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4. Numerical procedures for obtaining <P(e, i)>
Based on the results in Sect. 3, we develop numerical procedures for efficiently obtaining <P(e, i)>. By using the framework of Hill’s equations, one can reduce the degrees of freedom of particle motion, as described in Paper I. Nevertheless, they are still too numerous to compute orbits densely and uniformly over the five-dimensional phase space (e, i, b, τ_s, ω_s) of initial conditions. Furthermore, we frequently find orbits which comes very close to the protoplanet or which revolve complicatedly many times around the protoplanet. In these cases, an enormously long computation time is needed to calculate them.
To avoid these difficulties, we introduce the following three simplifications:
(i) We neglect n-recurrent orbits in evaluating <P(e, i)> when n≧3; that is, we stop an orbital calculation if the particle does not collide with the protoplanet even after two close encounters.
(ii) When the planetesimal crosses the sphere surface of the two-body approximation, the two-body formula determines whether or not a collision occurs.
(iii)We omit orbital computations in some regions of τ_s, which depend on a given set of e, i, and b; it is found empirically that there are no collision orbits in these regions.
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