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Migration occurs because of an interchange of angular momentum between disk particles and planets, such that the total angular momentum of the system remains constant. Depending on the relative angular momenta of these components, a planet can either undergo an outward or inward migration. Following a similar approach to Ida et al. (2000b) and Gomes (2004), this process can be analytically modeled with Rutherford Scattering. That is, during an encounter between two particles, the velocity vectors of the two particles instantaneously change. The Z-component of a body's angular momentum is given by   \begin {equation}  $H H  = \sqrt{ a(1-e^2) } \cos i$ i  \end {equation}