 # Document 677348

```原始惑星系円盤の

○系外惑星の直接撮像

○原始惑星系円盤形成の数値計算

よる分裂

(Carson et. al. 2013)

Toomre s Q parameter
Q<1

1.Q.2
:音速
:エピサイクル振動数
:面密度
(local instability, Toomre 1969)

(global instability Takahara 1976, 1978, Iye 1978)
●
Shock 加熱
● 重力トルクによる角運動量輸送

ガス降着

Cooling time
tcool =

(Gammie 2001, Rice et al. 2005, 2014, Meru and Bate2012 等)
Kepler time として冷却過程をモデル化
dE
E
1
=
,
dt
tcool

<
crit
⇠ 30 で分裂
(Meru and Bate 2012)

Machida et al. 2010 : 断熱・分裂する

urface
density,
v vis isvelocity,
EE
is is
internal
energy
perper
unit
area,
P isP the
vertically
surface
density,
velocity,
internal
energy
unit
area,
is
the
e thin disk approximation. We assume an ideal gas equationvertically
of state,
ressure, Φ is the gravitational potential, ΛC is the cooling rate per unit area. We
pressure,
Φ is the We
gravitational
potential,
Λin
the cooling
rate per unit
area.
We
C isthis
pecific
heat.
γ
=
5/3
calculation.
The
temperature
from the thin disk approximation. We assume an ideal gas equation of state,
Φ from the thin disk approximation.
We assume an ideal gas equation of state, (1.4)
P = (γ − 1)E,
Setup (FARGO)
= (γ − 1)E,
(1.4)

P
=
(γ
−
1)E,
(1.4)
Open
boundary
H= 5/3 in this calculation.
of specific heat. We
γ
The
temperature
T =
,
(1.5)
@⌃
he ratio
of specific heat.
We
γΣ
= 5/3 in this calculation. The temperature
k

B
+
r
·
(⌃v)
=
0
the ratio of specific heat. We adopt γ = 5/3 in this calculation. The rtemperature
= 20, 1000AU
@t
✓ kB isµm
◆H HPBoltzmann
olecular weight,
the
constant and mH is (1.5)
hydrogen
µm
P,
@v T =
(1.5)
H ,PrP
k=Bkµm
Σ

+ vT· T=
rv
=
⌃r
Σ ,is modeled asM
(1.5)
B Λ
⇤ = 0.5M
= 2.34. The cooling
rate
follows
(Hubeny,
1990;
@t
C
kB Σ
nhemolecular
weight,
kB kis
the
constant
mhydrogen
mean molecular
weight,
theBoltzmann
Boltzmann constant
andand
mH is
H is hydrogen
B is
the
mean molecular
weight,
kB is the Boltzmann constantM
and
mH=is0.34
hydrogen , 0.38
04);
0.24M
0.28M
P
=
(
1)E

disk
we
µ
=
2.34.
The
cooling
rate
Λ
is
modeled
as
follows
(Hubeny,
1990;
pt µ = 2.34. The cooling rate ΛCC is modeled as follows (Hubeny, 1990;
✓
◆
eoodman,
µ
=
2.34.
The
cooling
rate
Λ
C is modeled as follows (Hubeny, 1990;
2004); @E
r
,
2004);
8
τ
Goodman,
2004);
4 = Pr · v
エネルギー
+4 r
·T
(Ev)
Text = 150[K]
ΛC = σ(T
−
)
(1.6)
8
τ
@t
1
2
4 ext 4 1 2
1[AU]
− Text ) τ1 2 + τ1√τ τ +
(1.6)

84
2
44
44 τ + √ τ3 +
3
=
σ(T
− )Text
4) 1
3 2
ΛC = ΛCσ(T
−
T
3 1
ext
2+
1
1
√
2
3
τ
τ
+
√ τ3 + 2 3
3
τ
+
4
4
3
3
3/7
(1.6)
(1.6)
(Chiang
and
Goldreich 1997)
=
5/3
the
Stefan-Boltzmann
constant,
T
is
the
equilibrium
temperature
due to the due to the
Boltzmann constant, Textextis the equilibrium temperature

the the
Stefan-Boltzmann
constant,
Text
isoptical
the equilibrium
temperature
due to the
rom
central
star,
and
τ
=
κ
Σ
is
the
depth
of
the
disk.
