論文 / 著書情報
Article / Book Information
題目(和文)

埋没惑星によって引き起こされるガスの流れ、ダスト動力学、及び原
始惑星系円盤の構造

Title(English)

Gas flow, dust dynamics, and protoplanetary-disk substructuresinduced by embedded planets

著者(和文)

桑原歩

Author(English)

Ayumu Kuwahara

出典(和文)

学位:博士(理学),学位授与機関:東京工業大学,
報告番号:甲第12323号,
授与年月日:2023年3月26日,
学位の種別:課程博士,
審査員:井田 茂,中本 泰史,佐藤 文衛,玄田 英典,奥住 聡

Citation(English)

Degree:Doctor (Science),
Conferring organization: Tokyo Institute of Technology,
Report number:甲第12323号,
Conferred date:2023/3/26,
Degree Type:Course doctor,
Examiner:,,,,

学位種別(和文)

博士論文

Type(English)

Doctoral Thesis

Powered by T2R2 (Science Tokyo Research Repository)

T OKYO I NSTITUTE OF T ECHNOLOGYD OCTORAL T HESIS

Gas flow, dust dynamics, andprotoplanetary-disk substructures inducedby embedded planets

Author:Ayumu K UWAHARA

Supervisors:
Prof. Shigeru I DA,
Specially Appointed AssociateProf. Hiroyuki K UROKAWA

A thesis submitted in fulfillment of the requirements
for the degree of Doctor of Sciencein theDepartment of Earth and Planetary Sciences

January 27, 2023

iii

TOKYO INSTITUTE OF TECHNOLOGY

Abstract
Department of Earth and Planetary Sciences
Doctor of Science
Gas flow, dust dynamics, and protoplanetary-disk substructures induced byembedded planets
by Ayumu K UWAHARA

Planets form in protoplanetary disks. The outcome of planetary growth depends on the environment of disks. The current observed period-mass distribution of exoplanet suggests the dominance of low-mass planets. Recent high-spatialresolution observations have revealed dust substructures in disks such as rings andgaps, which do not always correlate with gas. The interaction between the planet
and the disk would be a key for interpreting the diversity of exoplanets and observations of protoplanetary disks. A growing planet embedded in a disk interacts withthe surrounding disk gas and induces three-dimensional gas flow. Small dust grains
are abundant in a disk, and they are sensitive to the gas flow. Planets accrete thesesmall dust grains. Because gas flow induced by embedded planets could affect dust
dynamics, it potentially affects the growth of planets and forms the observed dust
substructures in disks.
In this thesis, we investigate dust dynamics influenced by the planet-induced gas
flow and the potential of the planet-induced gas flow to sculpt the rings and gaps
in the dust profiles. Assuming a nonisothermal, inviscid sub-Keplerian gas disk,
we first perform three-dimensional hydrodynamical simulations, which resolve thelocal gas flow past a planet. We then numerically integrate the equation of motion for
dust particles to calculate the trajectories of dust influenced by the planet-induced
gas flow. We used the obtained hydrodynamical simulation data to calculate thegas drag force acting on the dust particle. Finally, we compute the steady-state dust
surface density by incorporating the influences of the planet-induced gas flow intoa one-dimensional dust advection-diffusion model.
We find that the gas flow induced by embedded planets affect dust dynamics
around planetary orbits and the spatial distribution of dust changes significantly.
Throughout this thesis, the key physics is the outflow of the gas. A substantial
amount of gas from the disk enters the gravitational sphere of the planet (inflow)
and exits (outflow). The outflow induced by the embedded planet occurs in theradial direction to the disk.
The outflow of the gas toward the outside of the planetary orbit inhibits the radial
drift of dust, leading to dust accumulation (the dust ring). The outflow toward theinside of the planetary orbit enhances the inward drift of dust, causing dust depletion around the planetary orbit (the dust gap). Under weak turbulence (αdiff . 10−4 ,where αdiff is the turbulence strength parameter), the gas flow induced by the planetwith & 1 M⊕ (Earth mass) generates the dust ring and gap in the distribution ofsmall dust grains (. 1 cm) with a radial extent of ∼ 1–10 times the gas scale heightaround the planetary orbit without creating a gas gap and pressure bump.

iv
We find that the response of the spatial distribution of dust to the planet-induced
gas flow varies significantly depending on the location of the planet within the disk,
resulting in the differences in the growth efficiency of planets. The dust gap formation by the planet-induced gas flow is susceptible to occur in the inner region of thedisk (. 1 au), where the growth of the planet via the accretion of dust onto the planet
is suppressed due to the dust gap. The dust ring formation by the planet-induced
gas flow is susceptible to occur in the outer region of the disk (& 10 au), where theefficient growth of the planets via dust accretion would be achieved.
Our results suggest that: (1) the gas flow induced by embedded planets can beconsidered a possible origin of the observed dust substructures in disks whose dust
substructures were found not to correlate with those of the gas, and (2) the planetinduced gas flow could determine the architecture of planetary systems, which may
be helpful in explaining the current observed period-mass distribution of exoplanets in which the low-mass planets are more frequent than giant planets at < 1 au.
The dust rings and gaps formed by the planet-induced gas flow might be probed by
future observations using a next-generation Very Large Array (ngVLA) and a possible future extension to the Atacama Large Millimeter/submillimeter Array (ALMA).
Our future work will include more quantitative discussions of both a comparison between our results and ptoroplanetary disk observations using synthetic images andthe influence of the planet-induced gas flow on the planetary growth with additional
physics such as dust growth and planet migration.

v

Acknowledgements
My deepest appreciation goes to my supervisor, Shigeru Ida, for his constructive
suggestions and thoughtful guidance over the past five years. He also gave me theimpetus for liking a sake. I owe my deepest gratitude to Hiroyuki Kurokawa, whois the other supervisor. He have taught me what a scientific inquiry is all about. He
has also provided me with tremendous support on many occasions over the past
five years. The collaborations with him has always inspired me and taught me thejoy of planetary science.
I would like to express my great appreciation to Satoshi Okuzumi of Tokyo Institute of technology. He has accompanied me in discussions regarding my study
on many occasions and given me precise advices. His comments were crucial in improving my understanding of basic physics, which was the foundation for writing
this thesis, and in considering the future prospects. I would like to express my gratitude to Takayuki Tanigawa of National Institute of Technology, Ichinoseki College,
for his kind advice and precise suggestions based on his deep insight into fluid dynamics. His comments helped me to come up with a research theme directly related
to the content of this thesis.
I would further like to express my heartfelt thanks to the members of the Department of Earth and Planetary Science, Tokyo Institute of Technology, and especiallyto the members of Planetary Group. Looking back, I am amazed at how much fun I
had and how fulfilling and irreplaceable my days were. I was fortunate to be in thewonderful environment for five years.
Special thanks to “Wish Upon the Pleiades”, a precious work that gave me asupportive push forward.

vii

Contents
Abstractiii

Acknowledgementsv

1

General Introduction
1.1 Observations of planetary systems . . . . . . . . . . . . . . . . . . . . .
1.2 Observations of protoplanetary disks . . . . . . . . . . . . . . . . . . . .
1.3 Overview of planet formation . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Minimum mass solar nebula . . . . . . . . . . . . . . . . . . . .
1.3.2 Physical properties of protoplanetary disks . . . . . . . . . . . .
1.3.3 Planetary growth . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.4 Radial drift of dust . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.5 Accretion regime of particles . . . . . . . . . . . . . . . . . . . .
1.3.6 Overview of disk-planet interaction . . . . . . . . . . . . . . . .
1.4 Origins of dust rings and gaps in protoplanetary disks . . . . . . . . .
1.4.1 Planet-free mechanism . . . . . . . . . . . . . . . . . . . . . . . .
1.4.2 Planet-dependent mechanism . . . . . . . . . . . . . . . . . . . .
1.5 This thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
3
6
6
6
7
8
10
14
17
17
18
18

2

Gas flow induced by low-mass planets
2.1 Introduction of this chapter . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Model overview . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Dimensionless unit system . . . . . . . . . . . . . . . . . . . . .
2.2.3 Unperturbed disk . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.4 Three-dimensional hydrodynamical simulations . . . . . . . . .
2.2.5 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.6 Boundary condition and numerical resolution . . . . . . . . . .
2.2.7 Disk model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Three-dimensional planet-induced gas flow . . . . . . . . . . .
2.3.2 Isolated envelope . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Structure of the planet-induced gas flow at the midplane region
Dependence on the planetary mass . . . . . . . . . . . . . . . . .
Dependence on the headwind . . . . . . . . . . . . . . . . . . .
2.3.4 Geometry of the outflow . . . . . . . . . . . . . . . . . . . . . . .
Width of the outflow . . . . . . . . . . . . . . . . . . . . . . . . .
Outline of the outflow . . . . . . . . . . . . . . . . . . . . . . . .
Length of the outflow . . . . . . . . . . . . . . . . . . . . . . . .
2.3.5 Vertical structure of the outflow . . . . . . . . . . . . . . . . . .
2.3.6 Outflow speed at the midplane region . . . . . . . . . . . . . . .
Positions of inflow and outflow . . . . . . . . . . . . . . . . . . .
Analytical estimation of the outflow speed . . . . . . . . . . . .

21
21
23
23
24
25
25
26
27
27
31
31
33
35
37
38
39
39
42
42
42
46
47
49

viii
2.3.7 Velocity distribution of the outflow . . . . . . . . . . . . . . . .
2.3.8 Caveats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Discussion of this chapter . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Comparison to the previous studies . . . . . . . . . . . . . . . .
Atmospheric radius . . . . . . . . . . . . . . . . . . . . . . . . .
Isolated envelope . . . . . . . . . . . . . . . . . . . . . . . . . . .
Outflow speed . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Applicability of Bernoulli’s theorem . . . . . . . . . . . . . . . .
2.4.3 Angular momentum of the gas . . . . . . . . . . . . . . . . . . .
2.4.4 Implications for the dust motion under the influence of theoutflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56
58
60
60
60
62
62
64
70

3

Dust motion influenced by the planet-induced gas flow
3.1 Introduction of this chapter . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Model overview . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Three-dimensional hydrodynamical simulations . . . . . . . . .
3.2.3 Three-dimensional orbital calculation of dust particles . . . . .
3.2.4 Calculation of accretion rate of dust . . . . . . . . . . . . . . . .
Width of accretion window and accretion cross section . . . . .
Accretion rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Results overview . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Orbital calculations . . . . . . . . . . . . . . . . . . . . . . . . . .
Trajectories of dust particles in 2D . . . . . . . . . . . . . . . . .
Trajectories of dust particles in 3D . . . . . . . . . . . . . . . . .
3.4 Discussion of this chapter . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Accretion rate of dust onto a planet . . . . . . . . . . . . . . . .
3.4.2 Comparison to previous studies . . . . . . . . . . . . . . . . . .
3.4.3 Caveat: the uniform distribution of dust particles . . . . . . . .

75
75
77
77
78
79
82
82
83
84
84
84
84
86
91
91
94
95

4

Dust ring and gap formation by the planet-induced gas flow97
4.1 Introduction of this chapter . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.2.1 Model overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.2.2 Three-dimensional hydrodynamical simulations (the local domain) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.2.3 Dust trajectory simulations (the local domain) . . . . . . . . . . 101
Three-dimensional orbital calculation of dust particles . . . . . 101
Spatial distribution of dust in the local domain . . . . . . . . . . 101
4.2.4 Model for global dust distribution . . . . . . . . . . . . . . . . . 103
Spatial distribution of dust in the global domain . . . . . . . . . 103
Azimuthally and vertically averaged drift velocity of dust . . . 103
4.2.5 Simulation of dust surface density . . . . . . . . . . . . . . . . . 104
Steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Time evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Analytical estimation . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.2.6 Numerical settings: Dust trajectory simulations . . . . . . . . . 105
4.2.7 Numerical settings: Azimuthally and vertically averaged drift
velocity of dust . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

2.4

71

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4.3.1
4.3.2
4.3.3
4.3.4
4.3.5
4.3.6

4.4

Results overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Three-dimensional planet-induced gas flow . . . . . . . . . . . 109
Existence of the forbidden region . . . . . . . . . . . . . . . . . . 109
Dust motion within the forbidden region . . . . . . . . . . . . . 110
Width of the forbidden region . . . . . . . . . . . . . . . . . . . 112
Influence of planet-induced gas flow on the dust surface density114
Dust ring and gap formation by planet-induced gas flow: Flowshear regime . . . . . . . . . . . . . . . . . . . . . . . . 115
Dust ring formation by planet-induced gas flow: Flow-headwind
regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Dependence on planetary mass . . . . . . . . . . . . . . . . . . . 118
Dependence on turbulence strength . . . . . . . . . . . . . . . . 121
Discussion of this chapter . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.4.1 Caveat: the steady-state solution . . . . . . . . . . . . . . . . . . 125
4.4.2 Time evolution of the dust surface density . . . . . . . . . . . . 126
4.4.3 Analytical estimation of the drift velocity of dust influenced
by the planet-induced gas flow . . . . . . . . . . . . . . . . . . . 132
4.4.4 Simulation of dust surface density using the analytic formula . 137
Steady-state solution . . . . . . . . . . . . . . . . . . . . . . . . . 137
Time evolution of the dust surface density . . . . . . . . . . . . 140
4.4.5 Comparison to previous studies . . . . . . . . . . . . . . . . . . 143
4.4.6 Caveat: the spatial distribution of dust in the global domain . . 144
4.4.7 Caveat: non-migrating planet . . . . . . . . . . . . . . . . . . . . 145

5Discussion and outlook
147
5.1 Implications for the origins of the architecture of a planetary system . 147
5.1.1 Turbulence in a protoplanetary disk . . . . . . . . . . . . . . . . 147
5.1.2 Size distribution of dust . . . . . . . . . . . . . . . . . . . . . . . 148
5.1.3 Planet formation via pebble accretion in the planet-induced
gas flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.2 Implications for protoplanetary disk observations . . . . . . . . . . . . 153
5.2.1 Correlation between the spatial distribution of gas and dust . . 154
5.2.2 Kinematic features in a protoplanetary disk . . . . . . . . . . . . 155
5.2.3 Comparison to observed widths of dust rings and gaps . . . . . 156
5.2.4 Comments on individual protoplanetary disks . . . . . . . . . . 158
HD 143006 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
Oph 163131 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
HD 100546 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.3 Dust growth and sequential formation of planetesimals . . . . . . . . . 160

6

Summary163

A Additional hydrodynamical simulations
167
A.1 Numerical convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
A.2 Dependence on smoothing length . . . . . . . . . . . . . . . . . . . . . . 172
B Additional figures
179
B.1 Comparison between the analytical estimations of the drift velocity of
dust influenced by the planet-induced gas flow and numerical results . 179
Bibliography

183

1

Chapter 1

General Introduction
Protoplanetary disks are birthplaces of planets. The disk contains gas and dust. A
growing planet embedded in a disk interacts with the surrounding disk gas anddust, which influences the processes of planet formation. The interaction between
the planet and the disk would be a key for interpreting the diversity of exoplanetsand observations of protoplanetary disks.

