# How to calculate escape velocity and orbital velocity for a cubular planet?

Given a planet that is a cube, how would one calculate the escape velocity and the orbital velocity?

For a circular planet with an even distribution of mass (aka center of mass=center of volume) the escape velocity can be calculated like so:

$$v_{esc} = \sqrt{\frac{2GM}{R}} \,.$$

Is there a modification of this formula (or any other method) that would work for non-circular objects?

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It's also not possible in the real world to have a "substantially" large object maintain a cubical shape. – David White Sep 29 '15 at 1:51
I deleted some comments and their responses - remember that answers should be posted as answers! – David Z Sep 29 '15 at 3:03

The escape velocity can be found in the usual way from the gravitational potential, but the potential no longer has the pure radial dependence you get with a spherically symmetric assumption. You have to compute the integral $$V = -G \iiint \!\!\mathrm{d}x \, \mathrm{d}y \,\mathrm{d}z \, \frac{\rho(x,y,z)}{R} \,,$$ where $\vec{R}$ is the displacement from the point of integration to the point for which the potential s being computed and all three integrals runs over $[-s/2,s/2]$. That is a bit easier if you put the point of consideration on a symmetry line (through the center of opposed faces or opposed points) but it is pretty grotty any way around. Also recall that there is no particular reason to suspect that the density is either uniform or even nicely varying.