To obtain <P(e, i)>, we must numerically compute a number of orbits of planetesimals with various values of b, τ, and ω for each set of (e, i) and then examine whether they collide with the protoplanet or not.
In this section, we consider the ranges of b, τ, and ω to be assigned in orbital calculations, and give initial conditions for orbital integration.
From the above two characteristics, it follows that it is sufficient to examine only cases where 0≦ω≦π and b≧0.
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2. Initial conditions for orbital integration
To obtain <P(e, i)>, we must numerically compute a number of orbits of planetesimals with various values of b, τ, and ω for each set of (e, i) and then examine whether they collide with the protoplanet or not. In this section, we consider the ranges of b, τ, and ω to be assigned in orbital calculations, and give initial conditions for orbital integration.
One can see in Eq. (6) that Hill’s equations are invariant under the transformation of z→-z and that of x→-x and y→-y; on the other hand, a solution to Hill’s equations is described by Eq. (7). From the above two characteristics, it follows that it is sufficient to examine only cases where 0≦ω≦π and b≧0. Furthermore, we are not interested in orbits with a very small b or a very large b; an orbit with a very small b bends greatly and returns backward like a horse shoe ( Petit and Hénon, 1986; Nishida, 1983 ), while that with a very large b passes by without any appreciable change in its orbital element.
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