Gravitational potential of a homogeneous circular torus: a new approach

The integral expression for the gravitational potential of a homogeneous circular torus composed of infinitely thin rings is obtained. Approximate expressions for the torus potential in the outer and inner regions are found. In the outer region, the torus potential is shown to be approximately equal to that of an infinitely thin ring of the same mass; it is valid up to the surface of the torus. It is shown in a first approximation that the inner potential of the torus (inside a torus body) is a quadratic function of the coordinates. A method of sewing together the inner and outer potentials is proposed. This method provides a continuous approximate solution for the potential and its derivatives, working throughout the region.