How Does Orbital Radius Change with a Time-Varying Solar Mass?

[fizzwhiz]

Homework Statement

An approach to the problem of finding how orbital radius changes with solar mass is to solve the radial equation of motion for a gravitational force that has an explicit time dependence based on the assumed rate of mass loss. Show by numerically solving this equation of motion that one gets the result D proportional to 1/M.

Homework Equations

I'm not entirely sure what equation of motion the question requires. We can equate the gravitational force with the centripetal force,

mv²/D = GMm/D²

but I'm unsure as to how to include the time-dependent solar mass.

I think we can assume the changes in mass per orbit are small and uniform.

The Attempt at a Solution

I tried replacing the constant mass M in the above equation with M(t) and differentiating with respect to t, and using the angular momentum to express v in terms of D, but it's not really getting me anywhere.

Any advice on how to approach this problem would be greatly appreciated. :D
Thanks!

 

[tiny-tim]

welcome to pf!

hi fizzwhiz! welcome to pf!

i think they mean, what happens to the same planet as the solar mass changes?

so v won't be constant, but the angular momentum of the planet will