Runaway Growth During Planet Formation: Explaining the Size Distribution of Large Kuiper Belt Objects

Abstract

Runaway growth is an important stage in planet formation during which large protoplanets form, while most of the initial mass remains in small planetesimals. The amount of mass converted into large protoplanets and their resulting size distribution are not well understood. Here, we use analytic work, that we confirm by coagulation simulations, to describe runaway growth and the corresponding evolution of the velocity dispersion. We find that runaway growth proceeds as follows. Initially, all the mass resides in small planetesimals, with mass surface density σ , and large protoplanets start to form by accreting small planetesimals. This growth continues until growth by merging large protoplanets becomes comparable to growth by planetesimal accretion. This condition sets in when Σ/σ ∼ α 3/4 ∼ 10 −3 , where Σ is the mass surface density in protoplanets in a given logarithmic mass interval and α is the ratio of the size of a body to its Hill radius. From then on, protoplanetary growth and the evolution of the velocity dispersion become self-similar and Σ remains roughly constant, since an increase in Σ by accretion of small planetesimals is balanced by a decrease due to merging with large protoplanets. We show that this growth leads to a protoplanet size distribution given by N (>R) ∝ R −3 , where N (>R) is the number of objects with radii greater than R (i.e., a differential power-law index of 4). Since only the largest bodies grow significantly during runaway growth, Σ and thereby the size distribution are preserved. We apply our results to the Kuiper Belt, which is a relic of runaway growth where planet formation never proceeded to completion. Our results successfully match the observed Kuiper Belt size distribution, they illuminate the physical processes that shaped it and explain the total mass that is present in large Kuiper Belt objects (KBOs) today. This work suggests that the current mass in large KBOs is primordial and that it has not been significantly depleted. We also predict a maximum mass ratio for Kuiper Belt binaries that formed by dynamical processes of α −1/4 ∼ 10, which explains the observed clustering in binary companion sizes that is seen in the cold classical belt. Finally, our results also apply to growth in debris disks, as long as frequent planetesimal-planetesimal collisions are not important during the growth.

Using Kuiper Belt Binaries to Constrain Neptune's Migration History

“…These binary formation scenarios are therefore inconsistent with reduction of the mass in the Kuiper Belt by two orders of magnitude or more in an entirely size-independent fashion, for example, in one large scattering event. Analytic work and numerical coagulation simulations that model the growth of KBOs indeed find that only about α 3/4 ∼ 10 −3 of the initial disk mass is converted into large KBOs suggesting that Σ/σ ∼ 10 −3 (Kenyon & Luu 1999;Kenyon & Bromley 2004;Schlichting & Sari 2011), which is consistent with the surface density values used in this section and the conclusion that v < v H .…”

“…This trend in the binary fraction as a function of heliocentric distance is in agreement with observations that find a binary fraction that is more than a factor of two lower in the excited and hot classical populations, which most likely formed closer to the Sun and were scattered to their current locations, than in the cold classical population, which most likely formed in situ at about 40 AU. An analogous calculation for the L 3 mechanism of binary formation yields the scaling relation sp(s) ∝ a γ −1/4 , where we used Σ/σ ∼ α 3/4 (Schlichting & Sari 2011). In Section 5, we explore the expected binary fraction as a function of current dynamical class in more detail.…”

Planetesimals in Debris Disks of Sun-Like Stars

“…In the Kuiper belt region, the solid mass of the so-called Minimum Mass Solar Nebula (MMSN) is ∼10 M ⊕ (Hayashi 1981;Weidenschilling 1977), while the mass in large Kuiper belt objects is estimated to be 0.1 M ⊕ (see, e.g., Gladman et al 2001;Bernstein et al 2004). This large difference, however, is explained by current models where the formation of large planetesimals has a very low efficiency (Bromley & Kenyon 2006;Schlichting & Sari 2011).…”

“…Inferred initial masses for a broken power-law size distribution. We investigate two particular forms, motivated by coagulation simulations by Kenyon & Bromley (2008) and Schlichting & Sari (2011), respectively. Other parameters adopted are e = 0.1, Δa/a = 0.1, and the hard strength law.…”

Trans-Neptunian Objects as Natural Probes to the Unknown Solar System

“…In addition to sourcing the Centaurs and SPCs, a fraction of these lost Trojans will have left scars af- ter impacts on planets or the satellites of the giant planets (Horner and Lykawka, 2010a). 7.4 Probing the dynamical signatures of early solar system massive planetesimals In general, planet formation models are based on the accretion of planetesimals according to the main stages of runaway growth and oligarchic growth (Kenyon, 2002;Rafikov, 2003;Goldreich et al, 2004b;Schlichting and Sari, 2011). Several massive planetesimals (or planetoids) are expected to have existed in the disk during the late stages of planet formation.…”