星形成ゼミ
担当：SFN#341 24-28
2021/06/25
総研大 D2 竹村英晃

SFN#341 24-28
24. Thermal Wave Instability as an Origin of Gap and Ring Structures inProtoplanetary Disks
25. Multicolor Variability of Young Stars in the Lagoon Nebula: Driving
Causes and Intrinsic Timescales
26. Tree-based solvers for adaptive mesh refinement code FLASH - II:radiation transport module TreeRay
27. No impact of core-scale magnetic field, turbulence, or velocity
gradient on sizes of protostellar disks in Orion A

28. TIMES I: a Systematic Observation in Multiple Molecular Lines Toward theOrion A and Ophiuchus Clouds

arXiv:2105.13852v1 [astro-ph.EP] 28 May 2021

3 University Observatory, Faculty of Physics, Ludwig-Maximilians-Universität München, Scheinerstr. 1, 81679 Munich, Germany

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4 Exzellenzcluster ORIGINS, Boltzmannstr. 2, D-85748 Garching, Germany

Thermal Wave Instability as an Origin of Gap and Ring Structures(Received April 1, 2021; Revised May 6, 2021; Accepted May 28, 2021)
in Protoplanetary Disks
Submitted to ApJL
Takahiro Ueda, Mario Flock, Tilman Birnstiel
ABSTRACT
Recent millimeter and infrared observations have shown that gap and ring-like structures are common
in both dust thermal emission and scattered-light of protoplanetary disks. We investigate the impactof the so-called Thermal Wave Instability (TWI) on the millimeter and infrared scattered-light imagesof disks. We perform 1+1D simulations of the TWI and confirm that the TWI operates when the disk
is optically thick enough for stellar light, i.e., small-grain-to-gas mass ratio of & 0.0001. The mid-planetemperature varies as the waves propagate and hence gap and ring structures can be seen in bothmillimeter and infrared emission. The millimeter substructures can be observed even if the disk is fully
optically thick since it is induced by the temperature variation, while density-induced substructures
would disappear in the optically thick regime. The fractional separation between TWI-induced ringand gap is r/r ⇠ 0.2–0.4 at ⇠ 10–50 au, which is comparable to those found by ALMA. Due tothe temperature variation, snow lines of volatile species move radially and multiple snow lines areobserved even for a single species. The wave propagation velocity is as fast as ⇠ 0.6 au yr 1 , which
can be potentially detected with a multi-epoch observation with a time separation of a few years.

原始惑星系円盤のミリ波と赤外線の観測において、Thermal
Wave Instability（TWI）が与える影響を、
Keywords: protoplanetary disks — planets and satellites: formation
1+1Dシミュレーションで調べた。
1. INTRODUCTION

tures have been detected even at wavelengths where
the disk is fully
optically
thick (e.g., Carrasco-González
円盤が星の輻射について光学的に厚い場合（small-grain-to-gas
mass
ratio≥0.0001のときなど）、TWI
The recent ALMA observations have revealed that axet al. 2019; Macias et al. 2021).
isymmetric gap and ring structures are common in theがはたらくようになり、ミリ波と赤外線の両方でギャップを見られるようになる。
The recent infrared observations have shown that the
dust continuum emission of protoplanetary disks (Angap and ring-like structures are also common in scatギャップは温度の違いによって見られるので、光学的に厚い場合にも見られる。
drews et al. 2018). The origin of these structures istered light images (Avenhaus et al. 2018; Garufi et al.
still unclear, but several formation mechanisms have⁄𝑟~ 0.2-0.4でALMAの観測結果と整合的。
リングとギャップの間隔は10-50auの位置において、∆𝑟2018).
Although it is unclear if the substructures inbeen proposed such as dust filtering associated with ascattered light are associated with the (sub-)millimeter
planet-induced gap (e.g.,lineが移動し、単一の物質においても複数のsnow
Pinilla et al. 2012; Dipierro
揮発性の物質のsnowlineが生じることになる。
substructures,theprevalenceof substructures in bothet al. 2015), snow lines (Zhang et al. 2015; Okuzumi
dust and gas disk raises the questions how these sub波の伝播速度は~yr-1で、multi-epoch
observationで観測的に確かめられるかもしれない。
et al. 2016; Pinilla 0.6
et al.au2017),
instability induced bystructures are created and howhttp://arxiv.org/abs/2105.13852v1
they are associated with
dust-gas interaction (Takahashi & Inutsuka 2014, 2016)

背景•

原始惑星系円盤に対するALMA観測によって、ギャップやリング構造が見つかっているが (Andrews et al.
2018)、起源はいまだわかっていない。

•

•

•

惑星由来(e.g., Pinilla et al. 2012; Dipierro et al.22015)

U•

Snow line (Zhang et al. 2015; Okuzumi et al. 2016; Pinilla et al. 2017)•

ダストとガスの相互作用による不安定性 (Takahashi & Inutsuka 2014, 2016)•

円盤表面に小さなバンプができると、中心星側は照らさ

ceives more stellar light and the shadowed outer side
the bump receives insufficient flux. At the illuminate•
磁気流体力学的効果 (Flock et al. 2015; Ruge et al. 2016; Krapp et al. 2018; Riols et al. 2020)
side, the disk surface pu↵s up further as the mid-plan
Ø ほとんどのメカニズムにおいて、密度の変化からsubstructureを作るが、光学的に厚い波長でもtemperature increases, resulting in further decrease
substructureが観測されている(e.g., Carrasco-Gonz
́alez et2019;
MaciasSince
et al. the2021)。
the temperature
at al.
thebackside.
heating e
ficiency depends on(Avenhaus
the grazing
angle,
which
is etdefine赤外線の散乱光でもリングやギャップのような構造が観測されている
et al.
2018;
Garuﬁ
as the angle between the incident stellar light and th
al. 2018)。
disk surface, the bump responsible of increase the gra
リング/ギャップ構造を作れるメカニズムが Thermal Wave Instability (TWI; DʼAlessio et al. 1999;
ing angle moves inward and the next bump is generate
Dullemond 2000; Watanabe & Lin 2008; Siebenmorgen& Heymann
2012;
Ueda1).
et al.
just behind theformer one(FigureIn 2019)。
this letter, w
れて、反対側は影になる。
•