1.1

Observations of planetary systems

More than two decades have passed since the first discovery of an exoplanet orbitinga main-sequence star, 51 Peg b (Mayor and Queloz, 1995) and the number of confirmed exoplanets has now reached around 5000 and continues to increase1 . As thenumber of detections of exoplanets increases, the period–mass and period–radius
distribution has shown diversity of exoplanets (Fig. 1.1). About half of Sun-like stars
harbor close-in super-Earths with orbital periods of less than 85 days, radii of 1–4 R⊕(Earth radii), and masses of 2–20 M⊕ (Earth masses; Fressin et al., 2013; Weiss andMarcy, 2014). About 1% of Sun-like stars host hot Jupiters (. 0.1 au), while around
10% stars harbor cold Jupiters (∼ 1–10 au) with a peak of occurrence rate of between
2 and 3 au (Johnson et al., 2010; Fernandes et al., 2019). These latter planets referred
to as gas giants with radii of ∼ 10–20 R⊕ and masses of ∼ 100–4000 M⊕ . The occurrence rate of the planets with an orbital distance of ∼ 30–300 au is as low as 1%
at most (Bowler, 2016). The current observed period-mass distribution of exoplanets suggests that low-mass planets are more frequent than giant planets (Cumming
et al., 2008; Howard et al., 2010; Mayor et al., 2011; Wright et al., 2012; Dong andZhu, 2013; Petigura et al., 2013; Zhu and Wu, 2018; Fernandes et al., 2019). The exoplanets have distinct properties compared with the planets in the Solar System interms of masses, sizes, compositions and orbital properties. For instance, the strong
correlation between super-Earths and cold Jupiters have been reported, where cold
Jupiters seem to be commonly found with super-Earths (Zhu and Wu, 2018; Bryan
et al., 2019). In the Solar System, however, Jupiter does not accompany super-Earths.
Debate over the origin of exoplanetary systems is still ongoing. The diversity of
exoplanetary systems may indicate that the process of planet formation and environment in the exoplanetary systems differ from those in the Solar System. Planets
form in protoplanetary disks. Thus, the studies of protoplanetary disks are essential
for understanding the planet formation process.
1 http://exoplanet.eu/catalog/

2

Chapter 1. General Introduction

F IGURE 1.1: Distribution of exoplanets. The gray dots represents the exoplanets. We note that this plot includes the planets
whose masses are estimated by their radii. Data obtained fromhttp://exoplanets.org/

1.2. Observations of protoplanetary disks

1.2

3

Observations of protoplanetary disks

Since the protoplanetary disks are birthplaces of planets, studies on their environments lead to an understanding of the planet formation process. Recent high-spatialresolution observations of protoplanetary disks have provided clues to understandthe planet formation process. Let us first summarize the tracers of protoplanetary
disk. The three traces are used to observe disks: the scattered light, thermal continuum emission, and the spectral line emission.
1. The scattered light, which is obtained when micronmeter(µm)-sized dust in
the uppermost layer of the disk reflects light emitted from the central star. The
scattered light is observed at optical and near-infrared wavelengths. The scattered light image shows the distribution of µm-sized dust grains, which depends on the radial and vertical distribution of dust in the disk.
2. Thermal continuum emission emitted by millimeter(mm)-sized dust grains at
the midplane region of the disk. The dust continuum emission is observedat radio wavelengths. In these wavelengths, the stellar emission is opticallythin, but the thermal continuum from ∼mm-sized dust peaks at ∼mm wavelengths.The dust mass in the disk can be inferred from the observed flux ofmm continuum emission.
3. The spectral line emission from (sub-)mm rotational transitions of tracer molecules
show the distribution of disk gas. Hydrogen is thought to be the most abundant in the disk, but it does not emit in a wide range of the disk because it does
not have a permanent dipole moment. Thus, the spectral line observations relyon the rare molecular species such as CO.
Though important paramerters such as the mass, size, density, and temperature of
the protoplanetary disk, in this thesis, we focus especially on the spatial distributionsof gas and dust in the disk. Observations using different wavelengths are crucial tounderstand these structures of protoplanetary disks. For instance, the sizes of gas
and dust disk are obtained separately from the spectral line and continuum emissions. A comparison of the size of the gas and dust disks shows that the gas disks
are on average about twice as large as the dust disks (Andrews et al., 2011; Ansdell
et al., 2018; Jin et al., 2019; Trapman et al., 2020). This may caused by radial drift ofdust (described later in Sect. 1.3.4).
In recent years, the sensitivity and angular resolution of telescopes have improved remarkably. The Atacama Large Millimeter/submillimeter Array (ALMA)
has the capability to resolve disks at milliarcsecond (0.01 arcsec), corresponding to a
∼ 3–5 au scale in nearby star forming regions (. 150 pc). The ALMA observations
have revealed substructures in dust profiles (hereafter dust substructures) in bothyoung and evolved protoplanetary disks (e,g., ALMA Partnership et al., 2015; Andrews et al., 2018; Huang et al., 2018a; Long et al., 2018; Marel et al., 2019). Figure
1.2 summarizes the dust substructures obtained from recent high-spatial-resolution
observations and shows the various morphologies of dust substructures. The dust
substructures can be classified into three broad categories.
1. Rings-gaps
The axisymmetric bright and dark contours in Figs. 1.2a and b show the regions where the dust is accumulated (the dust ring) and depleted (the dust
gap; Avenhaus et al., 2014; ALMA Partnership et al., 2015; Follette et al., 2015;
Andrews et al., 2016; Pinilla et al., 2017; Monnier et al., 2017; Plas et al., 2017;

4

Chapter 1. General Introduction
Huang et al., 2018a; Long et al., 2018; Marel et al., 2018; Pinilla et al., 2018a;
Pinilla et al., 2018b; Kudo et al., 2018; Guzmán et al., 2018; Van Terwisga etal., 2018; Villenave et al., 2019). These dust substructures are most commonly
found in protoplanetary disk observations. We focus especially on the ringsand gaps in the dust profiles in Chapter 4. About half of the disks accompanied with ring-like structures have only a single ring, and others have multiple
rings (e.g., ALMA Partnership et al., 2015). The large central cavities have beenfound in the transition disks where the inner disks are depleted in dust relative
to the outer disk (Fig. 1.2a). The dust rings have been found in a wide range ofdisk radii, with a peak frequency around ∼ 10–50 au from the central star (Baeet al., 2022). Due to limitations of the angular resolution, it is difficult to detect rings in the inner region of the disk. A next-generation Very Large Array(ngVLA) will provide the capability to probe the inner few au regions (Ricci etal., 2018; Harter et al., 2020; Ilee et al., 2020; Burrill et al., 2022). The observed
widths of the rings range from a few au to 80 au. The position and width of therings have not been correlated with the the mass of the host star. Compared todisks without rings, those with rings have been found to have predominantly
larger dust masses, indicating that more dust is retained in the disk for longer
periods of time (Marel et al., 2019). Dust gaps have also been found at a wide
range of radii from ∼ 10–120 au (Long et al., 2018). The observed gap widths
range from a few to ∼ 50 au.
2. Spirals
The spiral-like structures are less common than the dust rings and gaps. The
spirals have been observed across a wide range of radii from ∼ a few to 1000
au (Muto et al., 2012; Grady et al., 2012; Avenhaus et al., 2014; Benisty et al.,2015; Pérez et al., 2016; Benisty et al., 2017; Akiyama et al., 2016; Huang et al.,2018b). Spirals with two arms are the most commonly found.
3. Arcs (or crescents)
The non-axisymmetric dust concentrations are referred to as arcs (Casassus etal., 2013; Fukagawa et al., 2013; Marel et al., 2013; Pérez et al., 2014; Kraus et al.,2017; Cazzoletti et al., 2018; Pérez et al., 2018; Isella et al., 2018). These structures are rare compared to rings-gaps and spirals (Francis and Marel, 2020).
The arcs are discovered at various radii from a few tens of au to ∼ 200 au fromthe star. The arc structure is also thought to be part of a ring around the central
cavity. The brightness contrast in disks with arcs ranges from ∼ 2–100 (Casassus et al., 2013; Marel et al., 2013; Pérez et al., 2014; Pinilla et al., 2014; Tanget al., 2017; Der Marel et al., 2021).

Although these dust substructures are likely to be found in the outer region of large,
bright protoplanetary disks (Andrews et al., 2018), the survey which covers a wide
range of brightness has reported that all disks that are large enough compared tothe spatial resolution are structured (Long et al., 2018). Thus, the dust substructures
seem to be ubiquitous.

1.2. Observations of protoplanetary disks

F IGURE 1.2: Gallery of dust substructure morphologies. Figure takenfrom Andrews, 2020. It should be noted that this figure contains bothmm continuum and near-infrared scattered light images. The nameof each object is omitted.

5

6

1.3

Chapter 1. General Introduction

Overview of planet formation

Planet formation is a complex multi-step process. It can be classified into two broadstages: from µm-sized dust grains to killometer(km)-sized planetesimals, and fromplanetesimals to planets.

1.3.1

Minimum mass solar nebula

One of the most important aspects of studying the planet formation process is thedisk model. While various models have been proposed, the standard model is theso-called minimum mass solar nebula (MMSN) model (Hayashi, 1981). The MMSN
model is a model to reproduce the minimum disk mass distribution to build thecurrent Solar System. In this model, the gas surface density is given by
ΣMMSN= 1700
g

r
1 au

!−3/2
g cm−2 ,(1.1)

where r is the radial distance from the central star. The disk temperature profile isgiven byr
TgMMSN = 280
1 au

!−1/2

L∗
L

!1/4
K,

(1.2)

where L∗ and L are the stellar and the Solar luminosity, respectively.

1.3.2

Physical properties of protoplanetary disks

In this section, we consider the fundamental physics of protoplanetary disks. Hereafter, we denote r- and z-direction as radially outward (away from the star) andvertical directions, respectively (Fig. 1.3). We first consider the vertical structure of
the protoplanetary disk. The radial force balance in a disk is described later in Sect.
1.3.4. Assuming a non-self gravitating disk, the vertical hydrostatic equilibrium of
the gas can be described bydp= −ρg Ω2 z,
dz

(1.3)

√
where p is the pressure, ρg is the gas density, and Ω = GM∗ /r3 is the Keplerian
angular velocity with G and M∗ being the gravitational constant and the mass of thecentral star, respectively. We assumed z  r in Eq. (1.3). Assuming an isothermal
equation of state,p = ρg c2s ,where cs is the isothermal sound speed, we integrate Eq. (1.3) and obtain" 2 #1 z
ρg = ρ0 exp −
.
2 H

(1.4)

(1.5)

1.3. Overview of planet formation

7

In Eq. (1.5), ρ0 is the gas density at the midplane and H is the gas scale height at
which the mass density of the disk gas is 1/e of the midplane density,
H=

cs
.
Ω

(1.6)

The disk gas accretes onto the central star due to angular momentum transport indisks. One of the possible origin of the mechanisms of angular momentum transport
is gas turbulence (Shakura and Sunyaev, 1973). For simplicity, we consider isotropic
turbulence. Following Shakura and Sunyaev, 1973, we assume that the turbulence
within the disk can provide an effective viscosity. Assuming the maximum scaleand speed of turbulent cells correspond to the sound speed and gas scale height, theviscosity can be described byν = αacc cs H,(1.7)

where αacc is a dimensionless turbulent parameter, which describes the global efficiency of angular momentum transport of the gas.

