }{\partial r}\biggl[ \biggl(\frac{\Rho (\zeta)}{r^2}\biggr) \biggr] + \biggl[ \frac{c_s^2 t}{r^2} \biggr] m(\zeta) </math>

<math>~\Rightarrow ~~~ \frac{\partial U}{\partial t} - (c_s U) \frac{\partial U}{\partial r} </math>


<math>~\frac{c_s}{\Rho} \biggl[ \biggl( \frac{\partial \Rho}{\partial r}\biggr) -\frac{2\Rho}{r} \biggr] + \biggl[ \frac{c_s^2 t}{r^2} \biggr] m(\zeta) \, .</math>

Now, from the functional dependence of <math>~m(\zeta)</math> on <math>~\Rho(\zeta)</math> and <math>~U(\zeta)</math>, we have,

<math>~\frac{\partial m}{\partial r}</math>


<math>~ \biggl[ U + \frac{1}{\zeta} \biggr] \frac{\partial\Rho}{\partial r} + \Rho \biggl[ \frac{\partial U}{\partial r} - \frac{1}{\zeta^2} \frac{\partial \zeta}{\partial r}\biggr] </math>



<math>~ \biggl[ U + \frac{1}{\zeta} \biggr] \frac{\partial\Rho}{\partial r} + \Rho \biggl[ \frac{\partial U}{\partial r} + \frac{1}{r \zeta} \biggr] \, , </math>


<math>~\frac{\partial m}{\partial t}</math>


<math>~ \biggl[ U + \frac{1}{\zeta} \biggr] \frac{\partial\Rho}{\partial t} + \Rho \biggl[ \frac{\partial U}{\partial t} - \frac{1}{\zeta^2} \frac{\partial \zeta}{\partial t}\biggr] </math>



<math>~ \biggl[ U + \frac{1}{\zeta} \biggr] \frac{\partial\Rho}{\partial t} + \Rho \biggl[ \frac{\partial U}{\partial t} - \frac{1}{t \zeta} \biggr] \, . </math>

Hence, the first two governing equations become, respectively,

<math>~\Rho </math>


<math>~(r\zeta) \biggl\{ \biggl[ U + \frac{1}{\zeta} \biggr] \frac{\partial\Rho}{\partial r} + \Rho \biggl[ \frac{\partial U}{\partial r} + \frac{1}{r \zeta} \biggr] \biggr\} </math>



<math>~ \biggl[ r\zeta U + r \biggr] \frac{\partial\Rho}{\partial r} + (r\zeta \Rho ) \frac{\partial U}{\partial r} + \Rho </math>



<math>~ \biggl[ \zeta U + 1\biggr] \frac{\partial\Rho}{\partial r} + (\zeta \Rho ) \frac{\partial U}{\partial r} \, ; </math>

<math>~\Rho (\zeta) U(\zeta)</math>


<math>~ \Rho\biggl[ U + \frac{1}{\zeta}\biggr] + t \biggl\{ \biggl[ U + \frac{1}{\zeta} \biggr] \frac{\partial\Rho}{\partial t} + \Rho \biggl[ \frac{\partial U}{\partial t} - \frac{1}{t \zeta} \biggr] \biggr\} </math>

<math>~\Rightarrow ~~~0</math>


<math>~ t \biggl\{ \biggl[ U + \frac{1}{\zeta} \biggr] \frac{\partial\Rho}{\partial t} + \Rho \biggl[ \frac{\partial U}{\partial t} \biggr] \biggr\} </math>

<math>~\Rightarrow ~~~0</math>


<math>~ \biggl[ \zeta U + 1 \biggr] \frac{\partial\Rho}{\partial t} + (\zeta \Rho) \frac{\partial U}{\partial t} \, . </math>

Now, we can use these two relations to replace derivatives of <math>~\Rho</math> with derivatives of <math>~U</math> — or visa versa — in the third governing relation. In the first case, we obtain,

<math>~ \frac{\Rho}{c_s} \biggl[\frac{\partial U}{\partial t} - (c_s U) \frac{\partial U}{\partial r} \biggr]</math>


<math>~\biggl( \frac{\partial \Rho}{\partial r}\biggr) -\frac{2\Rho}{r} + \frac{\Rho^2}{r} \biggl[\zeta U + 1\biggr] </math>



<math>~ \frac{\Rho^2(\zeta U + 1)}{r} -\frac{2\Rho}{r} - \biggl( \frac{\partial U}{\partial r}\biggr) \biggl[ \frac{\zeta \Rho}{(\zeta U + 1)} \biggr]</math>

