Surface gravity of rotating dumbbell shapes

We investigate the problem of determining shape a rotating celestial object - e.g., comet or an asteroid under its own gravitational field. More specifically, we consider symmetric with respect to one axis such as dumbbell that rotates around second perpendicular symmetry axis. assume can be modeled incompressible fluid constant mass density, which is regarded first approximation aggregate particles. In literature, field body often described multipolar expansion involving spherical coordinates (Kaula, 1966). this work describe in terms cylindrical coordinates, are most naturally adapted body, and express potential generated by simple formula elliptic integrals. An equilibrium occurs when energy rotational kinetic at surface balance each other out. Such derived solution optimization problem, found via variational method. give example where apply method two-parameter family shapes, find approximate numerical solutions corresponding problem.