破壊を考慮した惑星形成

イエナ大学
小林 浩

Wednesday, July 14, 2010

©2009 Google - 画像 ©2009 TerraMetrics, 地図データ ©2009 PPWK, Tele Atlas -google map
Wednesday, July 14, 2010

©2009 Google - 画像 ©2009 TerraMetrics, 地図データ ©2009 PPWK, Tele Atlas -google map
Wednesday, July 14, 2010

内容
• 惑星形成における破壊• cratering v.s. catastrophic• 破壊を考慮した惑星形成• まとめ• 議論：ガス惑星形成
Wednesday, July 14, 2010

惑星形成標準モデル
105-6yr105-7yr

106-8yr

109yr

Wednesday, July 14, 2010

破壊の重要性• 惑星形成後期になると、微惑星の衝突速度が原始惑星の重力により大きくなり、微惑星は衝突•破壊を起こす。

• 破片は次々に衝突を繰り返し、小さくなる。
• 小さい天体はガス抵抗により消失する。
• その結果、衝突カスケードは天体の面密度を減少させる。
Wednesday, July 14, 2010

破壊の種類
catastrophic

cratering
QD*

Wednesday, July 14, 2010

破片のモデル
catastrophic
me
m10.5

differential number

1cratering

¡
4mL
m1

eje
cta
m

ïE

remnant
0.01

1q

Eimp / m1QD*
Wednesday, July 14, 2010

100mL

mass (m)m1ïPe破片のモデル
catastrophic
me
m10.5

differential number

1cratering

¡
4mL
m1

eje
cta
m

ïE

remnant
0.01

1q

Eimp / m1QD*
Wednesday, July 14, 2010

100mL

mass (m)m1ïPe破片のモデル
catastrophic
me
m10.5

differential number

1cratering

¡
4mL
m1eje
ctam

ïE

remnant
0.01

1q

Eimp / m1QD*
Wednesday, July 14, 2010

100mL

mass (m)m1ïPe破片のモデル
catastrophic
me
m10.5

differential number

1cratering

¡
4mL
m1eje
ctam

ïE

remnant
0.01

1q

Eimp / m1QD*
Wednesday, July 14, 2010

100mL

mass (m)m1ïPe面密度減少• cratering と catastrophicのどちらが重要?

• 一回の衝突破片の分布の影響は?
解析的に面密度減少時間を導出し、モデル依存性を明らかにする！
Wednesday, July 14, 2010

基礎方程式
質量保存則∂mns (m) ∂F (m)+=0∂t∂m

質量フラックスF (m)=−ΩK+ΩK� ∞m� m0(Tanaka et al. 1996)
dm1

dm1

� ∞0
� ∞
0

dm2 m1 f (m, m1 , m2 )Pcol ns (m1 )ns (m2 )dm2 m1 [1 − f (m, m1 , m2 )]Pcol ns (m1 )ns (m2 ),m1 f (m,m1,m2): m1とm2の衝突の結果、m1から生成される質量mより小さい天体の総質量
Wednesday, July 14, 2010

減少時間

Wednesday, July 14, 2010

減少時間
仮定： v2/QD* >> 1 かつ v2/QD* ∝ m-p

Wednesday, July 14, 2010

減少時間
仮定： v2/QD* >> 1 かつ v2/QD* ∝ m-p

Σ0Σ=
1 + t/τdep

Wednesday, July 14, 2010

減少時間
仮定： v2/QD* >> 1 かつ v2/QD* ∝ m-p

Σ0Σ=
1 + t/τdep
13

τdep

Wednesday, July 14, 2010�

�−α+1mmaxv(mmax )=�2(2 − α) Σ0 h0 ΩK
2QD (mmax )����−11× − ln � +s1 (α) + s2 (α)
.
2−b2

減少時間
仮定： v2/QD* >> 1 かつ v2/QD* ∝ m-p

Σ0Σ=
1 + t/τdep
τdep

Wednesday, July 14, 2010

衝突時間1�
�−α+123mmaxv(mmax )=�2(2 − α) Σ0 h0 ΩK2QD (mmax )����−11× − ln � +s1 (α) + s2 (α).
2−b減少時間
仮定： v2/QD* >> 1 かつ v2/QD* ∝ m-p

Σ0Σ=
1 + t/τdep
τdep

Wednesday, July 14, 2010

衝突時間
(e.g., Wyatt et al. 2007)
1
�
�−α+1
23
mmaxv(mmax )=�2(2 − α) Σ0 h0 ΩK
2QD (mmax )����−11× − ln � +s1 (α) + s2 (α).
2−b減少時間
仮定： v2/QD* >> 1 かつ v2/QD* ∝ m-p