1.3.3

Planetary growth

In a protoplanetary disk, µm-sized dust grains grow by collisional sticking as a firststep of planet formation (e.g., Weidenschilling and Cuzzi, 1993). The outcome ofcollisions depends on the collisional velocity. The relative velocities between particles are determined by various physics, such as Brownian motion, differential drift
motions, and gas turbulence. These also depend on dust sizes. In the early stage ofdust growth, µm-sized dust grains can stick together due to low-velocity collision(e.g., Meakin, 1991). The collisional velocity increases as the particles grow. Whenthe collisional velocity exceeds the critical velocity for collisional growth, collisions
among dust particles do not result in sticking, but rather in bouncing or fragmentation (Blum and Wurm, 2008; Güttler et al., 2010). The collisional growth of dust
particles would halt when dust grows up to mm- to centimeter(cm)-sized particles(Zsom et al., 2010; Birnstiel et al., 2012). As dust grains become aerodynamical large
particle, they begin to drift inward (Sect. 1.3.4; Weidenschilling, 1977a).
To form km-sized planetesimals from ∼mm- to cm-sized grains, additional mechanisms are needed to proceed further growth of dust particles. The streaming instability is a possible mechanism to overcome this problem (Youdin and Goodman,
2005; Johansen and Youdin, 2007). The dense dust filaments form via the streaming
instability, and then the planetesimals form via gravitational collapse of filaments.
After the formation of planetesimals, the subsequent growth of bodies proceeds
via accretion of solid onto a protocore. One of the major planet formation theories
currently being considered is the planetesimal accretion model. In this model, thebuilding blocks of the planets are km-sized planetesimals. As the planetary embryos accrete planetesimals and become massive enough, their gravitational focusing becomes more significant, and then the massive embryos grow faster than othersmaller ones. This phase is called the runaway growth process (Kokubo and Ida,
1996). After the runaway growth phase, the largest embryos (protoplanets) growin an oligarchic manner, while most planetesimals remain small: this stage is called
the oligarchic growth process (Kokubo and Ida, 1998). The disk gas accretes onto acore formed in this way, and the total mass of the planet exceeds critical core mass(∼ 10 M⊕ ). Runaway gas accretion is then triggered and the planet evolves into a

8Chapter 1. General Introduction

gas giant (e.g., Mizuno, 1980; Pollack et al., 1996; Ikoma et al., 2000). In the planetesimal accretion model however, it is difficult to form gas giants within the lifetimeof the disk at distances larger than ∼ 5 au, unless the disk was massive (Kobayashi
et al., 2011).
Pebble accretion is a new model of planet formation (e.g., Ormel and Klahr, 2010;
Lambrechts and Johansen, 2012), which may overcome several problems remaining
in the theory based on planetesimal accretion (Kokubo and Ida, 2000). In protoplanetary disks, µm-sized dust grains grow by coagulation (e.g., Weidenschilling andCuzzi, 1993). As dust grains become aerodynamical large particle, they begin todrift inward (Sect. 1.3.4; Weidenschilling, 1977a). The drifting particles referred toas pebbles, whose size lies typically in mm to cm range. When the drifting pebbles
reach the orbit of (proto)planets, a fraction of them accretes due to the gravity of thebodies and the gas drag acting onto pebbles. Accreting pebbles onto the protocores
can form cores massive enough to trigger gas giant formation within the lifetime ofgas disks (e.g., Lambrechts and Johansen, 2012). When a growing planet reaches acertain critical mass, which referred to as pebble isolation mass, the planet generates
a pressure bump outside its orbit. The drifting pebbles are brought to a halt at thepressure bump, and then the planet is isolated from pebble accretion (Lambrechts
and Johansen, 2014).
The accretion regime of particles (planetesimals or pebbles) depends on the massof the embryo and the stopping time of a particle. We summarize the the accretion
regime of particles later in Sect. 1.3.5.

1.3.4

Radial drift of dust

In protoplanetary disks, when the dust particles grow up to ∼mm- to cm-sized dust
grains, they begin to drift inward. This is the so-called radial drift of dust in disks.
In this thesis, the radial drift of dust is a key physics. Here we summarize the mechanism of the radial drift of dust as below.
Let us consider a particle and the gas in a disk. When the gravity of the central
star and centrifugal force acting on the particle are balanced, the particle rotates inthe disk at the Keplerian velocity. In this section, we revisit the motion of gas anddust in the disk. For simplicity, we consider only the midplane region of the disk
and ignore the turbulence of the gas (Sect. 1.3.2). In a frame centered at the central
star, the equations of motion for the gas and dust are described by
dvg
ρ vd − vg
GM∗
∇p
=− d
− 3 r−,dt
ρg tstoprρg
vd − vg
dvd
GM∗=−− 3 r,
dt
tstop
r

(1.8)
(1.9)

where vg is the gas velocity, vd is the dust velocity, ρd is the mass density of dust,
and tstop is the stopping time of dust (Sect. 1.3.5), respectively. Hereafter, we assume
ρd /ρg  1, i.e., we ignore the backreaction from dust to gas.
Equation (1.8) suggests√ that the disk gas rotates at a different speed from the
Keplerian velocity, vK = GM∗ /r, due to the pressure gradient term (the third termin the right-hand side of Eq. (1.8)). The deviation of the rotation speed of the gas
from Keplerian rotation can be described by
vφ,g = (1 − η )vK ,(1.10)

1.3. Overview of planet formation

9

where
1
η=−
2



cs
vK

2

d ln p,d ln r

(1.11)

is a dimensionless quantity characterizing the global pressure gradient of the diskgas (Nakagawa et al., 1986). For instance, in the MMSN model (Sect. 1.3), η has on
the order of O(10−3 ). Thus, the deviation of the rotational speed of the gas from the
Keplerian speed is very small. When the pressure gradient term is negative, the disk
gas rotates slower than the Keplerian velocity (sub-Keplerian motion of the gas). In
the following paragraph, we assume dp/dr < 0 across the entire region of the disk.
The dust particles tend to rotate with the Keplerian velocity because of the absence of the pressure gradient. However, the the dust particles experience a headwind of the gas, ηvK , and thus receive the gas drag force (the second term in theright-hand side of Eq. (1.9)), leading to the loss of angular momentum of dust particles. Assuming a steady state and |vr,d |, |vφ,d − vK |  vK , where vr,d and vφ,d are
the radial and azimuthal component of the dust velocity, we obtain the steady-statesolution for the dust velocity (Nakagawa et al., 1986):
2tstop Ωv ≡ vdrift ,1 + (tstop Ω)2 hw
vhw
vφ,d = vK −,1 + (tstop Ω)2
vr,d = −(1.12)(1.13)

where
vhw = ηvK ,(1.14)

is the headwind of the gas. From Eq. (1.13), the azimuthal speed of the particle
deviates from the Keplerian speed due to the headwind of the gas. The dust particles
drift from the outer to the inner region of the disk (Eq. (1.12); Fig. 1.3), where the drift
speed is proportional to the headwind of the gas.

10

Chapter 1. General Introduction

F IGURE 1.3: Schematic illustration of the protoplanetary disk. The
blue-shaded region represents the disk gas. The whit dots correspondto dust. The figure displayed at the upper right corner is an enlarged
view, where the forces acting on the dust and gas are schematicallyillustrated. The solid curve is the schematic illustration of the dust
motion within the disk. The x-, y-, and z-aixs are the axes of a frame
co-rotating with the planet (Sect. 1.3.5).

1.3.5

Accretion regime of particles

When the particles reach the orbit of (proto)planets, a fraction of them accretes onto
the planets. In this section, we classify the accretion regime of particles according
to the stopping time of particles and the mass of the accreting planets. In particular,
we aim to describe quantitatively a process of the so-called pebble accretion, which
is one of the main focuses in Chapters 3 and 5.

1.3. Overview of planet formation

11

We consider a planet embedded in a disk, but we ignore the interaction between
the planet and the disk gas at this stage. Hereafter we denote x- and y-direction
as radially outward (away from the star) and azimuthal (the orbital direction of theplanet) directions, respectively (Fig. 1.3). In a frame co-rotating with the planet,
since the Keplerian velocity is proportional to Ω ∝ r −2/3 , the Keplerian rotation canbe approximated by a linear shear velocity, −3/2Ωx (the velocity difference causedby the different orbital radii around the central star), in the vicinity of the planetary
orbit. From Eqs. (1.12) and (1.13), the x- and y-component of the velocity of particles
can be described by
2tstop Ωv ,1 + (tstop Ω)2 hw
3vhw
vy,d = − Ωx −.
2
1 + (tstop Ω)2v x,d = −(1.15)(1.16)

We first focus on the following three timescales: the stopping time, encounter
time, and the settling time.
1. Stopping time
The stopping time represents the time it takes for the particle to follow the
motion of the gas. In the limit of tstop → 0, the trajectories of particles coincidewith the gas streamlines, while in the limit of tstop → ∞ the particle are inmotion completely independent of the gas. The stopping time depends on therelation between the particle radius and the mean free path of the disk gas, andcan be described by

ρ• s
9


(1.17) ρg cs (s ≤ 4 λmfp : Epstein regime)
tstop =
4ρ• s29(s > λmfp : Stokes regime),(1.18)
 9ρ c λ4
g s mfp
where ρ• is the internal density of the particle, s is the radius of the particle,
λmfp is the mean free path of the gas, λmfp = µmH /ρg σmol = 1.44 cm ( a/1 au)11/4
with µ, mH and σmol being the mean molecular weight, µ = 2.34, the mass ofthe proton, and the molecular collision cross section, σmol = 2 × 10−15 cm2(Chapman and Cowling, 1970; Weidenschilling, 1977a; Nakagawa et al., 1986).
The mean free path of the gas proportional to λmfp ∝ ρg−1 , which means the
stopping time is independent of the gas density in the Stokes gas drag regime.
2. Encounter time
For simplicity, we only consider the midplane region of the disk. We define thedifference between the orbital radii of the planet and the particle as bd . Let 2bd
be the distance over which a particle passes near the planet with a velocity vy,drelative to the planet in the y-direction, the encounter time can be described bytenc =

2bd.
vy,d(1.19)

When the shear velocity dominates the relative velocity of the particle, |vy,d | ∼|3/2Ωx |, the encounter time can be estimated by tenc ∼ Ω−1 . When the headwind of the gas dominates the relative velocity of the particle, |vy,d | ∼ |vhw |,the encounter time can be estimated by tenc ∼ 2bd /vhw .

12Chapter 1. General Introduction
3. Settling time
The settling time represents the time required for a particle with the terminal
velocity vsettle to settle in the gravitational field of the planet. We assume theparticle at the distance bd from the planetary orbit. Given the force balance
between the gas drag and the gravity of the planet acting on the particle,
GMpl
bd2

=

vsettle,tstop(1.20)

where Mpl is the mass of the planet. Form Eq. (1.19), the settling time can be
described bytsettle =bd
vsettle=bd3.
GMpl tstop(1.21)It should be noted that the gas drag force is essential for the settling of particles. In the limit of tstop → 0, Eq. (1.21) implies tsettle → 0. However, in thislimit the accretion of particles onto the planet occurs in the three-body regime(described later in this section). Hence, it is necessary to investigate in detail
whether settling in the intuitive sense occurs (see the following paragraphs).
From tstop , tenc , and tsettle , we can classify the accretion regime of particles onto theplanet as follows.
1. Ballistic regime
When tenc < tsettle , the accretion of particles onto the planets occurs in theballistic fashion (Fig. 1.4a). From Eqs. (1.19) and (1.21), this condition can be
described by
GMpl
bd2

<

vy,d
.
2tstop(1.22)

The above equation represents a situation where the particles pass through theplanet before they accrete onto the planet. The ballistic accretion of particles
occurs when the mass of the planet is small. When the planetary mass is small,the influence of the planet’s gravity on the motion of particles is negligible.
Thus, in this case, the relative motion of the particles to the planet is hardly
affected. The accretion cross section is scaled by the geometric cross section of
the planet. Planetary growth in the ballistic regime is inefficient because mostof the particles pass through the planet.
2. Settling regime
When tstop < tenc and tsettle < tenc , the accretion occurs in the settling regime
that is referred to as pebble accretion (Fig. 1.4b; Ormel, 2017). In this regime,
the gas drag plays an important role. As the planet grows and its mass increases, the settling time decreases (Eq. (1.21)). Particles passing near the planet
experience both the gravity of the planet and gas drag as they slow down andeventually accrete onto the planet. From the settling condition, the accretioncross section can be estimated as
R St1/3 (tenc ∼ Ω−1 ),(1.23)
sHillbd ∼2GMpl tstop
(tenc ∼ 2bd /vhw ),(1.24)
vhw

1.3. Overview of planet formation

13

where RHill is the Hill radius of the planet (the size of the gravitational radius
described later in Sect. 1.3.6). In the settling regime, the accretion cross sectionis scaled by the size of the gravitational radius of the planet.
3. Three-body regime
When tenc < tstop , the accretion occurs in the three-body regime (Fig. 1.4c).
This regime is referred to as planetesimal accretion. The motion of particles
is determined by the three-body interaction among the particle, planet, andthe central star. Although many particles enter the gravitational radius of theplanet, most of them experience the gravitational scattering due to the absenceof the gas drag.

F IGURE 1.4: Trajectories of particles in a frame co-rotating with theplanet. The orange and blue solid lines correspond to the trajectories
of particles which accreted and did not accrete onto the planet, respectively. Different panels show different accretion regimes. The mass ofthe planet and the dimensionless stopping time are different in eachpanel. Panel a shows the ballistic regime. The black dot denotes theplanet. Panel b shows the settling regime. The dashed-dotted circle corresponds to the Bondi radius of the planet (described later in
Sect. 1.3.6). Panel c shows the three-body regime. The dashed circle
corresponds to the Hill radius of the planet (described later in Sect.
1.3.6). We considered a specific case where the particles approach tothe planet from the front in the direction of the orbital motion of theplanet to highlight the differences among three regimes.

In this thesis, we consider the planetary growth via pebble accretion (Chapter 3).
To characterize pebble accretion more quantitatively, let us estimate the planetary
mass at which pebble accretion operates and the important range of the stopping
time. The early stages of planetary growth proceeds in the ballistic regime becauseof the small mass of the planet (Fig. 1.4a). In the ballistic regime, the encounter time
can be written as tenc ∼ 2bd /vhw . From Eqs. (1.19) and (1.24), the transition from theballistic to the settling regime occurs whenMpl =v3hw tstop,
8G' 2.4 × 10

−4

(1.25)

M⊕

a
1 au

3/2 

M∗M

−1/2 tstop Ω
1


(1.26)

where M is the solar mass and we assumed the MMSN model and used η = 1.8 ×
10−3 (r/1 au)−1/2 . The contribution of the shear velocity becomes dominant as theplanet grows. Then the encounter time can be written as tenc ∼ Ω−1 . From the14

Chapter 1. General Introduction

settling condition, we obtain the upper limit for the stopping time as
tstop . Ω−1 .

(1.27)

Based on the above discussions, we can conclude that pebble accretion refers to thephenomenon in which aerodynamically small particles (tstop < Ω−1 ) accrete onto
the planet due to the gas drag and the pull of the gravity of the planet with Mpl &10−4 M⊕ . This suggests the importance of the gas flow induced by an embedded
planet (later in Sect. 1.5).

