I have the following formula:
$L^2 = C \cdot a \cdot (1- e^2)$
I know that the PDF of $a$ is proportional to $a^\beta$ and the PDF of $e$ is proportional to $e$:
$PDF(a) \propto a^\beta$
$PDF(e) \propto e$
$C$ is a positive real constant, $a \in [0.1, 100]$ and for my case $\beta=-1$ (but could be any real number); finally $e \in [0,1]$. These proptionalities come out a few assumptions about the system that is described by the equation for $L^2$ (which is an equation for the angular momentum $L$, where $a$ is the semi major axis and $e$ is the eccentricity of Kepler orbits in an isotropic distribution)
How is $L^2$ distributed ? I can find an answer by simply sampling from e and a and fitting the histogram of $L^2$, this yields $L^2 \propto -1$, but it isn't immediately clear to me why that should be the case.