天文学概論（第５回）

星惑星形成２～太陽系形成論～

東京工業大学　佐々木貴教

古典的太陽系形成論モデル（京都モデル）・原始惑星系円盤（原始太陽系円盤）・微惑星から原始惑星、そして惑星へ地球の形成・初期進化

星惑星形成２～太陽系形成論～

京都モデルの基本概念円盤仮説・惑星系は原始惑星系円盤から形成される・円盤はガスとダストから構成される

微惑星仮説・ダストの集積によって微惑星が形成される・微惑星の集積によって固体惑星が形成される・固体惑星にガスが降り積もることによって　ガス惑星が形成される　　　　　　　　　　　　 [林忠四郎 他, 1985]

原始惑星系円盤（原始太陽系円盤）

分子雲から主系列星への進化

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原始惑星系円盤分子雲コア

分子雲コアの収縮　 重力と遠心力のつりあい原子惑星系円盤が形成

原始太陽系円盤の２つのモデル

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京都モデル（林モデル） キャメロンモデル

地球型惑星・巨大ガス惑星・巨大氷惑星の作り分け　→ 太陽系形成に関しては、京都モデルの方に軍配

原始太陽系円盤の組成一般に円盤質量の99%はガス（水素・ヘリウム）残りの1%がダスト（固体成分）

・現在の太陽系の惑星の固体成分（約10-4M太陽）　　→ すりつぶして円盤状にならす・固体成分の約100倍の質量のガス成分を加える

最小質量円盤モデル（京都モデル）

原始太陽系円盤の初期質量は約10-2M太陽重力と遠心力の釣り合いから半径は約100AU

原始太陽系円盤の質量分布

太陽からの距離（AU）

面密度（g/cm

2 ） ガス成分

固体成分

ガス成分：水素・ヘリウム固体成分：ダスト（岩石・金属鉄・氷）

2.7AU以遠では水蒸気が凝縮　　　↓氷ダストの分だけ面密度が上昇する

snow line

微惑星から原始惑星、そして惑星へ

太陽系の構成メンバー

地球型惑星　　水星　　金星　　地球　　火星

巨大ガス惑星　　　木星　　　土星

巨大氷惑星　　天王星　　海王星

太陽系形成標準理論（林モデル）

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巨大氷惑星形成

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���������� 微惑星の形成

ダストの合体成長　→ 微惑星形成

微惑星の円盤が形成

不明な点が多い重力不安定で形成？乱流が成長を妨害するダストの合体成長？中心星に落下する衝突で破壊される乱流渦中で形成？氷の昇華で密度上昇？

微惑星の合体成長数kmサイズの微惑星が形成

互いに衝突・合体を繰り返し成長

↓

暴走的成長　大きい粒子ほど成長が速い

秩序的成長　全ての粒子が同じ速度で成長

多体問題専用計算機 GRAPE多体（微惑星）の重力計算　→ 計算量が膨大になる

粒子間相互作用の部分だけを専用計算機で計算したい　→ GRAPE 誕生！

GRAPE-6 と 牧野淳一郎教授

20 KOKUBO AND IDA

FIG. 3. Snapshots of a planetesimal system on the a–e plane. The circlesrepresent planetesimals and their radii are proportional to the radii of planetesi-mals. The system initially consists of 3000 equal-mass (1023 g) planetesimals.The numbers of planetesimals are 2215 (t = 50,000 years), 1787 (t = 100,000years), and 1322 (t = 200,000 years). In the t = 200,000 years panel, the filledcircle represents a protoplanet (runaway body) and lines from the center of theprotoplanet to both sides have the length of 5rH.

the mass ratio keeps increasing when ! <!2. If ! >!2, meanmass should be similar to themaximummass, so that the increasein the mass ratio would stall soon. Note that our definition ofrunaway growth does not necessarily mean that the growth timedecreases with the mass of a body, but that the mass ratio of anytwo bodies increases with time as shown below.The evolution of the distributions of the RMS eccentricity and

the RMS inclination is plotted in Fig. 6. Let us focus on the massrange 1023 "m" 1024 g. The values for the mass range larger

FIG. 4. Time evolution of the maximum mass (solid curve) and the meanmass (dashed curve) of the system.

than this range are not statistically valid since eachmass bin oftenhas only a few bodies. First, the distribution tends to relax to adecreasing function of mass through dynamical friction among(energy equipartition of) bodies (t = 50,000, 100,000 years).Second, the distributions tend to flatten (t = 200,000 years). Thisis because as a runaway body grows, the system ismainly heatedby the runaway body (Ida and Makino 1993). In this case, theeccentricity and inclination of planetesimals are scaled by the

FIG. 5. The cumulative number of bodies is plotted against mass att = 50,000 years (dotted curve), 100,000 years (dashed curve), and 200,000years (solid curve). A runaway body at t = 200,000 years is shown by a dot.