照らされた側は、mid-planeの温度が上がるにつれてさらに膨らみ、反対側は温度がさらに下がる。

•

加熱効率は中心星からの放射と円盤表面のなす角に依存し、バンプは内側に移動し、バンプの後ろに新しいバンプが生まれる。

Illuminated frontside

Shadowed outer side

puff upshrink down

4
thermal (3)timescale is constantM⇤ = 1M , T⇤ = 4000 K, resultinginL⇤=={1
1L exp. The✓◆temperature
continuum
F
( of
2⌧
Tm
, of themis(Tm )}2 m (r)whereztheheightdisk
surfacewhere the radial
ricisdisk.
The
mid-planeTapassively
s
m
1/2
⌃
z
epends onFigure
the grazing
angle,
which
definedd
1. Schematic
view of thethermalwavewith
instability.
T/r.
Therefore,
the ase↵ectivetemperatureassociatedtheexternal
heatm
4p⇢d (r, z)= disk isexp, = {1(6)
on. We peroptical
depth
for
the
stellarlight
reaches
unity,
(T
heatedcalculatedwiththeenergyequation
2
F
(r)
exp
(
2⌧
(T
)}
T
,
(4)
⇤
⇤ ) isemm
軸対象の円盤についての1+1Dシミュレーション(Watanabe
&
Lin
2008)
e
gle between
theincident
stellar
light
and
the
The frontsideingof theperturbed
disk
surfacereceives
morestel- area filling
2h
2⇡h
would
be
broken
at
outer
regio
d
d
is
set
as
T
=
10
K.
The
total
emittingeemonstrate
Ueda
et
al.
the
Planckmean
opacityusing
thestellartemperature
lar
flux
and
pu↵s
up,
while
the
shadowedoutersideshrinks
ace, the bump responsible
ofofincrease
the grazwe luminosfocus mainly on the interm
Rand
Lconfactor
dust
, can
be
writtenas the stellar radius and+grains,
1of dust
kBwhere
⌃A
@T
⇤ are
where
hdThe
is superheatedthe scale heightg sdiskm ⇤whichhe observedtodeterminetheBlack-bodyradiation.
As is sometimes
down(solidline).
perturbation
propagatesinward
as
a
=
2(F
F
+
F
),
(1)
𝛾 the
= 1.4
moves
inward and the next bump is generateds#me•infrared
エネルギー方程式"2(andTWI operates anity,H external
respectively,Tof au)
thewheresuperZiss is the temperature
1)µm@t
tributesto theopacityandassumed
to
be
the
same
light
and
the shadowed
outer
side
of
incident
flux,
respectively,
andarewrittennd1
thermalwave(dottedlines).
approximated
with
sin
↵
where
↵
is
the
grazing angle,pnd the former one (Figure 1). In this letter, we
0 grains
0hydrostatic
3 /µm
heateddust
atthe disk surface andis the equilibrium for the
as
the
gas
scale
height
h
=
k
T
r
GM
.
A
(r)
=
1
exp

(T
)⇢
(r,
z
)dz
,
(5)
At
the
illuminatedasn insufficient
Section •2. flux.
g
B
m
H
⇤
s
s
⇤
d
𝑅∗ = 2.08𝑅⊙ 、 𝑀∗ =where
1𝑀⊙ 、 𝑇∗ = 4000
Kadiabatic、 𝐿∗ = 1𝐿、k𝑇we
𝑇$ = 10 K
followBoltzthe more accurate determination,
see Ap⊙but# ：表面のダストの温度、index,⌧Following Watanabe
Using
the dust distribution gi
zsassumption(r)is theB is theStefan-Boltzmann
constant.
& Lin(6) is based on=an1.4
that thediskacepu↵s up furtherEquation
as the mid-plane
Discussion
L⇤C inAsWatanabe
4R⇤ & Lin (2008). The dust density
pendixmann
constant,
µ
=
2.34
the
of +
disk asvertical
depth for the d• 中心星からの放射のﬂux：
Fin=vertical{1 isweexp( molecular2⌧(T(2008),
setthestellarparametersR,⇤(2)= 2.08Roptical,s (r)mradials )} weightisinhydrostatic
equilibriumthe
direction.
ases, resultinginfurtherdecrease
in
2
where
z
is
the
heightofthedisk
surfacewherethedistribution⇢d (r,3⇡rz) 3is givenbypropose the TWI ass a newformationmechanismof the8⇡gas rsurface
wave instability.
gas,
m
is
the
atomic
mass
unit,
⌃
is
the
as
H when the M
g4000 K, resulting in L⇤ =calculated, T⇤=(T
1L . The⇤ = 1M
This
assumption
is valid
dynamical
timescalet the backside.
Since
the
heatingef✓◆
opticaldepth inforthe stellar
lightreaches
unity,
andring
structures
dust
millimeter
continuum⇤ (T⇤ ) is)} T 4 ,ダストからの放射のﬂux：
2ceives more•gap
stelFthree={1 Ftemperature
exp
(
2⌧
(3)
Z zs (r)1density.
The
fluxes,,F
andF,
representthem (r)e↵ective
m
m
⌃
z
m
s
m
e
associatedwiththeexternal
heatd⌦asK wellis shorter
than
thermaltimescale
given
by
n the
grazing
angle,
which
definedthePlanckmeantheopacityusing
the Westellartemperature
p
⇢
(r,
z)
=
exp
,
(6)
emission
as isthescatteredlight
emission.
peruterside
shrinksd4
incident stellar
flux,
releasedenergyemitted
from
dust
⌧m (T
m (
m2; r) =
F
(r)
=
{1
exp
(
2⌧
(T
)}
T
,
(4)
2h
ing
is
set
as
T
=
10
K.
The
total
emittingarea
filling2⇡hemm•
外部からの放射のﬂux：
e
e
een
the
incident
stellar
light
and
the
d
d
determine ofthetheBlack-bodyAissometimes
ates inwardform
as a 1+1D tosimulations
TWI
demonstrates0+ 1 andkradiation.
⌃
B gfactorof superheateddusthgrains,
Asscale, can bewrittenas disk which contpotential
=
. the
(7)
ump responsible
ofTWI
increase
theagrazth withapproximated
sin
↵
where
↵and
grazing
angle,
whereheightof dust
3is
that
the
can
be
originofobserved
where
Rthe
L
are
the
stellarradiusandluminosd is the⇤T⇤2(1) µmHof the TWI"#
mward andgap
the andnextringbump
is generatedbut
we
followinthe
moreaccurate
determination,
seetributes
Ap- Z 1to theofopacity
and isassumed to be the samestructuresbothmillimeterandinfraredity,respectively,
Ts is
the
temperature
the super(2008). We
0
0p
3/2
3
A
(r)
=
1
exp

