In a previous column, we obtained numerical solutions of the relativistic motion of a particle orbiting a spherical, nonrotating central mass. In this supplement, we consider the more general case of a rotating central mass. This case is important because it is likely that the gravitational collapse that produces a black hole would yield one with nonzero rotational angular momentum ([2]).

As before, we assume that the central mass (the black hole) is at the origin of a rectangular perifocal coordinate system ([1]). We also assume that the axis of rotation of the central mass is oriented along the z-axis (see Fig. 1 of Ref. 1). Given a test particle's initial position and velocity and a time of flight, our goal is to find its orbit. If the central mass does not rotate, the gravitational field is spherically symmetrical, the test particle's motion is in the orbital plane, and the perifocal frame is convenient because the value of z can be assumed to be zero. In contrast, if the central mass does rotate, its gravitational field is asymmetrical, and the test particle does not remain in the orbital plane. However, we shall see that the perifocal frame is still convenient.

1. S. C. Bell, "A numerical solution of the relativistic Kepler problem," Computers in Physics 9, 281 (1995).
2. R. M. Wald, General Relativity, University of Chicago Press (1984).
3. D. F. Lawden, An Introduction to Tensor Calculus, Relativity and Cosmology, third edition, John Wiley & Sons, New York (1982).
4. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, second edition, Cambridge University Press (1992).
5. S. C. Bell, P. P. Rao, and M. A. Ginsburg, ``Monte Carlo Analysis of the Titan/Transfer Orbit Stage Guidance System for a Planetary Mission,'' Journal of Guidance, Control, and Dynamics 18, 121 (1995).
6. S. Chandrasekhar, The Mathematical Theory of Black Holes, Oxford University Press, New York (1983).