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Poisson and continuity equation for collapsing polytropes

  1. Aug 12, 2012 #1
    Hello everybody!
    I am using in my studies this beautiful book by Kippenhahn & Weigert, "Stellar Structure and Evolution", but I have some problems about collapsing polytropes (chapter 19.11)...
    After defining dimensionless lenght-scale z by:
    [itex]r=a(t)z[/itex]
    and a velocity potential [itex]\psi[/itex]:
    [itex]\frac{\partial r}{\partial t}=v_r=\frac{\partial \psi}{\partial r}[/itex]
    the authors rewrite the Poisson equation:
    [itex]\frac{1}{z^2}\frac{\partial}{\partial z}(z^2\frac{\partial \psi}{\partial z})=4\pi G\rho a^2[/itex]
    but I think there should be the gravitational potential [itex]\phi[/itex] instead of [itex]\psi[/itex], in fact performing a simple dimensional analysis shows that the left hand side is a square lenght over time, while the right hand side is a square lenght over square time, so I think the equation is wrong... Am I right? Did I miss something? :confused:
    Help please!
    Thanks!
     
    Last edited: Aug 12, 2012
  2. jcsd
  3. Aug 14, 2012 #2
    Ok, i got through it, and there is a mistake, indeed. The function in the differential equation is [itex]\Phi[/itex], the gravitational potential, and not the velocity potential [itex]\psi[/itex]... I found the correct formula... in the following page :biggrin: :biggrin: :biggrin:
    Life lesson: always read until end of chapter! (or paragraph at least...)
     
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