1. OVERVIEW Meteoroid mass flux, projectile arrival modeling Ejecta distributions: single particle and full ring layer Ballistic transport and redistribution: the math Structural evolution: model results for inner ring edges Compositional evolution due to MB & BT Motivation and observations The “2-point” model: initial and extrinsic composition Radial profiles under MB & BT compared with data Basics of MB & BT

2. Meteoroid bombardment and Ballistic Transport Main references used here: Morfill et al 1983 Icarus; Ip 1983 Icarus; Durisen 1984; “Planetary Rings” book D89: Durisen et al 1989 Icarus 80, 136: post-bombardment transport theory, mainly illustrated by isotropic ejecta distributions; first full development CD90: Cuzzi & Durisen 1990 Icarus 84, 467: bombardment modeling; the “realistic” ejecta distribution; impacts and spoke formation; mass flux D92: Durisen et al 1992 Icarus 100, 364: refined transport theory and ejecta yields, cleaner development, realistic ejecta distributions avoid high-order cancellations; structure of inner ring edges; exposure age of the rings Goertz & Morfill 1988 Icarus 74, 325; E&M forces and radial structure Durisen 1995 Icarus 115, 66: radial structure as BT instability?; Durisen et al 1996 Icarus 124, 220: torques and mass loading CE98: Cuzzi & Estrada 1998 Icarus 132, 1; evolution of ring composition due to MB; radial compositional profiles due to BT; (C, CD vs. A,B)

3. Meteoroid bombardment and Ballistic Transport Other useful references: Humes (1980) JGR 85, 5841; Pioneer measurements of mass flux in outer SS Landgraf et al (2002) AJ 123, 2857 Kruger et al (2001) P&SS 49, 1303 Sremcevic et al (2003) P&SS 51, 455; (2005) P&SS53, 625 Northrop and Hill; Northrop and Connerney; Connerney and Waite Love & Brownlee (1995) Science 262, 550 Gruen et al (1985) Icarus 62, 244 Colwell 1993 Icarus 106, 536

4. The Mass Flux Morfill & Goertz: 4 E-15 g cm-2 s-1 at Saturn’s ring Ip: E-16 g cm-2 s-1 Unfocussed, 1-sided Cuzzi & Durisen: 5E-17 g cm-2 s-1 Unfocussed, 1 sided Cuzzi & Estrada: 5E-17 g cm-2 s-1 Unfocussed, 1 sided 1 AU: Gruen et al 1985 Rings absorb their own mass in the age of the solar system CD90

5. Incident projectiles: focussing and aberration zero obliquity, no directional deviation by GF, no “shadowing” Assumptions: CD90 Vp=Vp=

6. Incident projectiles: focussing and aberration zero obliquity, no directional deviation by GF, no “shadowing” Assumptions: CD90 Vp=Vp=

7. Incident projectiles: focussing and aberration zero obliquity, no directional deviation by GF, no “shadowing” Assumptions: CD90 Vp=Vp=

8. Incident projectiles: focussing and aberration zero obliquity, no directional deviation by GF, no “shadowing” Assumptions: CD90 Vp=Vp=

9. Impact rates and torques as functions of longitude CD90

10. I, U Single-scattering limit: ejecta don’t produce ejecta V ej The Ejecta Distribution S(…) T(…) CD90 at infinity

11. I, U Single-scattering limit: ejecta don’t produce ejecta V ej The Ejecta Distribution S(…) T(…) CD90 at infinity

12. Single particle “albedo” and “phase function” CD90

13. Ejecta distribution functions vs. ring longitude average CD90 VsVs sun

14. Ballistic transport: Basic concepts (Durisen et al 1984, 1989, 1992) Gross erosion time t g : Throw distance x: Ejecta velocity distributions: yr Ejecta Yield Y: or pure powerlaw Powerlaw with “knee”

15. Ballistic transport: Basic concepts (Durisen et al 1984, 1989, 1992) Gross erosion time t g : Throw distance x: Ejecta velocity distributions: yr or pure powerlaw Powerlaw with “knee”

16. Ballistic transport: Basic equations (D89) Viscous evolution; the “effective” optical depth (D92) ; Where fits to Wisdom & Tremaine “local + non-local” viscosity Dynamical effects dominated by largest particles (~  /3)

17. Ballistic transport: The nitty-gritty (D89) Probability of absorption of ejecta at re-impact location r int Loss integrals Gain integrals ( ; ) ; ; ; Where..

