Poisson and continuity equation for collapsing polytropes

AmenoParallax

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?
Help please!
Thanks!

AmenoParallax

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
Life lesson: always read until end of chapter! (or paragraph at least...)