暴走的成長の様子

平均値

最大の天体

微惑星の暴走的成長　→ 原始惑星が誕生する

20 KOKUBO AND IDA

FIG. 3. Snapshots of a planetesimal system on the a–e plane. The circlesrepresent planetesimals and their radii are proportional to the radii of planetesi-mals. The system initially consists of 3000 equal-mass (1023 g) planetesimals.The numbers of planetesimals are 2215 (t = 50,000 years), 1787 (t = 100,000years), and 1322 (t = 200,000 years). In the t = 200,000 years panel, the filledcircle represents a protoplanet (runaway body) and lines from the center of theprotoplanet to both sides have the length of 5rH.

the mass ratio keeps increasing when ! <!2. If ! >!2, meanmass should be similar to themaximummass, so that the increasein the mass ratio would stall soon. Note that our definition ofrunaway growth does not necessarily mean that the growth timedecreases with the mass of a body, but that the mass ratio of anytwo bodies increases with time as shown below.The evolution of the distributions of the RMS eccentricity and

the RMS inclination is plotted in Fig. 6. Let us focus on the massrange 1023 "m" 1024 g. The values for the mass range larger

FIG. 4. Time evolution of the maximum mass (solid curve) and the meanmass (dashed curve) of the system.

than this range are not statistically valid since eachmass bin oftenhas only a few bodies. First, the distribution tends to relax to adecreasing function of mass through dynamical friction among(energy equipartition of) bodies (t = 50,000, 100,000 years).Second, the distributions tend to flatten (t = 200,000 years). Thisis because as a runaway body grows, the system ismainly heatedby the runaway body (Ida and Makino 1993). In this case, theeccentricity and inclination of planetesimals are scaled by the

FIG. 5. The cumulative number of bodies is plotted against mass att = 50,000 years (dotted curve), 100,000 years (dashed curve), and 200,000years (solid curve). A runaway body at t = 200,000 years is shown by a dot.

軌道長半径 [AU]

軌道離心率

質量 [1023 g]

時間 [年]

FORMATION OF PROTOPLANETS FROM PLANETESIMALS 23

FIG. 7. Snapshots of a planetesimal system on the a–e plane. The cir-cles represent planetesimals and their radii are proportional to the radii ofplanetesimals. The system initially consists of 4000 planetesimals whose to-tal mass is 1.3! 1027 g. The initial mass distribution is given by the power-law mass distribution with the power index ! = "2.5 with the mass range2! 1023 #m# 4! 1024 g. The numbers of planetesimals are 2712 (t = 100,000years), 2200 (t = 200,000 years), 1784 (t = 300,000 years), 1488 (t = 400,000years), and 1257 (t = 500,000 years). The filled circles represent protoplanetswith mass larger than 2! 1025 g and lines from the center of the protoplanet toboth sides have the length of 5rH.

there is no supply of small bodies. Figure 9 shows the snapshotof another run at t = 500,000 years. Two large protoplanets withmass 1–3! 1026 g with an orbital separation of about 10rH areformed. The two protoplanets contain 38% of the system mass.

FIG. 8. The number of bodies in linear mass bins is plotted for t = 100,000,200,000, 300,000, 400,000, and 500,000 years.

In Fig. 10, we plot the maximum mass and the mean mass ofthe system against time. The time evolution of the RMS eccen-tricity and inclination of the system is plotted in Fig. 11. Thevalues are scaled by the reduced Hill radius of the maximumbody. The reduced Hill radius is given by h= rH/a. After about20,000 years, the scaled RMS eccentricity and inclination are al-most constant, $e2%1/2 & 6hmax and $i2%1/2 & 3.5hmax. This resultagrees with the estimation of Ida and Makino (1993). They esti-mated the equilibrium eccentricity and inclination of a planetes-imal system perturbed dominantly by a protoplanet under gasdrag. They found that when the mass of a protoplanet becomes

寡占的成長の様子軌道離心率

各場所で微惑星が暴走的成長　→ 等サイズの原始惑星が並ぶ

寡占的成長とよぶ

＝

各軌道での原始惑星質量 [kg] 形成時間 [yr]