The
Rosseland
R
an-Boltzmann
constant,
Tis
is (Hubeny
the
equilibrium
temperature
due to
the
extthe
1990)
tral
star,
and
τ
=
κ
Σ
optical
depth
of
the
disk.
The
Rosseland
thegiven
central
τ = κR Σ is the optical depth of the
disk. The Rosseland✓

◆
yfrom
κR is
by star, and R
Optical
depth
central
star,
and τ = κR Σ is the optical depth of the disk. The Rosseland r
ity
κ
is
given
by
R
en by
!
"2
12/7
⌃/r
exp
T
given by
2 −1
!
"
κR =!κ10
[cm
g ]
(1.7)
2
r
"
disk
T
10[K] 2
2 −1
"2 [cm
κR =!κ10T
g ]
(1.7)
−1
10[K] [cm22 g−1
r
=
250[AU]
T
disk
κ
=
κ
]
=0.05
R
10
ng approximates
result of Semenov
et g
al. (2003)
in T ! 200 K, and almost (1.7) (1.7)
κRto=theκ10
[cm
]
10[K]
10[K]
(Semenov
et assume
al.
ing
approximates
to
the
result
of Semenov
et 2003)
al. (2003)
TQ
!⇠
200
and almost
ture of the disk is smaller than
200
K. We
Text asinfollows
(cf.
2K,Chiang
( Q&⌘ cs ⌦/(⇡G⌃) )
ature
than 200 K. We assume T\$ext as follows (cf. Chiang &
997); of the disk is smaller
#
mates
to to
thethe
result
etal.
al.(2003)
(2003)
T 200
! 200
K, and
almost
!
"
oximates
resultofofSemenov
Semenov
et
in in
T !
K, and
almost

c
(T
)Ω
s
ext
epi
¯
Q == 0.34
Mdisk
0.28M
πGΣini

¯!1
Q

¯"2
Q
Q ! 0.6

τ

β

Qcrit ∼ 0.6
1
tcool =
−1
< 30Ω
と比較
Q
!
0.
and the scale height c /ΩT
evaluated
at
the
center
of
the
spiral.
The
width
of
th
∼T
s
ext

τ
β = tcool Ω
< 30Ω
−1
Qcrit ∼ 0.6
c
(T
)Ω
¯=
Q
πGΣ
¯!1
Q
β
τ
Q ! 0.6
¯"2
Q
comparable to the scale height. The right panel of the second row shows the lin
the spiral arm. The line mass of the spiral arm is approximately given by c2s /G.
s ext
epi 2
t=6285
line mass is smaller
than yr
the critical line mass 2cs /G, Optical
the spiral isdepth
supported by th
30
against the self-gravity in the directionini
perpendicular
to the spiral arm. 70
The left
65
25 arm and the velocity along the s
the third row shows the pitch angle of the spiral
60
subtracted the azimuthal averaged rotation velocity.
The right panel of the
20
55third r
s
50 The
the distribution of the Toomre’s Q parameter 15
at the center of the spiral arm.
45
value of Q is about 0.5. The left panel of the 10
bottom row shows the normalized
n
40
35 norm
time βnet , the normalized cooling time βcooling and the optical depth τ . the
5
30
cooling time βnet and the normalized cooling time βcooling are defined as follows;
¯"2
Q
Q ! 0.6
β
0
βnet
25
0 20 40 60 80 100 120
E
=
Ω,
s [AU]
τ
Normalized Cooling Time
β
βcooling = EΩ
3
ΛC
!
1 2
τ
4
+
√1 τ
3
8σT 4 τ
+
2
3
"
.
The right panel of the bottom row shows the epicycle frequency, angular frequency
#
Kepler frequency estimated
star mass GM∗ /r3 .
< critfrom
⇠ the
30 central
を満たすが分裂しない。

Mdisk = 0.38
0.28M

Q
Mdisk =0.34
0.28M
1.1
1.05
1
0.95
0.9
0.85
0.8
0.75
0
Mdisk =0.38
0.28M
Q

1.6
1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
0
20
50
40
60 80 100 120
s [AU]
100 150
s [AU]
200
250
T ∼ Text

c
(T
)Ω
s
ext
epi
¯

πGΣini 1.1
¯!1
Q
Q
Mdisk =0.34
0.28M
Q=
¯"2
Q

Q
Mdisk =0.38
0.28M
Q=0.6
1.05
1
0.95
0.9
0.85
0.8
0.75
0
1.6
1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
0
20
50
40
60 80 100 120
s [AU]
100 150
s [AU]
200
250
Q
!
