ECCENTRIC DISK MODEL Following the approach of Hughes, Factor, Chiang et al., we model an eccentric circumstellar disk as a series of apse-aligned ellipses. We model the mass distribution within each ellipse using a mass-on-a-wire approach. In this document, we derive the mass distribution for a single ellipse as a function of orbital position. To start, we assume that the ith ellipse we have chosen to model has a total mass of mi. The linear density of this wire, λi, with units [g/cm], is proportional to the amount of time spent in that section of the orbit. This means that the density will be largest at apoapse, when a test particle in the orbit is traveling the slowest. Conversely, the density will be smallest at periapse, when the test particle is moving fastest. The linear density is uniquely defined as a function of orbital position. We can either measure the orbital position in terms of path length along the perimeter of the ellipse, s, where s = 0 denotes periapse, or, we can measure orbital position as a function of true anomaly[1] θ, which is the same as the angle f in Meredith’s notes. Once we have defined λi, we can check that we have the right definition by integrating the linear density over the perimeter of the elliptical orbit–we should recover mi. [1] http://spiff.rit.edu/classes/phys440/lectures/ellipse/true_anomaly.gif