<math>~ \Rightarrow ~~~ \frac{1}{r} \biggl[ \Rho^2(\zeta U + 1) - 2\Rho \biggr] </math>


<math>~ \frac{\Rho}{c_s} \biggl[\frac{\partial U}{\partial t}\biggr] - (\Rho U) \frac{\partial U}{\partial r} + \biggl( \frac{\partial U}{\partial r}\biggr) \biggl[ \frac{\zeta \Rho}{(\zeta U + 1)} \biggr] </math>

<math>~ \Rightarrow ~~~ \biggl[ \Rho(\zeta U + 1) - 2 \biggr] </math>


<math>~ \frac{r}{c_s} \biggl[\frac{\partial U}{\partial t}\biggr] + \biggl( \frac{\partial U}{\partial r}\biggr) \biggl[ \frac{r \zeta }{(\zeta U + 1)} - (rU) \biggr] </math>



<math>~ \frac{t}{\zeta} \biggl[\frac{\partial U}{\partial t}\biggr] + r \biggl( \frac{\partial U}{\partial r}\biggr) \biggl[ \frac{\zeta - U (\zeta U + 1)}{(\zeta U + 1)} \biggr] \, . </math>

And, given that,

<math>~\frac{\partial U}{\partial t}</math>


<math>~\biggl( \frac{dU}{d\zeta} \biggr) \frac{\partial \zeta}{\partial t} = \biggl( \frac{dU}{d\zeta} \biggr)\frac{c_s}{r} = \biggl( \frac{dU}{d\zeta} \biggr)\frac{\zeta}{t} \, ;</math>


<math>~\frac{\partial U}{\partial r}</math>


<math>~ \biggl( \frac{dU}{d \zeta} \biggr) \frac{\partial \zeta}{\partial r} = - \frac{c_s t}{r^2} \biggl( \frac{dU}{d \zeta} \biggr) = -\frac{\zeta^2}{c_st} \biggl( \frac{dU}{d \zeta} \biggr) \, ,</math>

we can rewrite this as an ODE of the form,

<math>~ \biggl[ \Rho(\zeta U + 1) - 2 \biggr] </math>


<math>~ \biggl(\frac{d U}{d\zeta}\biggr) -\zeta \biggl( \frac{d U}{d\zeta }\biggr) \biggl[ \frac{\zeta - U (\zeta U + 1)}{(\zeta U + 1)} \biggr] </math>

<math>~\Rightarrow ~~~ [ \Rho(\zeta U + 1) - 2 ](\zeta U + 1) </math>


<math>~ \biggl(\frac{d U}{d\zeta}\biggr) \biggl\{(\zeta U + 1) -\zeta \biggl[ \zeta - U (\zeta U + 1) \biggr]\biggr\} </math>



<math>~ \biggl(\frac{d U}{d\zeta}\biggr) \biggl[ \zeta^2U^2 + 2\zeta U + 1 - \zeta^2 \biggr] </math>

<math>~ \Rightarrow ~~~ \frac{d U}{d\zeta} </math>


<math>~ \frac{ [\Rho(\zeta U + 1) - 2 ](\zeta U + 1)}{ [ (\zeta U + 1)^2 - \zeta^2 ] } \, .</math>

In the second case, we obtain,

<math>~\frac{c_s}{\Rho} \biggl( \frac{\partial \Rho}{\partial r}\biggr) + \frac{\zeta}{t} \biggl[ \Rho (\zeta U + 1 ) - 2\biggr] </math>


<math>~ \frac{\partial U}{\partial t} - (c_s U) \frac{\partial U}{\partial r} </math>



<math>~ -\biggl[ \frac{\zeta U +1}{\zeta \Rho} \biggr] \frac{\partial \Rho}{\partial t} + (c_s U) \biggl[ \frac{\zeta U + 1}{\zeta\Rho} \biggr] \frac{\partial \Rho}{\partial r} </math>

<math>~\Rightarrow ~~~ \frac{\zeta}{t} \biggl[ 2- \Rho (\zeta U + 1 ) \biggr] </math>


<math>~ \biggl\{ \frac{c_s}{\Rho} - (c_s U) \biggl[ \frac{\zeta U + 1}{\zeta\Rho} \biggr] \biggr\}\frac{\partial \Rho}{\partial r} + \biggl[ \frac{\zeta U +1}{\zeta \Rho} \biggr] \frac{\partial \Rho}{\partial t} </math>

<math>~\Rightarrow ~~~ \frac{\zeta^2 \Rho}{c_s t} \biggl[ 2- \Rho (\zeta U + 1 ) \biggr] </math>


