Matching the shape and spin of 1620 Geographos

Now that we have found tidal disruption outcomes which match Geographos's ellipticity and spin rate, we can take a closer look at the resultant shapes of the rubble piles themselves. Our goal is to find distinctive features which match comparable features on Geographos, and which are possibly antithetical with a collisional origin. To make sure we can resolve these features, we have used a rubble pile containing nearly twice the number of components as before (491 particles). Fig. 4 shows this body going through a M-class event with the following encounter parameters: P = 6 hours prograde, $q = 2.1 R_\oplus$, and $v_\infty = 8$ km s^-1.

fig4

Fig. 4a shows the asteroid before encounter. The spin vector is normal to the orbital plane and points directly out of the page. The asteroid's equipotential surface (to which a liquid would conform) is a function of local gravity, tidal, and centrifugal terms. At this stage, it hugs the outer surface of the rubble-pile.

Fig. 4b shows the body shortly after perigee passage. Here, the equipotential surface becomes a more elongated ellipsoid with its longest axis oriented towards Earth. Differential tidal forces, greatest at perigee, and centrifugal effects combine to set the particles into relative motion, producing a landslide towards the ends of the body. Particles above the new angle of repose roll or slide downslope to fill the ``low spots'', and thereby further modify the body's potential. As a consequence, the rubble pile is elongated and, as the planet pulls on the body, its rotation rate altered. The action of the Earth stretches the model asteroid and, by pulling on the distorted mass, spins it up, increasing its total angular momentum. Mass ejection occurs when the total force on a particle near the asteroid's tips is insufficient to provide the centrifugal acceleration needed to maintain rigid-body rotation.

Fig. 4c shows the latter stages of the landslide. Particles near the tips are swept backward in the equatorial plane by the asteroid's rotation. The material left behind frequently preserves this spiral signature as cusps pointing away from the rotation direction. Note that these cusps are easy to create but difficult to retain with identical spherical particles at this resolution; we believe that real rubble piles, with rough or craggy components, would more readily ``freeze'' in position near the ends. Particle movement along the long axis is not uniform; shape changes, increased angular momentum, and mass shedding cause one side of the body to become bow-like. This effect produces a convex surface along the long axis and a ``hump''-like mound of material on the opposite side.

Fig. 4d shows the final shape of the object. The spin ( $P_{\rm rem}=5.03$ h) and ellipticity ( $\varepsilon_{{\rm rem}} = 0.65$) are virtually identical to Geographos (Fig. 1). The shapes of the two ends are, surprisingly, not symmetric. We believe this is caused by the starting topography, which can play a decisive role in the effectiveness of tidal deformation. The strength of tidal and centrifugal terms depends on each particle's position ([Hamilton and Burns 1996]), such that some particles lie further above the local angle of repose than others. Since our model asteroid, like real ECOs, is neither a perfect ellipsoid nor a readily-adaptable viscous fluid, the new distorted shape is influenced by the body's granular nature (i.e., friction and component size affect the strength of the landslide). Hence, particles leak more readily off one end than the other, often accentuated by limited particle movement before the rubble-pile reaches perigee. The end that sheds more mass frequently becomes elongated, tapered, and narrow when compared to the stubbier antipode. The overall final shape of the body is much like that of a ``porpoise'' or ``schmoo''. A comparison between Fig. 1 and Fig. 4 shows a good match; all of Geographos's main features have been reproduced.