Linking the collisional history of the main asteroid belt to its dynamical excitation and depletion
Introduction
The collisional and dynamical history of the main belt is strongly linked with the growth and evolution of the planets, with the events occurring during this primeval era recorded in the orbits, sizes, and compositional distributions of the asteroids and in the hand samples of meteorites. Like archaeologists working to translate stone carvings left behind by ancient civilizations, the collisional and dynamical clues left behind in or derived from the main belt, once properly interpreted, can be used to read the history of the inner Solar System. From a practical standpoint, this means we not only have to understand how asteroids have been scarred and fragmented over time by hypervelocity impacts but also how gravitational perturbations from planets, planetary embryos, and the solar nebula have affected the orbital distribution of the survivors.
The foundation for the work presented in this paper is Bottke et al. (2005) (hereafter B05), who used numerical simulations to track how the initial main belt size–frequency distribution was affected over time by collisional evolution. The constraints used in their modeling work, namely the wavy-shaped main belt size–frequency distribution, the quantity of asteroid families produced by the disruption of parent bodies over the last 3–4 Gyr, the cratered surface of Asteroid (4) Vesta, and the relatively constant crater production rate of the Earth and Moon over the last 3 Gyr, were wide ranging enough for them to estimate the shape of the initial main belt size–frequency distribution as well as the asteroid disruption scaling law , a critical function needed in all codes that include fragmentation between rocky planetesimals. In contrast to previous efforts (e.g., Durda et al., 1998), B05 demonstrated that the scaling laws most likely to produce current main belt conditions were similar to those generated by recent smoothed particle hydrodynamic (SPH) simulations (e.g., Benz and Asphaug, 1999). They also showed there was a limit to the degree of main belt comminution that could have taken place over Solar System history; too much or too little would produce model size distributions discordant with observations. In fact, they suggested that if the current main belt were run backward in time, it would need the equivalent of roughly 7.5–9.5 Gyr of “reverse collisional evolution” to return to its initial state, a time interval roughly twice as long as the age of the Solar System (4.6 Gyr).
To explain this puzzling result, B05 argued the extra comminution had to come from a collisional phase occurring early in Solar System history (∼3.9–4.5 Ga) when the primordial main belt held significantly more diameter bodies than it contains today. The extra material in this population would have been removed by dynamical rather than collisional processes. What B05 left out of their simulations, however, was a way to account for the excitation and elimination of main belt material by dynamical processes. They did this for two reasons. The first was that including dynamical processes into their simulations would have increased their number of unknown parameters, enough that obtaining unique solutions for and the initial shape of the main belt size distribution would have been impossible. The second is that they wanted to avoid locking themselves into a dynamical evolution scenario that would likely become obsolete as planet formation models increased in sophistication (e.g., next generation models will likely include the effects of dynamical friction between planetary embryos and planetesimals, gas drag on planetesimals, planetary migration, and the nature and effects of the so-called Late Heavy Bombardment; Hartmann et al., 2000, Gomes et al., 2005; see Section 2.2.4). Instead, B05 used a more approximate approach that retained the essential aspects of the problem but made simplifying assumptions (e.g., they assumed that most main belt comminution occurred when collisional probabilities and impact velocities were comparable to current values; a massive population experiencing comminution over a short interval is mathematically equivalent to a low mass population undergoing comminution over an extended interval). Thus, rather than modeling the collisional and dynamical history of the main belt over 4.6 Gyr, B05 instead tracked how a population comparable in mass to the current main belt would evolve over timescales longer than the age of the Solar System, with the extra simulation time compensating for a (putative) early phase of comminution occurring when the main belt was both excited and massive.
Using insights provided by B05, we are now ready to try a more ambitious model that directly accounts for both collisional and dynamical processes over the last 4.6 Gyr. Here we combine results from the best available dynamical model of early main belt evolution (Petit et al., 2001) with the collision code and best fit results of B05. As we show below, the results from this hybrid code can be used to estimate the initial mass of the population in the primordial main belt, constrain the formation timescale of Jupiter, and produce a more refined understanding of the nature of the dynamical depletion event that scattered bodies out of main belt population early in Solar System history. Thus, we argue the main belt provides us with critical clues that can help us probe the nature of planet formation events in the inner Solar System.