Σ0Σ=
1 + t/τdepτdep衝突時間(e.g., Wyatt et al. 2007)1��−α+123mmaxv(mmax )=�2(2 − α) Σ0 h0 ΩK2QD (mmax )����−11× − ln � +s1 (α) + s2 (α).
2−bα = (11+3p)/(6+3p)

Wednesday, July 14, 2010

(e.g., OʼBrien and Greenberg 2003)

s1 v.s. s220010060v2−p∝mQ�D

11 + 3pα=
6 + 3ps2(e.g. OʼBrien and Greenberg 2003;
Kobayashi and Tanaka 2009)20

s1

10621.65 1.7 1.75 1.8 1.85

αWednesday, July 14, 2010

s1 v.s. s220010060v2−p∝mQ�D

11 + 3pα=6 + 3ps2

(e.g. OʼBrien and Greenberg 2003;
Kobayashi and Tanaka 2009)

20

s1

1062

1.65 1.7 1.75 1.8 1.85α

破片の分布などは、τdep を変えない。
Wednesday, July 14, 2010

mass depletionΣ0Σ=
1 + t/τdep�−α+12mmaxv(mmax )=(2 − α)2 Σ0 h0 ΩK
2Q�D (mmax )����−11× − ln � +s1 (α) + s2 (α)
.
2−b13

τdep

Wednesday, July 14, 2010�mass depletionΣ0Σ=1 + t/τdep�−α+12mmaxv(mmax )=(2 − α)2 Σ0 h0 ΩK
2Q�D (mmax )����−11× − ln � +s1 (α) + s2 (α)
.
2−b13

τdep

Wednesday, July 14, 2010

�

cratering v.s. catastrophic
200v2−p∝m
Q�D

s2

100
60
20
10
6etacr(e.g. OʼBrien and Greenberg 2003;
Kobayashi and Tanaka 2009)

catastrophic

2
111 + 3pα=
6 + 3pgnir

1.65 1.7 1.75 1.8 1.85

α4-50 倍crateringの方が効く
Wednesday, July 14, 2010

統計的シミュレーション
個数質量Wednesday, July 14, 2010

統計的シミュレーション
個数質量Wednesday, July 14, 2010

破壊しきい値
1er

at
ining

er

10410

6

Time [yr]Wednesday, July 14, 2010g10ï4at

初期総質量: MEarth10

ï3crv = 3 km/s

10

ï2th(Benz and Asphaug 1999)

10

ï1cr

�−0.38wi

� r= 3.5 × 10
1 cm ��� r �1.36
ρ
1+0.3[erg/g]−31 g cm
1 cm
7100noQ�D

面密度/初期面密度

シミュレーションとの比較

108

破壊しきい値
1er

at
ining

er

10410

6

Time [yr]Wednesday, July 14, 2010

g

10

ï4at

初期総質量: MEarth10

ï3crv = 3 km/s

10ï2th(Benz and Asphaug 1999)

10

ï1cr

�−0.38wi� r= 3.5 × 10
1 cm ��
� r �1.36
ρ
1+0.3[erg/g]−31 g cm
1 cm
7

100noQ�D

面密度/初期面密度

シミュレーションとの比較

108

惑星形成での破壊

Wednesday, July 14, 2010

統計的シミュレーション

Wednesday, July 14, 2010

統計的シミュレーション• 質量座標を細かいビンに切り、ビンの中の天体の数の衝突による進化を計算する。

Wednesday, July 14, 2010

統計的シミュレーション• 質量座標を細かいビンに切り、ビンの•Wednesday, July 14, 2010

中の天体の数の衝突による進化を計算する。
ビンの天体の速度(離心率、傾斜角)も天体の質量分布の進化に応じて変化すすので、同時に計算する。

統計的シミュレーション• 質量座標を細かいビンに切り、ビンの••Wednesday, July 14, 2010

中の天体の数の衝突による進化を計算する。
ビンの天体の速度(離心率、傾斜角)も天体の質量分布の進化に応じて変化すすので、同時に計算する。
動径方向にもメッシュを切り、天体のガス抵抗の移動による変化も計算する。

惑星形成中の質量分布0 yearlog10 N> r302010５.2AU0ï4ï202log10( r / 1 km )Wednesday, July 14, 2010

4

惑星形成中の質量分布3.2 x 104 year

log10 N> r302010

５.2AU0ï4ï202log10( r / 1 km )Wednesday, July 14, 2010

4

惑星形成中の質量分布2.6 x 105 year

log10 N> r302010５.2AU0ï4ï202log10( r / 1 km )Wednesday, July 14, 2010

4

惑星形成中の質量分布5.0 x 105 year

log10 N> r30

2010

５.2AU0ï4

ï202log10( r / 1 km )Wednesday, July 14, 2010

4

惑星形成中の質量分布1.2 x 106 year

log10 N> r30

2010

５.2AU0ï4ï202log10( r / 1 km )Wednesday, July 14, 2010

4

破壊を考慮した惑星形成
破片

微惑星
原始惑星

τgrow
τdep
消失
Wednesday, July 14, 2010

Kobayashi and Tanaka (2009)

