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Jeremy Emmett edited Analytical Estimates.md
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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
$H = \sqrt( a(1-e^2) ) \cos (I)$
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.
In this particular simulation, the mass of Neptune was 4.5e-5, and the disk particle mass was approximately 3.33e-6 (about 7.4% the mass of Neptune). All bodies were initially placed on circular orbits ranging between a =0 and a = 1, with Neptune at a radius of a=0.5. Therefore, particles which were initially in the vicinity of Neptune had a angular momentum of approximately 0.5*3.33e-6*v_nep, whereas Neptune had an angular momentum of approximately 0.5*4.5e-5*v_nep. These differ only by a factor of the difference in their masses, so all particles which were initially near Neptune had an H smaller than that of Neptune. Therefore, we would expect to have observed an initial inward migration of Neptune. In order for Neptune to reverse this trend, and eventually undergo an outward migration, the angular momentum (and thus the orbital velocity) of the disk particles in the vicinity of Neptune would have to increase substantially. In other words, For particles near Neptune, we would have to see, through some process, a depletion in the number of those with H less than H_nep and an increase in the number of those with H greater than H_nep.
