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 Incoming links: Inclined comet collision time

Suppose the inclined population samples its arguments of periapsis and nodal orientations, or that sufficiently slow precession supplies this sampling. Near the planet the vertical amplitude is

H ≃ a_{p} I

. For uniformly distributed vertical phase,

z = H sin ψ

[π H^{2} - z^{2}]^{- 1}

, so the midplane volume probability is

Σ_{c} / (π H)

In the genuinely three-dimensional regime

R_{p} ≪ H ≪ a_{p}

π R_{p}^{2}

. The relative velocity differs from the coplanar value only at order

I^{2}

for the specified high-speed encounters. Therefore the inclined comet collision time is

t_{col, I} ≃ \frac{H}{R _{p}^{2} Σ _{c} ( a _{p} ) v _{rel}} ≃ \frac{π ^{2} I Q ^{3/2} a _{p}^{2} 1 - q / a _{p}}{R _{p}^{2} G M _{⋆} 3 - 2 2 q / a _{p}} .

At fixed

ρ_{p}, M_{⋆}, I

the leading scaling is

Q^{3/2} M_{p}^{- 2/3} a_{p}^{2}

, with no leading

q

dependence when

q ≪ a_{p}

. The displayed factor retains the first dependence on

q / a_{p}

. Also

t_{col, I} / t_{col} ≃ 2 a_{p} I / R_{p}

in this averaging convention.

Inclination alone does not specify a collision time for one fixed orbit. If its orbital nodes miss the planet's orbit, a tilted Kepler orbit can have no collisions at all. The finite result is an orientation-averaged population result, or requires precession or scattering to sample those orientations. When

a_{p} I ≲ R_{p}

, the three-dimensional estimate crosses over to the coplanar result instead of tending to zero.