If the earth's mass were doubled what it is in what ways would the moon's orbit be different

One equation that gives us information about how a satellite moves around an object is the following one:

v^2 = (G*M)/R

Where V is the velocity of the satellite, G is the gravitational constant, M is the mass of the Earth in this case, and R is the distance between the satellite and the Earth.

If you double M, there are a range of probabilities, where the extremes are:

The velocity of the moon does not change:

In this situation we have:

v^2 = (G*2M)/R'

where R' is the new distance, if we want to restore the previous equation, here we must have R' = 2R, this means that the distance between the Earth and the moon is also doubled.

The other extreme is that the distance is not changed:

(v')^2 = (G*2M)/R = 2*v^2

where v' is the new velocity  

Then we have that v' = √2*v will be the new velocity of the moon.

And there are the possibilities where both velocity and distance changes, which are a lot more.