Recent results

There are two major computer simulation studies based on the CAGC model. The ARC/UCSC group examines the issue of the conditions for which Jupiter could have formed with a low mass core and a short formation timescale (HBL05). The Bern group considered the effects on giant planet formation with the inclusion of migration and protoplanetary evolution in the core accretion formation model (Alibert et al. 2005). Highlights of these two studies are presented below.

The ARC/UCSC group's simulations of the growth of Jupiter were computed for three parameters shown in Paper 1 to affect planet formation. The opacity produced by grains in the protoplanet's atmosphere was varied, and two different values (10 g cm2 and 6 g cm2) were used for the initial planetesimal surface density in the solar nebula. Additionally, halting the solid accretion at selected core-mass values during the protoplanet's growth was studied. Decreasing the atmospheric opacity due to grains emulates the settling and coagulation of grains within the protoplanetary atmosphere, and halting the solid accretion simulates the presence of a competing embryo. The effects of these parameters were examined in order to determine whether gas runaway can still occur for small-mass cores on a reasonable timescale (Ikoma et al. 2000).

The nomenclature for the simulations that denotes the parameters used in the computations is in the following form: a-opacity-cut, where a is the initial surface density of planetesimals in the solar nebula with values 10 or 6 g cm2; opacity is denoted by either L for grain opacity at 2% of the interstellar value, H for the full interstellar value, or V for a variable (temperature dependent: T < 350 K ramping up to the full interstellar value for T > 500 K) grain opacity; and cut specifies the core mass (in units of M®), at which the planetesimal accretion rate is turned off. For cases with no solid accretion cutoff, cut is set to to. As an example, the model labelled 10LTO signifies that the simulation was

146 O. Hubickyj: Core accretion model

10 0

10 0

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Figure 2. a) The masses (units of M®) of the four basic cases are plotted as a function of time (units of million years). The solid line denotes the mass of solids and the dashed line denotes the mass of gas. b) The masses of the baseline case and the associated cutoff cases are plotted as a function of time. Units and line designations are the same as in a).

computed with ainit,z = 10 g cm2, the grain opacity is 2% of the interstellar value, and there was no solid accretion cutoff.

Four series of simulations have been computed in this latest study. Each series consists of a run computed through the cooling and contracting of the protoplanet (i.e., Fig. 1), plus up to three runs with a cutoff of planetesimal accretion at a particular core mass. For these four basic cases, mass as a function of time is plotted in Figure 2a. The reduced grain opacities produce formation times that are less than half of that for models computed with full interstellar grain opacity values (see curves labelled 10Hnd 10LTO in Fig. 2a). These models illustrate that the time spent in Phase 2 is decreased by ^60% for models with the grain opacity set to 2% of the interstellar value. Therefore, another model was computed to determine if there was a temperature range for which the grain opacity had the most influence on the evolution time. This model (10^was computed with the grain opacity set to 2% of the interstellar value for temperatures < 350 K and to the full interstellar value for temperatures >500 K, with interpolation in the intermediate region. The result of this calculation shows there is little difference from the model computed with the 2% interstellar value for the full temperature range. The reduction of opacity due to grains in the upper portion of the envelope with T < 500 K has the largest effect on the lowering of the formation time (see curves labelled 10H10LTO, and 10^ in Fig. 2a). This result is profoundly important, especially with reference to the grain settling work of Podolak (2003), who has developed a numerical model for the growth and sedimentation of grains in a protoplanetary atmosphere, coupled with a procedure

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Figure 2. a) The masses (units of M®) of the four basic cases are plotted as a function of time (units of million years). The solid line denotes the mass of solids and the dashed line denotes the mass of gas. b) The masses of the baseline case and the associated cutoff cases are plotted as a function of time. Units and line designations are the same as in a).

for calculating the opacity at each depth. These simulations indicate grain opacity values that are lower than the 2% ISM grain values we used in our most recent calculations.

