Abstract
We model the long-term evolution of the Hilda collisional family located in the 3/2 mean-motion resonance with Jupiter. Its eccentricity distribution evolves mostly due to the Yarkovsky/YORP effect and assuming that (i) impact disruption was isotropic and (ii) albedo distribution of small asteroids is the same as for large ones, we can estimate the age of the Hilda family to be 4+0−1 Gyr. We also calculate collisional activity in the J3/2 region. Our results indicate that current collisional rates are very low for a 200-km parent body such that the number of expected events over gigayears is much smaller than 1. The large age and the low probability of the collisional disruption lead us to the conclusion that the Hilda family might have been created during the late heavy bombardment (LHB) when the collisions were much more frequent. The Hilda family may thus serve as a test of orbital behaviour of planets during the LHB. We have tested the influence of the giant-planet migration on the distribution of the family members. The scenarios that are consistent with the observed Hilda family are those with fast migration time-scales ≃0.3–3 Myr, because longer time-scales produce a family that is depleted and too much spread in eccentricity. Moreover, there is an indication that Jupiter and Saturn were no longer in a compact configuration (with period ratio PS/PJ > 2.09) at the time when the Hilda family was created.
Highlights
There are many independent lines of evidence that the orbits of planets of the Solar system were not the same all the time, but that they have changed substantially over billions of years
Subsequent evolution of the synthetic Hilda family due to the Yarkovsky/YORP effect is computed up to 6 Gyr in order to estimate the time-span needed to match the observed family even though the family cannot be older than 4 Gyr
In the first test we compute an evolution of the synthetic Hilda family during planetary migration phase for the following parameter space: aJi = (5.2806 and 5.2027) au, aSi = (8.6250, 8.8250, 9.3000) au, eJi = (0.065, 0.045), eSi = (0.08, 0.05), τ mig = (0.3, 3, 30, 300) Myr, edampJ = 10−11, edampS = 10−11.1 The values of aJi and aSi correspond to period ratios PS/PJ from 2.09 to 2.39, i.e. the giant planets are placed already beyond the 2:1 resonance, since the 2:1 resonance crossing would destroy the Hilda family (Broz & Vokrouhlicky 2008)
Summary
There are many independent lines of evidence that the orbits of planets of the Solar system were not the same all the time, but that they have changed substantially over billions of years. In order to study the detailed distribution of the bodies librating inside the resonance, we have to use pseudo-proper resonant elements defined as approximate surfaces of sections (Roig, Nesvorny & Ferraz-Mello 2002), i.e. the intersection of the trajectory with a plane defined by These conditions correspond to the maximum of the semimajor axis a over several oscillations and the minimum of the eccentricity e or the inclination I. 2–4 of them are likely to be interlopers, because there is a very small number of background asteroids in the surroundings of the family (see Fig. 3) Recall that the whole J3/2 population exhibits a bimodal distribution of slopes, i.e. it contains a mixture of C- and D-type asteroids
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