Capture Probability under the Influence of Adhesion
Nicole Albers
Laboratory for Atmospheric and Space Physics, University of Colorado at Boulder, USA
Abstract
The influence of particle aggregation on particle ensembles in tidal environments such as planetary rings is generally quantified by the Roche limit. A collision model for binary adhesive, viscoelastic
particles yields a size and velocity dependent coefficient of normal restitution as well as a critical impact speed. The latter demarcates restitution from aggregation and assumes a power­law roughly inversely proportional to the square root of the effective mass for particles larger then decimeters in size. Smaller grains show aggregation even for impact speeds of m/s. Accounting for collisional aggregation, three­body orbit simulations were used to obtain the mutual capture probability as a function of particle size as well as orbit location. Contrary to an expected general enhanced likelihood of capture, the possibility of collisional aggregation based on surface adhesion paints a far more complex picture. In accordance with analytical estimates on aggregate stability, mutual capture of two same­sized smaller grains is generally possible as close as the Saturn C ring and clearly a function of actual grain size. The accumulation of regolith layers is possible all throughout the ring system since smaller grains easily stick to larger ones. Rubble pile formation is likely and their destruction and constant rebuilding may be responsible for the transient features observed in the F ring structure.
Model: ● homogeneous, solid spheres of radius R
● impact speed << speed of sound
● viscoelastic deformation ● surface energy of strength 
The equation of motion for the effective normal deformation (compression) reads [1]:
t 
3
1/ 2
3/ 2
3/ 4
meff ̈ AH  ̇H  −  6   H Reff  = 0
2
(1st) inertia (2nd) viscoelasticity (3rd) elasticity (4th) adhesion Given a certain impact speed a vimp = ̇0
collision may result in aggregation or restitution. In ad 0
the first case, a finite compression and a F ad =−3   Reff /2
“sticking” bond remain. Binary Contact Dynamics
The amount of dissipated energy is measured in terms of the coefficient of restitution which completely characterizes the collision. Collisional timescales are about milliseconds. N
v after
= −
v before
restitution:  N  0
aggregation:  N = 0
The normal restitution coefficient as a function of impact speed allows to distinguish aggregation from restitution while being a measure for the dissipated relative kinetic energy (Fig. 1). Adhesion works in favor of small particles and generally reduces the contact's elasticity by about 20%. This impact model is in fair agreement with collision experiments of icy grains at low temperatures in terms of maximum impact speed for possible aggregation and dynamical T = 1
behavior. Tangential friction has not been considered, i.e. .
Figure 1
Capture Probability
Orbit Dynamics of the Three­Body Problem
In order to study the influence of adhesion on the orbit dynamics of grains in the gravity field of a central massive body, the capture probability under the influence of mutual gravity and inelastic collisions is evaluated. In the approximation of the three­body problem, the relative motion of two grains is described the framework of Hill's equations [2]. Monte­Carlo Simulations are performed to solve and evaluate single trajectories and compute the weighted capture probability
Capture Probability =
Figure 2: Outcome of a collision in the three­body problem for the same initial trajectory but varying restitution coefficients. The relative motion of each encounter is quite different. Capture Rate
Collision Rate
If a collision is detected either a constant coefficient of restitution is assumed and the velocities corrected accordingly or the complete binary collision dynamics (as illustrated in Fig. 1) is solved. In case of restitution only the grain's mutual gravity can allow for a mutual capture. However, if the collision results in aggregation the entire relative kinetic energy is dissipated and a “sticking” bond of given strength additionally keeps the grains together. Tidal forces have to outbalance this bond to prevent capture. How important is adhesion?
Figure 4 shows results for both, circular and eccentric, relative orbits. Adhesion clearly enhances capture for small grains or for smaller ones being caught by larger ones. Note, that in the most left pictures gravitational capture is already considered and changes with respect to remaining plots are purely due to considering a contact dynamics as introduced above. Figure 3: A direct comparison between results and simulations performed by Ohtsuki (1993, Fig. 4 and Eq.(3.21)) [3] shows a good agreement of the two numerical studies. Results on simple stability estimates and the presented capture probability promote the existence of regolith layers. Their thickness most probably varies with radial distance to the central body. The elasticity of follow­up collisions decreases with increasing layer thickness. Moreover, this implies a not only material and impact speed but also location dependent restitution coefficient.
Figure 4: Capture Probability for almost circular orbits (upper row) and according to a Rayleigh velocity distribution with parameter (lower row). c
Black squares denote a zero probability, whereas increasing shades of gray indicate the likelihood of capture at given orbit location and particle size. The most left plots are obtained for a constant restitution coefficient, while for all other the equation of motion for binary contacts is solved and if needed an established “sticking” bond taken into account. Color scales are the same in each row. Here, distances of the Saturnian ring system are exemplified and cover a range from D to F ring. Conclusions
●
Aggregation can be distinguished from restitution in terms of the coefficient of restitution . In the presence of adhesion collisions are N
generally less elastic than in a pure viscoelastic encounter. The contact model provides an essential tool to cover the collisional dynamics of granular particles in a more general context of coexisting aggregation and restitution. Adhesion crucially shapes the mutual capture of grains. Small grains are very likely to stick to each other or be swept by larger ones.
● Capture of small grains may enhance density variations in the rings, thereby increasing contrast and provinding an essential “glue” to stabilize accumulated temporary structures. ● Regolith layers are expected to be common. Their thickness and elasticity depend on the relative distance to the central body. References
●
[1] Albers, N. and Spahn, F. (2006), Icarus, 181, 292­301
[2] Nakazawa, K and Ida, S. (1988), Progress of Theoretical Physics Supplement, 96, 167­174
[3] Ohtsuki, K. (1993), Icarus, 106, 228­246
[4] Albers, N. (2006), PhD Thesis, University of Potsdam, Germany
This work has partially been done during my graduate studies at the University of Potsdam, Germany.