How to figure out the orbital period of a planet based on its size?

My physics professor didn't mention in class how to do this homework question! Help would be much appreciated, formulas would be very useful, too.





"The mass of a star is 1.2e31 kg and it performs one rotation in 37.1 day. Find its new period (in day) if the diameter suddenly shrinks to 0.19 times its present size. Assume a uniform mass distribution before and after."|||This isn't an orbital problem. It's simply conservation of angular momentum. Like an ice skater doing a spin.





Angular momentum = moment of inertia * angular velocity = constant





Therefore, angular momentum is inversely proportional to moment of inertia





period is inversely proportional to angular velocity





moment of inertia is proportional to the square of the diameter





Put it all together:


T ~ 1 / omega ~ I ~ d^2





So your final period is:


Tfinal = Tinitial * (dfinal / dinitial)^2





They give you the initial period and the ratio of the diameters. Plugnchug. Note that the mass isn't needed.