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:
$r=a(t)z$
and a velocity potential $\psi$:
$\frac{\partial r}{\partial t}=v_r=\frac{\partial \psi}{\partial r}$
the authors rewrite the Poisson equation:
$\frac{1}{z^2}\frac{\partial}{\partial z}(z^2\frac{\partial \psi}{\partial z})=4\pi G\rho a^2$
but I think there should be the gravitational potential $\phi$ instead of $\psi$, in fact performing a simple dimensional analysis shows that the left hand side is a square length over time, while the right hand side is a square length 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 $\Phi$, the gravitational potential, and not the velocity potential $\psi$... I found the correct formula... in the following page
Life lesson: always read until end of chapter! (or paragraph at least...)