1.3.6

Overview of disk-planet interaction

A planet embedded in a protoplanetary disk interacts with the surrounding disk gas.
In this section, we briefly summarize the response of gas disks to the gravity of theplanet in three categories: (1) when the influence of the gravity of the planet on thegas is negligible, (2) when the planet has grown enough to retain an atmosphere andinteracts gravitationally with the gas, which is the main focus of this thesis, and (3)when the planetary mass becomes sufficiently large to curve a gas gap in the disk.
We first introduce the following three length scales: the physical radius of the planet,
the size of the gravitational sphere of the planet and the disk gas scale height.
The physical radius of the planet can be written as
Rpl =

3Mpl
4πρpl

!1/3

3

' 6.58 × 10

Mpl
M⊕

!1/3

!−1/3

ρpl

km,

5 g/cm3(1.28)

where ρpl is the density of the planet. In this thesis, we define the radius of thegravitational sphere of the planet as follows:
Rgrav = min( RBondi , RHill ),(1.29)

where RBondi and RHill are the Bondi and Hill radii of the planet. The Bondi radius isdefined as the radius where the escape velocity of the gas from the planetary surface
is equivalent to the sound speed,
RBondi =

GMpl
,
c2s
5

' 3.98 × 10

Mpl
M⊕

!

L∗
L

!−1/4

r
1 au

!1/2
km,

(1.30)

where we assumed the temperature profile of the MMSN model (Eq. (1.2)), and used5cs = 10

r
1 au

!−1/4

L∗
L

!1/8
cm s−1 .

(1.31)

1.3. Overview of planet formation

15

The Hill radius is the radius of the region where the gravity of the planet acting onto
a particle is more dominant than that of stellar gravity,
!1/3

Mpl
RHill = a
3M∗

,
Mpl
M⊕

6

' 1.5 × 10

!1/3

M∗
M

!−1/3

a
1 au

!
km,

(1.32)

where a is the orbital distance of the planet from the central star.
The gas scale height is the height at which the mass density of the disk gas is 1/eof the midplane density,csM∗H≡' 0.03Ω
M
6' 4.5 × 10

!−1/2

M∗
M

L∗
L

!−1/2

!1/8

L∗
L

r
1 au

!5/4

!1/8 

au,
r
1 au

5/4
km,

(1.33)

where we used (Eq. (1.2)). We plot Eqs. (1.28), (1.30), (1.32), and (1.33) in Fig. 1.5.
From these length scales, we divide the disk-planet interaction into three broad
regimes as below.
1. When RBondi < Rpl , the influence of the gravity of the planet on the disk gas
is negligible. This corresponds to . 103 km-sized (or . 10−3 –10−2 M⊕ ) bodies(Fig. 1.5).
2. When Rpl < Rgrav < H (corresponding to 10−2 M⊕ . Mpl . 10 M⊕ at 1 au;
Fig. 1.5), the primordial atmosphere forms around the planet. In this thesis, werefer to these planets as “embedded planets” or “non-gas-gap-opening planets” and focus especially on these planets. When the planet is embedded in thedisk, the gravitational interaction between the gas and the planet results in theattraction of gas in the vicinity of the planet. The density waves are excited by
the planet (e.g., Goldreich and Tremaine, 1979), and the planet migrates radially in the disk (Tanaka et al., 2002). Hydrodynamical simulations have been
performed to investigate the disk-planet interaction, which resolve the region
where the density waves propagate (& Rgrav ; Paardekooper and Papaloizou,
2009; Dong et al., 2011). The gas flow in the region outside the gravitational
sphere of the planet has been well investigated (Goodman and Rafikov, 2001;
Rafikov, 2002; Paardekooper and Papaloizou, 2009; Dong et al., 2011; Mutoet al., 2010; Masset and Benítez-Llambay, 2016). Recent high-resolution hydrodynamical simulations that focus on the region close to the planet (. Rgrav )have revealed that embedded planets induces gas flow with a complex threedimensional (3D) structure (Ormel et al., 2015b; Fung et al., 2015; Cimerman
et al., 2017; Lambrechts and Lega, 2017; Kurokawa and Tanigawa, 2018; Kuwahara et al., 2019). The gas flow induced by embedded planets is a key physics
in this thesis. We return back to this topic later in Sect. 1.5.
3. When H < Rgrav , the tidal torque by a giant planet can remove gas from thevicinity of the planetary orbit (e.g., Goldreich and Tremaine, 1979), leading to
the formation of a gas gap in a disk (e.g., Lin and Papaloizou, 1986; Kley, 1999;

16

Chapter 1. General Introduction
Nelson et al., 2000; Crida et al., 2006; Duffell and MacFadyen, 2013; Fung et al.,2014; Kanagawa et al., 2015). We refer to these planets as “gas-gap-opening
planets” in this thesis. The 3D gas accretion flow onto a circumplanetary disk
from a protoplanetary disk occurs in this regime (e.g., Machida et al., 2008;
Tanigawa et al., 2012; Tanigawa et al., 2014)

F IGURE 1.5: Length scales. The red, blue, and yellow solid lines correspond to the physical radius of the planet, Rpl (Eq. (1.28)), Bondi
radius of the planet, RBondi (Eq. (1.30)), and Hill radius of the planet,
RHill (Eq. (1.32)), respectively. We assumed 1 au. The horizontal purple dashed line corresponds to the gas scale height,H (Eq. (1.33)). We
assumed the temperature profile of the MMSN model (Sect. 1.3), solar
mass star, and the solar luminosity.

1.4. Origins of dust rings and gaps in protoplanetary disks

1.4

17

Origins of dust rings and gaps in protoplanetary disks

The non-smooth features of protoplanetary disks introduced in Sect. 1.2 are thought
to be common. This was in disagreement with the traditional picture in which asmooth disk profile was assumed (e.g., Hayashi, 1981, Sect. 1.3). Given a smoothpressure profile in a disk, dust particles drift monotonically from the outer to theinner region of the disk (Eq. (1.12)). Several mechanisms—planet-free and planetdependent—have been proposed to explain the dust substructures in disks.

1.4.1

Planet-free mechanism

The planet-free mechanisms are related to perturbations to the disk gas pressure anddust motion in the radial direction of the disk (Dullemond et al., 2018).
- When the turbulent parameter is not constant across the disk (Eq. (1.7)), theefficiency of angular momentum transport changes where the turbulent parameter changes (e.g., dead zone; Gammie, 1996), leading to the formation ofpressure bumps. The dust particles accumulate at the edge of the dead zone
(Flock et al., 2015).
- The snowlines are though to be the regions where the size of dust changes. The
drift speed of dust depends on the size of dust and would change inside andoutside the snowlines, leading to dust accumulation. The ice recondensation(Zhang et al., 2015) and dust sintering (Okuzumi et al., 2016) are considered tobe the growth mechanisms of dust at the snowlines.
- In the region where the dust to gas ratio exceeds unity, a pressure bump forms
due to the backreaction from the dust to the gas. This suggests that dust accumulations would lead to further dust concentration (the so-called self-induced
dust pileup; Gonzalez et al., 2015; Gonzalez et al., 2017).
- In the disk, the balance of the pressure gradient, Coriolis force, and the selfgravity of gas and dust results in the formation of an azimuthal zonal flowof gas and the dust accumulates by the self-gravity of itself. Due to the selfgravity, the secular gravitational instability is triggered and creates annular
substructures in the disk (Takahashi and Inutsuka, 2014; Takahashi and Inutsuka, 2016; Tominaga et al., 2018; Tominaga et al., 2019; Tominaga et al., 2020).
- Maxwell stress fluctuations in disks causes the fluctuations in the azimuthal
velocity of the gas (zonal flows; Johansen et al., 2009; Bai and Stone, 2014),
leading to the formation of axisymmetric pressure bumps in the disk and axisymmetric dust substructures. The zonal flow caused by the fluctuations ofMaxwell stress develops in the region where only ambipolar diffusion operates, where the dust substructures could from due to the magnetically coupleddisk wind (Suriano et al., 2017; Suriano et al., 2018).
- When the surface layer of the disk is perturbed and a small bump is generated, the bump receives more stellar flux and puffs up. On the other hand, theregion outside the bump is shadowed and the temperature of the shadowed
region decreases. The thermal perturbations propagates inward and generatepressure bumps, leading to the formation of dust substructures in disks (thermal wave instability; Watanabe and Lin, 2008; Ueda et al., 2019; Ueda et al.,2021; Wu and Lithwick, 2021; Okuzumi et al., 2022).

18

1.4.2

Chapter 1. General Introduction

Planet-dependent mechanism

The above mechanisms do not assume planets in disks. Protoplanetary disks arebirthplaces of planet. Thus, disk–planet interaction is the other mechanism for interpreting the dust rings and gaps in disks. Paardekooper and Mellema, 2004 were thefirst to discover that planets can open dust gaps in gas disks. A planet embedded ina disk excites the density waves. These waves steepen to shocks after they propagate at a certain distance (Goodman and Rafikov, 2001), leading to the formation of
specific vorticity. The changes in the specific vorticity leads to net radial mass flux,
and then a gas gap opens around the planetary orbit (Lubow, 1990; Goodman andRafikov, 2001; Rafikov, 2002; Muto et al., 2010). Dust particles can be trapped at
the edge of the gas gap, the so-called dust filtration mechanism, forming a dust ring
and a gap (e.g., Rice et al., 2006; Paardekooper and Mellema, 2004; Paardekooperand Mellema, 2006; Paardekooper, 2007; Pinilla et al., 2012; Zhu et al., 2012; Zhu
et al., 2014; Dong et al., 2015; Dong et al., 2017a; Dong et al., 2018; Kanagawa et al.,2018; Weber et al., 2018; Pérez et al., 2019; Kanagawa et al., 2021). Gas-gap-opening
planets have long been a plausible candidate to explain the dust rings and gaps indisks.
In the case of the dust substructure created by a gas gap, one would expect the
spatial distribution of dust substructures to be correlated with that of gas. However,this is not always the case. Recently, the Molecules with ALMA at Planet-forming
Scales (MAPS) ALMA Large Program revealed no significant correlation between
the spectral line emission of CO and mm continuum except the MWC 480 system(Zhang et al., 2021; Jiang et al., 2022). The MAPS program focused on five disks inwhich dust substructures were detected. This result poses a challenge for the dust
substructure formation model predictions by gas-gap-opening planets for several
systems.
Giant planets could be detected by the Hα emission lines, because of the gas
accretion flow onto the planet and the circumplanetary disk from the protoplanetary
disk (e.g., Lovelace et al., 2011; Tanigawa et al., 2012; Gressel et al., 2013; Szulágyi
et al., 2014). The planet-induced Hα emission has only been detected at the positionof PDS 70b and c (Wagner et al., 2018; Haffert et al., 2019).
Moreover, rings and gaps seem to be ubiquitous in the outer region of large,
bright protoplanetary disks(& 10 au; e.g., Andrews et al., 2018), but the occurrence
rate of giant planets per star is limited to 10% (Sect. 1.1; Johnson et al., 2010; Fernandes et al., 2019). The period-mass distribution of putative giant planets inferred fromthe observed dust gap widths is not reconciled with the current observed periodmass distribution of giant planets (Bae et al., 2018; Lodato et al., 2019; Disk Dynamics Collaboration et al., 2020). Although this discrepancy might be solved byconsidering large-scale migration and planetary growth (Lodato et al., 2019), it is
not conclusive that most dust substructures are derived from interactions between
disks and gas-gap-opening planets.

1.5

This thesis

Given that protoplanetary disks are birthplaces of planet, the dust ring and gap
formation by disk-planet interactions is a plausible mechanism. However, several
challenges still remain (Sect. 1.4). The ambiguous relationship between the spatial
distribution of gas and dust could be solved by considering low-mass planets that
do not carve a deep gas gap in a disk. The lack of Hα detections may indicate that

1.5. This thesis

19

low-mass, low-accreting planets dominate the exoplanet distribution (Huélamo etal., 2022).
The purpose of this thesis is to investigate dust dynamics under the influenceof the planet-induced gas flow and the potential of the planet-induced gas flow to
sculpt the rings and gaps in the spatial distribution of dust. In this thesis, we focus
on the embedded planets (low-mass, non-gas-gap-opening planets). The masses ofthese planets span the two mass ranges: (1) the mass at which the accretion regime ofparticles is in the settling regime (Sect. 1.3.5), where the efficient growth of the planets via pebble accretion would be achieved (e.g., Ormel and Klahr, 2010; Lambrechts
and Johansen, 2012), and (2) the mass at which the planet interacts gravitationallywith the surrounding disk gas and induces gas flow with a complex 3D structure(Sect. 1.3.6; Ormel et al., 2015b; Fung et al., 2015; Fung et al., 2019; Lambrechtsand Lega, 2017; Cimerman et al., 2017; Kurokawa and Tanigawa, 2018; Kuwahara
et al., 2019; Béthune and Rafikov, 2019; Moldenhauer et al., 2021; Moldenhauer et al.,2022).
Throughout this thesis, the key point is the outflow of the gas, which occurs in
the radial direction to the disk. The disks contain abundant mm- to cm-sized dust
particles (e.g., Testi et al., 2003). These particles are sensitive to the gas flow. Oncethe radial outward gas flow disrupts the motion of the particles, the inward drift ofparticles could be inhibited, leading to the formation of a dust substructure. The outflow would affect the growth of the planet via pebble accretion. A high-resolution
simulation is required to capture the detailed 3D structure of the gas flow within the
Bondi and Hill spheres, because the planet-induced gas flow is a local phenomenon.
Thus, the influences of the 3D planet-induced gas flow on dust dynamics and thespatial distribution of dust in disks are missing in previous studies.
In Chapter 2, we investigate the nature of the protoplanetary disk gas flow aroundan embedded planet. We perform 3D hydrodynamical simulations, which resolve
the local gas flow past the planet. We show that a substantial amount of gas fromthe disk enters the gravitational sphere of the planet (inflow) and exits (outflow),
tracing the surface of a planetary atmosphere. Based on the analytical arguments forthe outflow, we discuss the dependence of the outflow speed on the planetary massand the headwind of the gas.
In Chapter 3, we investigate the influence of the planet-induced gas flow on dust
dynamics in the vicinity of the planet. We numerically calculate the trajectories ofdust particles using the hydrodynamical simulation data obtained in Chapter 2. Weshow that the trajectories of dust in the planet-induced gas flow differ significantly
from those in the unperturbed flow. We discuss the influence of the planet-induced
gas flow on the accretion rate of dust onto the planet.
In Chapter 4, we investigate the influence of the planet-induced gas flow on
the spatial distribution of dust in disks. We numerically compute the dust surface
density by incorporating the influences of the planet-induced gas flow into a onedimensional dust advection-diffusion model. We show that the gas flow induced
by a non-gas-gap-opening planet generates dust accumulation (the dust ring) anddepletion (the dust gap).
In Chapter 5, we discuss the implications for planet formation theory, observations of protoplanetary disks, and an outlook on future work.
Chapter 6 summarize the main results of this thesis.