To verify our results, we made numerous comparisons between our model's predictions and the available constraints, with our results showing an excellent match between the two. These predictions include estimates of: (i) the post-accretion size distribution found among main belt planetesimals, (ii) the disruption scaling law controlling asteroid breakup events, (iii) the collisional lifetime of main belt asteroids, and (iv) the main belt and NEO population size distributions over a size range stretching from centimeter- to Ceres-sized objects over the last 4.6 Gyr. Using our NEO population model, we also investigated the nature of small craters on the terrestrial planets and whether they were formed by primary impacts or by secondary impacts produced by ejecta from large craters. For those wishing to see a detailed summary of our results before reading the paper, please turn to our conclusions in Section 6.
Here we provide a brief outline for this paper. In Section 2, we discuss the current state of the art in main belt dynamical evolution models, and closely examine the results described by Petit et al. (2001), whose model does the best job thus far of explaining the available dynamical constraints. In Section 3, we show how the dynamical excitation and depletion of the main belt, as well as the effects of Yarkovsky thermal forces, are included in our model. We also review our principal model constraints. In Section 4, we discuss several trial runs of our model while also presenting our large-scale production runs. In Section 5, we discuss the numerous implications of our results, which extend into many issues of main belt and NEO evolution. Finally, in Section 6, we present our conclusions.
One additional comment should be made here. In B05, we reviewed the history of main belt collisional evolution models. Since that time, an additional model has been submitted (O'Brien and Greenberg, 2005) and another published (Cheng, 2004). Between the two, the O'Brien and Greenberg (2005) model is more similar to one presented here, though there are several distinct differences in the choices made for initial conditions, model components and constraints, methods, and in the treatment of the dynamical evolution of the primordial main belt. The most important difference between these two papers and ours, however, is that our results were constrained by the observed distribution of asteroid families formed over the last 3.5 Gyr whose parent bodies had diameter (see Section 3.4 and B05). We were literally driven to the results described below by these data, which allowed us to eliminate many unrealistic input parameters. We believe any future models of main belt evolution will need to be similarly tested in order to obtain unique solutions.
Section snippets
Previous work
To determine the nature of the dynamical excitation and depletion event that affected the main belt, we first need to understand the dynamical constraints provided by main belt asteroids (a recent review of this issue can be found in Petit et al., 2001, Petit et al., 2002). One powerful constraint concerns the putative depletion of material in the main belt zone. While the main belt only contains of material today, it may have once held an or more of material Weidenschilling, 1977,
Modeling the collisional evolution of the main belt size distribution
To model the evolution of the main belt as completely as possible, we integrated the dynamical results described in Section 2.2 (Petit et al., 2001) with the collisional evolution code CoEM-ST (B05). Our modified code, called CoDDEM (for collisional and dynamical depletion model), is described below in two parts. In the first part, we briefly describe our nominal model (CoEM-ST). In the second part, we describe the modifications needed to allow our model to account for the dynamical ejection of
Initial conditions
In this section, we describe the input parameters needed by CoDDEM to track the evolution of the main belt. The principal unknowns affecting the DDE in our model are the timescale of Jupiter's formation and the initial size distribution of the main belt population. Using results from B05, we assume the shape of the latter is constrained to a fairly narrow range of values, though its magnitude is treated as a free parameter. Another important unknown extensively investigated by B05 is the
Discussion and implications
Here we briefly outline the issues discussed in this section. In Section 5.1, we explore the disruption history of the main belt and speculate on the possible origin of asteroids believed to be derived from differentiated parent bodies (e.g., M-type Asteroid 16 Psyche, V-type Asteroid 1459 Magnya). In Section 5.2, we compare our predicted collisional lifetimes to the cosmic ray exposure ages of stony meteorites and to other lifetime estimates in the literature. In Section 5.3, we compare our
Conclusions
Here we summarize the main conclusions of this paper.
It has long been known that the main belt zone is significantly depleted in mass compared to expectations from standard solar nebula estimates (i.e., vs , respectively). The best available dynamical model to explain this mass loss (Petit et al., 2001) suggests that the post-accretion main belt went through three broad phases of evolution:
Phase 1. An early phase lasting a few Myr where planetesimals and planetary embryos both
Acknowledgements
We thank Rick Binzel, Rick Greenberg, Bob Grimm, Alfred McEwen, Patrick Michel, Ed Scott, Kleomenis Tsiganis, and Stu Weidenschilling for valuable discussions and input to this study. We also thank referees Erik Asphaug and Dave O'Brien for their helpful and constructive reviews. Research funds for William Bottke were provided by NASA's Origins of Solar Systems Program (Grant NAG5–10658), Discovery Data Analysis Program (Grant NNG04GA75G), and Near-Earth Object Observations Program (Grant
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