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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   $H = \sqrt( \sqrt{  a(1-e^2) ) }  \cos (I)$ (i)$  where a = semi-major axis, e = eccentricity, and i = inclination. If the planet is placed on a circular orbit, then particles with an H larger than that of the planet will drive an outward migration of the planet. This is because, when crossing near the planet, their tangential velocities are larger than the circular velocity of the planet, thus 'kicking' it forwards into a higher orbit. Particles with an H smaller than that of the planet, however, will drive an inward migration of the planet. This is because, when crossing near the planet, their tangential velocities are smaller than the circular velocity of the planet, thus 'dragging' it backwards into a lower orbit.  Though the disk has some 'thickness' due to the fact that particles were initially given non-zero z-components in their positions, and therefore begin on trajectories that are inclined, we will assume a simple model in which all particles follow perfectly circular orbits in the plane of the system ($e,i = 0$)  In this case, all angular momentum is in the z-component, with a magnitude of a*m*v for each particle.