(T
)⇢
(r,
z
)dz
,
While
the
dynamical
timescaleincreases
as
r
,
the
rmer
(Figure1).
In
this
letter,we• one
加熱されたダスト粒子のﬁlling
factorss⇤d
C in model
Watanabe& Lin (2008).
The2.dust
density
asdiskthe surface
gas scaleandheightishgthe= kB T(5)images. Thependix
simulation
is describedin Section
heateddust
grainsatthem r /µmH GM⇤ .
y symmet3/2z(r)s
thermaltimescaleis constant⌃g / constant.
r
andFollowing
distribution
is Stefan-Boltzmann
given whenby3. Discussion
chanismofthesimulated
Equation Watanabe(6) is basedThe
resultsare⇢shown
Sectiond (r, z) in& on
Linan assumption that the diska passively
1/2
T
/
r
.
Therefore,
theassumptionofhydrostatic
✓
◆
m
where
z is the heightof thedisk
where ,theinradial
ter continuum
is inhydrostatic
equilibrium
the vertical direction.
andview
conclusion
are in wave
Section
4 and(2008),
5.
we
set theparameters
as surface
R⇤ =
2.08R
uation. Schematicof the thermal
instability.
⌃z 2s stellar
d
would
be
brokenatouter
region(&50
au).
However,p
opticaldepthstellarlight
unity,⇤ (Tis•of the
ダストの密度分布⇢d (r,morez) = stelexp, for(6)assumption
ssion.
We
perThiswhen
thetimescale⇤ ) dynamicalM⇤ = 1M, T⇤ =
K,theresultingin L⇤reaches= is1Lvalid
. The
ide
perturbed
disk surface receives
2 4000
2h
2⇡h
1
d
d
we
focus
mainlyontheintermediate
region(⇠10–50
2. MODEL
the Planck mean
opacityusing
stellartemperatured demonstrate⌦K isshorterthantheheatthermal
timescale given bye↵ective temperatureassociatedwiththe theexternalndpu↵s up, while the shadowed outerside shrinks
(1)
e ),
au)
where
the
TWI
operates
and
hence
we
assume
the
where
hThe
the
scale
heightwhich
condetermine
theTheBlack-bodyradiation.
ofd theline).observed
The perturbation
propagatesinwardasing
a is setoftod is TWIs is sometimes
asdustTe =disk
10 K.
total emitting
areaAfilling
2.1.
simulations+1
k B ⌃ghydrostatic
equilibrium
fortheentireregionofthedisk.
aveand(dottedtributes to the opacityfactor
and isto bethegrains,
same
=
(7)
approximatedwithsin ↵ Awhere↵tthiswritten
the grazing
angle, 3 .
erinfrared• lines).
円盤が垂直方向に静水圧平衡にあると仮定
ofassumed
superheateddust
ass , can bep2(1) µmH Tm
We perform
time-dependent
simulations
of
the
TWI
3
the
BoltzUsingthedust
distribution
givenbyEquation(6),
the#as the gas scale height hg = kbut
Tmwe
r /µm
.
followtheaccurate determination,
see Apd in Section 2.
H"GMZ⇤ more&' & Lin B(2008).
1
followinga modelgiven
byWatanabeWe•→dynamicaltimescaleΩ
よりも𝑡が短いときに成り立つ（≤
50
au）。
ight
of
disk
vertical
optical
depth
for
the
dust
thermalemissionis()% on an assumption
Equation (6) is based
theWhilediskpendixinthatWatanabe& theLin dynamical(2008).
dust
00The
n 3. Discussion
increases as r3/2 , theAs (r)=symmet1 Cexp
s (T, timescale(5) density⇤ )⇢d (r, z )dz
solve
a
1+1D
energyequationforanaxiallygas
surfaceasis in hydrostaticindistribution
the vertical⇢ddirection.
(r,zz)
byhe TWI
as a new calculated
formationmechanismequilibriumof the(r)is giventimescale
is constant when ⌃g / r 3/2 ands thermal
ric
disk.
The
mid-planetemperatureTofapassively
Zis zvalid
m the dynamical timescale
presentthe✓◆ the assumption of hydrostatic
s (r)
This
assumption
whenring structuresin dustmillimeter
continuumTm /⌃r 1/2 . Therefore,
2
0
0
z
where
z
is
the
heightofthedisk
surfacewheretheradialdheateddisk
iscalculatedwiththeenergyequation
1 ⌧minstability.
dtic(Tm ; r) =
thermal(r, z )dz ,given(8)•viewdust
光学的厚み
m (Tms)⇢dthe scatteredthermal⌦wave
is shorter
thantimescaleby⇢d (r, z)= p be broken
exp at outer, region (6)asfromwell
asofthelightemission.
Wethe
perwould(& 50 au). However,K0
optical
depth
for
the
stellarlight reaches
unity,2h⇤2d(T⇤ ) is2⇡hdperturbed
disk
surfacereceivesmore
stel1D simulations of thedemonstrate+ 1
we focus mainly on the intermediate region (⇠ 10–50+ 1TWIkB ⌃and
g @Tm
kB),⌃
the
Planckmean
opacityusing
the stellar temperature
g
p,
while
the
shadowedoutersideshrinks
=
2(F
F
+
F
(1)
e
= s&,⁄- m where(7)h. d is the scaleheighttheof TWI
dust operates
disk which
TWI can be a potential
originof @t
thetthobserved
au)
where
andconhence we assume the
3
2(
1)
µm
𝑟
𝑟
H
2(to determine1) µmH the
Tm Black-bodyradiation. As is sometimes
ns of
the TWI propagates inwardThe
perturbationas a&and isassumed to forbe thethe entire
same region of the disk.
ガスの面密度Σ* = 1700−
gto
cmthe opacityring• structuresin both millimeterand infrared exp tributeshydrostatic
pequilibrium
ed lines).
approximatedwith sin ↵ 3/2
where ↵ is thegrazing
angle,
Lin(2008). We
au𝑟
3.
theBoltzgas scale
kB Tmdistribution
r /µmH GMgiven
wheremodel= 1.4
is theindex,
the
While
theadiabatic
dynamical
timescaleas r height, Using
thehg =the dust
The simulation
is describedin Section
2. k isasincreases
by Equation (6), the⇤.
but weB follow the more accurate
determination, see Apxially symmet3/2
Equationan assumptionthat
diskthermal emission ismannµ = 2.34
the ismolecularof⌃disktimescaleconstantweightwhen
r is based
and on optical
lated results
areconstant,
shownthermal
in Section
3.isDiscussion
vertical
depth forthethedustg /(6)
pendixCin
Watanabe&Lin(2008).
The
dust
densityof a passively
1/2
isgas
in surfacehydrostatic
equilibrium
direction.
gas,
mH is theatomicmass⌃gthe
is the/ 5.
r 𝑓.-*,0. Σ
Therefore,
assumptionof hydrostatic
usion
in Section
4Tm
and
• are
ダストの面密度* , unit,𝑓.-*,0：小さなダストのダストガス比（0.01
ー 0.0001）
calculated asin the vertical
distribution ⇢d (r, z) is given byasequationa new formation mechanism of theZ zs (r) timescaleThisis valid when the dynamical
density. Thewould
three befluxes,Fs , atFmouterand F
brokenregion(& assumption
50 theau). However,e , represent✓◆
ctures in dust
millimeter
continuum
12
2. MODEL⌦Kfromis shorter
than
the thermal
given⌃d 10–50
incident
stellarflux, released
energydust (⇠⌧mz(Tmtimescale; r) =
mby(Tm )⇢d (r, z 0 )dz 0 ,(8)
we focus
mainly on
the emitted
intermediate
region
p
⇢
(r,
z)
=
exp
,
(6)
the
scatteredlight
emission.
We
perd+ F ),(1)
0
2

