1. Do 100 km asteroids spin- fission into binaries? Keith Holsapple University of Washington Seattle, WA

2. Really: The theories of shape v. spin limits: implications for binary fission What guidance do theories give us on interpretation of observations? What guidance on expectations of code calculations? Binaries may form when spin limits are exceeded

3. Binary Workshop Meudon, FR Mai, 2008 Hierarchy of Theories: 1.Ellipsoidal shapes Fluids Idealized “Rubble piles” “Real” Rubble piles Cohesive bodies 2.Other shapes Fluid Dumb-bells, Fission, Binaries and other multiple bodies

4. Binary Workshop Meudon, FR Mai, 2008 Permissible states must satisfy: 1.Equilibrium throughout 2.Stress-free on surface 3.Any material constraints (strength) 4.And stability

5. Binary Workshop Meudon, FR Mai, 2008 Ellipsoidal: fluid and rubble piles Fluid: no shear stresses, Maclaurin and Jacobi shapes Idealized rubble pile: pressure dependent strength Shear strength pressure Tan(  )

6. Binary Workshop Meudon, FR Mai, 2008  Range of spin limits Rigid, surface spin-off 2.3 hr

7. Binary Workshop Meudon, FR Mai, 2008 Summary: Ellipsoids, fluids and ideal RP Fluids limited in shape RP have much larger, but definite range of spins at any given shape There is no size scale to the problem: same solutions at any scale

8. Binary Workshop Meudon, FR Mai, 2008 The data

9. Binary Workshop Meudon, FR Mai, 2008 So there is a size dependence… why do the observed spins fall off at large sizes??  And why are large slow spinning bodies round??

10. Binary Workshop Meudon, FR Mai, 2008 Why do all >30 km asteroids have a lower limit spins than the smaller ones?? Their strength is lower? Or the 800 pound gorilla reason Say What?? You want me to spin??

11. Binary Workshop Meudon, FR Mai, 2008 Is everything big required to be on the Jacobi curve?? Astronomers routinely ASSUME Jacobi shapes are required, and solve for things like mass density: –2003 EL61 Or infer density profiles –E.g. Ceres 

12. Binary Workshop Meudon, FR Mai, 2008 Why? All materials relax over time??? –So asteroids would, if they lasted long enough, all become the Jacobi ellipsoids? –But they don't last that long?? –Or a certain pressure threshold must be exceeded for relaxation? Or is there another reason? The strength is constant in time, but smaller for large bodies?

13. Binary Workshop Meudon, FR Mai, 2008 Recall that: The spin limit does depend on friction angle  

14. Spin limits versus Friction angle..  Angle of Friction Scaled Limit Spin 35˚, Scaled spin =0.92 10˚, Scaled spin =0.47

15. Binary Workshop Meudon, FR Mai, 2008 Maybe large asteroids have a lower friction angle??

16. Binary Workshop Meudon, FR Mai, 2008 Tensile strength Cohesion ANGLE OF FRICTION Data

17. Binary Workshop Meudon, FR Mai, 2008 The real strength of rocks (not the Mohr-Coulomb idealized model) Pressure Shear Strength The strength envelope 11 22 33 Effective friction angle decreases with size..

18. Binary Workshop Meudon, FR Mai, 2008 So The effective angle of friction does decrease substantially as the body size (and pressure) increased, and that would account for a decreasing spin limit… So does this idea work? Well, sort of, with a little encouragement

19. Binary Workshop Meudon, FR Mai, 2008  =50˚  =43˚  =17˚ The Hoek Brown strength model for "rock masses" 600 km asteroid  =40˚

20. Binary Workshop Meudon, FR Mai, 2008 So if the friction angle changes from about 40˚ to 17˚, then: There could well be a difference approaching 2:1 in the spin limits as the asteroid diameter approaches 600km or larger…

21. Binary Workshop Meudon, FR Mai, 2008 So, (by reaching a little) we could get a limit curve that looks like:

22. Binary Workshop Meudon, FR Mai, 2008 Ellipsoids, Cohesive: can spin a lot faster..

23. Binary Workshop Meudon, FR Mai, 2008 Other shapes Known for fluids (binary stars), remember there is no size scale in the theory..

24. Binary Workshop Meudon, FR Mai, 2008 10 hr Equilibrium Shapes of Fluid bodies

25. Binary Workshop Meudon, FR Mai, 2008 The dumb-bell -> binary branch

26. The pear-shaped sequence (terminates in mass shedding)

27. The binary sequence

28. The hamburger sequence

29. Binary Workshop Meudon, FR Mai, 2008 Code calculations The challenge: can they reproduce the known fission mechanisms for fluid bodies? Should not RP bodies admit those solutions (as well as others)??

30. Binary Workshop Meudon, FR Mai, 2008 More study required.. (If we knew what we were doing, it wouldn't be called research..

31. Binary Workshop Meudon, FR Mai, 2008 Range of limits  =0.8  Range of limits  =1 Range of spin limits

32. Binary Workshop Meudon, FR Mai, 2008