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 Incoming links: Dust halo

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 316 / 2 / vi / Solution Created 2026-10-03 Updated 2026-10-05

Assume negligible fragment release speed relative to a circular parent planetesimal. Its initial speed is

v_{b} = G M_{⋆} / r_{b}

, but radiation pressure reduces the grain's gravitational parameter to

(1 - β) G M_{⋆}

. Its specific orbital energy and specific angular momentum are

ε_{0} = \frac{G M _{⋆}}{r _{b}} (β - \frac{1}{2}), h_{0}^{2} = G M_{⋆} r_{b} .

For

0 \leq β < 1/2

, the resulting dust orbit released from a circular parent ring has

a_{0} = \frac{( 1 - β ) r _{b}}{1 - 2 β}, e_{0} = \frac{β}{1 - β}, q_{0} = r_{b}, Q_{0} = \frac{r _{b}}{1 - 2 β} .

Release is at periapsis: the inherited tangential speed exceeds the new circular speed. Low-

β

grains remain near the dust birth ring; higher bound

β

gives increasingly eccentric orbits extending well outside it. Uniform release longitude produces an approximately axisymmetric dust halo, with long residence times near apoapsis. At

β = 1/2

the conservative release orbit is parabolic; above this radiation-pressure blowout threshold it is initially unbound. For

β \geq 1

, zero or repulsive effective gravity also prevents a bound ellipse.

Figure 1.
Dust release orbits and inspiral time under Poynting–Robertson drag
. Left: representative orbits released at one point on the birth ring, including the parabolic release limit. Right: the formal averaged lifetime of bound grains as a function of the radiation-pressure coefficient.

The left panel shows one release longitude; rotating it around the ring fills the halo. At longer times Poynting–Robertson drag also carries bound grains inward, so an initially outward-spanning release distribution does not imply an empty inner region in a continuously supplied, collision-free population.

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