Kokubo and Ida (2002)

最終原始惑星質量timescale [yr]1010

τgrow >> τdep

7odep

no more growth

6ogrow
510 24
101026M [g]原始惑星質量[g]Wednesday, July 14, 20101028

最終原始惑星質量timescale [yr]1010

τgrow >> τdep

7odep

no more growth

6ogrow
510 24
101026M [g]原始惑星質量[g]Wednesday, July 14, 20101028

破壊込みの惑星成長
原始惑星(M)の成長
M
dM=dtτgrow (M, Σ)

面密度(Σ)の減少

Kokubo and Ida (2002)

ΣdΣ
=−dt
τdep (M, Σ)
Kobayashi and Tanaka (2009)
Wednesday, July 14, 2010

惑星の最終質量
林モデル
ガスの寿命
1千万年
１億年後

final embryo mass [M ]計算条件

10ssamnolatio

is

ガス無し

0.1

シミュレーション

0.01

0.001123

4 5 6 7 8 910

distance [AU]Wednesday, July 14, 2010

ガス中

12030

結論
• 衝突カスケードによる面密度減少時間を導出。

• Crateringはcatastrophicより4-50倍重要である。

• 天体消失時間を用いて原始惑星質量を見積もった。

• 最終質量は火星質量程度。
Wednesday, July 14, 2010

ガス惑星形成

Wednesday, July 14, 2010

ガス惑星とデブリ円盤

RESEARCH ARTICLES

Optical Images of an Exosolar
Planet 25 Light-Years from Earthfind common proper motion candidate sources,
we observed Fomalhaut with the Keck II 10-m
telescope in 2005 and with HST in 2006 (9). InMay 2008, a comprehensive data analysis revealed that Fomalhaut b is physically associatedwith the star and displays orbital motion. Followup observations were then conducted at Gemini
Observatory at 3.8 mm (9).
Fomalhaut b was confirmed as a real astrophysical object in six independent HST observations at two optical wavelengths (0.6 mm and
0.8 mm; Fig. 1 and table S1). It is comoving
with Fomalhaut, except for a 0.184 T 0.022 arc
sec (1.41 T 0.17 AU) offset between 2004 and2006 (DT = 1.73 years) corresponding to 0.82 T
0.10 AU year−1 projected motion relative toFomalhaut (9). If Fomalhaut b has an orbitthat is coplanar and nested within the dust
belt, then its semimajor axis is a ≈ 115 AU,
close to that predicted by Quillen (7). An object
with a = 115 AU in near-circular Keplerianmotion around a star with mass 2.0 times thatof the Sun has an orbital period of 872 years
and a circular speed of 3.9 km s–1. The six
Keplerian orbital elements are unconstrainedby measurements at only two epochs; however,by comparing the deprojected space velocity+1.1(5.5–0.7km s–1) with the circular speed, we•

惑星形成時の破片がデブリ円盤を作る(e.g., Kenyon and Bromley 2008)•A
ガス惑星とデブリ円盤が観測されている(Kalas et al. 2008)

Paul Kalas,1* James R. Graham,1 Eugene Chiang,1,2 Michael P. Fitzgerald,3 Mark Clampin,4Edwin S. Kite,2 Karl Stapelfeldt,5 Christian Marois,6 John Krist5Fomalhaut, a bright star 7.7 parsecs (25 light-years) from Earth, harbors a belt of cold dust with astructure consistent with gravitational sculpting by an orbiting planet. Here, we present opticalobservations of an exoplanet candidate, Fomalhaut b. Fomalhaut b lies about 119 astronomical units (AU)from the star and 18 AU of the dust belt, matching predictions of its location. Hubble Space Telescopeobservations separated by 1.73 years reveal counterclockwise orbital motion. Dynamical models of theinteraction between the planet and the belt indicate that the planet’s mass is at most three times that ofJupiter; a higher mass would lead to gravitational disruption of the belt, matching predictions of its location.
The flux detected at 0.8 mm is also consistent with that of a planet with mass no greater than a few timesthat of Jupiter. The brightness at 0.6 mm and the lack of detection at longer wavelengths suggest thatthe detected flux may include starlight reflected off a circumplanetary disk, with dimension comparable
to the orbits of the Galilean satellites. We also observe variability of unknown origin at 0.6 mm.