シミュレーション

結果（温度、円盤の厚み）

4

0. 25 au yr-1@10 au

Ueda et al.

0. 24 au yr-1@10 au𝑡̃ = 𝑡⁄𝑡(),1 ,𝑡(),1 = 41.2 yr

0. 14 au yr-1@10 au

6.4𝑡!",$ ≈ 264 yr
2.4𝑡!",$ ≈ 99 yr

3.2𝑡!",$ ≈ 132 yr

Figure 2. Time evolution of the mid-plane temperature (top) and the height of disk surface (bottom) for the model with
• 𝑓=.-*,00.01の場合、𝑡̃ = 10auにある波が𝑡̃ = 11.2version
で10ofauに達している。→速度~0.
625 evolution
au yr-1 of
fd2g,s0.01=(left),
0.001 (center)
andで400.0001(right). An animatedthis figure provides the whole time
the TWI from t̃ = 0 to 19.8.

•

速度は中心にいくにつれ遅くなり、≤0.3 auで消える（速度はダストガス比に依存）。

disk thinner and hence cooler, the TWI can grow only
plane temperature at 5 and 50 au for the models with
fd2g,s= 0.01, 0.001 and 0.0001. At 5 au, the mid-planeat more inner region for smaller fd2g,s . At 50 au, we• 温度の変動のタイムスケールもダストガス比に依存。
temperature varies from ⇠ 30 K to ⇠ 80–90 K for all
clearly see that the amplitude of the temperature varimodels. The time separation of each peak is larger foration decreases with fd2g,s and almost vanishes whensmaller fd2g,s . This is because smaller fd2g,s makes the
fd2g,s = 0.0001. This trend indicates that the TWI can結果（輻射輸送）
𝑓!"#,% = 0.01•

RADMC-3Dでモデル画像→地球から140 pcの距離にあるとして0.03”のビームでスムージング•

ダストは次の分布をもつとする•

小さなダスト：最大半径は10 μm、ダストガス比は
𝑓.-*,0•

大きなダスト：最大半径は1 mm、ダストガス比は
80.01 − 𝑓.-*,0Ueda et al.

Figure 4. Synthesized images of the thermal wave instability observed at= 1.65 µm (left), 870 µm and 3.1 mm for the mode•3。
with fd2g,s = 0.01. The images are convolved
with a beam size of 0.00 03 which is denoted with a filled white circle at bottom lef
ミリ波のリングとギャップの強度比は~3、赤外線では~10•ギャップとリングの間隔は、0.2-0.4r•

can potentially detect the radial motion of the
wave. The substructures induced by the othermechanisms would move more slowly. For exam温度の差によってリング/ギャップが生じるので、円盤が光学的
ple, substructures associated with planet-induced
gap would migrate with a velocity slower than theに厚くても観測できる。
type I migration (e.g., Kanagawa et al. 2018).

in each panel. The simulated intensity atstellar flux attenuation.
gap•= 1.65 µm is multiplied by a square of the radial distance to compensate for the

で観測結果とほぼ同じ。

3. The gap and ring structure should appear both
密度差起源のリング/ギャップは光学的に厚い円盤では見えなくなる。

in millimeter dust continuum and infrared scattered light. The intensity variation is more pronounced for the scattered light. The other mech-

substructures between the TWI-induced ring andgap, which makes the apparent separation smaller
4.3. Caveats

Here we summarize our assumptions and discuss thepotential e↵ect on the TWI. First, we assumed that thedisk is in hydrostatic equilibrium in vertical direction
This assumption would be broken in outer region (> 50
au). The hydrodynamical motion of the disk gas might
potentially suppress the TWI at outer region, although結果（snow line）

𝑓!"#,% = 0.01 TWI• snow lineの位置が時間変化する。
•

•

•

10

Figure 5. Radial locations of the snow lines of volatile
species at t̃ = 0 and 11 for fd2g,s = 0.01.

