Abstract
In the Jovian planet region, after the formation of protoplanets, they begin to capture the surrounding nebular gas gravitationally and become gaseous giant planets such as Jupiter and Saturn (Mizuno 1980, Bodenheimer and Pollack 1986, Ikoma et al. 2000). The formation time of these gaseous planets depends strongly on a solid core mass (i.e., a protoplanet's mass). Especially, Ikoma et al. (2000) pointed out that, in order to form massive gaseous envelopes of the present Jovian planets around protoplanets within the nebular life time, a protoplanet's mass must exceed a critical mass. However, we find that a typical mass of a protoplanet formed in the Jovian planet region in the minimum mass solar nebula model is smaller than the above-mentioned critical mass. This means that a protoplanet formed in the minimum mass solar nebula cannot become the present gaseous giant planets alone.
One of the possible ways to resolve this difficulty in the formation of Jovian planets is to consider the collision and accretion between protoplanets. If we think about this possibility, we are confronted with a difficult problem. In the presence of the nebular disk, a protoplanet system is expected to be prevented from undergoing an orbital instability (and, as a matter of course, collisions between protoplanets), since the nebular gas has an effect of suppressing eccentricities of protoplanets through gravitational and hydrodynamical interaction (Adachi et al. 1976, Ward 1988, Artymowicz 1993). Thus, we need to investigate the orbital stability of a protoplanet system in the nebular disk in detail.
In the present study, we investigated the orbital stability of a protoplanet system in the nebular gas through orbital calculations, for the cases where the masses of protoplanets are equally 3 times as massive as that of the earth. The tidal (gravitational) interaction between a protoplanet and the nebular gas is taken into account as a drag force proportional to the random velocity of a protoplanet. In numerical simulations, five equal-mass protoplanets are distributed with an equal separation distance and their initial eccentricities and inclinations are set to be zero.
Through long term orbital calculations, we obtained the following results:(1) The logarithm of the orbital instability time of a system without a gas disk increasesin proportion to the separation distance between protoplanets.(2) In the presence of the nebular gas, the instability time of a system becomes extremely large compared with the orbital instability time in the absence of the gaseous disk by the effect of the drag force due to the nebular gas and the system substantially doesn't experience an orbital instability, when the separation distance is larger than a critical separation distance. (3) The value of a critical separation distance becomes large with a decrease in the surface density of the nebular gas.
Furthermore, we obtained a semi-analytical expression for a critical separation distance, partly using our simulation results. Finally, applying our results to the orbital stability of a protoplanet system in the Jovian planet region, we found that in the presence of the minimum mass solar nebula, the orbital instability of a protoplanet system never occurs in the Jovian planet region. Considering that, in order to form the gaseous giant planets, the remaining gaseous disk around protoplanets must be as massive as the minimum mass solar nebula, we concluded that the protoplanets did not experience orbital instabilities in the formation process of Jovian planets.
