Pebble accretion

In pebble accretion the accretion of objects ranging from cm's up to meters in diameter onto planetesimals in a protoplanetary disk is enhanced by aerodynamic drag. The rapid growth of the planetesimals via pebble accretion allows for the formation of giant planet cores in the outer Solar System before the dispersal of the gas disk. A reduction in the size of pebbles as they lose water ice after crossing the ice line and a declining density of gas with distance from the sun slow the rates of pebble accretion in the inner Solar System resulting in smaller terrestrial planets, a small mass of Mars and a low mass asteroid belt.

Pebbles ranging in size from cm's up to a meter in size are accreted at an enhanced rate in a protoplanetary disk. A protoplanetary disk is made up of a mix of gas and solids including dust, pebbles, planetesimals, and protoplanets. Gas in a protoplanetary disk is pressure supported and as a result orbits at a velocity slower than large objects. The gas affects the motions of the solids in varying ways depending on their size, with dust moving with the gas and the largest planetesimals orbiting largely unaffected by the gas. Pebbles are an intermediate case, aerodynamic drag causes them to settle toward the central plane of the disk and to orbit at a sub-keplerian velocity resulting in radial drift toward the central star. The pebbles frequently encounter planetesimals as a result of their lower velocities and inward drift. If their motions were unaffected by the gas only a small fraction, determined by gravitational focusing and the cross-section of the planetesimals, would be accreted by the planetesimals. The remainder would follow hyperbolic paths, accelerating toward the planetesimal on their approach and decelerating as they recede. However, the drag the pebbles experience grows as their velocities increase, slowing some enough that they become gravitationally bound to the planetesimal. These pebbles continue to lose energy as they orbit the planetesimal causing them to spiral toward and be accreted by the planetesimal.

Small planetesimals accrete pebbles that are drifting past them at the relative velocity of the gas. Those pebbles with stopping times similar to the planetesimal's Bondi time are accreted from within its Bondi radius. In this context the Bondi radius is defined as the distance at which an object approaching a planetesimal at the relative velocity of the gas is deflected by one radian; the stopping time is the exponential timescale for the deceleration of an object due to gas drag, and the Bondi time is the time required for an object to cross the Bondi radius. Since the Bondi radius and Bondi time increase with the size of the planetesimal, and the stopping time increases with the size of the pebble, the optimal pebble size increases with size of planetesimal. Smaller objects, with ratios of stopping times to Bondi times less than 0.1, are pulled from the flow past the planetesimal and accreted from a smaller radius which declines with the square root of this ratio. Larger, weakly coupled pebbles are also accreted less efficiently due to three body effects with the radius accreted from declining rapidly between ratios of 10 and 100. The Bondi radius is proportional to the mass of the planetesimal so the relative growth rates is proportional to mass squared resulting in runaway growth. The aerodynamic deflection of the gas around the planetesimal reduces the efficiency of pebble accretion resulting in a maximum growth timescale at 100 km.

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