Exoplanets in our Backyard 2020 (LPI Contrib. No. 2195) 3061.pdf PEBBLE ACCRETION BY BODIES ON ECCENTRIC ORBITS AND THE MASS RATIOS OF EXOPLANETS Steven Desch1, Alan Jackson1, Chuhong Mai1, and Jessica Noviello1 1School of Earth and Space Exploration, Arizona State University, PO Box 871404, Tempe AZ, 85287-1404 ([email protected] ). 2/3 Introduction: In the last decade, several lines of tion in the Hill regime scales as dMp/dt ~ Mp [14], so evidence—from our own solar system and from studies the mass ratios between embryos should decrease with of exoplanets—have converged to suggest that growth time very slowly, making it difficult to reconcile with of planets is extraordinarily fast. In our own solar sys- the peas-in-a-pod result. We propose a modification of tem, Hf-W dating of Mars shows it was an embryo pebble accretion involving embryos on eccentric orbits formed between 1 and 3 Myr after CAIs (calcium-rich, that leads to embryos growing to similar sizes. aluminum-rich inclusions) [1]. The isotopic dichotomy of the solar system [2], reduced water content inside Growth on circular orbits: Growth of embryos the snow line [3] and the concentration of CAIs in car- on circular orbits proceeds as follows. Pebbles with St bonaceous chondrites [4] have all been explained by ~0.01-0.1 entering an embryo’s Hill sphere with radius invoking formation of Jupiter’s ~20-30 M core in < 1 R = a (M / 3M )1/3 are accreted with high efficiency, E H p ¤ Myr. Ingassing of nebula gas into the magma oceans of assuming they have scale height < RH. Pebbles are the Earth [5,6] and Theia [7] explain their low-D/H assumed to sweep into the Hill sphere at velocity VH = hydrogen reservoirs and their He and Ne isotopic R Ω , Ω = (GM /a3)1/2, due to Keplerian shear across H K K ¤ abundances. This requires them to achieve masses of the Hill radius. The embryo sweeps up pebble mass at several × 0.1 ME in < 3 Myr. In exoplanetary systems, a rate dMp/dt = 2 RH Σp VH, where Σp is the surface many super-Earths accreted substantial hydrogen-rich density of pebbles (St~0.01–0.1 solids). This yields -2 2/3 -1 atmospheres [8,9], requiring them to grow to several dMp/dt ~ 47 (Σp / 1 g cm ) (Mp / 1 ME) ME Myr at 1 Earth masses within < 3 Myr [10]. The gaps in young AU. This is such a high accretion rate, an embryo will protoplanetary disks (sometimes ~ 105 yr, like HL Tau) quickly (~100 years) sweep up the mass in its torus, -2 1/3 observed by the Atacama Large Millimeter Array, may 2πa (2 RH) Σp = 0.0047 (Σp / 1 g cm ) (Mp / 1 ME) be due to massive (tens of Earth masses) planets [11]. ME , and will grow only if a pebble flux can feed its -2 Overall, planets in our solar system and others demand torus. [A surface density Σp = 1 g cm of pebbles cor- -6 -1 accretion rates of at least 10 ME yr . responds to 20% of all solids if the gas surface density -2 Recent astrophysical modeling suggests a path for is 1000 g cm .] The radial drift of pebbles is dMp/dt = 2 rapidly turning micron-sized dust into planets thou- (2πa) Σp Vpr, where Vpr = -St (ηVK) / (1+St ), and η = 2 2 sands of km in diameter. Fast coagulation produces -(C / VK ) d lnP / d lnr [16]. But only a fraction of the- millimeter-sized particles (precursors of compactified se, (2RH/Vpr)(VH/2πa) (which is < 1 if St ≥ 0.01 and dust, or melted chondrules). For conditions in the inner Mp ≤ 1 ME), are accreted before drifting through the disk, their Stokes number (the ratio of aerodynamic annulus entirely, yielding the same result: dMp/dt = 2 -3 2/3 stopping time to orbital time) is St ~ 10 . Turbulence RH Σp VH. Thus dMp/dt ~ Mp and larger planets then concentrates these into aggregates up to ~10 cm in should grow faster than small ones. Integrating the -2 size [12], with St ~ 0.01 – 0.1. These particles are in- growth equation, assuming Σp = 1 g cm at 1 AU, a 0.5 5 termittently but rapidly collected into ~100-km objects ME planet would grow to 13.2 ME in only 10 yr, and a by streaming instability [13]. The very largest of these 2.0 ME planet would grow to 22.6 ME in the same time. objects, those several ×100 km in diameter, can grow rapidly by pebble accretion. This aerodynamic process Growth on eccentric orbits: Growth by pebble accre- relies on drag to slow particles (especially those with tion is faster for an embryo on an eccentric orbit in- St ~ 0.01 – 0.1) in the vicinity of a growing planet, stead of a circular one. In a frame co-moving with the allowing it to capture almost all such particles entering embryo, the embryo makes epicyclic orbits with radial the planet’s Hill sphere [14]. Growth at rates of at least excursions ±ae, where e is the eccentricity. The em- -6 -1 10 ME yr appear possible. bryo and pebbles have relative velocity ~ eVK = eaΩK, As successful as pebble accretion is, the theory ap- several km/s. Analogous to the circular case, immedi- pears incomplete at present. An important constraint ately after being put onto an eccentric orbit, embryos from observations is that exoplanets in multiple-planet grow as dMp/dt = 2RH Σp eVK, accreting a high fraction systems (observed in transits by Kepler and followed of the pebbles in its epicyclic torus in one orbit [17]. up by radial velocity measurements) appear to be simi- Thereafter it grows at the rate pebbles in the annulus lar in orbital period ratios and in size, and therefore between a(1-e) and a(1+e) can drift into the torus. also mass [15]. The growth of planets by pebble accre- Pressure support of gas makes it orbit at an azimuthal Exoplanets in our Backyard 2020 (LPI Contrib. No. 2195) 3061.pdf -2 1/2 1/2 velocity ηVK/2 slower than the embryo, and the azi- 4.26 (Σp / 1 g cm ) (e / 0.1) ME, both within ~5 × muthal velocities of pebbles differ from Keplerian by ~ 104 yr. Once an embryo’s orbit has circularized, it will 2 -3 (-ηVK/2) / (1 + St ) [16]. Typically η ~ 10 ; in the continue to accrete at the circular pebble accretion rate, -3 models of [4], η =3 × 10 . Within a time 2πa / (ηVK/2) dMp/dt = 2 RH Σp VH. ~ 2η-1 orbits, the embryo sweeps up the entire annulus of area ≈ 2πa (2ae). The mass accretion rate while it is System Architecture: In systems with embryos sweeping up the annulus is dMp/dt = (ae) Σp (ηVK) / growing by pebble accretion on circular orbits, dMp/dt 2 2/3 (1+St ). A correction would have to be made if the Hill ~ Mp , and the mass ratio of two embryos tends to sphere were small enough that the pebbles could cross decease with time, but slowly. In the example above, the torus width in less than one orbit, i.e., if 2RH / over 0.1 Myr, the mass ratio of two embryos decreased 3 (ηVK/2) < 2π / ΩK, or Mp < [157 η] ME ~ 0.0034 ME; from (2.0)/(0.5) = 4.0, to (22.6)/(13.2) = 1.7. Embryos but as long as an embryo is at least this large, it will scattered onto eccentric orbits will accrete by pebble sweep up pebbles at the rate dMp/dt ≈ (ae) Σp (ηVK). accretion more rapidly than if they were on circular An embryo will sweep up the entire pebble mass of the orbits, at rates independent of mass, at least until their -2 annulus, ΔM = (2πa)(2ae) Σp = 0.05 (Σp / 1 g cm ) (e / eccentricities damp. In the one example above, with St 3 0.1) ME in less than 10 yr. > e/2π, the mass ratio of two embryos decreased from As with circular orbits, continued embryo growth (2.0)/(0.5) = 4.0, to (10.3)/(10.1) = 1.02. In the other relies on radial drift of pebbles. Pebbles are brought case, with St < e/2π, the mass ratio decreased from into the annulus at a rate dMp/dt = (2πa) Σp Vpr, where (2.0)/(0.5) = 4.0, to (4.26)/(3.80) = 1.12. Embryos are 2 Vpr = -St (ηVK)/(1+St ). The fraction of these that are only transiently on eccentric orbits, after being scat- accreted is 100% if St < e/2π, and dMp/dt ≈ 2πa Σp St tered; but while they are, the small embryos tend to ηVK; but is e/(2πSt) if St > e/2π, and dMp/dt = ae Σp grow more rapidly and catch up in mass with the larger ηVK, as before. Either generally exceeds the circular embryos. By the time they each circularize, they will orbit pebble accretion rate = 2 RH Σp VH, because the have attained masses much more similar than if they embryo can sweep up pebbles from a larger area, and grew at the circular orbit pebble accretion rate. The because pebbles are swept up with greater efficiency. tendency of exoplanets in the same system to have similar masses [8] may best be explained if a signifi- Embryos may accrete at the circular orbit rate until cant fraction of their growth occurs after they are scat- they are scattered onto an eccentric orbit.