[PAE22-07] Pebble Accretion by Planets on Eccentric Orbits and the Mass Ratios of Exoplanets
Keywords:exoplanets, Theia, pebble accretion
Introduction: Planets in our solar system and others grew remarkably fast. Super-Earth exoplanets have thick H2/He atmospheres [1,2], requiring growth to several ME within < 3-10 Myr [3]. Hf-W dating of Mars shows it mostly formed between 1 and 3 Myr [4]. Low-D/H hydrogen reservoirs in Earth [5,6] and Theia [7,8] and solar He, Ne [9] demand ingassing of solar nebula gas into the magma oceans of the planetary embryos comprising them; these must have grown to ~0.4 ME in < 3 Myr [3]. Pebble accretion [10], an aerodynamic process allowing a planet to capture most particles with Stokes numbers St ~0.01–0.1 in its Hill sphere of radius RH, best explains this growth.
Super-Earths in multi-exoplanet Kepler systems appear similar in orbital period ratios and in size and mass, like “peas in a pod” [11]. Likewise proto-Earth and Theia were probably similarly sized embryos. But most pebble accretion models predict that planets grow at rates dMp/dt ~ Mp2/3 [10] and the mass ratios should decrease with time only slowly, unlike the peas-in-a-pod result. We propose that planet growth may be dominated by the short intervals of time spent by a planet on an eccentric orbit following a scattering event, and that this would produce similar masses.
Circular vs. eccentric orbits: A planet on a circular orbit sees its mass Mp increases at a rate dMp/dt = 2 RH2 Σp Ω, where Σp is the surface density of pebbles (St~0.01–0.1 solids), yielding dMp/dt ~ 47 (Σp / 1 g cm-2) (Mp / 1 ME)2/3 ME Myr-1 at a=1 AU. Assuming Σp = 1 g cm-2 at 1 AU, in 105 yr a 0.5 ME planet would grow to 13.2 ME, but a 2.0 ME planet to 22.6 ME. The mass ratio would decrease from 4.0 to 1.7. Planets on orbits with eccentricity e grow faster: in a co-rotating frame, the planet makes epicyclic orbits with radial excursions ±ae, moving relative to the gas at speeds ~ e a Ω, several km/s. We can show that if e > 2πSt ~10-1, then dMp/dt = ae Σp ηVK, where VK = a Ω and η = -(C2/ VK2) d lnP / d lnr ~ 10-3 [12]. This generally exceeds the circular orbit pebble accretion rate, because the embryo can sweep up pebbles from a larger area, and because pebbles are swept up with greater efficiency.
If scattered onto an eccentric orbit, planets accrete faster, until their orbits damp on timescales τ = 0.1 (Mp/ 0.5 ME)-1 Myr due to disk torques [13]. The growth rate dMp/dt is proportional to e, but e is damped at rates de/dt =-e/τ proportional to Mp. Integrating the coupled differential equations, we find that being scattered onto an orbit with eccentricity e will cause a 0.5 ME embryo to reach mass Mp = 3.80 (Σp / 1 g cm-2)1/2 (e / 0.1)1/2 ME, and a 2.0 ME embryo to reach mass Mp = 4.26 (Σp / 1 g cm-2)1/2 (e / 0.1)1/2 ME, before circularizing within about ~5 × 104 yr. After a burst of rapid growth while on eccentric orbits, the mass ratio drops from 4.0 to 1.12, much closer to 1. Growth primarily while on eccentric orbits may explain why exoplanets in a system have similar masses [11].
References: [1] L. M. Weiss and G. W. Marcy (2014) ApJL 783, L6-12. [2] B. J. Fulton et al. (2017) AJ 154, 109-127. [3] A. Stokl et al. (2015) A&A 576, 87-96. [4] N. Dauphas and A. Pourmand (2011) Nature 473, 489-492. [5] L. Hallis et al. (2015) Science 350, 795-797. [6] J. Wu et al. (2018) JGR 123, 2691-2712. [7] K. Robinson et al. (2016) GCA 188, 244-260. [8] S. J. Desch and K. L. Robinson (2019) Chemie der Erde 79, 125546 (2019). [9] C. Williams and S. Mukhopadhyay (2019) Nature 565, 78-81. [10] M. Lambrechts and A. Johansen (2012) A&A 544, 32-44. [11] L. Weiss et al. AJ (2018) 155, 48-59. [12] T. Takeuchi and D. N. C. Lin (2002) ApJ 581, 1344-1355. [13] M. Morris, et al. (2012), ApJ 752, 27-43.
