Angular momentum of an accretion disk I need to plot the time evolution of the total angular momentum in an accretion disk. This confuses me because I thought this should be constant, since angular momentum has to be conserved?
I'm given the angular velocity $\Omega=(GM/R^3)^\frac{1}{2}$ where $M$ is the mass of the central object, and that the disk is made up of annuli of matter lying between $R$ and $R+ΔR$ with mass $2πRΔRΣ$, where $Σ(R,t)$ is the surface density at time $t$ (I calculated the surface density numerically at different times in the previous question, so I assume this has to be used in my answer).
So
a) Why does total angular momentum change?
b) How do I know what function of $R$, $Ω$, and $Σ$ represents total angular momentum?
c) I also need to plot the position of the "peak angular momentum surface density ($R^2ΩΣ$) as a function of time". Do I have to differentiate $R^2ΩΣ$ w.r.t. R and set equal to zero in order to find a maximum, and what does this actually represent?
 A: a) The total angular momentum of the entire system would be conserved in reality. However, the mechanisms by which angular momentum is conserved in accretion disks are rather complicated (viscosity and the creation of turbulent eddies). So if your model is very simple then the angular momentum may not be conserved.
b) Consider a ring of radius $r$ and mass $m$, the moment of inertia is
$$I=mr^2$$
and the angular momentum is
$$L=I\Omega$$
where $\Omega$ is the angular velocity. So this allows you to compute the angular momentum for a rotating ring. Then integrate to compute it for the entire disk.
c) The peak angular momentum density at time $t$ is
$$ P(t)=\max_{R>0} R^2\Sigma(R,t)\Omega(R) $$
If you have an analytical expression for $\Sigma(R,t)$ then we can possibly obtain $P(t)$ analytically. Otherwise you will need to iterate over all $R$ and find the maximum value numerically.
What you should find is that $P(t)$ initially decreases and then levels out at a constant value (possibly exponentially). This will show that mass density peaks have a tendency to 'spread out'. The function will also allow you to quantify the rate at which a newly formed accretion disk equilibrates.
