On the Calculation of the Gravitational Force of Axi-symmetric Infinitely Thin Disks

摘要

We describe an efficient method for the numerical calculation of the gravitational force of axi-symmetric infinitely thin disks. The evolution of a self-gravitating astrophysical disk is affected by the force acting in the plane of the disk. Because of the long evolution times, the calculation of the dynamics of such a disk requires high numerical accuracy of the intermediate computations. Numerical codes which correctly satisfy the conservation of mass and angular momentum are used (see, e.g., Norman et al.). The gravitational force enters the dynamical equations via the source terms and must be calculated with sufficient precision. We present a computational scheme which meets the requirements of numerical accuracy, computational efficiency, and vectoriza-tion. The most crucial point is the regularization of the singularity of the integral kernel. The separate treatment of the singular point commonly used is avoided. Our numerical scheme is based on piecewise approximations of the surface density by parabolas. The time consuming computation of the elliptical integrals is omitted by a matrix representation of the integral operator. The problem of the singularities of the integrands is removed by a particular representation of the approximation parabolas. Since astrophysical disks are often not isolated, but surrounded by exterior masses, some exterior surface density distributions are included.