0.6
2 2
Ω th
plitude ofπGM
surface density of the spiral is small. Thus we 4R
define
”渦状腕”の自己重力不安定性
c
region wherec the surface density is 0.85 times peak surface densit
L
ω
˜ = k˜2 −
2
2
s
2
˜
˜
˜
˜
˜
[K0 (k/2)L
(
k/2)
+
K
(
k/2)L
(
k/2)]
k
+
−1
1
0
τ
epi
2
s
[ /(L/cs )]2
(振動数)2

arm is larger 渦状腕
than 20∼AU
since the temperature is high and the s
e two parameters
2
β
(Line
mass
ne mass of the分散関係
spiral arm
is comparable
to
c
GML = s /G and
) the Q value
f≡ 2 ,
s larger30than 0.75. Thus the resultcsthat this spiral does not fra
Q
∼
0.6
crit
25
terion for
the fragmentation Q ! 0.6.
Since the temperature of th
2RΩepi
20
Q=1 l ≡
. 最大成長波長 2L
cs ∼ βnet ! 3 in satisfied in t
rate is also
large. As the result, βcooling
15
10
cansuggests
rewrite
normalized
dispersion cooling
relation安定
also
that the normalized
time itself is not related
Q=0.6
5
gmentation.
0
2
2
2
2
˜
˜
˜
˜
˜
˜
ω
˜ = k − πf [K0 (k/2)LQ=0.4
−1 (k/2) + K1 (k/2)L0 (k/2)]k + l .
-5

0
1
2
3
4
5
6
7
kLthe surface density of the ring as Gaussian
sume that the structure of
Inner

: リングの幅）
is 0.5R. Then
Toomre’s
Q parameter is
simulations performed in this chapter,
thel surface density of the in
cs Ω
Q=
=√
.
after the calculations start. πGΣ
Thus results
8πf of our simulations
[ /(L/cs )]2
(振動数)2
s both spiral
arms satisfy
condition for the
cooling
time suggested
by
Mes
arms.
Therefore,
the the
axisymmetric
mode
cannot
grow
in
the
Q
!
0.6
2 2
4R
Ωspir
plitude
of
surface
density
of
the
spiral
is
small.
Thus
we
define
th
πGM
2).
This
result
suggests
that
the
condition
for
the
Q
parameter
in
the
L
epi
2non-axisymmetric
2
2
˜
˜
˜
˜
˜
mode
may
grow
in
the
spiral
arms.
Sinc
ω
˜ = k˜ −”渦状腕”の自己重力不安定性
[K
(
k/2)L
(
k/2)
+
K
(
k/2)L
(
k/2)]
k
+
0
−1
1
0
2
eregion
essential
thanc2sthe
for the cooling
c
where
the condition
surface density
is 0.85time.
times peak
surface
densit
s
τ
arms are small
(!
0.2),
we
the
conditions
for
the
instab
∼AU

arm is larger 渦状腕
than
20
since
the
temperature
is
high
and
the
s
Chapter
1
Condition
for
the
fragmentation
of
d
e
two
parameters
e critical Q value is roughly
estimated
!
0.6
and
the
mo
2
β
(Line
mass
ne
of the分散関係
spiralon
arm
is comparable
GML to
= cs /G and
) the Q value
.4 mass
Dependence
the
opacity
f≡
, analogy with filament (
three
times
the
width
of
the
ring
by
2
csthat this spiral does not fra
scooling
largerrate
than
0.75.
Thus
the
result
30
offormed
the disks
depends
on arm
the opacity.
Since
the
most
of the
reg
agments
are
in
the
spiral
of
the
model
1,
but
are
not
fo
Q
∼
0.6
crit
Appendix
the
conditi
25 A). The spiral arm in model 1 satisfies
−1
terion
for
the
fragmentation
Q
!
0.6.
Since
the
temperature
of th
2RΩ
epi
s are optically
thick,
the
cooling
rate
is
proportional
to
κ
.
Thus
the
20
Q=1
l
≡
.
2L

5
does
not
satisfy
the
condition.
We
test
the
condition
by
c
rate
is
also
large.
As
the
result,
β
∼
β
!
3
in
satisfied
in
t
ﬃcient
when
κ
is
small.
