The model

To investigate the effects of planetary tides on ECOs (cf., [Solem and Hills (1996)]), we have used a sophisticated N-body code to model Earth flybys of spherical-particle aggregates ([Bottke et al., 1997,Bottke et al., 1998,Richardson et al., 1998]). Our goal in this section is to determine whether Geographos-like shapes are common by-products of tidal disruption. Model details, analysis techniques, and general results described in Richardson et al. (1998). For brevity, we only review the basics here.

The particles' motions are tracked during the encounter using a 4th-order integration scheme that features individual particle timesteps (Aarseth 1985). This method allows us to treat interparticle collisions rigorously, with a coefficient of restitution included to produce energy loss (i.e., friction); previous models usually assumed elastic or perfectly inelastic collisions. Note that if energy dissipation is not included, clumps formed by gravitational instability are noticeably less tightly bound ([Asphaug and Benz 1996]).

The code is capable of modelling tidal disruption over a range of rubble pile shapes, spin rates, spin-axis orientations, and hyperbolic trajectories. To verify the code was accurate enough to realistically model shape changes, we consulted two experts in granular media, J. Jenkins of Cornell University, and C. Thornton of Aston University, UK. Based on their suggestions, we checked our code against some standard diagnostic tests in their field. For our first test, we numerically modeled spherical particles being dropped into a pile along a flat surface. Our results showed that we were able to reproduce an empirically-derived angle of repose. For a second test, we examined the pre- and post-planetary encounter particle configurations of our rubble piles to determine whether their shapes were artifacts of a crystalline lattice structure (i.e., ``cannonball stacking''). Our results showed that lattice effects are nearly unavoidable in rubble pile interiors, especially when same-sized spherical particles are used, but that the outer surfaces of our rubble piles had essentially randomized particle distributions. Thus, based on our success with these tests and the positive comments of the granular media experts, we have some confidence that our N-body code yields reasonable results.

Our model rubble piles had dimensions of 2.8 x 1.7 x 1.5 km, our choice for a representative ECO shape (Richardson et al. 1998), and bulk densities of 2 g cm^-3, similar to the estimated densities for Phobos and Deimos ([Thomas et al., 1992]). Note that this value may be overly-conservative, given the 1.3 g cm^-3 density found for Mathilde. Individual particles have densities of 3.6 g cm^-3, similar to ordinary chondritic meteorites ([Wasson 1995]). For most test cases, our rubble pile consisted of 247 particles, with each particle having a diameter of 255 m. Same-sized particles were chosen for simplicity; future work will investigate more plausible particle size-distributions. Cases deemed interesting were examined further using rubble piles with 491 same-sized particles. In these instances, particle densities were modified to keep the aggregate's bulk density the same as before. We found that the change in resolution did not significantly modify the degree of mass shedding, the final shape, or the final spin rate of the model asteroid, though it did make some shape features more distinctive.

The tidal effects experienced during a rubble pile's close approach to Earth are determined by the rubble pile's trajectory, rotation, and physical properties. To investigate such a large parameter space, Richardson et al. (1998) systematically mapped their outcomes according to the asteroid's perigee distance q (between 1.01 and 5.0 Earth radii), approach speed $v_\infty$ (between 1.0 and 32 km s^-1), rotation period P (tested at P = 4, 6, 8, 10, and 12 h for prograde rotation, P = 6 and 12 h for retrograde rotation, and the no-spin case $P= \infty$), spin axis orientation (obliquity varied between $0^\circ$ and $180^\circ$ in steps of $30^\circ$), and orientation of the asteroid's long axis at perigee (tested over many angles between $0^\circ$ and $360^\circ$). We discuss the outcomes, especially those pertaining to Geographos, below.