We investigated the impact of the lineが存在し、Thermal Wave
ひとつの分子についても複数のsnow
stability (TWI) on the millimeter and infrared emissi
リングやギャップを作りうる。
of disks using the 1+1D simulations of disk temperatu
evolution.
confirm that the TWI operates when t• snow We
lineの位置を調べるときには、円盤のdisk is optically thick enough for stellar light, i.e., t
温度を正確に求める必要がある。
dust-to-gas mass ratio of small grains is & 0.0001. T
温度の変化のタイムスケールは、radial
driftのタ
mid-plane temperature varies as the waves
propagaand hence gap and ring structures can be seen in bo
イムスケールよりも短いので、ダストの焼結由来
millimeter and infrared emission even if the dust aのギャップ/リング形成は同時は起きない。
gas surface density have no substructures. Since t
substructures are induced by the temperature variatio
ただし、昇華と再凝縮によってダストサイズに変the millimeter substructures can be seen even if the d化が起きる可能性があり、substructureの形成に
is completely optically thick, while the density-inducsubstructures would disappear for optically thick dis
つながりうる。
Ueda et al.
The fractional separation
between the TWI-induced ri
and gap is typically ⇠ 0.2–0.4 at ⇠ 10–50 au, althou
additional substructures might form within them, e
by snow lines. Due to the temperature variation, snlines of volatile species move radially and multiple snlines are observed even for a single species, which wou
a↵ect the disk chemistry. The wave propagation veloc
is as fast as ⇠ 0.6 au yr 1 , which can be potentially dtected with a multi-epoch observation with a time searation of a few years. The TWI might be stabiliz
by the radial energy di↵usion and/or hydrodynami
motion of the disk gas, which are not fully taken inaccount in this work. The two-dimensional radiati

arXiv:2106.00148v1 [astro-ph.GA] 31 May 2021

Hsi-Wei Yen,1 Bo Zhao,2, 3 Patrick M. Koch,1 and Aashish Gupta4, 1•

Academia Sinica
of Astronomy and Astrophysics,
11F of Astro-Math
Bldg, 1, Sec.
4, Roosevelt Rd, Taipei 10617,
No impactofInstitutecore-scale
magnetic
ﬁeld,
turbulence,orTaiwanvelocity
Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, L8S 4M1, Canada
gradientMax-Planck-Institut
on sizesof protostellar
disks
in Orion A
für extraterrestrische
Physik (MPE), Garching,
Germany, 85748
1

2

3

4

Graduate Institute of Astronomy, National Central University, 300 Zhongda Road, Zhongli, Taoyuan 32001, Taiwan

Hsi-Wei Yen, Bo Zhao, Patrick M. Koch, Aashish Gupta

ABSTRACT
We compared the sizes and fluxes of a sample of protostellar disks in Orion A measured with the ALMA
0.87 mm continuum data from the VANDAM survey with the physical properties of their ambient environmentson the core scale of 0.6 pc estimated with the GBT GAS NH3 and JCMT SCUPOL polarimetric data. We did
not find any significant dependence of the disk radii and continuum fluxes on a single parameter on the corescale, such as the non-thermal line width, magnetic field orientation and strength, or magnitude and orientationof the velocity gradient. Among these parameters, we only found a positive correlation between the magnitudeof the velocity gradient and the non-thermal line width. Thus, the observed velocity gradients are more likely
related to turbulent motion but not large-scale rotation. Our results of no clear dependence of the disk radii onthese parameters are more consistent with the expectation from non-ideal MHD simulations of disk formationin collapsing cores, where the disk size is self-regulated by magnetic braking and diffusion, compared to othersimulations which only include turbulence and/or a magnetic field misaligned with the rotational axis. Therefore, our results could hint that the non-ideal MHD effects play a more important role in the disk formation.
Nevertheless, we cannot exclude the influences on the observed disk size distribution by dynamical interaction
in a stellar cluster or amounts of angular momentum on the core scale, which cannot be probed with the current
data.

原始星周りの円盤の形成メカニズムを調べるために、円盤のパラメータ（サイズ、線幅、磁場）の相関関
Keywords: Star formation (1569), Interstellar magnetic fields (845), Star forming regions (1565), Protostars係を調べた。

(1302), Circumstellar disks (235)•

protostellar sources but not inACUPOLのデータを使用。
the others (Li et al. 2014a;
1. INTRODUCTION
ターゲットはOrion
A分子雲で、ALMA（VANDAM survey）、GBT、JCMT•

速度勾配と非熱的な線幅のみで正の相関が見られ、他では相関関係は得られなかった。
The growth of a protostellar disk is related to the trans-••

Tsukamoto et al. 2020; Zhao et al. 2020a).
Protostellar disks around young protostars are potential sites of planet formation (Manara et al. 2018;
fer of mass and angular momentum from its parental dense
Sheehan & Eisner 2018; Segura-Cox et al. 2020). Large
非理想MHD効果が円盤形成に重要な役割を担っていると考えられる。
core to the vicinity of its central protostar via collapsing masamples of protostellar disks have been observed in therterial. In hydrodynamics, where the angular momentum of
現状のデータでは、円盤のサイズ分布に対する星団内の力学相互作用やコアスケールの角運動量の影響に
mal dust continuum emission (e.g., Segura-Cox et al. 2018;
collapsing material in dense cores is more or less conserved,
Cieza et al. 2019; Maury et al. 2019; Tobin et al. 2020).
ついて調べることはできない。
the size of a protostellar disk is proportional to the rotation
Keplerian rotation of several protostellar disks around
http://arxiv.org/abs/2106.00148v1

背景•Class 0/I天体の周りのケプラー回転する円盤の観測例はあるが、そのサイズは1桁以上異なる。

•

このサイズの違いは円盤の時間進化や円盤のサイズの成長を抑制するプロセスによるかもしれないが、詳しくはよくわかっていない。

•

円盤の成長と母体となる分子雲コアから原始星近傍へ質量や角運動量を運ぶプロセスとは関係がある。

•

角運動量が保存する流体力学的には、円盤のサイズはコアの角速度や円盤の質量に比例するはず。

•

母体の分子雲コアの回転軸と磁場の向きが揃っている場合において理想MHD的にはmagnetic brakingによって円盤の成長が抑制される。
•

向きが揃っていない場合にはmagnetic brakingの効率が下がり、大きな円盤形成される。

•

磁場の拡散が起こる非理想MHDの効果によっても、大きな円盤を形成することができる。

•（観測的には）どのメカニズムが円盤の形成と成長に大切なのかはわかっていない。

•

原始星周りの円盤のサイズ、ガスの運動、磁場の関係を調べることで、円盤の形成と成長のメカニズムに関する手がかりが得られると思われる。
•

→本研究

データ
ALMA 0.87 mm
singlestrophysical Journal, 890:130 (92pp), 2020 February 20

multiple system

The Astrophysical Journal, 890:130 (92pp), 2020 February 20•

サンプル数：56個（原始星は合計79個）•

Class 0：25個、Class I：13個、flat：18個•

single：26個、multiple system：30個•

半径はガウシアンでフィットした際の2σで定義し、分Tobin et al.