The
result
of
the
model
1,
8,
9,
10,
and
12
sugges
s
cooling
net
15
est unstable wavelength of the gravitational instability derived from
f can
them
fragment
andcase
the
others
do
not安定
fragment.
Then
we fin
mentation
in the
where
thecooling
opacity
is
small
andisthe
10occurs
rewrite
normalized
dispersion
relation
also
suggests
that
the
normalized
time
itself
not
related
2 1/2
c linear stability
analysis
2π(c
/Ω)Q/(1
+
(1
−
Q
)
)
is
longer
tha
s
Q=0.6
5other parameters are not changed. Figure 1.8 shows the structu
ent
if
the
agmentation
of
the
spiral
arms
is
valid
for
the
all
spiral
arms
gmentation.
arms.
Therefore,
the
axisymmetric
mode
cannot
grow
in
the
a
0
2
2
2 spiral
2
˜
˜
˜
˜
˜
˜
l arm in ω
dose not
fragment.
In
the
spiral
arm,
Q
>
0.6
is
˜the=model
k −8,πfwhich
[K0 (k/2)L
(
k/2)
+
K
(
k/2)L
(
k/2)]
k
+
l
.
−1
1
0
Q=0.4
-5
nd
Figure
1.7,
the
normalized
cooling
times
β
is
smaller

cool
d,
non-axisymmetric
mode
may
grow
in
the
spiral
arms.
Since
the
s the result that
spiral3 arm4 is not
is consistent with our criterio
0
1this 2
5 fragment
6
7

ms
the
condition
for
thethe
cooling
time
Mo
arms
are small
(!
0.2),
wethe
conditions
for
instability
kL
sume
that
the
structure
surface
density
of 1the
ring
as
Gaussian
the satisfy
opacity
is
larger
thanof
the
opacity
of
the
model
bysuggested
athe
degree
ofby
magni
Inner
normalized
cooling
time
βcoolQ∼parameter
βnet "
10.the
In the
case
theinopacity
is
he
critical
Q
value
is roughly
estimated
!
0.6where
and the
most

suggests
that
the
condition
for2L
Q parameter
the uns
spi
で
is
0.5R. Then
Toomre’s
is
l arms
cannot
shrink
enoughoftothe
satisfy
the
Qanalogy
< 0.6 in with
the region
a few(Inutsu
times la
ut
three
times
the
width
ring
by
filament
the condition
for
the
cooling
time.
simulations
performed
in
this
chapter,
thel surface density of the in
cΩ

s is important for the fragmentation of
width.
In
this
sense
the
eﬃcient
cooling
= in model
= √1 satisfies
.
e Appendix A). The spiral Q
arm
the condition, an
calculations
start.
Thus
results
of our simulations

8πfworks.
, after
and ourthe
criterion
is consistent
withπGΣ
the previous
Opacityに対する依存性
Chapter 4 Condition for the fragmentation of di

50
45
Σ100 [g cm-2]
40
35
Fragmentation

25
20
15

10
0.0001

No
fragmentation

0.01
0.1
κ10 [cm2 g-1]
1

e 4.13: Classification of the simulation results on the κ10 −Σ100 plane. Filled circles
Opacityに対する依存性
Chapter 4 Condition for the fragmentation of di

50
45
Σ100 [g cm-2]
40
35
30
25
20
15

Fragmentation

0.0001

No
fragmentation

0.01
0.1
κ10 [cm2 g-1]
1

e 4.13: Classification of the simulation results on the κ10 −Σ100 plane. Filled circles
Opacityに対する依存性
Chapter 4 Condition for the fragmentation of di

50
45
Σ100 [g cm-2]
40
35
30
25
20
15

Fragmentation

c
(T
)Ω
s
ext
epi
¯
Q=
πGΣini
¯!1
Q
10
0.0001

No
fragmentation

0.001
0.1
1
Q

κ10 [cm2 g-1]

e 4.13: Classification of the simulation results on the κ10 −Σ100 plane. Filled circles
まとめ
•

•
これまで、円盤が分裂する条件として円盤の冷却率が重要だ
と考えられてきが、この条件は他の数値計算結果と矛盾す
る。
•

した。円盤が分裂する条件は円盤に形成された渦状腕中で
Q<0.6で与られることがわかった。この結果はリングの線形解

•
Opacity が大きく冷却しにくい円盤ほど分裂に必要な質量は大
きい。断熱まで含めた広いパラメータで分裂条件はQ<0.6で与
られる。
```