Stochastic Model for Regolith
Growth and Pollution on
Saturn’s Moons and Ring
Particles
LW Esposito and JP Elliott
LASP, University of Colorado
8 October 2007
Overview
• Regolith model
• Stochastic approach and model test
• Results: Regolith depth distribution and
expectation value
• Compare to Quaide and Oberbeck’s
lunar regolith model
• Implications: meteoritic pollution, ring
spectrum, age of Saturn’s rings
Regolith Model
Consider an infinite slab of depth, D
The regolith depth at time t: h(t)
For a moonlet or ring particle, D corresponds to the diameter.
Physical approach
• Meteorites strike surface element
• If the impact penetrates the regolith, it
breaks and excavates new material
• For any impactor size distribution, only
impactors larger than a(h) will penetrate
a regolith of present depth h(t)
• The ejecta are emplaced on the surface
uniformly: every surface element is as
likely to recapture ejecta
Mathematical approach
• Take h(t), regolith depth, as a stochastic
variable
• This is a Markov chain: discrete values of h
are the states of the chain; transitions occur
when a meteorite strikes; transition
probabilities can be calculated from the mass
flux and size distribution
• We do not need to know the exact strike
location, just that the strikes are uniformly
distributed
• D drops out, since the probability of a strike
and the area its ejecta cover both scale as D2
Test case
For an impactor size distribution that is a
power law of index 3, we can solve the
differential equation for h(t), assuming
all material is excavated:
h(t) = Hmax[1 - exp(-t/T0)]
Hmax = H1 amax
T0 =
H max
Ý/ 
FGYm
3
4a
4b
Realistic case for Saturn
• Use Cuzzi and Estrada (1998) impactor
size distribution
• Compare to Quaide and Oberbeck
(1975) lunar regolith model
• Our Markov chain model result gives
depth within a factor of 2 of their values
for 106 < t < 109 years
8b
CE98 impactor size distribution
7a
CE98 impactor size distribution
7b
Implication: short lifetimes
• For CE98 mass flux, T0 = 4 x 104 years
for a 5cm diameter particle!
• Because the regolith shields the particle
as it grows, it can last longer: our result
is 107 years for a 5cm particle
• For a 1m meter particle, the lifetime is
1010 years
This implies young rings?
The fractional pollution of the regolith, fp,
is given by
fp =
FG mÝt / 
h(t)
fp is 0.01 in 107 years, a rough upper limit
from ring observations at microwave

Volume pollution rate
But the volume pollution rate can be much slower,
since we have:
fp(vol) =
Ý
2 fG m
t

For ring particles larger than 1m, the pollution darkens
mostly the outer surface. When such larger objects are
disrupted, new material can cover the ring surfaces, and
the visible pollution will be reset, toward the value

predicted by the volume rate.
Estimating ring age from the
volume pollution rate
For a ring system with surface mass
densi ty, , we have
fp(vol) =
8
2
10 g /cm / year t

So, fp(vol) =0.01 and  = 100g/cm2
gives t=108 years, consistent with CE98

Ring age
• Ring particles of cm diameter should develop
reflectance spectra showing 1% pollution in
107 years: this is a problem even if the rings
formed a few hundred million years ago!
• An alternate conclusion is that the ring mass
is underestimated: then, continuing recycling
could renew their composition, consistent
with upper limits on non-icy material
• A ring must be young for many reasons
• Density wave analysis (Colwell) and
dynamical arguments (Stewart) indicate
larger B ring mass. See their talks!
Conclusions
• A stochastic regolith model shows the
surfaces of ring particles could be 1%
meteoritic after only 107 years
• If rings recycle ejecta, the volume
pollution rate is better: 1% meteoritic in
108 years for standard values of ring
mass and impact flux
• For larger mass or lower flux, the ring
age could be proportionately greater