地球軌道 1×1024 7×105

木星軌道 3×1025 4×107

天王星軌道 8×1025 2×109軌道長半径 [AU]

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原始惑星の質量 [地球質量]

軌道長半径 [AU]

地球型惑星　原始惑星同士の合体

巨大ガス惑星　原始惑星のガス捕獲

巨大氷惑星　原始惑星そのまま

snow line

ジャイアントインパクト

原始惑星同士の巨大天体衝突を繰り返し, 現在の惑星へ

ジャイアントインパクトの様子

軌道長半径 [AU]

軌道離心率

planets is hnM i ’ 2:0 ! 0:6, which means that the typical result-ing system consists of two Earth-sized planets and a smallerplanet. In thismodel, we obtain hnai ’ 1:8 ! 0:7. In other words,one or two planets tend to form outside the initial distribution ofprotoplanets. In most runs, these planets are smaller scatteredplanets. Thus we obtain a high efficiency of h fai " 0:79 ! 0:15.The accretion timescale is hTacci " 1:05 ! 0:58# $ ; 108 yr. Theseresults are consistent with Agnor et al. (1999), whose initial con-ditions are the same as the standard model except for !1 " 8.

The left and right panels of Figure 3 show the final planets onthe a-M andM–e, i planes for 20 runs. The largest planets tend to

cluster around a " 0:8 AU, while the second-largest avoid thesame semimajor axis as the largest, shown as the gap around a "ha1i. Most of these are more massive thanM%/2. The mass of thelargest planet is hM1i ’ 1:27 ! 0:25M%, and its orbital elementsare ha1i ’ 0:75 ! 0:20 AU, he1i ’ 0:11 ! 0:07, and hi1i ’0:06 ! 0:04. On the other hand, the second-largest planet hashM2i’ 0:66 ! 0:23M%, ha2i ’ 1:12 ! 0:53AU, he2i ’ 0:12 !0:05, and hi2i ’ 0:10 ! 0:08. The dispersion of a2 is large, sincein some runs, the second-largest planet forms inside the largestone, while in others it forms outside the largest. In this model, wefind a1 > a2 in seven runs.

Fig. 2.—Snapshots of the system on the a-e (left) and a-i (right) planes at t " 0, 106, 107, 108, and 2 ; 108 yr for the same run as in Fig. 1. The sizes of the circlesare proportional to the physical sizes of the planets.

Fig. 3.—All planets on the a-M (left) and M–e, i (right) planes formed in the 20 runs of the standard model (model 1). The symbols indicate the planets first(circles), second (hexagons), third (squares), and fourth (triangles) highest in mass. The filled symbols are the final planets, and the open circles are the initialprotoplanets in the left panel. The filled and open symbols mean e and i in the right panel, respectively. [See the electronic edition of the Journal for a color versionof this figure.]

KOKUBO, KOMINAMI, & IDA1134 Vol. 642

長い時間をかけて原始惑星同士の軌道が乱れる　→ 互いに衝突・合体してより大きな天体に成長

巨大天体衝突による月形成

原始地球に火星サイズの原始惑星が衝突飛び散った破片が地球の周囲に円盤を形成円盤中で月が誕生！

巨大ガス惑星の形成

原始惑星に円盤ガスが暴走的に流入 → ガス惑星へ

ガス捕獲による巨大ガス惑星形成原始惑星は重力により周囲の円盤ガスを捕獲・10地球質量以下 → 大気圧で支えられて安定に存在・10地球質量以上 → 大気が崩壊・暴走的にガス捕獲

軌道付近に残っているガスを全て加速度的に捕獲　→ 急激に質量を増し木星・土星へと成長する

巨大ガス惑星の形成の様子

and goes downward (upward) according to the Keplerian shearmotion. The shocks (crowded contours near the Hill radius) canbe seen in the upper right and lower left regions near the proto-planet. In this model, the shock front almost corresponds to theHill radius (Fig. 4, left). When the gas approaches the proto-planet, the streamlines are bent by the gravity of the protoplanet.According to Miyoshi et al. (1999), the gas flow is divided intothree regions: the pass-by region ( xj jk rH), the horseshoe region(xP rH and yj jk rH), and the atmospheric region (rP rH). Notethat although Miyoshi et al. (1999) classified the flow pattern intheir two-dimensional calculation, their classification is useful fora global flow pattern in three dimensions. In the pass-by region,the flow is first attracted toward the protoplanet and then causes ashock after passing by the protoplanet. At the shock front, thedensity reaches a local peak and the streamlines bend suddenly.On the other hand, the gas entering the horseshoe region turnsaround because of the Coriolis force and goes back. The outer-most streamlines in the horseshoe region (i.e., the streamlinespassing very close to the protoplanet) pass through the shockfront, whereas the gas on the streamlines far from the protoplanetdoes not experience the shock. In the atmospheric region, the gasis bound by the protoplanet and forms a circumplanetary diskthat revolves circularly around the protoplanet in the prograde(counterclockwise) direction.

