- #1
Hiero
- 322
- 68
The idea is that, if we take the rotating frame of the fluid-planet so that it is at rest, then the sum of gravity and the centrifugal force must be perpendicular to the surface.
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 = <Nx, Ny> = <ω2x - GMx/(x2+y2)3/2, -GMy/(x2+y2)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/√(x2+y2) + 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?
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 = <Nx, Ny> = <ω2x - GMx/(x2+y2)3/2, -GMy/(x2+y2)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/√(x2+y2) + 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?