If the Sun were to suddenly become a black hole of the same mass, what would the orbital periods of the planets be?

Question

I am interested in theoretical and practical considerations.

Answer 1

It would be exactly the same, (atleast in Newtionian picture), no gravitational fields outside planets radius would change. The easiest way to see this I think is to use the gravitational analogoue of Gauss law.

Since we have spherical symmetry in both cases Int G dA = G*4*pi*r^2 ~ M
So G is constant.

See http://en.wikipedia.org/wiki/Gauss's_law