Although the streamlines on the z ! 0 plane (Fig. 4, left) aresimilar to those in recent two-dimensional calculations (e.g., Lubowet al. 1999; Tanigawa & Watanabe 2002a), there are important

differences. In the two-dimensional calculations, a part of the gasnear the Hill sphere can accrete onto the protoplanet. Lubow et al.(1999) showed that only gas in a narrow band distributed from thelower left to the upper right region against the protoplanet fory < 0 spirals inward toward the protoplanet, passes through theshocked region, and finally accretes onto the protoplanet (for de-tails, see Figs. 4 and 8 of Lubow et al. 1999). On the other hand,in our three-dimensional calculation, gas only flows out from theHill sphere and thus does not accrete onto the protoplanet on themidplane. The left panel of Figure 4 shows that although gas flowsinto the Hill sphere, some of the gas flows out from the centralregion. The right panel of Figure 4 is a three-dimensional view atthe same epoch as the left panel. In this panel, only the stream-lines flowing into the high-density region of rTrH are drawnfor z " 0 and are inversely integrated from the high-density re-gion. This panel clearly shows gas flowing into the protoplanet inthe vertical direction.To investigate the gas flowing into the protoplanet system in

detail, in Figure 5 we plot the streamlines at the same epoch asFigure 4, but with different grid levels (l ! 3, 5, and 7). In thisfigure, each of the top panels shows a three-dimensional view,while each of the bottom panels shows the structure on the crosssection in the y ! 0 plane. Note that, in the bottom panels, thestreamlines are projected onto the y ! 0 plane. Figure 5a showsonly the streamlines in a narrow bundle flowing into the proto-planet system. This feature is similar to that shown in two-dimensional calculations (Lubowet al. 1999;Tanigawa&Watanabe

Fig. 4.—Structure around the Hill sphere for model M04 on the midplane (left) and in three dimensions, shown in bird’s-eye view (right). The gas streamlines (redlines), density structure (color), and velocity vectors (arrows) are plotted in both panels. The dashed circle in the left panel represents the Hill radius. The size of the domainis shown in each panel.

MACHIDA ET AL.1226 Vol. 685

Fig. 1.—Time sequence for model M04. The density (color scale) and velocity distributions (arrows) on the cross section in the z ! 0 plane are plotted. The bottompanels (l ! 3) are 4 times the spatial magnification of the top panels (l ! 1). Three levels of grids are shown in each top (l ! 1, 2, and 3) and bottom (l ! 3, 4, and 5) panel.The level of the outermost grid is denoted in the top left corner of each panel. The elapsed time tp and the central density !c on the midplane are denoted above each of thetop panels. The velocity scale in units of the sound speed is denoted below each panel.周囲の円盤ガスが原始惑星の重力圏内に捕獲される

巨大氷惑星の形成

円盤散逸後に原始惑星が形成 → ガスを纏えず氷惑星へ

地球の形成・初期進化

地球内部の層構造の形成

鉄

岩石

マントル

コア

重力分離マントル・コア分化

地球内部の冷却内核の形成

火山活動地殻の形成

地球の大気・海洋の形成原始太陽系星雲ガスからの大気捕獲　水素・ヘリウムに富む還元的な大気

惑星内部からの脱ガス大気の付加　微惑星中の揮発性物質の取り込み　　　　　　↓表面にマグマオーシャンが形成　　　　　　↓大気中の水蒸気が凝結　全球的に雨が降り海が形成

地球の炭素循環（負のフィードバック）

温暖化

降水量増加風化・浸食量増加

二酸化炭素は海底に固定大気中の二酸化炭素除去

降水量減少風化・浸食量減少

火成活動で二酸化炭素付加大気中の二酸化炭素が増加

寒冷化

地球のプレートテクトニクス

生命と地球の共進化

酸素発生型光合成生物の誕生・進化　→ CO2 地球大気からO2 地球大気へ