Surface of Sections Applet
Involved Physics

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To calculate orbits in the applet, we first assume some analytic gravitational potential for the galaxy. The potential used is the general logarithmic potential:


In this potential, rc represents a "core radius" and q represents the flattening of the potential on the y axis. For example, rc = 0 gives the singular logarithmic potential, and q = 1 gives a round potential. Smaller values of q give more flattened potentials. In models where the central black hole is turned on, there is an additional acceleration provided by a black hole of mass M = 0.01 and gravitational softening epsilon = 0.01. We calculate the gravitational acceleration from:

We integrate orbits using a 4th order Runge-Kutta integrator. Because the RK timestep gets smaller when the potential gradient gets larger, it takes more calculation time to follow the orbit when the star is close to the center of the galaxy (particularly in the singular model, or a model with a black hole). Because of this increase in computing time, the star appears to slow down when it passes through the center. Realize that this is only due to the longer computing time; in reality, the star actually speeds up when it passes near the center, since the accelerations are large.