bout 15% of nearby stars are surroundedby smaller bodies that produce copious
amounts of fine dust via collisional erosion (1). These “dusty debris disks” are analogs
to our Kuiper Belt and can be imaged directly
through the starlight they reflect or thermalemission from their dust grains. Debris disks
may be gravitationally sculpted by more massive
objects; their structure gives indirect evidence
for the existence of accompanying planets [e.g.,(2, 3)]. Fomalhaut, an A3V star 7.69 pc fromthe Sun (4), is an excellent example: A planetcan explain the observed 15 AU offset between
the star and the geometric center of the belt,
as well as the sharp truncation of the belt’s inneredge (3, 5–7). With an estimated age of 100 to300 million years (My) (8), any planet around
Fomalhaut would still be radiating its formation heat and would be amenable to direct detection. The main observational challenge is that
Fomalhaut is one of the brightest stars in thesky (apparent visual magnitude mV = 1.2 mag);to detect a planet around it requires the use of
specialized techniques such as coronagraphy toartificially eclipse the star and suppress scatteredand diffracted light.
Detection of Fomalhaut b. Coronagraphic
observations with the Hubble Space Telescope(HST) in 2004 produced the first optical image of Fomalhaut’s dust belt and detected sev-eral faint sources near Fomalhaut (6). Fomalhaut’s
proper motion across the sky is 0.425 arc sec
per year in the southeast direction, which meansthat objects that are in the background will appear to move northwest relative to the star. To

破壊(デブリ円盤)を考慮したガス惑星形成が重要
1

Wednesday, July 14, 2010

Astronomy Department, University of California, Berkeley, CA94720, USA. 2Department of Earth and Planetary Science,
University of California, Berkeley, CA 94720, USA. 3Institute ofGeophysics and Planetary Science, Lawrence Livermore National
Laboratory, Livermore, CA 94551, USA. 4Exoplanets and Stellar

Kalas et al. Science 322,1345 (2008)

Fig. 1. HST coronagraphic image of Fomalhaut at 0.6 mm, showing the location of Fomalhaut b (whitesquare) 12.7 arc sec radius from the star and just within the inner boundary of the dust belt. All the otherapparent objects in the field are either background stars and galaxies or false positives. The fainter lower halfof the dust belt lies behind the sky plane. To obtain an orientation with north up and east left, this figure

Downloaded from www.sciencemag.org on October 14, 2009CORRECTED 2 JANUARY 2009; SEE LAST PAGE

ガス惑星形成の臨界コア質量

Mcore ≥ 10M⊕で静水圧平衡の大気を持てなくなりガス惑星形成 (e.g., Ikoma et al. 2001)
Wednesday, July 14, 2010

ガス惑星形成の臨界コア質量

Mcore ≥ 10M⊕で静水圧平衡の大気を持てなくなりガス惑星形成 (e.g., Ikoma et al. 2001)
Wednesday, July 14, 2010

ガス惑星形成の臨界コア質量

Mcore ≥ 10M⊕で静水圧平衡の大気を持てなくなりガス惑星形成 (e.g., Ikoma et al. 2001)
Wednesday, July 14, 2010

ガス惑星形成の臨界コア質量

Mcore ≥ 10M⊕で静水圧平衡の大気を持てなくなりガス惑星形成 (e.g., Ikoma et al. 2001)
Wednesday, July 14, 2010

原始惑星質量at 5 AU
惑星質量 [MEarth]10010臨界質量見積もりと同程度の天体ができる。
りあス
ガ

10.1

し無ス

0.010.0010.1

ガ
1

10

円盤質量 [0.0０8 Msun]
5-30AU
Wednesday, July 14, 2010

原始惑星質量at 5 AU
惑星質量 [MEarth]10010

臨界質量見積もりと同程度の天体ができる。
りあス
ガ

10.1

し
無ス

0.010.0010.1

ガ
1

臨界惑星核質量は超えられない。
10

円盤質量 [0.0０8 Msun]
5-30AU
Wednesday, July 14, 2010

大気の効果246810log rp [cm]Inaba and Ikoma (2003)
Wednesday, July 14, 2010

Rc: collisional radius
Rcore: target radius
rp: projectile radius小さい天体の重要性
落下時間 [年]Kenyon and Bromley (2009)
108106104102at 5 AU

010 0
10Wednesday, July 14, 2010

10210410

天体半径[cm]

6smin = 1cm

小さい天体の重要性
落下時間 [年]Kenyon and Bromley (2009)
108106104102at 5 AU

010 0
10Wednesday, July 14, 2010

10210410

天体半径[cm]

6smin = 1cmガス惑星のための惑星核形成
at 5 AU
惑星質量[MEarth]

10010

臨界質量

10.1
0.01
0.001
0.1

110

円盤質量 [0.008 Msun]
Wednesday, July 14, 2010

今後の研究• 大気の効果でガス惑星はできる(Inabaet al. 2003)。

• どのような条件で大気の効果は有効か。
• 大気の効果を含めた、原始惑星の質量はどのように表現されるか。

Wednesday, July 14, 2010

Thank you for your attention.
Danke für Ihre Aufmerksamkeit.
ご清聴ありがとうございました。

Wednesday, July 14, 2010