Orbits in Extended Mass Distributions: General Results and the Spirographic Approximation

This paper explores orbits in extended mass distributions and develops ananalytic approximation scheme based on epicycloids (spirograph patterns). Wefocus on the Hernquist potential which provides a good model for manyastrophysical systems, including elliptical galaxies, dark matter halos, andyoung embedded star clusters. For a given potential, one can readily calculateorbital solutions as a function of energy and angular momentum using numericalmethods. In contrast, this paper presents a number of analytic results for theHernquist potential and proves a series of general constraints showing thatorbits have similar properties for any extended mass distribution (including,e.g., the NFW profile). We discuss circular orbits, radial orbits, zero energyorbits, different definitions of eccentricity, analogs of Kepler's law, thedefinition of orbital elements, and the relation of these orbits to spirographpatterns (epicycloids). Over much of parameter space the orbits can beadequately described (with accuracy better than 10%) using the parametricequations of epicycloids, thereby providing an analytic description of theorbits. As an application of this formal development, we find a solution forthe orbit of the Large Magellanic Cloud in the potential of our Galaxy.Comment: 40 pages including 9 figures; accepted to The Astrophysical Journa