**IF YOU DON'T WANT TO SEE THE ANSWER DO NOT READ THIS POST**fuck this forums lack of spoiler tags

Sorry. It's a bit rushed. It might not make sense but its the answer none the less.

You're on your own to find the period of orbit though. This is just the major part of the problem.

First lets define some variables:

Since the density of the expanded sun is uniform we can use

to find the mass of the sun within earths orbit.

is the density of the sun when the radius is the orbital radius of earth

or

while

is the density of the sun at radius

solving for

we get

and so:

Now using the equations

and using

because the earth is experiencing a centripetal force. setting these equal to each other we can solve for

:

and therefore