Spheroidal potentials and gravitational attraction by a rod and a disc

We show that the gravitational Newtonian potential of a disc and a rod on an external point are respectively oblate and prolate confocal spheroids. The end points of the rod are the fixed foci on the symmetry axis of the prolate spheroid. The circle with the radius of the disc is the locus of the foci in the equatorial plane of the oblate spheroid. We show the connection between the potential of a disc and the potential of a full oblate spheroid. The acceleration caused by the rod and the disc on an external point is a simple expression in the parameters of the meridian ellipse through that point. In the equatorial plane, spheroidal potentials are central and their first order approximation is a J2 problem for which we obtained the following results. The energy integral can be interpreted as the trajectory in pedal coordinates. For circular orbits, Kepler's third law is adjusted with the distance to the center of gravity. The parameters p and a, representing the angular momentum and energy, remain different from each other and the radius of the circular orbit. The precession of the orbit is approached via a split up of the angular momentum. The exact equatorial trajectory is a rotating 4th degree oval. The oval is obtained by adding a constant segment to the radius vector of an ellipse. The attraction of rods and discs could have applications for galaxies and large structures in the universe. The attraction of a disc is a first approximation for the attraction of an oblate ellipsoid.