Super-Earths in multi-exoplanet Kepler systems appear similar in orbital period ratios and in size and mass, like “peas in a pod” [11]. Likewise proto-Earth and Theia were probably similarly sized embryos. But most pebble accretion models predict that planets grow at rates dMp/dt ~ Mp2/3 [10] and the mass ratios should decrease with time only slowly, unlike the peas-in-a-pod result. We propose that planet growth may be dominated by the short intervals of time spent by a planet on an eccentric orbit following a scattering event, and that this would produce similar masses.
Circular vs. eccentric orbits: A planet on a circular orbit sees its mass Mp increases at a rate dMp/dt = 2 RH2 Σp Ω, where Σp is the surface density of pebbles (St~0.01–0.1 solids), yielding dMp/dt ~ 47 (Σp / 1 g cm-2) (Mp / 1 ME)2/3 ME Myr-1 at a=1 AU. Assuming Σp = 1 g cm-2 at 1 AU, in 105 yr a 0.5 ME planet would grow to 13.2 ME, but a 2.0 ME planet to 22.6 ME. The mass ratio would decrease from 4.0 to 1.7. Planets on orbits with eccentricity e grow faster: in a co-rotating frame, the planet makes epicyclic orbits with radial excursions ±ae, moving relative to the gas at speeds ~ e a Ω, several km/s. We can show that if e > 2πSt ~10-1, then dMp/dt = ae Σp ηVK, where VK = a Ω and η = -(C2/ VK2) d lnP / d lnr ~ 10-3 [12]. This generally exceeds the circular orbit pebble accretion rate, because the embryo can sweep up pebbles from a larger area, and because pebbles are swept up with greater efficiency.
If scattered onto an eccentric orbit, planets accrete faster, until their orbits damp on timescales τ = 0.1 (Mp/ 0.5 ME)-1 Myr due to disk torques [13]. The growth rate dMp/dt is proportional to e, but e is damped at rates de/dt =-e/τ proportional to Mp. Integrating the coupled differential equations, we find that being scattered onto an orbit with eccentricity e will cause a 0.5 ME embryo to reach mass Mp = 3.80 (Σp / 1 g cm-2)1/2 (e / 0.1)1/2 ME, and a 2.0 ME embryo to reach mass Mp = 4.26 (Σp / 1 g cm-2)1/2 (e / 0.1)1/2 ME, before circularizing within about ~5 × 104 yr. After a burst of rapid growth while on eccentric orbits, the mass ratio drops from 4.0 to 1.12, much closer to 1. Growth primarily while on eccentric orbits may explain why exoplanets in a system have similar masses [11].
References: [1] L. M. Weiss and G. W. Marcy (2014) ApJL 783, L6-12. [2] B. J. Fulton et al. (2017) AJ 154, 109-127. [3] A. Stokl et al. (2015) A&A 576, 87-96. [4] N. Dauphas and A. Pourmand (2011) Nature 473, 489-492. [5] L. Hallis et al. (2015) Science 350, 795-797. [6] J. Wu et al. (2018) JGR 123, 2691-2712. [7] K. Robinson et al. (2016) GCA 188, 244-260. [8] S. J. Desch and K. L. Robinson (2019) Chemie der Erde 79, 125546 (2019). [9] C. Williams and S. Mukhopadhyay (2019) Nature 565, 78-81. [10] M. Lambrechts and A. Johansen (2012) A&A 544, 32-44. [11] L. Weiss et al. AJ (2018) 155, 48-59. [12] T. Takeuchi and D. N. C. Lin (2002) ApJ 581, 1344-1355. [13] M. Morris, et al. (2012), ApJ 752, 27-43.