The CAGC computer model

At the time of this conference, there are four groups that have computer models based on the CAGC formation of gas giant planets: the collaborators at NASA-Ames Research Center and University of California at Santa Cruz (referred to ARC/UCSC group); the group in Japan; the group in Bern, Switzerland; and G. Wuchterl. The first three groups use a similar technique based on a modified stellar structure evolution code, and Wuchterl uses a fully hydrodynamical computer code to model the evolving protoplanet. The discussion of the computer code technique will be concentrated on the one used by the ARC/UCSC group, and variations on this work by others will be noted and described.

The ARC/UCSC code consists of three main components:

(1) The calculation of the rate of solid accretion onto the protoplanet with an updated version of the classical theory of planetary growth (Safronov 1969) to calculate the rate of growth of the solid core. The gravitational enhancement factor, which is the ratio of the total effective accretion cross section to the geometric cross section, is an analytical expression that was derived to fit the data from the numerical calculations of Greenzweig & Lissauer (1992), consisting of a large number of three-body (Sun, protoplanet, and planetesimal) orbital interaction simulations.

(2) The calculation of the interaction of the accreted planetesimals with the gas in the envelope (Podolak et al. 1988), which determines whether the planetesimals reach the core, are dissolved in the envelope, or a combination of the two. Calculations of trajectories of planetesimals through the envelope result in the radius in the envelope at which the planetesimal is captured (required to compute the accretion rate of the plan-etesimals), and the energy deposition profile in the envelope (required for the structure computation).

(3) The calculation of the gas accretion rate and evolution of the protoplanet, under the assumption that the planet is spherical and that the standard equations of stellar structure apply. The conventional stellar structure equations of conservation of mass and energy, hydrostatic equilibrium, and radiative or convective energy transport are used. The energy generation rate is the result of the accretion of planetesimals and the quasi-static contraction of the envelope.

The following assumptions were applied in the computer simulation:

(1) The growing protoplanet is a lone embryo, which is surrounded by a disk consisting of planetesimals with the same mass and radius. There is an initially uniform surface density in the region of the protoplanet.

(2) The protoplanet's feeding zone is assumed to be an annulus extending to a radial distance of about four Hill-sphere radii on either side of its orbit (Kary & Lissauer 1994), which grows as the planet gains mass. Planetesimals are spread uniformly over the zone and do not migrate into or out of the feeding zone.

(3) The equation of state is nonideal and the tables used are based on the calculations of Saumon et al. (1995), interpolated to a near-protosolar composition of X = 0.74, Y = 0.243, Z = 0.017. The opacity tables are derived from the calculations of Pollack et al. (1985) and Alexander & Ferguson (1994).

(4) The capture criterion includes planetesimals that deposit 50% or more of their mass into the envelope during their trajectory.

(5) Once the mass and energy profiles in the envelope have been determined, the planetesimals are assumed to sink to the core, liberating additional energy in the process.

(6) In order to account for the depletion of the planetesimal disk by accretion onto neighboring embryos, the rate of planetesimal accretion near gas runaway is limited to its value at crossover.

The inner and outer boundary conditions are set at the bottom and top of the envelope, respectively. The core is assumed to have a uniform density and to be composed of a combination of ice, CHON, and rock, depending on the conditions in the nebula in which the planet forms. The outer boundary condition of the protoplanet is applied in three ways, depending on the evolutionary stage of the planet (as noted in Section 10.1 and described in full in BHL00).

The calculations start at t = 104 yr with a core mass of 0.1 and an envelope mass of 10~9 M®. These initial values were chosen for computational convenience, and the final results are insensitive to initial conditions. Near the end of the evolution of the protoplanet, the gas accretion rate is limited by the ability of the solar nebula to supply gas at the required rate. When this limiting rate is reached, the planet contracts inside its accretion radius (evolution enters the transition stage). The supply of gas to the planet is eventually assumed to be exhausted as a result of tidal truncation of the nebula, the removal of the gas by effects of the star, and/or the accretion of all nearby gas by the planet. The planet's mass levels off to the limiting value defined by the object that is being modeled. Since this process is not yet modeled in the ARC/UCSC code, the gas accretion rate onto the planet is assumed to reduce smoothly to zero as the limiting mass value is approached. The planet then evolves through the isolated stage, during which it remains at constant mass.