Tobin et al.

解できない場合はビームの1σを上限値として設定。
6

truncation radius
(a)

(b)

・Class 0

・Class I▼unresolved◉singleFigure 2. Comparisons between the disk radii measured with the ALMA 0.87 mm continuum data and (a) the projected separations and (b) thebolometric temperatures of the protostars. If the disks are not resolved with the synthesized beam of 0.!! 11 (43 au), the upper limits of their radii
are plotted as triangles. Red dots are the Class 0 protostars, and blue dots are Class I and flat-spectrum protostars. For a given protostar, theseparation is defined as its distance to the protostar nearest to it. Single protostars are also plotted here with the separations calculated with thesame definition, and they are labelled with open circles. In (a), a dashed line presents the expected truncation radius as a function of separation
in a binary system with equal masses and a non-eccentric binary orbit, and a vertical dotted line denotes a separation of 800 au. In (b), a verticaldashed line denotes a bolometric temperature of 70 K, which is adopted to separate Class 0 and more evolved protostars.

•

天体間距離<800 auの円盤は連星の相互作用の影響を受けている可能性あり。→サンプルから除く（残り50個）

This difference within a factor of two has also been foundin the synthetic observations of the numerical simulationsof a Keplerian disk formed in a magnetized collapsing core
(Aso & Machida 2020). Therefore, we expect that othermechanisms, such as initial angular momentum distributions, magnetic field, and/or turbulence in parental dense
cores (e.g., Terebey et al. 1984; Basu 1998; Li et al. 2014a;
Hennebelle et al. 2016; Wurster & Lewis 2020), are more
important in determining the disk radius distribution in our
sample.

•結果は連星の相互作用の影響を受けていない。

•

Tbolを進化の指標として用い、70 Kを境界に設定。

•

進化に伴う円盤サイズの減少は確認できない。

•

同じ進化段階（Tbol）でも、円盤サイズの大きさには1桁程度の開きがある。

4.2. Dependence on turbulence and magnetic field•

ダストの成長はミリ波連続波で見える円盤のサイ

Figure 3a compares the disk radii with the non-thermalline widths, which are expected to trace the levels of the turbulenceon the core scale. There is no correlation between
3. Example images from ALMA (left) at 0.87mm and the VLAFigure(right) 4.
at Example
9mm toward
imagesselectedfrom ALMA
sources.(left)
The attop0.87mm
row showsandHOPS-409,
the VLA (right)
a Class
at 9mm
0 protostar
toward selected wide multiple systems. The top row shows HOPS-182 (also
the turbulence
and theradii,
neithertheandentiren apparent resolved disk at 0.87 and 9mm. HOPS-361-C, a high-luminosity
known as L1641N),Class 0whichsource,
is is
made
shown
up of
in atheclose
middleand row
wideandmultipleappears
system.
disklike
Theatbrighter
0.87mm.
sources, HOPS-182-A
anddisk-B, bothappearresolved in
at 0.879mm.subAer, HOPS-361-C is resolved into a close binary system by the VLA
binaryat 9mm.
Class I The
source,brighter
HOPS-170,source also
is shown
exhibitsin thean extended
middle panel.
free–free
Bothjetcomponents
at this wavelength.
of HOPS-170 show apparent resolved disks that are likely viewed face-on. The
sampleor only for the young protostars with T bol < 70 K

Tobin et al. 2020

Map res. ~ 43 au

Pearson correlation3 coefficients between the disk radii andthese parameters of the turbulence and the magnetic fields.
The absolute values of the correlation coefficients are mostly
less than 0.1 and at most 0.17. We also computed the Spearman’s rank correlation4, which is more sensitive to a nonlinear correlation. No significant correlations were foundamong these parameters, and the confidence levels of having correlations are all lower than 50%, no matter we consider the entire subsample or only the young protostars withT bol < 70 K in the subsample.
We also compared the disk radii with the misalignments
between the rotational axes of the disks and the magnetic
field orientations on the core scale projected on the plane of
the sky. The rotational axes of our sample disks are definedas the axes perpendicular to the disk major axes measured
with the ALMA observations, and the misalignments of 0◦
and 90◦ mean the magnetic fields are parallel and perpendicular to the rotational axes, respectively. The disk radii in our
entire subsample do not show any dependence on the mis-ズを決める主要なプロセスではない。

