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
(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 ), spin axis orientation
(obliquity varied between
and
in steps of ),
and orientation of the asteroid's long axis at perigee (tested over many
angles between
and ). We discuss the outcomes,
especially those pertaining to Geographos, below.