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For a population with randomized vertical phases, orbital inclination

I

, and

R_{p} ≪ a_{p} I

, the effective volume probability near the planetary plane is proportional to

Σ_{c} / (a_{p} I)

. The collision time then scales as

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

, at fixed stellar mass and planetary mass density. An individual inclined Kepler orbit whose orbital nodes do not intersect the planet's orbit need not collide at all.

Orbital plane Created 2026-10-06 Updated 2026-10-14

The plane containing a central-force Kepler orbit, perpendicular to its conserved specific angular momentum. Its orientation can be specified by an orbital inclination and an orbital node. Perturbing forces can change that orientation.

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.

An encounter can occur only near an orbital node of the comet's inclined orbit. Its tangential velocity projected into the planet's plane is reduced by

cos I_{c}

, so the relative-speed calculation becomes

Δ v_{p c}^{2} = v_{p}^{2} [3 - \frac{a _{p}}{a _{c}} - 2 \frac{a _{c}}{a _{p}} (1 - e_{c}^{2}) cos I_{c}] .

The missing projected component is a vertical relative velocity, so the moon encounter is three-dimensional:

b

becomes a vector in the impact plane, and a collision also requires the trajectory to pass close to the moon's orbital plane. Once the total asymptotic speed is used, the two-body gravitational focusing formulae in parts (v) and (vi) retain their form, but the geometrical collision probability is lower.