18. Inner edges: ramps, irregular structure Ballistic transport: some results (D92)

19. Inner edges: ramps, irregular structure “knee” at v ej =12m/s “powerlaw”; v ej =4 m/s

20. Ballistic transport and ring structure - summary Inner ring edges (B, A) can be sharpened by realistic ejecta parameters Irregular structure can grow with scale  ~ 4xr ~ 100km for certain combinations of ejecta yield, velocity, & viscosity “Ramps” form inside inner edges; analytical explanation points to broad ejecta angular distribution and powerlaw velocity distribution Numerical results can be scaled to other combinations of incident mass flux, ring viscosity, ejecta yield, and ejecta velocity distribution Refinements such as combinations of rapid, prograde “cratering” ejecta plus slower, retrograde “disruptive” ejecta may be needed to reconcile edges, ramps, and irregular structure (Durisen 1995)

21. Ring compositional evolution (“pollution”) Large incoming mass flux of “primitive” material (30% C, 30% rock) implies rings might absorb their own mass in age of the solar system Only a tiny mass fraction of highly absorbing material can drive the albedo of icy particles to much lower values than observed - young rings? (Doyle et al 1989); ring particles at J, U, N are dark Structural evolution under ballistic transport indicates exposure ages of ~ 10 2 t g ~ 10 7 - 10 8 years; what compositional effects in that time? C ring and Cassini Division particles known to be darker and less red than A and B ring particles since Voyager results (Smith et al 1981, Cooke 1991, Doyle et al 1989, Dones et al 1993) Similarity in structure and composition between (C, CD) and (A,B) Main rings (A,B) are known to be primarily water ice ( > 90%) from microwave and near-IR observations; particle albedos 0.45 - 0.55

22. Estrada & Cuzzi 1996; Estrada et al 2003 B-C boundary

23. Estrada & Cuzzi 1996; Estrada et al 2003 B-C boundary “2 point” Benchmarks

24. Pure “ 2 point” pollution model (no transport) Meteoroid mass flux & absorption probability A (  ) - after aberration - from CD90 and D92; allow for impact destruction of absorber since where ~ 0.08 /

25. Particle regolith radiative transfer model Connects mass fractions of icy and non-icy material to ring particle albedos (as functions of wavelength) Regolith grain albedo from Hapke theory for refractive index n i Overall ring particle albedo from multiple scattering theory Analytical solution in similarity regime (van de Hulst) assuming single scattering; Observed I/F can be inverted from observations to determine ni B and ni C directly! Having both ni B and ni C, can then solve directly for ni o and ni e, ( the initial, intrinsic composition and the composition of the extrinsic pollutant), as functions of wavelength, and compare with candidate materials For nearly transparent material, only imaginary indices really matter

26. Having ni B and ni e, of course, allows us to find f eB and thus T/t g, if we knew  ' and Y o Extrinsic pollution of initially reddish material by fairly neutral material naturally explains how optically thinner, less massive regions become darker and less red than massive adjacent regions.

27. BUT WAIT, THERE’S MORE ! !

28. Radial structure in ring composition by Ballistic Transport Cuzzi & Estrada 1998

30. Model results for radial profiles of ring composition under BT

32. Opacity profile: uncertain! Showalter & Nicholson 1990 Density waves

33. Effects of uncertain opacity (mass-optical depth relationship)

34. Summary (AT LAST!) Meteoroid bombardment and ballistic transport probably important Many important parameters of the process are poorly known: (retention efficiency, ejecta yield, ejecta velocities) * More lab or numerical studies would be important here Improved Cassini observations of opacity, particle size, etc are critical Improved Cassini observations of the mass flux are essential, if we are ever going to convert models into an absolute age date Modeling structural and compositional evolution together may help eliminate uncoupled parameters as unknowns (using shapes of profiles) Current models seem to provide a natural explanation for coupled structural and compositional properties (bright red vs dark neutral) Radial compositional profiles suggest exposure ages ~ few 10 2 t g, comparable to inferences from structural evolution (ramps, edges, etc)