Reshaping rubble piles with planetary tidal forces

To quantify shape changes, we measured the length of each rubble pile's axes after encounter ( $a_1 \ge a_2 \ge a_3$), calculated the axis ratios ( $q_2 \equiv a_2 / a_1$ and $q_3 \equiv a_3 / a_1$), and defined a single-value measure of the remnant's ``ellipticity'' ( $\varepsilon_{{\rm rem}} \equiv 1 - \frac{1}{2} (q_2 + q_3)$). For reference, our progenitor has $\varepsilon _{{\rm rem}} = 0.43$ and Geographos has a value of $\varepsilon _{\rm rem} = 0.64$.

Sampling a broad set of parameters to map tidal disruption outcomes, Richardson et al. (1998) identified 195 S, B, or M-class events produced with a $\varepsilon _{{\rm rem}} = 0.43$ rubble pile. Fig. 2 shows this set with ellipticity plotted against the fraction of mass shed by the progenitor during tidal disruption.

fig2

We find that, in general, S-class events tend to yield lower ellipticity values; only 2 of the 79 outcomes are likely to have a Geographos-like elongations ( $\varepsilon_{{\rm rem}} > 0.60$). The mean value of $\varepsilon_{{\rm rem}}$ for the S-class events is 0.22 with standard deviation $\sigma$ = 0.14. The near-spherical shapes produced by S-class events are a by-product of gravitational instabilities in the fragment chain which readily agglomerate scattered particles as they recede from the planet.

B-class events do not show a simple trend with respect to ellipticity, though these values tend to increase as the degree of mass shedding decreases. We find that 5 of 40 outcomes have Geographos-like $\varepsilon_{{\rm rem}}$ values. The mean value of $\varepsilon_{{\rm rem}}$ for all 40 B-class events is 0.45 ($\sigma = 0.14$), very close to the starting ellipticity of 0.43.

M-class events are most effective at increasing $\varepsilon_{{\rm rem}}$ and creating Geographos-like shapes, probably because tidal torques must first stretch and/or spin-up the rubble pile before particles or clumps can be ejected near the ends of the body. Fig. 2 shows 23 of 76 M-class events with Geographos-like $\varepsilon_{{\rm rem}}$ values. Overall, the 76 outcomes have a mean $\varepsilon_{{\rm rem}}= 0.54$ with $\sigma = 0.10$. Thus, getting a Geographos-like ellipticity from a M-class disruption is less than a 1 $\sigma$ event, decent odds if such disruptions (and $\varepsilon _{{\rm rem}} = 0.43$ progenitors) are common.