Simulations of small solid accretion on to planetesimals in the presence of gas

MNRAS 472, 3543–3553 (2017) doi:10.1093/mnras/stx1964 Advance Access publication 2017 August 2 Simulations of small solid accretion on to planetesimals in the presence of gas A. G. Hughes‹ and A. C. Boley‹ Department of Physics and Astronomy, University of British Columbia, 2329 West Mall, Vancouver, BC V6T 1Z4, Canada Accepted 2017 July 28. Received 2017 July 28; in original form 2017 June 5 The growth and migration of planetesimals in a young protoplanetary disc are fundamental to planet formation. In all models of early growth, there are several processes that can inhibit grains from reaching larger sizes. Nevertheless, observations suggest that growth of planetesimals must be rapid. If a small number of 100 km sized planetesimals do manage to form in the disc, then gas drag effects could enable them to efficiently accrete small solids from beyond their gravitationally focused cross-section. This gas-drag-enhanced accretion can allow planetesimals to grow at rapid rates, in principle. We present self-consistent hydrodynamics simulations with direct particle integration and gas-drag coupling to estimate the rate of planetesimal growth due to pebble accretion. Wind tunnel simulations are used to explore a range of particle sizes and disc conditions. We also explore analytic estimates of planetesimal growth and numerically integrate planetesimal drift due to the accretion of small solids. Our results show that, for almost every case that we consider, there is a clearly preferred particle size for accretion that depends on the properties of the accreting planetesimal and the local disc conditions. For solids much smaller than the preferred particle size, accretion rates are significantly reduced as the particles are entrained in the gas and flow around the planetesimal. Solids much larger than the preferred size accrete at rates consistent with gravitational focusing. Our analytic estimates for pebble accretion highlight the time-scales that are needed for the growth of large objects under different disc conditions and initial planetesimal sizes. Key words: hydrodynamics – planets and satellites: formation – planets and satellites: physical evolution. 1 I N T RO D U C T I O N The planet formation process is critically dependent on the coagulation of dust grains into larger bodies. The time-scale for significant solid growth is thought to be a few million years, based on the chronologies of meteorites (Villenueve, Chaussidon & Libourel 2009), the fraction of inferred protoplanetary discs relative to cluster ages (Mamajek et al. 2009), and the incidence of gas giant planets (Cumming et al. 2008). The latter provides a constraint on core-nucleated instability (Pollack et al. 1996) as a dominant formation model for giant planets, which requires the formation of a large solid core before the gaseous disc dissipates. Furthermore, morphological features in discs such as HL Tau (ALMA Partnership 2015) and TW Hydrae (Andrews et al. 2016) may indicate the presence of embedded planets, which if correct, would require the rapid growth of large solids in some discs.  E-mail: (AGH); (ACB) The early stages of planet formation rely on the growth of micron to submillimetre grains into planetary sizes. As grains reach submillimetre and millimetre radii, further growth may be hindered due to bouncing at low collisional speeds and fragmentation at higher collisional speeds (Blum & Wurm 2008; Testi et al. 2014). Regions of high solid density could potentially build planets or planetesimals through direct gravitational collapse (Goldreich & Ward 1973) if solids become concentrated due to some mechanism such as vertical settling (D’Alessio, Calvet & Hartmann 2001). However, vertical settling is opposed by turbulent mixing, which can lift grains away from the mid-plane of the disc (Dubrulle 1995), slowing or preventing significant grain growth. Furthermore, gas is expected to orbit at sub-Keplerian speeds due to the negative outward pressure gradient in the disc. This causes objects that are on Keplerian orbits to face a headwind, resulting in the loss of angular momentum for small solids. Without any mitigating factor, centimetre to metre-sized objects would consequently spiral into their stars on short time-scales (Adachi, Hayashi & Nakazawa 1976; Weidenschilling 1977). While spiral arms or other pressure perturbations could concentrate solids  C 2017 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society ABSTRACT 3544 A. G. Hughes and A. C. Boley 2 T H E O RY A N D M E T H O D S Gas in a protoplanetary disc is expected to orbit at sub-Keplerian speeds due to support from a negative gas pressure gradient. In the absence of such support, solids will orbit at Keplerian speeds. This velocity difference causes the solids to encounter a headwind with the gas, and alters the orbit of solids depending on size. Small solids quickly become entrained with the gas flow, while larger solids carry MNRAS 472, 3543–3553 (2017) Table 1. Disc and object properties used in the hydrodynamics simulations, following the flared disc profile outlined in equation (1) along with predicted best-accreted pebble size for each distance as found using equation (9). While most disc conditions change dramatically at different stellar separations, the relative velocity between the gas and the planetesimal stays roughly constant. r (au) ρ g g cm−3 T (K) v rel m s−1 s (cm) 10 3 1 0.3 0.1 3.2 × 10−12 95 170 300 550 950 54.8 54.7 54.7 54.6 54.6 0.0005 0.01 0.2 3.0 50.0 6.4 × 10−11 1.0 × 10−9 2.0 × 10−8 3.1 × 10−7 enough momentum that gas drag only affects their motion over very long time-scales. Throughout this paper, we assume a disc model with the following mid-plane profiles:  r −n ; ρ = ρ0 au  r −m ; (1) T = T0 au where r is the stellar separation measured in au, ρ 0 = 1 × 10−9 g cm−3 and T0 = 300 K are gas density and temperature values at 1 au. For our model, we use the exponents m R = 0.5 and n = 2.5. We take the gas to be ideal with P = μg T ρ and μ = 2.3 g mol−1 . At any location in the disc, the difference between the circular Keplerian orbit and the circular gas motion is given by   r dP GM GM − + (2) vrel = r r ρ dr ⎛ = vK ⎝1 −   (1 − (n + m) c0 vK 2 ⎞ r0  m ⎠ r (3) for circular Keplerian speed v K at r and temperature and density power-law exponents m and n, respectively. The isothermal sound speed at r0 = 1 au is given by c0 . The gas conditions at various stellar separations are shown in Table 1, along with the corresponding relative wind speeds. For our assumptions, the wind speed v rel stays roughly constant with stellar separation. As a solid experiences a relative wind, drag forces couple them to the gas. The characteristics of this coupling depend on the drag regime, which can be divided into two basic types: Epstein and Stokes. A solid falls into the Epstein regime when its radius is much smaller than the mean free path (...truncated)