An equilibrium problem on the sphere with two equal charges

We study the equilibrium measure on the two dimensional sphere in the presence of an external field generated by two equal point charges. The support of the equilibrium measure is known as the droplet. Brauchart et al. showed that the complement of the droplet consists of two spherical caps when the charges are small. When the charges are bigger the droplet becomes simply connected and we prove that the boundary of the droplet is mapped by stereographic projection to an ellipse in the plane. Moreover, we compute a mother body for the droplet that we derive from an equilibrium problem with a weakly admissible external field on the real line.