Size-Frequency Distribution of S-complex Implanted Asteroids.

Introduction: The main asteroid belt (MAB) is known for having a very small total mass (5x10-4 Earth masses) and for being primarily composed of S- and C-complex bodies [1]. Isotopic measurements on meteorites suggest C-complex bodies, many which are water/volatile-rich, are likely to have originated exterior to Jupiter’s orbit, whereas S-complex bodies, most which are water/volatile-poor, come from within it [2].The MAB’s low mass and taxonomical mixing can potentially be explained if the MAB was either originally empty [2] or never had much mass [3]. In this case, the currently observed MAB would result from limited implantation of S- and C-complex asteroids that were scattered from their source regions [2].  This proposition is in agreement with models suggesting that the solar system building blocks formed in concentric rings at various radial distances around the Sun [4] (models which were partly motivated by cosmochemical constraints [5]). This would naturally form a primordial MAB depleted in or devoid of mass [3].What remains to be explored is whether the size-frequency distribution (SFD) of the MAB could potentially be explained by the implantation of S- and C-complex asteroids. The power law slope (q) for MAB’s SFD for objects 100 km ≤ D ≤ 400 km in diameter is probably unchanged over solar system history [6]. Thus, the implanted asteroids in this size range should have a slope comparable to the MAB’s current slope. Here we focus on the implantation from S-complex asteroids originating in the terrestrial planet region.Methods:  We conducted terrestrial planet formation simulations starting from planetesimal-sized objects that were tracked over 5 Myr within the solar nebula [7]. We assumed the gas disk exponentially dispersed over timescales τgas of 0.5, 1, and 2 Myr. We gave our initial planetesimal population a cumulative SFD following N(>D)∝ D-q, with q = 0 (D = 100 km), 3.5, and 5 (100 km ≤ D ≤ 1,000 km). The total mass was 2.5 Earth masses. Planetesimals were radially distributed according to Σ ∝ r-x with x = 1 and 5.5 in the semi-major axis range 0.7 au ≤ a ≤ 1.5 au [4].The evolution of the planetesimals’ SFD was tracked with the code LIPAD [8] during their accretion phase; some eventually grow to become planetary embryos. Given that S-complex planetesimals are only expected to be implanted in the MAB after the solar nebula has dispersed [2], we compared our evolved SFD at the end of the simulation with that from S-complex asteroids in the current MAB [6, 9].Results: A compilation of our results is shown in Fig. 1. A large concentration of mass is needed near 1 au to form terrestrial planets and reproduce their orbits [4], so collisional evolution there is intense. As a result, our SFDs, regardless of x, q and τgas, rapidly reaches collisional equilibrium. In this state, the slope of our evolved SFDs broadly match that of the current MAB SFD in the range 100 km < D < 400 km.Conclusions:  Assuming that the evolved SFDs stay in collisional equilibrium after gas disk dispersal, and that implantation in the MAB is size-independent, we conclude that S-complex asteroids may indeed be objects that formed in the terrestrial planet region, i.e., within 1.5 au. That includes asteroid (4) Vesta [3].Additionally, we find that implantation efficiency  is likely to be