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

This paper explores orbits in extended mass distributions and develops an analytic approximation scheme based on epicycloids (spirograph patterns). We focus on the Hernquist potential ψ = 1/(1 + ξ), which provides a good model for many astrophysical systems, including elliptical galaxies (with an R1/4 law), dark matter halos (where N-body simulations indicate a nearly universal density profile), and young embedded star clusters (with gas density ρ ~ ξ-1). For a given potential, one can readily calculate orbital solutions as a function of energy and angular momentum using numerical methods. In contrast, this paper presents a number of analytic results for the Hernquist potential and proves a series of general constraints showing that orbits have similar properties for any extended mass distribution (including, e.g., the NFW profile). We discuss circular orbits, radial orbits, zero-energy orbits, different definitions of eccentricity, analogs of Kepler's law, the definition of orbital elements, and the relation of these orbits to spirograph patterns (epicycloids). Over a large portion of parameter space, the orbits can be adequately described (with accuracy better than 10%) using the parametric equations of epicycloids, thereby providing an analytic description of the orbits. As an application of this formal development, we find a solution for the orbit of the Large Magellanic Cloud in the potential of our Galaxy.