<math>~ \biggl[ \zeta - U (\zeta U + 1 ) \biggr] \frac{\partial \Rho}{\partial r} + \frac{1}{c_s}\biggl[ \zeta U +1 \biggr] \frac{\partial \Rho}{\partial t} \, . </math>

And, given that,

<math>~\frac{\partial \Rho}{\partial t}</math>


<math>~\biggl( \frac{d\Rho}{d\zeta} \biggr) \frac{\partial \zeta}{\partial t} = \biggl( \frac{d\Rho}{d\zeta} \biggr)\frac{c_s}{r} = \biggl( \frac{d\Rho}{d\zeta} \biggr)\frac{\zeta}{t} \, ;</math>


<math>~\frac{\partial \Rho}{\partial r}</math>


<math>~ \biggl( \frac{d\Rho}{d \zeta} \biggr) \frac{\partial \zeta}{\partial r} = - \frac{c_s t}{r^2} \biggl( \frac{d\Rho}{d \zeta} \biggr) = -\frac{\zeta^2}{c_st} \biggl( \frac{d\Rho}{d \zeta} \biggr) \, ,</math>

we can rewrite this as an ODE of the form,

<math>~ \frac{\zeta^2 \Rho}{c_s t} \biggl[ 2- \Rho (\zeta U + 1 ) \biggr] </math>


<math>~ - \frac{\zeta^2}{c_s t} \biggl[ \zeta - U (\zeta U + 1 ) \biggr] \frac{d\Rho}{d \zeta} + \frac{\zeta}{c_s t}\biggl[ \zeta U +1 \biggr] \frac{d\Rho}{d\zeta} </math>

<math>~\Rightarrow~~~ \zeta\Rho [ 2- \Rho (\zeta U + 1 ) ] </math>


<math>~ - \zeta\biggl[ \zeta - U (\zeta U + 1 ) \biggr] \frac{d\Rho}{d \zeta} + \biggl[ \zeta U +1 \biggr] \frac{d\Rho}{d \zeta} </math>



<math>~ \biggl\{( \zeta U +1 ) - \zeta [ \zeta - U (\zeta U + 1 ) ] \biggr\} \frac{d\Rho}{d \zeta} </math>



<math>~ [(\zeta U + 1)^2 - \zeta^2]\frac{d\Rho}{d \zeta} </math>

<math>~ \Rightarrow~~~ \frac{d\Rho}{d \zeta} </math>


<math>~ \frac{ \zeta\Rho [ 2- \Rho (\zeta U + 1 ) ] }{[(\zeta U + 1)^2 - \zeta^2] } \, . </math>

Thus, we are able to understand the origin of the pair of 1st-order ODEs, given above, that describe the connected relationship between the two quantities, <math>~\Rho</math> and <math>~U</math>.

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>:


<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>


<math>~\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>


where, <math>~Q_0</math> is a positive constant.


  • For, <math>\zeta \approx 0</math>:


<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>


<math>~\Rho \approx \Rho_0 - \zeta^2\biggl[\frac{1}{2} \Rho_0(\Rho_0 -2)\biggr] \, ,</math>


where, <math>~U_0</math> and <math>~\Rho_0</math> are positive constans.


  • As, <math>\zeta \rightarrow + \infty</math>:


<math>~U \approx (2m_0 \zeta)^{1 / 2} \, ,</math>


<math>~\Rho \approx \biggl( \frac{m_0}{2\zeta} \biggr)^{1 / 2} \, ,</math>


where, <math>~m_0</math> is a positive constant.


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 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>:


<math>~v_r(r,t) \approx - \frac{2r}{3(-t)} \, ,</math>


<math>~\rho(r,t) \approx \biggl[\frac{e^{Q_0}}{4\pi G}\biggr] \frac{1}{t^2} \, .</math>



  • For, <math>~r \gg c_s|t|</math> at any time:


<math>~v_r(r,t) \approx - c_s U_0\, ,</math>


<math>~\rho(r,t) \approx \biggl[\frac{c_s^2 \Rho_0}{4\pi G}\biggr] \frac{1}{r^2} \, .</math>



  • For, <math>~t > 0</math> and <math>~r \ll c_s|t|</math>:


<math>~v_r(r,t) \approx - \biggl( \frac{2m_0}{c_s} \biggr)^{1 / 2} \biggl( \frac{t}{r}\biggr)^{1 / 2} \, ,</math>


<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>



See Especially

Whitworth's (1981) Isothermal Free-Energy Surface

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