# Angular momentum of two rotating spheres

I am trying to calculate an instantaneous merger of two rotating spheres into one. Two spheres each rotating around their own axis of rotation (which are generally not aligned) and moving relative to each other. Imagine two entirely inelastically colliding planets.

While I can calculate their individual spin angular momentum and the orbital angular momentum, it is unclear to me how to combine this into one (spin) angular momentum for the resulting sphere. It seems to me I cannot simply add the four vectors together... or can I?

After merging the two into a single body maintaining the initial momentum, can I calculate the released energy (in the inelastic collision) simply as the difference in kinetic energy, using the same reference frame?

I realize this should be pretty trivial, but I have thus far not been able to neither derive nor locate and answer.

After the collision, the linear velocity of the final planet must be zero by conservation of momentum. Thus the final planet will have no orbital angular momentum. However, we know that the angular momentum must be conserved, so the planet must be spinning about sum axis, and you must have $\vec{L} = I \vec{\omega}$. Since you know what $\vec{L}$ is and you presumably know $I$, you know what $\omega$ must be.