21

Chapter 2

Gas flow induced by low-massplanets
2.1

Introduction of this chapter

Planets form in protoplanetary disks and interact with the surrounding disk gas. Recent hydrodynamical simulations have revealed that a non-gas-gap opening planet
embedded in a disk induces gas flow with a complex 3D structure (Ormel et al.,2015b; Fung et al., 2015; Fung et al., 2019; Cimerman et al., 2017; Lambrechts andLega, 2017; Kurokawa and Tanigawa, 2018; Kuwahara et al., 2019; Béthune andRafikov, 2019; Moldenhauer et al., 2021; Moldenhauer et al., 2022). Assuming an
isothermal equation of state, Ormel et al., 2015b performed 3D hydrodynamical simulations of the gas flow around a planet. They found that a substantial amount ofgas from the disk enters the gravitational sphere of the planet (inflow) and exits(outflow), causing atmospheric recycling.
The key point in this chapter is the outflow of the gas, which occurs in the radial
direction to the disk. As described in Chapter 1, the planet-induced gas flow could
affect the motion of small dust particles, which may not only delay the planetary
growth but could also affect the spatial distribution of dust in disks. The young
disks contain abundant millimeter- to centimeter-sized dust particles (e.g., Testi etal., 2003). These particles drift from the outer to the inner region of the disk due tothe global pressure gradient (Whipple, 1972; Weidenschilling, 1977a). Once the radial outward gas flow disrupts the motion of the particles, both the accretion of theparticles onto a planet and the inward drift of particles could be inhibited. The velocity of the planet-induced gas flow is a key parameter to investigate the influenceof the planet-induced gas flow on the motion of dust particles.
Assuming an isothermal, Keplerian gas disk, Fung et al., 2015 performed 3D hydrodynamical simulations and found that outflow of the gas at the midplane region
has the speed of the order of isothermal sound speed, ∼ cs . The properties of theplanet-induced outflow have been investigated by Kuwahara et al., 2019. The authors assumed an isothermal, Keplerian gas disk and performed 3D hydrodynamical simulations. From Bernoulli’s theorem along an outflow streamline, Kuwahara
et al., 2019 showed that the speed of midplane outflow increases with the planetary
mass and eventually exceeds the terminal speed of incoming dust. This suggeststhat the outflow could affect the motion of dust particles around the planetary orbit.
However, the assumptions, such as the isothermal condition and the Keplerianmotion of the gas should be relaxed to investigate the realistic influence of the planetinduced gas flow on the motion of dust particles. In the nonisothermal case, theatmospheric recycling has only been observed outside the isolated inner envelope(Cimerman et al., 2017; Lambrechts and Lega, 2017; Kurokawa and Tanigawa, 2018).
Thus, the properties of the outflow of the gas under nonisothermal condition could

22

Chapter 2. Gas flow induced by low-mass planets

differ from those under isothermal condition. Moreover, the sub-Keplerian motionof the gas due to the global pressure gradient affect the 3D structure of the planetinduced gas flow (Ormel et al., 2015b; Kurokawa and Tanigawa, 2018). How thenature of the planet-induced gas flow changes as a function of the planetary massand the magnitude of the deviation of the speed of the gas from Keplerian rotation
remain unclear.
In this chapter, we investigate the nature of the protoplanetary disk gas arounda non-gas-gap-opening planet. In Sect. 2.2 we describe our numerical methods. In
Sect. 2.3, we show the results obtained from a series of hydrodynamical simulations.
In Sect. 2.4, we discuss the implications for the dust motion under the influence ofthe planet-induced gas flow.

2.2. Method

2.2

Method2.2.1

Model overview

23

In this section, we provide an overview of our model. Our approach is summarized
in Fig. 2.1. Assuming a compressible, inviscid sub-Keplerian gas disk with the vertical stratification due to the vertical component of the stellar gravity, we performed
3D nonisothermal hydrodynamical simulations. Radiative cooling is implemented
by using the β cooling model (Gammie, 2001; Kurokawa and Tanigawa, 2018). The
self-gravity of gas is neglected. Hydrodynamical simulations were performed in theframe co-rotating with the planet (the black sphere in Fig. 2.1). These simulations
resolve the region’s local gas flow, including the planet’s Hill and Bondi spheres.

24

Chapter 2. Gas flow induced by low-mass planets

F IGURE 2.1: Schematic illustration of our model. We performed 3Dhydrodynamical simulations on the spherical polar grid (the black
sphere), which hosts a planet located at its center (the brown-filled
circle). We consider the planet on a fixed circular orbit with the orbital radius, a. We adopt unperturbed sub-Keplerian shear flow at
the outer boundary of the hydrodynamical simulations (the blue arrows). In our disk model, the Mach number of the headwind of thegas, Mhw , increases with the orbital radius (Eq. (2.21)).

2.2.2

Dimensionless unit system

Throughout all of our simulations, the length, times, velocities, and densities are normalized by the disk scale height, H, reciprocal of the orbital frequency, Ω−1 , soundspeed cs , and unperturbed gas density at the location of the planet, ρdisk , respectively.
Figure 2.2 summarizes the relation between dimensionless and dimensional quantities for the steady accretion disk model. While the mass of the disk gas is normalized by ρdisk H 3 , our simulations that neglect the self-gravity of the gas allow us to
introduce another normalization for the planetary mass (Ormel et al., 2015a) as given2.2. Method

25

by:
m=

Mpl
RBondi
=
,
H
Mth

(2.1)

where Mth is the thermal mass of the planet (Goodman and Rafikov, 2001):c3
Mth = s = M∗
GΩ



H
r

3
,

(2.2)

where M∗ is the mass of the central star and r is the orbital radius. The Hill radius
is given by RHill = (m/3)1/3 H in this unit. In this thesis, we define the radius of thegravitational sphere of the planet as follows (Eq. (1.29)):
Rgrav = min( RBondi , RHill ).

(2.3)

The relation between dimensionless planetary masses and dimensional ones for asteady viscous-accretion disk model is shown later in Fig. 2.2a. The range of dimensionless planetary mass in this chapter is m = 0.01–1, which corresponds to a planetwith ∼ 0.07–30 M⊕ orbiting a solar-mass star at ∼ 1–10 au.

2.2.3

Unperturbed disk

We first consider the unperturbed state of the gas disk. Hereafter we denote x-, y, and z-direction as radially outward (away from the star), azimuthal (the orbital
direction of the planet), and vertical directions, respectively (Fig. 2.1). The verticalstructure of the density distribution in the disk is given by!
z2
ρg,∞ = ρdisk exp −
.
(2.4)
2
The unperturbed gas velocity in the local frame is assumed to be the speed of the
sub-Keplerian shear,


3
(2.5)
vg,∞ = − x − Mhw ey ,
2
where

Mhw ≡

vhw
,
cs

(2.6)

is the Mach number of the headwind of the gas and ei is the unit vector in the idirection. The Mach number increases as the orbital distance increases for the steadyviscous-accretion disk model as shown later in Fig. 2.2b. The range of the Mach
number in this chapter is Mhw = 0.0–0.1, which covers a wide range of the disk(. 100 au).

2.2.4

Three-dimensional hydrodynamical simulations

A planet is embedded in an inviscid gas disk and is orbiting around a solar-massstar with a fixed orbital radius, a. Our simulations were performed in a spherical

26

Chapter 2. Gas flow induced by low-mass planets

polar coordinate co-rotating with a planet using the Athena++ code1 (Fig. 2.1; Whiteet al., 2016; Stone et al., 2020). Most of our methods of hydrodynamical simulations
are the same as described in detail in Kurokawa and Tanigawa, 2018, except for thesize of the inner boundary (Sect. 2.2.6).

2.2.5

Governing equations

The dimensionless governing equations for inviscid and compressible fluid in ourdimensionless unit are described as follows.
∂ρg+ ∇ · ρg vg = 0,
∂t

∂
∇p
+ vg · ∇ vg = −+ ∑ Fi ,
∂t
ρg
i


U (ρg , T ) − U (ρg , T0 )∂E
+ ∇ · vg ( E + p) = ρg vg · ∑ Fi −
,
∂t
β
i

(2.7)
(2.8)
(2.9)

where Fi is the external force, respectively. The external force consists of four elements: the Coriolis force Fcor = −2ez × vg , the tidal force Ftid = 3xe x − zez , theglobal pressure force due to the sub-Keplerian motion of the gas Fhw = 2Mhw e x ,and the gravitational force (Ormel et al., 2015a; Ormel et al., 2015b; Kurokawa andTanigawa, 2018; Kuwahara et al., 2019),!("
 2 #)
m
1
t
Fgrav = −∇ p
1 − exp −,(2.10)
22
2t
R + Rsminjpwhere R =x2 + y2 + z2 is the distance from the center of the planet, Rsm is thesmoothing length, and tinj is the injection time. Throughout chapters 2–5, the valueof Rsm is chosen to be equal to 10% of the radius of the gravitational sphere of theplanet, Rsm = 0.1Rgrav . We investigate the dependence of the smoothing length inAppendix A. Since the rapid increase of the gravity of the planet in the unperturbed
disk affects the results of the simulations (Ormel et al., 2015a), the gravity of theplanet is gradually inserted into the disk at the injection time, tinj . The internal andtotal energy densities are given by
p
,
γ−1
1
E = U + ρg v2g ,2U=(2.11)(2.12)

where γ = 1.4 is the adiabatic index.
Our simulations utilized the β cooling model, where the temperature T relaxes
toward the background temperature T0 with the dimensionless timescale β (Gammie, 2001). To determine an adequate β value for a certain planetary mass, we followed the discussion in Kurokawa and Tanigawa, 2018. Considering the relaxation
time β for the temperature perturbation whose wavelength is equal to the Bondi radius of the planet, we fixed β = 1 at m = 0.1 and assumed that β scales linearly with
1 https://github.com/PrincetonUniversity/athena

2.2. Method

27

the square of m, namely,
β=

2.2.6

m
0.1

!2
.

(2.13)

Boundary condition and numerical resolution

Since we considered an inviscid gas disk, we introduced the free-slip inner boundary
to prevent the loss of mass. We used varying sizes of the inner boundary, Rin , as afunction of the mass of the planet to follow a mass–radius relationship of the nongaseous part of the planet:
Rin ' 3 × 10−2 m1/3 Hρpl
5 g/cm3

!−1/3

M∗
1M

!1/3

r
0.1 au! −1.

(2.14)

Our hydrodynamical simulations are computationally expensive, but the computational time is reduced when we increase the size of the inner boundary. Thus, we
assumed r = 0.1 au in Eq. (2.14). The size of the inner boundary does not affect our
results. Thus, when we convert the dimensonless planetary mass into a dimensional
one, we can use an arbitrary orbital radius regardless of a fixed Rin .
The gas velocity and density have constant values, vg,∞ and ρg,∞ , at the outeredge of the computational domain of hydrodynamical simulations. For the azimuthal direction, a periodic boundary condition was introduced such that the relationship A(r, θ, φ) = A(r, θ, φ + 2π ) holds for an arbitrary scalar or vector quantity
A.
The numerical resolution is [log r, θ, φ] = [128, 64, 128]. We adopted a logarithmic
grid for the radial coordinate, which has a higher resolution in the vicinity of theplanet. The numerical convergence was confirmed in (Kuwahara et al., 2019).
We list our parameter sets in Table 2.1.

