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The formation of giant planets is explained by the nucleated instability model, in which a solid core captures a large amount of nebular gas when it grows to critical core mass. It is well known that critical core mass scarcely depends on the boundary conditions of the envelope, i.e., its distance from the central star and the density and temperature of the nebular gas. However, this is not the case when the envelope is wholly convective. Such a situation is realized if we consider the formation of giant planets close to central stars and/or in dense cool nebulae. In the present study, we extensively investigate the dependence of the critical core mass on the distance from the central star and on the density and temperature of the nebular gas ; we found that the critical core mass reduces to 2È3 at 0.1 AU in dense nebulae Mŵ ith a surface density about 20 times larger than that in the minimum-mass solar nebula model. This result suggests a possibility of in situ formation of the detected extrasolar giant planets close to the central stars.

We investigated the effects of a tidal interaction with a gas disk and the dynamical friction with a planetesimal disk on the orbital instability of a protoplanet system. Both effects are expressed as the drag force, which is proportional to the random velocity of a protoplanet. We calculated numerically the orbits of 5 protoplanets with the same separation distance under the drag-force effect and examined the orbital instability time under the drag force, $T_{\mathrm{inst}}^{\mathrm{df}}$. We found that $T_{\mathrm{inst}}^{\mathrm{df}}$ can become much larger than the instability time under the drag-free condition, and that the onset of the orbital instability is prevented when the separation distance exceeds a critical value. We obtained a relation between the critical separation distance and the surface density of the gas or planetesimal disk. By applying this relation, we found that, for the formation of terrestrial planets from a protoplanet system with a typical orbital separation (i.e., $\sim 10$ Hill radii), the surface density of the nebular gas must be reduced to about one-thousandth of that in the minimum-mass nebula model. Terrestrial planets would be formed after such a depletion of the solar nebula.

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We have investigated numerically the orbital instability of a protoplanet system while taking account of the gas-drag force due to the solar nebula. In the present work, we considered an equally spaced five-protoplanet (with the same mass of $1 \times 10^{-7} {{{M}_{\odot}}}$) system, in which their initial orbits are coplanar and circular, and assumed that the gas-drag force is proportional to the square of the relative velocity between the gas and a protoplanet. We first re-examined and confirmed that, under a gas-free condition, $\log_{10} T_{\mathrm{inst}}$ can be approximately written as a linear function of the initial orbital separation distance, $\Delta \tilde{a}_{0}$, where $T_{\mathrm{inst}}$ is the time of the orbital instability (i.e., the time of the first orbital crossing between any two protoplanets). Next, we investigated the instability time under the gas-drag effect, $T_{\mathrm{inst}}^{\mathrm{gas}}$, and found that $T_{\mathrm{inst}}^{\mathrm{gas}}$ suddenly becomes large compared with $T_{\mathrm{inst}}$, when ${\Delta \tilde{a}}_{0}$ is larger than a certain critical separation distance, ${({\Delta \tilde{a}}_{0})}_{\mathrm{crit}}$. Furthermore, we showed that ${({\Delta \tilde{a}}_{0})}_{\mathrm{crit}}$ can be described semi-analytically as a function of the gaseous density. From a function extrapolated with a density in the minimum mass nebula model, we estimated ${({\Delta \tilde{a}}_{0})}_{\mathrm{crit}}$ in the nebula as being about 10 Hill radius at 1 AU.