- #1

- 321

- 68

Take the planet to be rotating about the y axis at a rate ω. By symmetry we only need to work in 2 dimensions. So what we get is that the surface’s normal is directed along

**N**= <N

_{x}, N

_{y}> = <ω

^{2}x - GMx/(x

^{2}+y

^{2})

^{3/2}, -GMy/(x

^{2}+y

^{2})

^{3/2}>

So the goal is to find the surfaces which are everywhere perpendicular to

**N**. Well

**N**is curl-less and so can be written as the gradient of a scalar potential, and then the surfaces we wish to find will simply be equipotentials. The scalar field with gradient equal

**N**is GM/√(x

^{2}+y

^{2}) + 0.5(ωx)

^{2}but if we take this expression equal to a constant, we do not get the equation of an ellipse. (The 2D cross section of the surface should be an ellipse.) What went wrong?