2.2.7

Disk model

When we convert the dimensionless quantities into dimensional ones, we consider
the steady accretion disks with a constant turbulence strength (Shakura and Sunyaev, 1973, Sect. 1.3.2), αacc , including viscous heating due to the accretion of the gas
and irradiation heating from the central star (Fig. 2.2; Kusaka et al., 1970; Nakamoto
and Nakagawa, 1994; Oka et al., 2011; Ida et al., 2016). The dimensionless viscous alpha parameter, αacc , describes the global efficiency of angular momentum transportof the gas (Shakura and Sunyaev, 1973). We assumed αacc = 10−3 .
The disk midplane temperature is given by Tdisk = max( Tvis , Tirr ), where Tvis
and Tirr are temperatures determined by viscous heating and stellar irradiation (Garaud and Lin, 2007; Oka et al., 2011; Ida et al., 2016),
Tvis '200

M∗
1M

Tirr '150

L∗
1L

!3/10

!2/7

!−1/5
!2/5
!−9/10
αacc
Ṁ∗
r
K,
10−3
10−8 M /yr
1 au
!−1/7
!−3/7
M∗
r
K,
1M
1 au

(2.15)(2.16)

28

Chapter 2. Gas flow induced by low-mass planets

TABLE 2.1: List of hydrodynamical simulations. The following
columns give the simulation name, the size of the Bondi radius of theplanet, the size of the Hill radius of the planet, the size of the innerboundary, the size of the outer boundary, the size of the limited part ofthe calculation domain (3.3.2), the smoothing length, the Mach number of the headwind, the dimensionless cooling time, the injection
time, and the length of the calculation time, respectively. Simulations
with an asterisk are used to calculate the orbits and velocities of dust
particles influenced by the planet-induced gas flow (Chapters 3 and4).
Namem003-hw0,-hw001∗ ,-hw003∗ ,-hw01∗
m003-hw0,-hw001∗ ,-hw003∗ ,-hw01∗
m005-hw0,-hw001,-hw003,-hw01
m007-hw0,-hw001,-hw003,-hw01
m01-hw0,-hw001∗ ,-hw003∗ ,-hw01∗
m02-hw0,-hw001,-hw003,-hw01
m03-hw0,-hw001∗ ,-hw003∗ ,-hw01∗
m04-hw0,-hw001,-hw003,-hw01
m05-hw0,-hw001,-hw003,-hw01
m06-hw0,-hw001,-hw003,-hw01
m07-hw0,-hw001,-hw003,-hw01
m08-hw0,-hw001,-hw003,-hw01
m09-hw0,-hw001,-hw003,-hw01
m1-hw0,-hw001,-hw003,-hw01
m003-hw0-Rsm0,-hw001-Rsm0,-hw003-Rsm0,-hw01-Rsm0
m01-hw0-Rsm0(-Rsm3Rin),-hw001-Rsm0(-Rsm3Rin),-hw003-Rsm0(-Rsm3Rin),-hw01-Rsm0(-Rsm3Rin)
m02-hw0-Rsm0,-hw001-Rsm0,-hw003-Rsm0,-hw01-Rsm0
m03-hw0-Rsm0,-hw001-Rsm0,-hw003-Rsm0,-hw01-Rsm0
m04-hw0-Rsm0,-hw001-Rsm0,-hw003-Rsm0,-hw01-Rsm0
m05-hw0-Rsm0(-Rsm3Rin),-hw001-Rsm0(-Rsm3Rin),-hw003-Rsm0(-Rsm3Rin),-hw01-Rsm0(-Rsm3Rin)

RBondi
0.01
0.03
0.05
0.07
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

RHill
0.15
0.22
0.26
0.29
0.32
0.41
0.46
0.51
0.55
0.58
0.62
0.64
0.67
0.69

Rinn
6.46×10−3
9.32×10−3
1.11×10−2
1.23×10−2
1.39×10−2
1.75×10−2
2×10−2
2.21×10−2
2.38×10−2
2.53×10−2
2.66×10−2
2.78×10−2
2.9×10−2
3×10−2

Rout
0.5
0.5
1
1
5
5
5
5
5
5
5
5
5
5

Rhydro0.6Rout
0.6Rout
0.6Rout
Rout-

Rsm
0.1Rgrav
0.1Rgrav
0.1Rgrav
0.1Rgrav
0.1Rgrav
0.1Rgrav
0.1Rgrav
0.1Rgrav
0.1Rgrav
0.1Rgrav
0.1Rgrav
0.1Rgrav
0.1Rgrav
0.1Rgrav

Mhw
0, 0.01, 0.03, 0.1
0, 0.01, 0.03, 0.1
0, 0.01, 0.03, 0.1
0, 0.01, 0.03, 0.1
0, 0.01, 0.03, 0.1
0, 0.01, 0.03, 0.1
0, 0.01, 0.03, 0.1
0, 0.01, 0.03, 0.1
0, 0.01, 0.03, 0.1
0, 0.01, 0.03, 0.1
0, 0.10, 0.10, 0.10, 0.1β
9 × 10−4
0.09
0.25
0.49
1
4
9
16
25
36
49
64
81
100

tinj
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
1
1
1
1
1
1

tend
10
50
100
100
200
200
1000
1000
1000
1000
1000
1000
1000
1000

0.01

0.15

6.46×10−3

0.5

-

0

0, 0.01, 0.03, 0.1

9 × 10−4

0.5

10

0.1

0.32

1.39×10−2

5-0, 3Rin

0, 0.01, 0.03, 0.1

1

0.5

200

0.2

0.41

1.75×10−2

5-00, 0.01, 0.03, 0.1

4

0.5

200

0.3

0.46

2×10−25-

00, 0.01, 0.03, 0.1

9

0.5

200

0.4

0.51

2.21×10−2

5-00, 0.01, 0.03, 0.1

16

0.5

500

0.5

0.55

2.38×10−2

5-

0 , 3Rin0, 0.01, 0.03, 0.1

25

1

500

2.2. Method

29

where Ṁ∗ is the accretion rate of the disk and L∗ is the stellar luminosity. In thisthesis, we assume a solar-mass host star, M∗ = 1 M , the solar luminosity, L∗ =1 L , and the typical accretion rate of classical T Tauri stars, Ṁ∗ = 10−8 M /yr.
The aspect ratio of the disk is given byh≡H= max(hg,vis , hg,irr ),r(2.17)

where hg,vis and hg,irr are aspect ratios given by (Ida et al., 2016)hg,vis '0.027

M∗
1Mhg,irr '0.024

L∗
1L!−7/20!1/7!−1/10
!1/5
!1/20
αacc
Ṁ∗
r
,
10−3
10−8 M /yr
1 au
!−4/7
!2/7
M∗
r
.
1M
1 au

(2.18)(2.19)In this disk model, the mass of the planet and the Mach number of the headwind
can be described by (Fig. 2.2):
Mpl = mM∗ h3
' max 6.6m

Mhw =

r
1 au

3/20


, 4.6mr
1 au

6/7 !
M⊕ ,(2.20)

h d ln p
ηvK=−cs
2 d ln r
' max 0.0344

r
1 au

1/20


, 0.0333r
1 au

2/7 !
,(2.21)

where we assume d ln p/d ln r ' −2.55 for the viscous region and d ln p/d ln r '−2.78 for the irradiation region (Ida et al., 2016).

30

Chapter 2. Gas flow induced by low-mass planets

F IGURE 2.2: Relation between the dimensionless quantities and dimensional ones. Each line represents the planetary mass (top; 2.20)
and Mach number (bottom; Eq. (2.21)) as a function of the orbital radius.

2.3. Results

2.3

31

Results

The main subject of this chapter is to clarify the dependence of the planet-induced
gas flow on the planetary mass and the Mach number of the headwind of the gas.
Sections 2.3.1 and 2.3.2 summarize the characteristic 3D structure of the planetinduced gas flow obtained by 3D hydrodynamical simulations. Section 2.3.3 show
the structure of the planet-induced gas flow at the midplane region of the disk. Section 2.3.4 focus on the geometry of the planet-induced outflow of the gas at the midplane. In Sect. 2.3.5, we discuss the vertical structure of the outflow of the gas.
Sections 2.3.6 and 2.3.6 provide the analytical arguments for the outflow, and discuss the dependence of the planet-induced gas flow on the planetary mass and theMach number of the headwind of the gas.

2.3.1

Three-dimensional planet-induced gas flow

The gas flow around an embedded planet is perturbed by the planet, which forms
the characteristic 3D structure of the flow field (Ormel et al., 2015b; Fung et al., 2015;
Fung et al., 2019; Cimerman et al., 2017; Lambrechts and Lega, 2017; Kurokawa andTanigawa, 2018; Kuwahara et al., 2019; Béthune and Rafikov, 2019; Moldenhauer
et al., 2021; Moldenhauer et al., 2022). The fundamental features of the flow field are
as follows:
1. Gas from the disk enters the Bondi or Hill sphere (inflow) and exits (outflow;
the red lines of Figs. 2.3 and 2.4). We refer to these streamlines as recycling
streamlines. In the nonisothermal case, an isolated inner envelope (a boundatmosphere) emerges around a planet due to the buoyancy barrier (Sect. 2.3.2;
Kurokawa and Tanigawa, 2018). The buoyancy force prevent the inflow of thegas from reaching deeper regions in the envelope. Thus, the recycling flow
passes the planet, tracing the surface of the isolated envelope.
2. The horseshoe flow exists in the anterior-posterior direction of the orbital pathof the planet (the orange lines of Figs. 2.3 and 2.4). The horseshoe streamlines
have a columnar structure in the vertical direction (Figs. 2.3a and b).
3. The Keplerian shear flow extends inside and outside the orbit of the planet (thegreen lines of Figs. 2.3 and 2.4).
The 3D flow structure depends on Mhw (Ormel et al., 2015b; Kurokawa and Tanigawa, 2018). Following Kuwahara and Kurokawa, 2020b, we classify the planetinduced gas flow into the flow-shear and the flow-headwind regimes (Figs. 2.3 and2.4). Qualitatively, the planet-induced gas flow has a vertically rotational symmetricstructure in the flow-shear regime (Fig. 2.3). The outflow occurs in the second andfourth quadrants of the x-y plane. In the flow-shear regime, gas flows in at high latitudes of the gravitational sphere of the planet at its horseshoe turn. In other words,
the outflow streamlines originated from the widest horseshoe streamlines at highaltitudes (the transient horseshoe flow; Fung et al., 2015).
The symmetry is broken in the flow-headwind regime, where the outflow occurs
only in the fourth quadrant of the x-y plane (Fig. 2.4). As the Mach number of theheadwind of the gas increases, the horseshoe streamlines move to the negative direction in the x-axis. This is because the corotation radius for the gas moves toward thenegative direction of the x-axis. The corotation radius for the gas, where vg,∞ = 0, is32

Chapter 2. Gas flow induced by low-mass planets

given by2
xcor,g = − Mhw .
3(2.22)In Fig. 2.4, the horseshoe flow shifts significantly to the negative direction in the xaxis. Thus, the inflow of the gas originated from the horseshoe flow at high altitudes
no longer occur. The Bondi sphere, which indicates the size of the planetary atmosphere, is exposed to the headwind of the gas. In the flow-headwind regime, thedominant inflow of the gas occurs at the midplane region of the disk.
The transition from the flow-shear to the flow-headwind regime occurs when thedimensionless planetary mass drops below the flow transition mass (Kuwahara andKurokawa, 2020b):
mt,flow '

4
M .
15 hw

(2.23)

When mt,flow , the planet-induced gas flow is in the flow-headwind regime. In ourparameter set, only the case of m001-hw01 and m003-hw01 are classified as the flowheadwind regime, otherwise classified as the flow-shear regime based on the definition in Kuwahara and Kurokawa, 2020b.

F IGURE 2.3: Streamlines of 3D planet-induced gas flow around theplanet with m = 0.3 and Mhw = 0. The planet-induced gas flow
is in the flow-shear regime. The solid green, orange, and red lines
are the Keplerian shear, the horseshoe streamlines, and the recycling
streamlines (Ormel et al., 2015b; Kuwahara et al., 2019) (or transient
horseshoe streamlines (Fung et al., 2015)), respectively. The sphere is
the Bondi sphere of the planet, whose size is 0.3 H. The arrows represent the direction of the gas flow. Panels a and b: Perspective views of
the flow field. Panel c: x-y plane viewed from the +z-direction.

2.3. Results

33

F IGURE 2.4: Same as Fig. 2.3 but with m = 0.03 and Mhw = 0.1. The
planet-induced gas flow is in the flow-headwind regime. The sphereis the Bondi sphere of the planet, whose size is 0.03 H. The verticaldashed line in panel c corresponds to the x-coordinate of the positionof the corotation radius for the gas, xg,cor = −2/3Mhw .

The above mentioned characteristic 3D structure of the flow field does not change
significantly in the isothermal case (β → 0), except for the inner region of the gravitational sphere of the planet, < Rgrav . In short, the nonisothermal and isothermal
simulations differ only in terms of the existence of the isolated envelope. In theisothermal cases, there is no clear boundary demarcates the bound atmosphere fromthe background disk gas (Ormel et al., 2015b). The inflow of the gas can penetrate
the gravitational sphere of the planet, reaching more deeper regions in the envelope
than in the nonisothermal cases.

2.3.2

Isolated envelope

In the nonisothermal cases, an isolated envelope emerges within the gravitational
sphere of the planet (Fig. 2.5; Cimerman et al., 2017; Lambrechts and Lega, 2017;
Kurokawa and Tanigawa, 2018). The isolation of the inner part of the envelope iscaused by the buoyancy, which is originated from the entropy gradient (Kurokawa
and Tanigawa, 2018). Figure 2.5 shows the entropy structure around the planet.
A low-entropy region corresponds to the isolated envelope. We here consider the
motion of a parcel of gas along a recycling streamline.
1. The inflow of the gas is always subsonic in our parameter sets. Thus, thepressure of the gas inflow is identical to that of the surrounding gas, because
the timescale for the pressure equilibrium is much shorter than the dynamicaltimescale, ∼ Rgrav /cs  Ω−1 .
2. The temperature of the parcel of the inflow of the gas increases due to adiabatic
compression, leading to a reduction of the density of the gas parcel.

34

Chapter 2. Gas flow induced by low-mass planets
3. The density difference between the parcel of the inflow of the gas and surrounding gas exerts buoyancy force onto the parcel, leading to an isolation of
the inner part of the envelope from the recycling flow (Fig. 2.5).

The size of the isolated envelope (hereafter the atmospheric radius) can be estimated by the maximum radius where the azimuthal velocity is dominant in the
midplane region (Moldenhauer et al., 2022):

Ratm = max r | vφ |θ =π/2 > max(vr , vθ ) ,(2.24)
where vr , vθ , and vφ are components of the velocities in the radial, azimuthal, andpolar directions in a spherical polar coordinates centered at the planet. Figure 2.6
shows the atmospheric radius as a function of the planetary mass and Mach number.
We found that the size of the atmospheric radius scales with the Bondi or Hill radiusof the planet. The scaling law switches at m ∼ 0.1. The headwind of the gas reduces
the size of the atmosphere, which is consistent with the previous studies (Ormel,
2013; Moldenhauer et al., 2022). We introduce a fitting formula for the atmosphericradius as
Ratm,fit = min (C1 × RBondi , C2 × RHill ) ,(2.25)

where C1 and C2 are the fitting parameters. We set C1 = 0.7–1.0 and C2 = 0.3–0.4.

F IGURE 2.5: Same as Fig. 2.3 but the result obtained from m05-hw01.
The planet-induced gas flow is in the flow-shear regime. The colorγ
contour represents p/ρg , which is the measure of entropy. The black
sphere denotes the isolated envelope whose size is Ratm = 0.2 H. The
vertical gray lines correspond to the x-coordinate of the position of
the corotation radius for the gas, xcor,g = −2/3Mhw . The arrows represent the direction of the gas flow. Panels a and b: Perspective viewsof the flow field. Panel c: x-y plane viewed from the +z-direction.

