A Globally Shaken and Fractured Asteroid

with Brazilian Nut Effect

G.Tancredi, S.Bruzzone, S.Roland, S. Favre, A. Maciel

Depto. Astronomía, Fac. CienciasObservatorio Astronómico Los Molinos

URUGUAY

Gaspra (Galileo - ’91)

En color verdadero y color resaltando las variaciones de albedo superficial

Eros (NEAR - ’00)

Eros (NEAR - ’00)

MUSES – C / Hayabusa (eagle)

Itokawa

Asteroid structures

Solid with surface craters

Solid with big cracks

Rubble-piles covered by dust

Agglomerate of boulders

“Standard” Model for the formation of Itokawa

(Fujiwara et al. 2006)

A little bit of impact seismology

Richardson et al. (2005)

A little bit of dynamics …NEA and Mars crosserVery low inclination (i=1.6°)

Yoshikawa (2007)

Origin of Itokawa

A little bit of dynamics + collisions …

Collision probability Arbitrary

unitsCollision probability

Computed with a method similar to “Cube” (Liou 2006) but developed independently

Consequences of collisions:after a few equations …P

roje

ctil

e

Based on Richardson et al.

(2005), with corrections by

R. Gil-Hutton

Itokawa

GLOBAL SHAKING

Assuming a cumulative size distribution of projectiles with exp. -2.6:There would be 107 global shaking collisions before a catastrophic disruptive one.

Mean time between collisions:

Shaking (~cm projectile) Catastrophic (~10m proj.)At present: 80 yr 800 MyrDuring the Inner MB stage: 1 yr 10 Myr

The non-catastrophic collisions would fracture the asteroid and they would produce a frequent shaking of the entire body.

Brazilian Nut effect

Well known phenomenon in the physics of granular media and rock mechanics.

Kudrolli (2004)

Brazilian nut effect in asteroidsIt was proposed by Ausphaug et al. (2001) to explain the size-sorting of

Eros’ regolith; motions driven by surface gravitational slopes.

Global shaking could produce a Brazilian Nut effect in Itokawa’s entire body.

¿How do we calculate the total potential of an irregular body?

V

Gravitational Potential

Total Potential CentrifugGravTotal

Centrifugal Rotacional Potencial

Computation: Monte Carlo integration over Gaskell faceted shape model

What do we observe?

Regions of LOW POTENTIAL – SMALL BOULDERS

Regions of HIGH POTENTIAL – LARGE BOULDERS

ALETERNATIVE SCENARIO

Due to the past collisional evolution, the global shaking induced by impacts is the main reshaping and boulders relocation mechanism in Itokawa.

The size distribution should then be correlated with the total potential.

Where do we count boulders?

Counting boulders (1) The minimum size for detectable boulders was of apox. 5 px ~ 10 cm (in close-up images).

Head Bottom Body highlands

Counting boulders (2)Number of boulders counted per image

# 128 1688 47 482 434

# 192 / 2182 7 120

Transition zone Muses Sea

Over 5000 boulders were visually counted !!!

Thanks Santiago and Sebastián !!

Cumulative

Cumulative

Correlation between potential and size distribution

RegionPotential

(10-2 m2/s2) Slope

Head -1.08 -3.84

Bottom -1.23 -3.76

Body highlands -1.33 -3.73

Transition zone 1 -1.41 -3.99

Transition zone 2 -1.43 -4.45

Muses Sea -1.52 -5.62

Conclusions• The high frequency of collision regime experienced

by Itokawa could induce global shaking associated phenomena, like the Brazilian nut effect.

Thanks to Dr. M. Yoshikawa for introducing us the Hayabusa archive.

Potential high low

Size of boulders

large small

Slope of the distribution shallow steep

A numerical simulation is needed to study this effect.

Numerical methods applied to the study of granular media

• Asteroids are modelled as agglomerate of small pieces:

rubble-piles

• Three methods can be considered:

• Smoothed Particle Hydrodynamics (SPH)

• Interacting ellipsoids

• Granular Physics

Smoothed Particle Hydrodynamics(SPH)

Computational method to model a flux of particles. The evolution of each particle is given from the properties of the near by particles. The properties of the particles are smoothed with a characteristic “smoothing length”.

SPH simulation of the formation of asteroids families (Michel et al. 2002)

Interacting ellipsoids(Roig et al. 2003)

• Rigidi ellipsoids interacting with an artificial repulsive potential and disipative function

Granular Physics Granular materials reveal very different behavior under different

circumstances:

Excited (fluidized) granular material often resembles a liquid. It flows through pipes.

In other situations it behaves more like a solid (a heap of sand)

The numerical simulation of the evolution granular materials has been done recently with the Discrete Element Method (DEM).

DEM is a family of numerical methods for computing the motion of a large number of particles like molecules or grains under given physical laws.

Molecular Dynamics (MD), is a particular case of a DEM, when the particles are spherical molecules.

Interaction scheme

Normal direction Tangential direction

Linear Dashpot Force

Alternative force models: Viscoelastic Spheres

Main Limitations (I)Computer efficiency

“Bottle neck” of the method: Computation of the interacting forces for each particle at each timestep.

Simple approach: N(N-1)/2 operations per time step. But if the particles are equal spheres, each one interact

with at most 6. The number of operations should be 3N

Efficient methods to reduce the number of pairs to compute

Verlet Lists Method Link Cells Algorithm Lattice algorithm

Main Limitations (II)Timestep

The timestep should be:

t << duration of collisions

(typically 1/10-1/20 of duration of collisions)

Based on the Hertzian elastic contact theory the duration time of contact is v-1/5

Typically ~ 10-5 sec t~10-6 sec !!

Some experimentsfrom Computational Granular Dynamics

(Pöschel and Schwager 2005)

First experiment using DEM in impact process(Wada et al. 2006)

Vertical impacts on a target made of a large number of spherical particles under Earth gravity.

Our experiments Code from Computational Granular Dynamics

(Pöschel and Schwager 2005)

2D shaking box with 1800 small spherical particles of radius 4mm and one large particle of 25mm.

Floor and walls of spherical particles of radius 5mm

The elastic and physical parameters of the particles were chosen of typical rocks, and some of them are fixed by simulating a particle bouncing on the floor.

Simulating a seism The floor and the walls are vertically displaced

according to a predefined function of the form:

The process is repeated after a fixed number of seconds, depending on the settling time given by the gravity regime.

The simulations typically takes 1 day of CPU time for 100sec in PC 2.4GHz 8Gb RAM.

Earthg=9.8 m/s2

Max amp=3cmMoving floor

Beating Freq. = 30 HzTime between seism = 10s

Earthg=9.8 m/s2

Max amp=3cmMoving floor

Beating Freq. = 30 HzTime between seism = 10s

Erosg=6.1x10-3 m/s2

Max amp=3mmShaking box

Beating Freq. = 14 HzTime between seism = 30s

Erosg=6.1x10-3 m/s2

Max amp=3mmShaking box

Beating Freq. = 14 HzTime between seism = 30s

Itokawa g=9.1x10-5 m/s2

Max amp=0.5mmShaking box

Beating Freq. = 14 HzTime between seism = 60s

Itokawa g=9.1x10-5 m/s2

Max amp=0.5mmShaking box

Beating Freq. = 14 HzTime between seism = 60s

Preliminary conclusions of the DEM experiments

The Brazil nut effect does work in low-gravity regimes

Further work is needed to simulate the evolution of a rubble-pile due to frequent impact seismic events.