Potential energy of gravitationally interacting disk galaxies

Methods are developed for analysing the gravitational properties of disks having circularly symmetric distribution of matter. It is shown how this can be conveniently done by assuming that the surface density distribution may be approximated by a polynomial in ascending powers of the distance from the centre of the configuration. A theory has been developed to determine the gravitational potential of a single disk at any point in space in terms of the coefficients of the polynomial defining the surface distribution of matter, and the potential energy of two disks of arbitrary separation and orientation due to their mutual gravitational attraction. The basic functions, required for obtaining the potential in the plane of the disk and the mutual potential energy of two coplanar disks, have been tabulated. Two overlapping coplanar disks attract just like mass-points at a certain separation,r _c , of their centres. The force of attraction of disks is less than the force of attraction of mass-points having masses equal to the masses of the disks, if the separation of the centres is less thanr _c , and greater if the separation is greater thanr _c . For typical galaxies of equal radiiR,r _c ≈R.