Motivated by the interior models of Jupiter and Saturn by Guillot et al. (1997) and Saumon & Guillot (2004), who call for low solid mass cores for Jupiter and Saturn, the ARC/UCSC group studied the effect of the surface density of planetesimals and the effect of halting solid accretion on the formation of the protoplanet. Decreasing the surface density of planetesimals lowers the final core mass of the protoplanet, but increases the formation timescale considerably (see curves labelled 10L' and 6L' in Fig. 2a).

The effect of halting solid accretion is illustrated in Figure 2b. The plot shows the mass as a function of time for the baseline case 10L' and the three associated runs for which the solid planetesimal accretion is turned off at core masses 10, 5, and 3 M®. It is clearly demonstrated that the time needed for a protoplanet to evolve to the stage of runaway gas accretion is reduced, provided the cutoff mass is sufficiently large. The overall results indicate that, with reasonable parameters and with the assumptions in the ARC/UCSC CAGC model code, it is possible that Jupiter formed via the core accretion process in 3 Myr or less.

Migration is more than likely a viable aspect of gas giant planet formation in explaining the wide range of eccentricities and semi-major axes, like the hot-Jupiter type extrasolar planets (Jupiter-sized planets found in orbits very close to their central star), deduced from the observations of extrasolar planets. Alibert et al. (2005) incorporated migration into their core accretion computer simulation and examined the effects on giant planet formation. Their results show that the formation timescale of gas giants is much shorter, by a factor of 10, when migration is included compared to in situ formation. A migrating embryo starting at 8 AU will migrate to 5.5 AU and reach crossover mass in —1 Myr, whereas the same embryo at 5.5 AU without migration and without disk evolution (i.e., in situ formation) reaches crossover mass in —30 Myr, a factor of 10 greater. The reason for this speed-up due to migration is quite simple. In CAGC formation models, the long formation timescale depends on the presence of Phase 2 ocurring after the core is isolated at the end of Phase 1. A migrating embryo will never suffer isolation and will go directly from Phase 1 to Phase 3, reducing the formation time.

It should be noted that the in situ model computed by Alibert et al. (2005) should not be compared with the newer models in HBL05. In fact, the parameters chosen for the model discussed in Alibert et al. (2005) are those of a protoplanet growing in a pro-toplanetary disk which is twice that of a minimum-mass solar nebula—not three to four times as dense, as was used in the ARC/UCSC models.

Though most of the computational work is based on Jupter and Saturn models, the understanding derived from these computer simulations can be applied to extrasolar planets—as evidenced by the observation trait of high-metal planets. It is not unreasonable to apply the conclusions to extrasolar planets. The observational studies of extrasolar planets have shown that planets are discovered much more frequently around metal-rich stars (Gonzalez 1998; Santos et al. 2001, 2004; Fischer & Valenti 2003, 2005). While this trend has been found to be consistent with simplified core-accretion models (Ida & Lin 2004; Kornet et al. 2005), it has also been suggested that the correlation is a result of preferential migration of planets in high-metallicity disks into the period range where they are observed (Sigurdsson et al. 2003). Sozzetti (2004) suggests that there is a correlation between observed orbital period and the host star metallicity in the sense that the higher-metallicity stars are more likely to have short-period planets. This tendency would be consistent with the migration scenerio; however, the correlation is weak, and Santos et al. (2003) do not find it. On the theoretical side, the simple model of Livio & Pringle (2003) results in only a small difference in migration rates in metal-rich and metal-poor disks, not sufficient to explain the trend seen in Fischer & Valenti (2003). Thus, this correlation is more likely to be a result of the formation mechanism itself. Although a higher-metallicity planet has a higher opacity in the envelope that results in longer formation times, increases in opacity by only a factor of two have a very small effect on the time; factors of 50 or more in opacity changes are required to make significant differences in formation times. More study on this topic is necessary.

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