Potential surprises in Newtonian Gravity

Abstract

Consider a planet which has a weird shape, resembling, say, that of a diseased potato. Is it possible that the gravitational force exerted by this planet—which is distinctly non-spherical in shape—falls exactly as r ^-2 everywhere outside of it? The initial reaction of many physicists will be: “No, of course, not; you need a spherically symmetric distribution of mass to produce a 1/r ² force outside it”. Surprisingly, this is not true.