that the standard deviations tend to be underestimated by
The northern part of the Orion A molecular clouds has
13%, 6%, and 4% when N is 4, 7, 10, and the uncertainties in
The northern part of the Orion A molecular clouds has
13%, 6%, and 4% when N is 4, 7, 10, and the uncertainties inen observed in the polarized thermal dust emission at 850
thecalculated standard deviations due to the small number of
eenobserved
inthepolarized
thermaldust
emissionat850
4 450 µm by the SCUPOL survey using the Jamesthe calculated standard deviations due to the small number of
anddata points were estimated to be 39%, 29%, and 23%, respecm and 450 µm by the SCUPOL survey using the Jamesdata points were estimated to be 39%, 29%, and 23%, respecerk Maxwell Telescope (JCMT; Matthews et al. 2009), as
tively.
appliedtheseofcorrections
andanduncertainties
whenwith
Clerk
Maxwell
Telescope(JCMT;
Matthewsetal.
2009),
aseach
protostarwas
computedastionWeatlocation
detection,
only the pixels
GBT
NH
コアスケール~0.06
tively.
Wetheappliedthese thecorrections
and uncertainties
when
the strengths andpcのガスの運動
orientations of
own in Fig. 1. To3estimate
we measuredtheangulardispersionsof the magnetic
field
the strengths and orientations ofhown in Fig. 1. To estimate!
detections
were
includedinthecalculations.
Then
this
largewe measured the angular dispersions of the magnetic fieldmagnetic field on the core scale around our sample protoorientations
on the corescale in our was
sampleprotostars
based
kT kour sample protohe magnetic field on the core scale 2aroundscale magnetic
subtracted
fromthe ori速度勾配
orientations
on thefieldcoreorientation
scale in our sample
protostars
basedδVmapsrs, we retrieved the δV
Stokesnt = IQU
ob − at 850, µm of OMC- (1)onthenumbersofthepolarization
detections
associatedwithtars, we retrieved the Stokes IQU mapsmatµm of OMCof that
detection.
Finally,
we calculated
NH850
3
on entation
the numbersof polarization
the polarization
detections
associatedwithnd -3 obtained with the JCMT SCUPOL
surveyfrom the
Wang,
J.-W.
et
al.
(2019),
wethenofsubthem.
As
discussedinand -3 obtained with the JCMT SCUPOL
surveyfromthethestandard
deviationoftheresidualorientations
theet thenal. 19932
al. (2019), we
sub-pothem. As discussed in Wang, J.-W. etGoodman.
These
Stokes
IQU
UBA
Polarimeter
LegacyCatalogue
コア周囲の33pixの値を使用（1
pix~4200
au）
2
""
where
δV
is
the
observedNH
line
width,kistheBoltz3tractedthe uncertainties
the polarization
orientations
dueob
3. These Stokes IQU
CUBA Polarimeter
Legacy Cataloguelarizations
detectionsinwithin
15 from thesample protostar,""tractedthe uncertainties
in the polarization
orientations
due
psmannwereconstant,smoothedand
to m
have
an
angularresolutionof20"" 3 .
isantheangularmolecular
weightofof20
NH
to
the
noise
in
the
data
(δθ
)
from
the
measuredangulardisσ
3maps were (a)smoothed to NH
have
resolution
andobtainedtheangulardispersion
ofthemagnetic
field(b)
to the noiseJCMT
in the data850(δθσ ) fromthe measured
angular dis-oriatthews
etmeasured
al. 2009).theWemagnitude
extracted polarization
detections
磁場
μm
We
also
and
directionofthevelocpersion(δθ)
toestimatetheintrinsic
angulardispersionsofob on the core scale.
Matthews
etNH
al. 2009). We extracted polarization detections
entationspersion(δθGBTob ) to estimate the intrinsic angular dispersions ofmity
theStokes
IQU
maps
with
the
detection
criteriaofthe3
gradient
onIQU
the coreeach protostar.
magneticfield(δθBstandard) as,
rom theStokesmapsscale
witharoundthe detection
criteria ofWe
thefol- theWe notethatorientations
the calculated
the magnetic
fieldorientations(δθB ) as, deviation can be bi方向
nal-to-noise
ratiosof
StokesI
andpolarized
intensities
lowed the method
al. polarized(1993)
and850
fittedμm
a linJCMT
ignal-to-noise
ratiosinofGoodman
Stokes I etand
intensities
ased due to the small"number
included. We" 2 of detections
hear
above
three.
Then
we
inferredthemagneticfieldorien2
velocity
gradient
to3
by3
pixelsaroundtheprotostellar
δθ
−
δθ
.
δθ
=
ob withσN 2numbers of data(2)B samplesoth above three. Then we inferred the magnetic field2
generated simulated
points◦ orienδθ
−
δθ
(2)
δθ
=
ob
σ .
B
ons
by
rotatingthepolarization
orientations
by
90
.
The
◦
positioninthecentroidline-of-sightvelocitymap.
ations by rotating the polarization orientations by 90 . The
randomly drawn from a Gaussian distribution, and calculated
Finally, we derived the magnetic field観測ノイズ
strengths on the coreerred magnetic field orientations are shown in Fig. 1b.
Finally,
we derived
the magnetic
field strengths
on thenferred magnetic field orientations are shown in Fig. 1b.
the standard
deviationsof the simulated
samples.
Wecorefound
3.3. JCMT polarized
850weµm computed
continuum mean
scale projected on the plane of the sky (Bpos) in our samFrom the polarization
detections,scaleon theplane of tend
the sky(Bpos) in our sam- by
From the polarization detections, we computed mean
thatprojected
the standard
deviations
to beunderestimatedple protostars天球面に射影した磁場強度
with the Davis–Chandrasekhar–Fermi method
okesTheQ andU intensities
averaged
overA3 by
3 pixels aroundnorthern
part
of
the
Orion
molecular
cloudshas
ple13%,
protostars
with
Davis–Chandrasekhar–Fermi
method intokes Q and U intensities averaged over 3 by 3 pixels
around6%,（チャンドラセカールフェルミ法）
and
4%the
when
N is 4, 7, 10, and the uncertainties""(Davis1951;
Chandrasekhar
&
Fermi
1953)
as,
chbeensampleprotostar,
where
the
pixel
size
is
10
,
and
only
""
observedin thewherepolarized
thermalemission
at 850(Davis
1951; Chandrasekhar& Fermidue1953)
as,small number ofach sample
protostar,
the pixel
sizedust
is 10, and only
the calculated
standard deviations
to thepixelswith
detections
were
includedinthecalculations.
#
µm
and
450
µm
by
the
SCUPOLsurvey
using
the
James
he pixels with detections were included in the calculations.
nt
data points were estimated #to beδV
39%,
29%, and 23%, respec(3)Bpos = ξ 4πρ δVnt,
omClerk
the mean
StokesQ and (JCMT;U intensities,
the meanmagMaxwell
Telescope
Matthewsetal.
2009),as= ξcorrections
4πρδθB ,and uncertainties (3)rom the mean Stokes Q and U intensities, the mean magtively. We appliedBpos
these
when
δθ
icshown
field orientation
on
the
core
scale
was
derivedforeach
B
estimate
thescalestrengths
and orientations
Fig. 1. To on
the meanthe
andangularstandard
deviation ofof theetic fieldinorientation
the core
was derived
for each of which
weare
measureddispersions
the diameters
magnetic field
𝜉 =to
0.5
where
ξ
was
adoptedtobe0.5
correctforinhomogeneous
otostar.
In
oursample,
there
are
56
protostars
withnearbythemagnetic
field
on
the
core
scale
aroundoursampleprotoof
the
dense
cores
in
Orion
A
(Lane
et
al.
2016;
Kirk
et al.based
where
ξ
was
adopted
to
be
0.5
to
correctforinhomogeneous
rotostar. In our sample, there areorientations
on the fieldcore scalein our structures
sample protostars"" 56 protostars with nearby
andcomplexmagnetic
anddensityalong
thearization
detections
within
15
and
thus
with
the
mea"" maps at 850 µm of OMC- 2017),
stars, we retrieved
thewithin
Stokes15
IQU
respectively.
Withtheestimatedcolumndensityfromandoncomplex
magnetic
field
and density
structures
along
the
𝜌：Stokes
Iマップよりolarization
detections
and thus with the meathe
numbersofthepolarization
detections
associated
with
line
of
sight
(Ostriker
et
al.
2001)
and
ρ
is
the
volumedened2 meanmagneticfieldorientations
on
the
core
scale.
-3 obtained
JCMT SCUPOL
surveyfrom the theline
continuum
emissionin
andassumedalong
theof sight(Ostriker
etal. the2001)
andet ρal.lengthis(2019),the volume
denuredandmeanmagneticwith
fieldtheorientations
on the corescale.
Wang,
J.-W.
we thensubthem.
Asdiscussed
sity.
δV
is
from
the
GAS
NH
data.
We
estimated
ρ
from
the
2nearby polariza𝛿𝑉：アンモニア観測より
nt
3
For
the
34
protostars
with
more
than
three
2(
line
of
sight,
thevolumedensitywasderivedforeachsample.
These
StokesIQU
SCUBA
Polarimeter
LegacyCatalogue
sity.
δV
is
from
the
GAS
NH
data.
We
estimated
ρ
from
the
nt the uncertainties in3the polarization orientations due
For the 34 protostars withmore than three nearby polarizameantracted
intensityof the 850µm3.continuum
emissionon the core"" protostar
n mapsdetections
within15"" ,toalsoanmeasured
its angle dis""weandadoptedEq.
The
normalized
mass-to-flux
weresmoothed
have
angularresolution
of
20
mean
intensityof thethein850µm
continuum
emission
onangular
the coredison detections within 15 , we also measured its angle disto
the
noise
in
data
(δθ
)
from
the
measured
σ
scale
in
the
StokesI
maps
obtained
from
the
JCMT
SCUPOLsionofthemagneticfieldorientations
on
the
core
scale
ratio
(λ)
was
also
computedforeachsampleprotostaras,al. 2009).fieldWe orientations
extracted polarization
detections(Matthews
scale
in the(δθ
Stokesmaps mass-to-ﬂux
obtainedfrom theJCMTdispersions
SCUPOL ofersionof theetmagnetic
on the corescalepersiontoIestimate
the intrinsic
angularnormalized
ratioob )sample
survey.
Foreachprotostar,wefirstmeasuredthe op0.06
pc.
To
removecontributions
bylarge-scale
magnetic
B offield thethe ToStokes
IQUcontributions
maps with thecriteria
survey.
For eachfieldsampleprotostar,wefirstmeasuredthe opf from0.06 pc.
removeby detection
large-scale
magneticthemagneticorientations
(δθ
)
as,
Class 0√ Σ BClass I, Flat spec. tical depth of the 850 µm continuum, assuming that the dust
dsignal-to-noise
structures, we ratios
adoptedthemethodsimilartounsharpof Stokes
I and similar
polarized
intensities
λ=2πcontinuum,G, assuming that the(4)tical depth of the 850µmdust
eld structures, we adopted
the method
to unsharp"
temperature
is
the
same
as
the
kinetic
temperature
measuredB
pos
Pattleetal.
(2017)
sking,
asintroducedin,forexample,both above
three. Thenthe Pattle
magneticorien2 temperature2
temperature is the same Bas=the δθkinetic
measured(2)et al.field(2017)
masking,as introduced
in,weforinferred
example,. optical
ob − δθσ thewith the GAS NH data.δθThen
we converted
depth