2.3. Results

35

F IGURE 2.6: Atmospheric radius as a function of the planetary massand the Mach number of the headwind of the gas. The square symbols are obtained from hydrodynamical simulations. Different colors
correspond to different Mach numbers, Mhw . The gray- and pinkshaded regions are given by the fitting formula for the atmosphericradius, Ratm,fit .

2.3.3

Structure of the planet-induced gas flow at the midplane region

Hereafter we analize the nature of the outflow of the gas. Figures 2.7 and 2.8 show
the gas flow structure at the midplane region of the disk. An embedded plant perturbs the surrounding gas at the typical scale of the gravitational sphere of theplanet, Rgrav . As mentioned in Sect. 2.3.1, the outflow occurs in the second andfourth quadrants of the x-y plane in the flow-shear regime (Fig. 2.7). Disk gas enters a sphere of typical radius ∼ Rgrav and flows out through near the L1 and L2
Lagrange points. The outflow occurs only in the fourth quadrant of the x-y plane in
the flow-headwind regime (Fig. 2.8). The Coriolis force swirls the inflow of the gas
at the midplane region from − x to + x direction. For instance, in Fig. 2.7, the speedof outflow has reached ∼ 30% of the sound speed. Such a fast gas flow has the potential to affect the motion of dust particles around the planet, which is investigatedlater in Chapter 3.

36

Chapter 2. Gas flow induced by low-mass planets

F IGURE 2.7: Flow structure around a planet at the midplane region of
the disk. The planet-induced gas flow is in the flow-shear regime. The
result was obtained from m03-hw0. The color contour represents theflow speed in the x-direction, v x,gas . The solid lines correspond to thespecific streamlines. The dotted and dashed-dotted lines correspondto the Bondi and Hill radii of the planet, respectively.

2.3. Results

37

F IGURE 2.8: Same as Fig. 2.7, but with the flow structure obtained from m003-hw01. The planet-induced gas flow is in the flowheadwind regime.

Dependence on the planetary mass
Figure 2.9 shows the structure of the planet-induced gas flow at the midplane fordifferent planetary masses. In Fig. 2.9, we set Mhw = 0. Thus we can focus on
the dependence of the planet-induced gas flow on the planetary mass. Since weneglect the effect of the headwind of the gas, the flow topologies are common in
all panels in Fig. 2.9. However, the gas velocity changes with the planetary mass. In
particular, v x,gas near the Bondi or Hill radius of the planet—the outflow speed in thex-direction, which is the main focus of this chapter—shows the complex dependence
on the planetary mass. When m . 0.5, v x,gas near the Bondi or Hill radius increases

38

Chapter 2. Gas flow induced by low-mass planetswith the planetary mass (Figs. 2.9a–c), but it seems to become weaker for a highermass planet (Fig. 2.9d). From the analytical point of view, we discuss again thedependence of v x,gas on the planetary mass later in Sect. 2.3.6.

F IGURE 2.9: Same as Fig. 2.7, but with the flow structures obtained
from m005-hw0 (panel a), m01-hw0 (panel b), m05-hw0 (panel c), andm07-hw0 (panel d), respectively. The black dashed circles denote thetwice the atmospheric radius, 2Ratm (Eq. (2.6)). The blue dashed,vertical, horizontal lines are described later in Sect. 2.3.4. The blue
dashed lines are the curves representing the outline of the outflow
of the gas (Eq. (2.32)). The verticals lines denote the x-coordinatesof the asymptote of the curve representing the outline of the outflow(Eq. (2.29)). The horizontal lines denote the y-coordinates of the maximum length of the outflow in the y-direction ((Eq. (2.33))).

Dependence on the headwind
The headwind of the gas breaks a vertically rotational symmetric structure of theplanet-induced gas flow. The transition of the gas flow field from the flow-shear
to the flow-headwind regime occurs smoothly. In Fig. 2.10, we fixed the planetary
mass, m = 0.2. Thus, Fig. 2.10 shows the dependence of the gas flow structurearound the planet on the headwind of the gas. In this figure, the planet-induced gas
flow is always in the flow-shear regime based on the definition in Kuwahara andKurokawa, 2020b, but it is useful to understand the tendency of the smooth transition of the flow regime. The corotaion radius for the gas, xcor,g shifts to the negative direction in the x-axis as Mhw increases, and then the horseshoe region which
formed at xcor,g also shifts to the − x direction. As shown in Fig. 2.10, the outflow
speed toward the inside (outside) of the planetary orbit becomes weak (strong) as
Mhw increases. From the analytical point of view, we discuss again the dependenceof v x,gas on the headwind of the gas later in Sect. 2.3.6.

2.3. Results

39

F IGURE 2.10: Same as Fig. 2.7, but with the flow structures obtained
from m02-hw0 (panel a), m02-hw001 (panel b), m02-hw003 (panel c), andm02-hw01 (panel d), respectively. The vertical dotted lines correspondto the corotation radii for the gas.

2.3.4

Geometry of the outflow

In this section, we present geometric arguments for the outflow of the gas. Here we
aim for a quantitative description of the properties of the planet-induced gas flow,such as length scales and morphologies of the outflow of the gas.
Width of the outflow
We first evaluate the width of the outflow of the gas in the x-direction. In this chapter, we define the width of the outflow as the region where the flow speed in thex-direction perturbed by the planet is more prominent compared to that of the unperturbed flow. The important length scales are the width of the subsonic region
and the half-width of the transient horseshoe flow.
We define the subsonic region as the region where the background gas velocity
with respect to the corotation radius for the gas is always subsonic, |vg,∞ | < cs . The
x-coordinate of the edge of the subsonic region with respect to the x-coordinate of
the corotation radius for the gas is given by2±
xsubsonic= ± H − | xcor,g |.
3(2.26)The other important length scale is the half-width of the transient horseshoe flow.
Let us start to consider an endmember case, Mhw = 0 (shear only case). As described in Sect.2.3.1, the outflow streamlines originated from the widest horseshoe
streamlines at high altitudes (the transient horseshoe flow; Fung et al., 2015). Thus,
the width of the outflow streamline at the midplane region could be constrained by
the half-width of the horseshoe flow in the far field, |y| → ∞. The half-width of thehorseshoe region is given by (Jiménez and Masset, 2017):√
1.05 m + 3.4m7/3.
wHS =1 + 2m2(2.27)The horseshoe region shifts to the negative direction in the x-axis as Mhw increases,
because the corotation radius for the gas shifts to the negative direction in the x-axis

40

Chapter 2. Gas flow induced by low-mass planets(e.g., Fig. 2.10). Outside the planetary orbit, the disk gas lying outside of the horseshoe region flows into the space where the horseshoe flow was originally locatedwhen Mhw = 0 (e.g., Fig. 2.5c). In other word, an empty space that is caused by
the shift of the horseshoe region is replenished by the recycling flow. We consider
that the x-coordinate of the position of the transient horseshoe streamline in the far
field (y → −∞) would shift to the positive direction in the x-axis by +| xcor,g |. Inside the planetary orbit, the x-coordinate of the position of the transient horseshoe
streamline at the midplane in the far field (y → ∞) would also shift to the negative
x-direction by −| xcor,g |. Based on these arguments, we define the x-coordinate of theposition of the transient horseshoe flow (the widest horseshoe flow at the midplane
region) as±
xtHS= ±wHS ± | xcor,g |(double-sign corresponds),(2.28)

where the plus and minus sign denote outside and inside the planetary orbit. Hereafter we omit the notation of the double-sign corresponds in each equation, unlessotherwise specified.
We plotted Eqs. (2.26) and (2.28) in Figs. 2.11 (where m = 0.1) and 2.12 (wherem = 0.5). When m & 0.3, the x-coordinate of the position of the transient horseshoe
streamline exceeds that of the subsonic region. From these figures, we found thatthe width of the outflow of the gas can be constrained by

±
±
±
wout
= ± min | xsubsonic, xtHS| .
(2.29)
Figures 2.11 and 2.12 explain our findings, which show the difference of flow speed
from the sub-Keplerian shear (Figs. 2.11a and 2.12a).
1. When | xtHS | < | xsubsonic | (m . 0.3; for simplicity, we omit the plus and minus
signs in this paragraph), the width of the outflow is constrained by the halfwidth of the transient horseshoe streamline. Considering the gas flow along
the recycling streamline (the streamline of the transient horseshoe flow), thegas velocity approaches the background sub-Keplerian shear when x → xtHS ,which leads to v x,gas → 0 (Fig. 2.11a).
2. When | xtHS | > | xsubsonic | (m & 0.3), the width of the outflow is constrainedby the width of the subsonic region. We consider again the gas flow along therecycling streamline. Tthe recycling streamline can intrude the supersonic region, | x | > | xsubsonic |, because its width is determined by xt,HS . However, thegas velocity perturbed by the planet approaches rapidly to that of the background sub-Keplerian shear velocity in the supersonic region due to the influence of the supersonic background flow (Fig. 2.12a).
We plotted Eq. (2.29) in Figs. 2.9 and 2.10. Equation (2.29) can serve as a useful
guide for the width of the outflow region in the x-direction, where the flow speed in
the x-direction perturbed by the planet is more prominent compared to that of theunperturbed flow.

2.3. Results

F IGURE 2.11: Flow structure around a planet with m = 0.1 at the midplane region of the disk. The color contours show the difference offlow speed from the sub-Keplerian shear (panel a) and the flow speed
in the x-direction, v x,gas (panel b). The results obtained from m01-hw0.
The solid lines correspond to the specific streamlines. The dasheddotted and dashed lines correspond to the half-width of the transient
horseshoe flow and the width of the subsonic region, respectively.

F IGURE 2.12: Same as Fig. 2.11, but with the results obtained fromm05-hw01.

41

42

Chapter 2. Gas flow induced by low-mass planets

Outline of the outflow
We found that the morphologies of the outflow at the midplane region can be outlined by the following fitting formulae:+
r =wouttan θ−
r =wouttan θ(3π/2 ≤ θ ≤ 2π ),(π/2 ≤ θ ≤ π ),(2.30)(2.31)

for a polar coordinates centered at the planet. In a local Cartesian coordinates corotating with the planet, Eqs. (2.30) and (2.31) can be written by
y =− q

x2
+ 2
(wout
) − x2H( x ) + q

x2
− 2
(wout
) − x2H(− x ),

(2.32)

where H( x ) is the Heaviside step function. The above curves represent the curve±with wout
as the asymptotes. We plotted Eq. (2.32) in Figs. 2.9 and 2.10. Equation(2.32) can be used for the outline of the outflow region, because v x,gas → 0 whenx → wout . We note that, when |wtHS | > | xsubsonic |, Eq. (2.32) does not coincide withthe recycling streamline in the far field, |y| → ∞, because the recycling streamline
can intrude the supersonic region in this case.
Length of the outflow
Based on Eq. (2.32), we evaluate the characteristic length scale of the outflow in they-direction. Similar to the width of the outflow, we define the length of the outflow
as the region where the flow speed in the y-direction perturbed by the planet ismore prominent compared to that of the unperturbed flow. Considering the gas
flow along the outline of the outflow given by Eq. (2.32), we define the y-coordinate±
at a position close enough to the asymptotes, wout
, as the characteristic length of theoutflow±L±y,out = y ( xcrit ),(2.33)±±
xcrit= C3 × wout.

(2.34)

where

We set C3 = 0.95 as a fitting parameter. We plotted Eq. (2.33) in Figs. 2.9 and 2.10.
Equation (2.33) can serve as a useful guide for the length scale of the outflow in they-direction.

2.3.5

Vertical structure of the outflow

The preceding section focus on the flow structure at the midplane region. In thissection, we analize the vertical structure of the outflow. The outflow extends in thevertical direction (Figs. 2.13 and 2.14). Figures 2.13 and 2.14 show that the outflow has the maximum speed near the midplane region of the disk, but it disappear
sharply at high altitudes, z & Ratm . As described in Sect. 2.3.1, the recycling flow
passes the planet, tracing the surface of the isolated envelope.

2.3. Results

43

F IGURE 2.13: Gas flow structure averaged in the y-direction within
the calculation domain of hydrodynamical simulation. The results
obtained from m02-hw0 (panel a) and m02-hw01 (panel b). The color
contour from red to black represents the x-component of the gas flow
velocity averaged in the y-direction within the calculation domain ofhydrodynamical simulation around a planet, hv x,gas iy . The vertical+−lines correspond to wout
and wout, respectively. The dashed circles denote 2Ratm (Eq. (2.24)). The solid gray lines correspond to the streamlines. Thanks to the symmetry of the system, we only focus on theregion where z ≥ 0.

F IGURE 2.14: Same as Fig. 2.13, but with the results obtained fromm05-hw0 (panel a) and m05-hw01 (panel b).

Based on the 3D structure of the recycling streamlines and Figs. 2.13 and 2.14,
we assume that the outflow speed in the x-direction decreases exponentially whenz > Ratm . Figures 2.15 and 2.16 compare our assumptions with the gas flow velocities obtained from hydrodynamical simulations, where the velocities of the gas flow

44

Chapter 2. Gas flow induced by low-mass planets

are averaged in the x- and y-directions within the calculation domain of hydrodynamical simulation. In these figures, we introduced the following fitting formula:
" 
2 #
z
±
±
,
(2.35)
hv x,gas i x,y = hv x,gas i x,y |z=0 × exp −Ratm
where hv±
x,gas i x,y |z=0 is the x-component of the gas flow velocity at the midplane
region averaged in the x- and y-directions within the calculation domain of hydrodynamical simulation,RRρg v x,gas |z=0 dxdy±
RR
hv x,gas i x,y |z=0 =.
(2.36)
ρg dxdy
In Eq. (2.36), the integration range in the y-direction covers the entire region of thedomain of hydrodynamical simulation, but we set the integration range from 0 to±
wout
in the x-direction.
Equation (2.35) shows excellent agreement with the results of hydrodynamical
simulations for lower-mass planets, m . 0.4 (Fig. 2.15). For higher-mass planets,
m & 0.5, Eq. (2.35) reproduces a decreasing trend of hv x,gas i x,y with altitude only inthe range of z < Ratm . For higher-mass planets, the meridional circulation of the gas
above the outflow leads to the complex behavior of the flow speed (Fig. 2.14). Thus,
Eq. (2.35) does not coincide with the obtained hydrodynamical simulation results athigh altitudes, z > Ratm (Fig. 2.16).