Figure 1 illustrates the typical nature of the simulations. The mass, luminosity, solid and gas accretion rates, and radii are plotted as a function of time. The solid core is accreted during Phase 1. This phase ends when the feeding zone is depleted of solids, thus the protoplanet reaches its isolation mass. Phase 2 is characterized by a steady rate of both solid and gas accretion, but with the gas rate being slightly greater. The duration of Phase 2 ends at the crossover point, when the gas mass is equal to the solid mass. It is evident that Phase 2 determines the overall timescale for the protoplanet to form, since this time is much longer than the times for the solid core to accrete and for the gas runaway to occur. Phase 3 is characterized by the gas runaway, which lasts until both the gas and solid accretion rates turn off. The protoplanet then cools and contracts to its presently observed state.

The Bern group's CAGC code (Alibert et al. 2005) is similar to the ARC/UCSC code except for the inclusion of the components that compute the evolution of the proto-planetary disk and the migration of the growing protoplanet. The disk structure and its evolution are based on the method of Papaloizou & Terquem (1999), which is in the framework of the a-disk formulation of Shakura & Sunyaev (1973). Migration occurs when there is a dynamical tidal interaction of the growing protoplanet with the disk, which leads to two phenomena: inward migration and gap formation (Lin & Papaloizou 1979; Ward 1997; Tanaka et al. 2002). For low-mass planets, the tidal interaction is a linear function of mass and the migration is Type I (i.e., inward migration with no gap opening). Higher-mass planets open a gap, leading to a reduction of the inward migration; this is referred to as type II migration. The rest of the procedure used by the Bern group is similar to that used by the ARC/UCSC group: the solid accretion rate uses the gravitational enhancement factor based on that of Greenzweig & Lissauer (1992); the interaction between the infalling planetesimal and the atmosphere of the growing planet is based on the work of Podolak et al. (1988); and, the standard planetary structure

60 e 40

en o

60 e 40

1 mxy j

; contract ;

4 capt surf :

r 1: 2 core

: cool : * Sc ~ x contract :

Figure 1. The mass (units of M® ), the luminosity (units of Lq ), the accretion rates (units of M®/year), and the radii (units of Bjup) are plotted as a function of time (units of million years) for the baseline case 10LX. The three phases of evolution and the cooling and contraction of the protoplanet are marked. Dotted line: Phase 1, solid line : Phase 2, second dotted line : Phase 3, dashed line : contraction and cooling.

and evolution scheme is applied. The rate for type I migration is a free parameter in the Alibert et al. (2005) calculations, therefore, the starting location of the embryo is adjusted for each choice of the migration rate, in order for the protoplanet to reach the crossover mass at 5.5 AU. The profile of the initial disk surface density, ainitjz, is the same for each of the migration rates, namely a power law <rinit,z r-2, which was chosen to correspond to a case with <rinit,z = 7.5 g cm2), which is about the twice the density of a minimum mass solar density (Paper 1).

The analysis of the gas flow in the envelope that was undertaken by Wuchterl (1991a,b, 1993) using a hydrodynamic code to study the flow velocity of the gas by solving an equation of motion for the envelope gas in the framework of convective radiation-fluid dynamics. This allows the study of the collapse of the envelope, of the accretion with finite Mach number, and of linear, adiabatic and nonlinear, non-adiabatic pulsational stability of the envelope. Furthermore, the treatment of convective energy transfer has been improved by calculations using a time dependent mixing-length theory of convection in hydrodynamics (Wuchterl 1995, 1999). Wuchterl starts his calculations at critical mass, Mcrit, and finds that instead of collapsing as Mizuno surmised from his models, these envelopes begin to pulsate, resulting in mass loss. A state of quasi-equilibrium is achieved after the protoplanet loses a large fraction of its envelope mass. A major result of the hydrodynamical studies is that the protoplanet may pulsate and develop pulsation-driven mass loss. This leaves a planet with a low-mass envelope and properties similar to Uranus and Neptune, but the model does not account for Jupiter and Saturn. His results are contrary to the linear stability analysis of Tajima & Nakagawa (1997), who find the envelope models in Bodenheimer & Pollack (1986) to be dynamically stable. This issue with the quasi-static models is still unresolved.

The group in Japan continues to work on all aspects of planet formation. Kokubo & Ida (1998, 2002) studied the formation of planets and planet cores by oligarchic growth. Ikoma et al. (2000, 2001) examined the effect of the opacity of the grains in the protoplanet's growing envelope and the rate of solid accretion on formation timescales. Ida & Lin (2004) analytically investigated the migration of planets.

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