NoteMeridional circulation of gas into gaps opened by giant planets in three-dimensional low-viscosity disks
Introduction
Understanding what sets the terminal mass of a giant planet in a runaway gas accretion regime is an open problem in planetary science. Runaway gas accretion is the third stage of the classical core-accretion scenario for the formation of giant planets (Pollack et al., 1996). We remind the reader that in stage I, a solid core of 5–10 Earth masses () is formed by planetesimal accretion (or possibly by pebble-accretion; see Lambrechts and Johansen, 2012, Morbidelli and Nesvorny, 2012). In stage II, the core starts to capture gas from the protoplanetary disk, forming a primitive atmosphere in hydrostatic equilibrium; the continuous accretion of planetesimals heats the planet and prevents the atmosphere from collapsing. In stage III the combined mass of core and atmosphere becomes large enough (the actual mass-threshold depending on opacity and energy input due to planetesimal bombardment) that the latter cannot remain in hydrostatic equilibrium anymore; thus the planet enters in an exponential gas-accretion mode, called runaway.
The accretion timescale in the runaway accretion mode is very fast and, once started, can lead to a Jupiter-mass planet in a few (see for instance the hydro-dynamical simulations in Kley, 1999, D’Angelo et al., 2003, Klahr and Kley, 2006, Ayliffe and Bate, 2009, Tanigawa et al., 2012, Gressel et al., 2013, Szulagyi et al., 2014) in a proto-planetary disk with mass distribution similar to that of the Minimum Mass Solar Nebula model (Weidenschilling, 1977, Hayashi, 1981). This rapid accretion mechanism can explain how giant planets form. On the other hand, there is no obvious reason for this fast accretion to stop. As its timescale is much shorter than the proto-planetary disk life time (a few My Haisch et al., 2001), it is unlikely that the disk disappears just in the middle of this process, raising the question why Jupiter and Saturn and many giant extra-solar planets did not grow beyond Jupiter-mass.
It is well known that giant planets open gaps in the protoplanetary disk around their orbits (Lin and Papaloizou, 1986, Bryden et al., 1999). Thus it is natural to expect that the depletion of gas in the planet’s vicinity slows down the accretion process. Still, all of the hydro-dynamical simulations quoted above that feature the gap-opening process show that the mass-doubling time for a Jupiter-like planet is not longer than .
However, these simulations may have been hampered by the assumption of a prescribed viscosity throughout the protoplanetary disk, or by significant numerical viscosity in the simulation scheme. It is expected that planets form in dead zones of the protoplanetary disk (Gammie, 1996), where the viscosity is much smaller than in numerical simulations, so that Jovian-mass planets could presumably open much wider and deeper gaps, with consequent inhibition of further growth (e.g. Thommes et al., 2008, Matsumura et al., 2009, Ida and Lin, 2004).
On this issue, it is worth stressing that there is quite of a confusion on the role of viscosity in gap opening. In a 2D disk isothermal model, Crida et al. (2006) showed that the width and the depth of a gap saturates in the limit of vanishing viscosity. This paper has been challenged recently by Duffell and MacFadyen, 2013, Fung et al., 2013, still for 2D disks. For a massive planet, the results of Duffell and MacFadyen actually agree with those of Crida et al., because the former group also finds that the gap has a saturated depth and width in the limit of null viscosity and they demonstrate that this result is not due to numerical viscosity. The actual disagreement is on the ability of small planets to open gaps in disks much thicker than their Hill sphere. Fung et al. address this case just by extrapolation of formulæ obtained for massive planets in viscous disks, so it is not very compelling. We believe that gap opening by small planets in Duffel and MacFadyen is due to the use of an adiabatic equation of state which, despite adopting a value for the parameter very close to unity, is not equivalent to the isothermal case (see Paardekooper and Mellema, 2008). The issue, however, deserves further scrutiny.
This controversy is nevertheless quite academic, because real disks have a 3 dimensional structure. Thus, in this Note we discuss gap opening in 3D disks and we focus on giant planets that are massive enough to undergo runaway gas accretion, i.e. Jupiter-mass bodies. In Section 2 we present the structure of the gaps and the gas circulation in their vicinity, using three-dimensional isothermal simulations. We also interpret the results with a simple intuitive model. From this model, we derive in Section 3 an estimate for the flow of gas into a gap in a layered disk (i.e. a disk that is viscous on the surface and “dead” near the mid-plane), that is in agreement with the numerical results of Gressel et al. (2013). Conclusions and discussion of the implications for terminal mass problem of giant planet are reported in Section 4.
Section snippets
Gaps in 3D disks
In the framework of a study on planet accretion (Szulagyi et al., 2014), we conducted 3D simulations of a Jupiter-mass planet embedded in an isothermal disk with scale height of 5% and -prescription of the viscosity (Shakura and Sunyaev, 1973). We adopted (viscous simulation) and (inviscid simulation). The latter simulation was conducted with two different resolutions, to change the effective numerical viscosity of the simulation code. The technical parameters of the simulations
Implications on planet’s accretion from the flow of gas into the gap
The results of the previous section seem to suggest that the flow of gas into the gap and the planet’s accretion rate have to vanish with viscosity. But remember that what governs the meridional circulation of gas in the gap’s vicinity is the viscous timescale near the disk’s surface. Protoplanetary disks cannot be fully inviscid: stars are observed to accrete mass, which implies that angular momentum has to be transported at least in part of the disk. A popular view is that of a layered disk,
Conclusions
In this paper we have analyzed the dynamics of gas near gaps, opened by planets in three dimensional proto-planetary disks, with particular emphasis on the small viscosity limit.
We observe a flow of gas into the gap at high altitude in the disk, at a rate dependent on effective viscosity (prescribed or numeric). We have explained this result with a simple analytic model, that assumes that the disk is in vertical hydrostatic equilibrium at all radii. Thus the radial profile of the gas volume
Acknowledgments
The Nice group is thankful to ANR for supporting the MOJO project. J. Szulágyi acknowledges the support from the Capital Fund Management’s J.P. Aguilar Grant. The computations have been done on the “Mesocentre SIGAMM” machine, hosted by the Observatoire de la Côte d’Azur. T.T. is supported by Grant-in-Aid for Scientific Research (23740326 and 24103503) from the MEXT of Japan. We also thank P. Duffell for open and frank discussions on gap opening by small planets in 2D disks.
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