天文学概論7

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V Eimp = - p dV V0 d ~ Eimp [g/cm3] f ~ N 2.40 f 3 2 dN Eroll Suyama et al. submitted to ApJ 10-4g

random numbers that produce initial distribution of planetesimals. Their results are qualitatively the same. We also performed simulations of 2-D accretion where orbits of planetesimals are confined to a plane and compared with 3-D a OLIGARCHIC GROWTH OF PROTOPLANETS 177 is the runaway stage, while most planetesimals remain small. The obtai typical orbital separation of protoplanets kept while grow- more ing is about 10rH. This value depends only weakly on the mass of protoplanets, the surface density of the solid mate- 4.1. R rial, and the semimajor axis. This self-organized structure is a general property of self-gravitating accreting bodies Sna in a disk when gravitational focusing and dynamical friction t5 are effective. 1b. T If we assume that the oligarchic growth continues till circle the final stage of planetary accretion, the mass of proto- numb planets is estimated by M 2ab. In the solar nebula model that is 50% more massive than the minimum mass creas model, the surface mass density of the solar nebula is tion, given by locall where 3/2 a the la 10 [g cm2] a 2.7 AU 1 AU larges (12) a 3/2 defini 4 [g cm2] a 2.7 AU. 1. It is 5 AU mass, Adopting this and b 10rH, we have M 0.2M and mass b 0.07 AU at 1 AU ( 10 g cm2), M 7M and away bFIG. AU The maximum massgof the), and M 17M and 2 3. at 7 AU ( 2.4 cm2 planetesimals (solid curve) and away their mean mass except the maximum (dashed curve) are plotted as a b 8 AU at 25 AU ( 0.36 g cm2), where M is the function of time. Earth mass. In the terrestrial planet region, the estimated times mass and the orbital separation of protoplanets are still smaller than the present planets. This may suggest that oligarchic growth does not continue till the final stage of FIG. 4. The same as Fig. 1 but for the system initially consists of planetary accretion in the terrestrial planet region. The 4000 equal-mass planetesimals (m 3 1023 g). The radius increase orbital separation may get larger in the terrestrial planet

1134 KOKUBO, KOMIN Fig. 2.—Snapshots of the system on the a-e (left) and a-i (right) planes at t ¼ 0, 1 are proportional to the physical sizes of the planets. planets is hnM i ’ 2:0 Æ 0:6, which means that the typical result- ing system consists of two Earth-sized planets and a smaller planet. In this model, we obtain hna i ’ 1:8 Æ 0:7. In other words, one or two planets tend to form outside the initial distribution of protoplanets. In most runs, these planets are smaller scattered planets. Thus we obtain a high efﬁciency of h fa i ¼ 0:79 Æ 0:15. The accretion timescale is hTacc i ¼ ð1:05 Æ 0:58Þ ; 108 yr. These results are consistent with Agnor et al. (1999), whose initial con-

1226 MACHIDA ET AL. 1.— Time sequence for model M04. The density (color scale) and velocity distributions (arrows) on the cross section in the z ¼ 0 plane are plotted. The bottom ˜ ¼ 3) are 4 times the spatial magniﬁcation of the top panels (l ¼ 1). Three levels of grids are shown in each top (l ¼ 1, 2, and 3) and bottom (l ¼ 3, 4, and 5) panel. l of the outermost grid is denoted in the top left corner of each panel. The elapsed time ˜p and the central density c on the midplane are denoted above each of the t ˜ ls. The velocity scale in units of the sound speed is denoted below each panel.

15 12 9 6 3 0 0.0001 0.001 0.01 0.1 1

random velocity of planetesimals is pumped up as high as the escape velocity of protoplanets. This high random veloc- On the ot ity makes the accretion process slow and inefficient and thus in circular o Tgrow longer. This accretion inefficiency is a severe problem HD 192263 with Æ1 e1 for in situ f Mdisk T cont <Tdisk Tgrow<Tdisk case. It is di slingshot m circular orb the magneti may be wea disks may b Terrestria Jovian plan planetary a key process systems. We confir holds in a Æsolid ¼ Æ1 ð ¼ 1=2; 3= tions. We d Fig. 13.—Schematic illustration of the diversity of planetary systems systems dep against the initial disk mass for < 2. The left large circles stand for central stars. The double circles (cores with envelopes) are Jovian planets, and the disk profile others are terrestrial and Uranian planets. [See the electronic edition of the growth time

earing continues through scattering. After orbital time scales and high inclinations. three categories: (i) hot Earth analogs interior to 00 million years the inner disk is composed Two of the four simulations from Fig. 2 the giant planet; (ii) Bnormal[ terrestrial planets the collection of planetesimals at 0.06 AU, a contain a 90.3 M] planet on a low-eccentricity between the giant planet and 2.5 AU; and (iii) M] planet at 0.12 AU, the hot Jupiter at 0.21 orbit in the habitable zone, where the temper- outer planets beyond 2.5 AU, whose accretion U, and a 3 M] planet at 0.91 AU. Previous ature is adequate for water to exist as liquid on has not completed by the end of the simulation. sults have shown that these planets are likely a planet_s surface (23). We adopt 0.3 M] as a Properties of simulated planets are segregated be stable for billion-year time scales (15). lower limit for habitability, including long-term (Table 1): hot Earths have very low eccentric- Many bodies remain in the outer disk, and ac- climate stabilization via plate tectonics (24). ities and inclinations and high masses because they accrete on the migration time scale (105 niscent of the recently discovered, close-in 7.5 M] years), so there is a large amount of damping planet around GJ 876 (25), whose formation is during their formation. These planets are remi- also attributed to migrating resonances (26). g. 1. Snapshots in time of the evolution of one simulation. Each panel of each body’s inclination on the y-axis scale. The color of each dot ots the orbital eccentricity versus semimajor axis for each surviving body. corresponds to its water content (as per the color bar), and the dark inner he size of each body is proportional to its physical size (except for the dot represents the relative size of its iron core. For scale, the Earth’s water ant planet, shown in black). The vertical ‘‘error bars’’ represent the sine content is roughly 10j3 (28).

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天文学概論7

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