データ

結果①

(a)

(b)

(c)

(d)

(e)

(f) Class 0 protostars

・Class 0
・Class I▼unresolved◉singleいずれも相関なし

in 3D space

結果②

•

回転は長軸方向の速度勾配をうみ、円盤と速度勾配のなす角は小さく揃っているはず。
9• 実際そうなっていない。

(a)

(c)

•

(d)

Figure 4. Comparisons between the disk radii measured with the ALMA 0.87 mm continuum data and (a) the magnitudes of the velocity

gradients measured with the GBT NH data, (b) the alignments between the disk
axes and the velocity gradients, and (c) the magnitudes• major
唯一相関関係が見られる。
速度勾配を長軸に射影しても、相関は見らof the velocity gradients projected onto the disk major axes as an assessment of possible rotational motion, on the core scale of 0.06 pc. Panel
れない。 (d) compares the magnitudes of the velocity gradients with the non-thermal line widths on the core scale measured with the GBT NH data.
3

3

Error bars present the 1σ uncertainties. If the disks are not resolved with the synthesized beam of 0.!! 11 (43 au), the upper limits of their radii
are plotted as triangles. Red dots are the Class 0 protostars, and blue dots are Class I and flat-spectrum protostars. Single protostars are labelledwith open circles.

This is because the disk radii are self-regulated by magnetic braking and diffusion of the magnetic field due to thealignment, and magnetic field strengths. In this case, thedisk radii could more depend on the magnetic diffusivities

議論
•

•

•
•

•
•

理想MHD
• 乱流と磁場ー回転軸間のmisalignmentによってmagnetic brakingの効率が落ち、円盤が大きくなる（シミュレーション結果より）。
• 観測結果と異なる。
非理想MHD• 乱流と磁場ー回転軸間のmisalignmentの円盤の成長へさほど影響せず、 magnetic
braking / diffusionによってサイズが決まる（シミュレーション結果より） 。
• パラメータ間の相関がないことと無矛盾。
• 円盤のサイズは、宇宙線の電離率やダストサイズなどから決まる磁気拡散係数に依存する可能性がある。
Class 0天体20個に対する観測では、磁場と円盤が揃っていない場合、エンベロープが大きな角運動量を持つ傾向にあることを示している。
本研究では、より小さなスケールにおいて、そのような傾向は見られていない。
• 原始星近傍の100 auスケールではmagnetic braking / diffusionが効率的になる。

コアスケールにおいて速度勾配と非熱的な線幅の間に相関が見られたので、速度勾配はlarge-scale motionではなく乱流によって生じていると考えられる。
Class 0天体に対する800-900 auの分解能の観測において、比角運動量 𝑗 のradial profile
は、𝑗 ∝ 𝑟%.'となる（剛体回転は𝑗 ∝ 𝑟 ( 、乱流は𝑗 ∝ 𝑟%.) 、Pineda et al. 2019）。
• 速度勾配や角運動量を正確に見積もるにはより高い分解能が必要。