On existence of out-of-plane equilibrium points in restricted three-body problem with oblateness

We analyze in this paper the existence of the "out-of-plane" equilibrium points in the restricted three-body problem with oblateness. From the series expansion of the potential function of an oblate asteroid, we show analytically all equilibrium points locate on the orbital plane of primaries and how artificial equilibrium points may arise due to an inappropriate application of the potential function. Using the closed form of the potential of a triaxial ellipsoid, we analytically demonstrate that the gravitational acceleration in $z$-direction is always pointing toward the equatorial plane, thus it could not be balanced out at any value of $z\neq 0$ and the out-of-plane equilibrium points cannot exist. The out-of-plane equilibrium points appear only when additional acceleration other than the gravitation from primaries is taken into account. We suggest that special attention must be paid to the application of the spherical harmonics expansion of potential to find the equilibrium points, especially when these points may be very close to the celestial body.