Expansion of the Potential of a Homogeneous Circular Torus in Terms of Geometrical Parameter

Abstract

The spatial potential of a homogeneous gravitating (or statically charged) circular torus is expanded in terms of geometrical parameter q. The first term of the expansion (coefficient of zero-power parameter q) coincides with the potential of a thin ring the mass of which is equal to the mass of torus and the radius of which is equal to the radius of the axial line. It is shown that the coefficients of terms with odd powers are zeros. Even terms of the expansion of the potential are analytically represented in terms of standard complete elliptic integrals. The corresponding series describes the potential of torus in entire space including internal region. The proposed method for representation of the potential makes it possible to obtain gravitational energy of a homogeneous circular torus. The method is used to calculate masses of two rings of the Chariklo asteroid. The mass of the internal ring is Mr1 ≈ 9.8 × 1018 g and the ratio of such a mass to the mass of asteroid is $$\frac{{{M_{r1}}}}{{{M_0}}}$$ ≈ 0.001. The results for the external ring are Mr2 ≈ 1018 g and $$\frac{{{M_{r1}}}}{{{M_0}}}$$ ≈ 10–4.