$2$ Interacting binary $(15$ pts$)$ Two stars with masses M$1$ and M$2$ are in a binary orbit with separation a$.$ Star $1$ reaches the end of its main sequence lifetime and expands to become a red giant. This expansion causes star $1$ to overflow its Roche lobe and begin to accrete matter onto star $2$ at rate M$9.$ Total mass M$1`$ M$2$ is conserved, so M$91$M$9$ and M$92`$M$9.$ a$.$ Derive a differential equation to describe how the separation evolves with time. $($You don$$t need to solve this equation.$)$ b$.$ If M$1$ M$2,$ do the stars grow closer together or further apart? c$.$ How does this affect the orbital period or angular velocity of the binary? Hint: Angular momentum is conserved in this system. The angular momentum of the binary is given by J $$ I where you can assume the moment of inertia is that of a sphere of uniform density, and the angular velocity can be comp