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

Abstract

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 ¼ 1/(1 þ ), which provides a good model for many astrophysical systems, including elliptical galaxies (with an R 1/4 law), dark matter halos (where N-body simulations indicate a nearly universal density profile), and young embedded star clusters (with gas density $ À1 ). 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.

Early Evolution of Stellar Groups and Clusters: Environmental Effects on Forming Planetary Systems

“…Here we can model the orbits by assuming that the total cluster potential has the form of the Hernquist profile; orbits in this general class of extended mass distributions have a similar form (for further details, see Adams & Bloch [2005], hereafter AB05). The mean flux can then be written in the form…”

Embedded star clusters and the formation of the Oort Cloud

“…The eccentricity, although in principle undefined since the orbit is not Keplerian, is computed as e = (Q − q )/(Q + q ). The latter is basically a means to describe how radial the orbit is, but is not uniquely defined (Adams and Bloch, 2005). The inclination is obtained from the angular momentum vector.…”

UV Radiation Fields Produced by Young Embedded Star Clusters

“…If we define M T to be the total mass enclosed within the scale radius r s , which is taken here to be the cluster radius R c (N ), then É 0 ¼ 4GM T /r s . Following previous treatments (Adams & Bloch 2005; APFM06), we define dimensionless energy and angular momentum variables where E and J are the ( physical) specific energy and specific angular momentum of the orbit. As shown in APFM06, the radiation flux averaged over an orbit is then given by the expression…”