1. 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

2. 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

3. 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. Regolith Model

4. 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

5. 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 D 2

6. 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) = H max [1 - exp(-t/T 0 )] H max = H 1 a max T 0 =

7. 3

8. 4a

9. 4b

10. 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 10 6 < t < 10 9 years

11. 8b

12. 7a CE98 impactor size distribution

13. 7b CE98 impactor size distribution

14. Implication: short lifetimes For CE98 mass flux, T 0 = 4 x 10 4 years for a 5cm diameter particle! Because the regolith shields the particle as it grows, it can last longer: our result is 10 7 years for a 5cm particle For a 1m meter particle, the lifetime is 10 10 years

15. This implies young rings? The fractional pollution of the regolith, f p, is given by f p = f p is 0.01 in 10 7 years, a rough upper limit from ring observations at microwave

16. f p (vol) = Volume pollution rate But the volume pollution rate can be much slower, since we have: 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.

17. Estimating ring age from the volume pollution rate For a ring system with surface mass density, , we have f p (vol) = So, f p (vol) =0.01 and  = 100g/cm 2 gives t=10 8 years, consistent with CE98

18. Ring age Ring particles of cm diameter should develop reflectance spectra showing 1% pollution in 10 7 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!

19. Conclusions A stochastic regolith model shows the surfaces of ring particles could be 1% meteoritic after only 10 7 years If rings recycle ejecta, the volume pollution rate is better: 1% meteoritic in 10 8 years for standard values of ring mass and impact flux For larger mass or lower flux, the ring age could be proportionately greater