Gas accretion by planetary cores
GGas accretion by planetary cores
Ben A. Ayliffe and Matthew R. Bate
School of Physics, University of Exeter, Stocker Road, Exeter, EX4 4QL
Abstract.
We present accretion rates obtained from three-dimensional self-gravitating radiationhydrodynamical models of giant planet growth. We investigate the dependence of accretion ratesupon grain opacity and core/protoplanet mass. The accretion rates found for low mass cores areinline with the results of previous one-dimensional models that include radiative transfer.
Keywords: planet formation, hydrodynamics, accretion, radiative transfer
PACS:
INTRODUCTION
The core accretion model of planet growth has been devised to explain the formation ofgiant planets such as Jupiter and Saturn [1, 2]. Through the coagulation of solid grains,planetesimals are built which then, through a so far uncertain process, bind to form largekm scale bodies. When such bodies collide they shatter, but more often than not are heldtogether by their self-gravity [3], allowing them to grow through collisions to reach manyEarth masses. These large solid masses exert a considerable gravitational tug on the solarnebula in which they are immersed, causing them to slowly gather a gaseous atmosphere.If the solid cores are sufficiently massive (> 10 M
Earth ) they may trigger runaway gasaccretion. This occurs when the accreted gaseous atmosphere obtains a mass similar tothe solid core, causing it to collapse from its hydrostatic state. The collapse leads to arapid in fall of material from the surrounding nebula to replace the condensing gas. It isbelieved that such a process allows the accretion of hundreds of Earth masses of gas.Three-dimensional hydrodynamical calculations examining the interaction of a pro-toplanet with a disc have usually assumed a locally isothermal equation of state, and/ordo not resolve the gas flow far inside the planet’s Hill radius [4–13]. Recent hydrody-namical calculations have also considered different equations of state and/or radiationtransport [10, 12, 14–16]. Fouchet & Mayer [17] include both radiative transfer and self-gravity, but once again the resolution of gas flow near to the planetary core is limited.In this work we present results from three-dimensional self-gravitating radiative transferhydrodynamical calculations that are able to resolve gas flow down to realistic core radii.We vary the core/protoplanet masses and the grain opacity of the nebula to examine theeffects that these properties have upon the gas accretion rates of the growing planets.
MODEL SETUP
We model 10 - 333 M
Earth solid cores/protoplanets embedded in a circumstellar discwith a surface density of 75 g cm − orbiting a 1M (cid:12) star at 5.2 AU. The initial surface a r X i v : . [ a s t r o - ph . E P ] M a y ensity has a Σ ∝ r − / profile, whilst the initial temperature profile of the disc is T ∝ r − , yielding a constant disc scaleheight with radius of H / r = .
05. We vary thegrain opacity of the disc from the interstellar grain opacity (IGO) down to 0.1% of thisvalue to examine the effects this has upon the disc cooling.To achieve high resolution near a protoplanet at reasonable computational cost wemodel only a small section of the protoplanetary disc centred on the planetary core. Oursection size is 5 . ± .
78 AU, and φ = ± .
15 radians; for details see [18].
RESULTS & DISCUSSION
The accretion rates for the different mass cores/protoplanets that we modelled underdiffering opacities are shown in the left panel of figure 1. The trend in accretion rateagainst core/protoplanet mass is as would be expected, increasing as the central bodiesmass increases. There is a reduction in accretion rates for the highest mass protoplanetscaused by the evacuation of a large gap in the encompassing circumstellar disc bythese protoplanets. Such a gap reduces the material available in the feeding zone ofthe protoplanet. The accretion rates also show an increasing trend with reduced grainopacities for the low mass cores. Reduced grain opacities allow the infalling materialto cool more readily via radiation, and so the thermal pressure supporting it againstcapture is reduced. The only exception to this is at the lowest grain opacity of 0.1%IGO for the lowest mass core where the radiation and gas fields interact so little that thetwo fields decouple, preventing effective gas cooling via radiation. For the high massprotoplanets the grain opacity has very little impact on the accretion rates as the processis gravitationally dominated.The right hand panel of figure 1 compares the accretion on to a 10M
Earth solid core inour models with the accretion onto 5 and 15 M
Earth cores in the one-dimensional modelsof Papaloizou & Nelson [19]. It can be seen that the results show good agreement, sug-gesting that for at least the phase of accretion preceding runaway growth, the transitionfrom one-dimensional models to three-dimensional models does not have a significantimpact on the accretion rates.Figure 2 illustrates the differences between radiation hydrodynamical and isothermalcalculations. Features in the vicinity of the planet that are well established under isother-mal conditions, such as spiral shocks and circumplanetary discs, are weakened in theradiative transfer calculations. The thermal pressure acts to smear out these features. Itcan also be seen in figure 2 that reducing the grain opacity leads to a model that is moresimilar to the isothermal case. The increased cooling seen at lower opacities reduces thethermal pressure, thus mitigating its impact on the shocks and circumplanetary discs.
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IGURE 1. a.
Protoplanet accretion rates. The asterisks mark the accretion rates for our SPH calcula-tions using a locally-isothermal equation of state, essentially providing upper limits to the accretion ratesobtainable with various protoplanet masses. The diamonds mark the results of [9], and the plus signsthose of [6], connected with solid lines. These calculations were also locally-isothermal but were globaldisc simulations performed using the ZEUS code. The SPH and ZEUS accretion rates are in reasonableagreement. The accretion rates from our self-gravitating radiation hydrodynamical SPH calculations usingcore radii of 1% of the protoplanet’s Hill radius are given using line types that denote the different grainopacities. Results are shown using standard interstellar grain opacities (IGO) (dotted), 10% IGO (short-dashed), 1% IGO (dot-dash), and 0.1% IGO (long dashed). The inclusion of radiative transfer substantiallylowers the accretion rates of low-mass protoplanets, but Jupiter-mass protoplanets have similar accretionrates to the locally-isothermal result, regardless of the grain opacity. The analytic approximation of [20]is shown by the solid curved line. b. Accretion rates versus the gas mass accreted by planetary cores for1% (thin lines) and standard (thick lines) grain opacities. We plot the results of [19] for 5 M
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IGURE 2.
Surface density plots for locally-isothermal calculations (left), and self-gravitating radiationhydrodynamical calculations using 1% IGO (centre), and standard IGO (right) calculations with ourstandard protoplanetary disc surface density. From top to bottom the protoplanet masses are 22, 33, 100,166 and 333 M