2.3. Results

F IGURE 2.15: The x-component of the gas flow velocity averaged
in the x- and y-directions within the calculation domain of hydrodynamical simulation around a planet, hv x,gas iy , as a function of altitude, z. The yellow dashed, blue solid, and black dotted lines
are obtained from hydrodynamical simulations, the fitting formula(Eq. (2.35)), and the atmospheric radius (Eq. (2.24)), respectively.

45

46

Chapter 2. Gas flow induced by low-mass planets

F IGURE 2.16: Same as Fig. 2.15, but with the results obtained fromm05-hw0 (panel a) and m05-hw01 (panel b).

2.3.6

Outflow speed at the midplane region

In this section, an analytic formula of the outflow speed is presented. Following
Kuwahara et al., 2019, we first analyze the inflow and outflow points where the gas
chiefly flows in and out. Next, the analytic solution of the outflow speed is derived
from Bernoulli’s theorem.

2.3. Results

47

Positions of inflow and outflow
When we analyzed the velocity of the flow field around the planets, especially theestimation of the speed of outflow, it was necessary to judge where the gas chiefly
flowed in and out. Figure 2.17 shows the vertical structure of inflow and outflow
around a planet with m = 0.2 at the meridian plane. Gas flows in from the verticaldirection of the z-axis and escapes near the midplane to mid-latitudes of the disk. In
Fig. 2.17, the gas seems to be descending from a height twice as high as the top of
the atmospheric radius, 2Ratm , or more.

F IGURE 2.17: Gas flow structure averaged in the y-direction within
the calculation domain of hydrodynamical simulation around aplanet. The results obtained from m02-hw0 (panel a) and m02-hw01(panel b). The color contour from purple to orange represents the radial component of the gas flow velocity averaged in the y-direction
within the calculation domain of hydrodynamical simulation arounda planet, hvr,gas iy . The dashed circles denote the twice the atmospheric radius, 2Ratm (Eq. (2.24)). Thanks to the symmetry of thesystem, we only focus on the region where z ≥ 0.

To obtain a quantitative description of the inflow and outflow points, we analyze
the mass flux of the gas. Figures 2.18 and 2.19 show the azimuthally averaged massflux for a certain spherical shell of radius R, hρg vr iφ , as a function of the altitude,z. In Figs. 2.18 and 2.19, we measured the mass fluxes of the gas that flows in andout spherical shells at various radii. Gas flows in where hρg vr iφ < 0 and flows out
where hρg vr iφ > 0, respectively. As we expected in Fig. 2.17, we found that gas
chiefly flows in and out of the twice the atmospheric radius, 2Ratm (Figs. 2.18 and2.19). Thus, we define the inflow and outflow points are located at the sphere ofradius 2Ratm .

48

Chapter 2. Gas flow induced by low-mass planets

F IGURE 2.18: Azimuthally averaged mass flux, hρg vr iφ , as a function of altitude, z. The results obtained from m02-hw0 (panel a) andm02-hw01 (panel b). Each solid line represents the changes of hρg vr iφ ;altitude is varied along with a certain radius. Different colors correspond to different radii. We highlight the important radii with yellow(RBondi ), blue (RHill ), and red (2Ratm ) solid lines, respectively. Gas
flows in where hρg vr iφ < 0 and flows out where hρg vr iφ > 0, respectively.

2.3. Results

49

F IGURE 2.19: Same as Fig. 2.18, but with the results obtained fromm04-hw0 (panel a) and m04-hw01 (panel b).

Analytical estimation of the outflow speed
An analytic solution of the outflow speed was derived from Bernoulli’s theorem. We
first return to the Euler equation (Eq. (2.8)):∂∇p+ vg · ∇ vg = −+ ∑ Fi .
∂t
ρg
i

50

Chapter 2. Gas flow induced by low-mass planets

In the Euler equation, the advection term can be rewritten as follows:!
v2g− vg × ωg ,(vg · ∇)vg = ∇2(2.37)

where ωg = (∇ × vg ) is the vorticity of the fluid. For a barotropic flow, we candefine the pressure function
P=

Z

dp
.
ρg

(2.38)

Since we focus on the subsonic flow, the pressure structure is determined to be inhydrostatic equilibrium with the gravity of the planet. The temperature is nearly
uniform due to the efficience radiative cooling (Fig. 2.20). We assume the isothermal
flow along the recycling streamlines. For isothermal flow, the pressure function canbe described by
P = c2s ln ρg .

(2.39)F IGURE 2.20: Streamlines of 3D planet-induced gas flow around theplanet with m = 0.2 and Mhw = 0. The planet-induced gas flow isin the flow-shear regime. The color contour represents p/ρg , whichis the measure of temperature. The black sphere denotes the isolated
envelope whose size is Ratm = 0.128 H. The arrows represent thedirection of the gas flow. Panels a and b: Perspective views of the flow
field. Panel c: x-y plane viewed from the +z-direction.

From Eqs. (2.37) and (2.39), the Euler equation can be rewritten as
!
v2g
∂vg
= −∇
+ P + ∑ Fi + vg × ωg ,
∂t
2
i
!
2
2
vg
3
z
m
= −∇
+ c2s ln ρg (z) − x2 + − 2Mhw x −+ vg × (ωg + 2ez ),222R(2.40)

2.3. Results

51

For a steady-state flow, we have

∇ B = vg × (ωg + 2ez ),(2.41)

where
B=

v2g

3
z2
m
+ c2s ln ρg (z) − x2 + − 2Mhw x − ,2
2
2
R(2.42)

is a Bernoulli’s invariant. Equation (2.41) indicates that ∇ B does not have components in directions of vg and ωg . In other words, ∇ B = 0 holds along streamlines
and vortex lines. Thus, we integrate Eq. (2.41) along a streamline or vortex line, andobtainB = Const.

(2.43)It should be noted that we considered that the radiative cooling occurs on a finite
timescale, β = (m/0.1)2 . Thus, the Bernoulli’s invariant is not strictly conserved
along the streamline. However, we found that Bernoulli’s theorem is still useful toestimate the outflow speed because of the efficience radiative cooling. We discuss
the applicability of Bernoulli’s theorem later in Sect. 2.4.2.
We picked up a streamline of the recycling flow. We set two points on the streamline, Pin and Pout , which correspond to the inflow and outflow point where the gas
chiefly flows in and out. We first consider the inflow point. We set
Pin = (0, 0, zin ).

(2.44)

From the analysis of the kinetic energy of gas flow in our simulations, we foundthat the first term in the right-hand side of Eq. (2.42) was negligible (Fig. 2.21; Funget al., 2015; Kuwahara et al., 2019). The second (the Bernoulli’s function) and fourth(the z-component of the tidal potential) terms in the right-hand side of Eq. (2.42)
would be canceled because the stellar gravitational potential energy balances ln ρgin hydrostatic equilibrium. Based on the analysis of the mass flux of the gas (Figs.
2.18 and 2.19), we set R = 2Ratm,fit . Thus, the Bernoulli’s invariant at the inflow
point can be described by
Bin = −

m
2Ratm,fit.

(2.45)

52

Chapter 2. Gas flow induced by low-mass planets

F IGURE 2.21: Same as Fig. 2.20, but the color contour represents the
kinetic energy of gas flow, v2g /2. The black sphere denotes the twice
the atmospheric radius, 2Ratm = 0.256 H, where the gas chiefly flows
in and out.

Next, we consider the outflow point. Since the outflow has the maximum speed
near the midplane region of the disk (Figs. 2.13–2.16), we set
±
±
±
Pout
= ( xout, yout, 0),(2.46)

where we assume the following
q relation based on the analysis of the mass flux of

2 + y2the gas (Figs. 2.18 and 2.19): xout
out = 2Ratm,fit . The x- and y-coordinates ofthe outflow points are determined by those of the intersections of the outline of theoutflow (Eq. (2.32)) and the circle of the twice the atmospheric radius, 2Ratm,fit . The
Bernoulli’s invariant at the outflow point can be described by

±
Bout
=

± 2
(vout
)
3 ± 2
m
±
− ( xout) − 2Mhw xout−.
22
2Ratm,fit(2.47)

Since Bernoulli’s invariant has the same value in the steady state at each point of
Pin and Pout , from Eqs. (2.45) and (2.47), we obtain the outflow speed
q
±
± 2
±
|vout | = 3( xout
) + 4Mhw xout.
(2.48)
From Eq. (2.32), which represents the outline of the outflow, we can estimate an±angle of the outflow at the outflow point. We set the angle to the x-axis as ϕout
. The
x-component of the outflow velocity is given by±±v±
x,out = | vout | cos ϕout ,(2.49)

2.3. Results

53

where
±
ϕout
= tan−1



dy
dx


,

±
x = xout


+ 2
− 2
x 2(wout) − x2x 2(wout) − x2dy=−3/2 H( x ) +3/2 H(− x ),
+ 2
− 2
dx
(wout
) − x2(wout) − x2(2.50)(2.51)

and±xout= ±2Ratm,fit cos

2Ratm,fit
±
wout


.

(2.52)

Figure 2.22 compares our analytic solutions of the x-component of the outflow
velocity with the results obtained from hydrodynamical simulations. Equation (2.49)
reproduces the maximum flow speed in the x-direction obtained from hydrodynamical simulations for a wide range of planetary mass and Mach number of the headwind of the gas. It should be noted that, in Fig. 2.22, we sampled the maximum−
outflow speed toward the outside (v+x,out ) and inside (v x,out ) of the planetary orbit
obtained from hydrodynamical simulations at the midplane within the limited re−++gion where wout
≤ x ≤ woutand L−
y,out ≤ y ≤ Ly,out . We found that the flow speed in
the x-direction obtained from hydrodynamical simulations does not have the maximum values at the intersections of the outline of the outflow (Eq. (2.32)) and thetwice the atmospheric radius, 2Ratm,fit . This is because the fitting formula for theoutline of the outflow does not trace an actual streamline which gives the maximum
flow speed at the midplane region. Thus, the point where |v x,gas | has the maximum±
value does not necessarily coincide with the assumed outflow point, Pout(see thered and black stars in Fig. 2.23).

54

Chapter 2. Gas flow induced by low-mass planets

F IGURE 2.22: Outflow speed at the midplane region of the disk asa function of the dimensionless planetary mass and the Mach number of the headwind of the gas. The star symbols are obtained
from hydrodynamical simulations, corresponding to the maximum
outflow speeds in the x-direction toward the outside (red) and inside (black) of the planetary orbit within the limited region where
−
+
+
wout
≤ x ≤ woutand L−
y,out ≤ y ≤ Ly,out . The red and black solid lines
correspond to the analytic solutions of the x-component of the outflow velocity derived from Eq. (2.49). The dashed lines correspond to
the analytic expressions of the flow speed in the x-direction obtainedby Ormel, 2013 (blue; Eq. (2.61)), Fung et al., 2015 (purple; Eq. (2.66)),
and Kuwahara et al., 2019 (yellow; Eq. (2.68)), respectively. The grayshaded region represents the dimensionless pebble isolation mass,
miso (Eq. (2.57)). The values of assumed fitting parameters C1 and C2
are shown in the lower-right corner of each panel, which determine
the atmospheric radius (Eq. (2.25)).

2.3. Results

55

F IGURE 2.23: Gas flow structure around a planet at the midplane region. These results were obtained from m05-hw0 (panel a) and m1-hw0(panel b). The star symbols correspond to the maximum outflow
speeds in the x-direction toward the outside (red) and inside (black)of the planetary orbit at the midplane region, which were plotted inFig. 2.22. The white dots correspond to the maximum values of theflow speeds at each radial grid of hydrodynamical simulations in the−+x-direction within the limited region where wout
≤ x ≤ wout
and−+Ly,out ≤ y ≤ Ly,out . The color contour represents the flow speed in
the x-direction, v x,gas . The gray solid lines correspond to the specificstreamlines. The blue dashed lines are the outlines of the outflow of
the gas (Eq. (2.32)). The black dashed circles denote 2Ratm (Eq. (2.24)).
The vertical and horizontal lines correspond to the width and lengthscale of the outflow (Eqs. (2.29) and (2.33)), respectively (see Sect.
2.3.4).

We first consider the dependence of the outflow speed in the x-direction on theplanetary mass. Our analytic solution indicates that the dependence of the outflow
speed in the x-direction on the planetary mass changes when m ∼ 0.1–0.2 and m ∼0.3. When m ∼ 0.1–0.2, a power low dependence of the atmospheric radius with theplanetary mass changes (Eq. (2.25)). When m & 0.3, the half-width of the transient
horseshoe flow exceeds that of the subsonic region (Eq. (2.29)).
From Fig. 2.22, the outflow speed in the x-direction increases with the planetary
mass when m . 0.2–0.3. From the analytical point of view, this trend can be understood as follows. For simplicity, we only consider the shear only case (Mhw = 0).
From Eq. (2.48), the outflow speed is proportional to |vout | ∝ xout ∝ Ratm,fit ∝ m, inthe low-mass regime (Eq. (2.25)). From Fig. 2.24, we found that the dependence ofthe cosine term in Eq. (2.49), cos ϕout , on the planetary mass is weak, ∼ m−1/3 or less.
Thus, the outflow speed in the x-direction is almost proportional to v x,out ∝ m2/3 .

56

Chapter 2. Gas flow induced by low-mass planets

The outflow speed in the x-direction has a peak at m ∼ 0.3, and then decreases
with the planetary mass when m & 0.3. In this case, the dependence of the outflow
speed on the planetary mass is |vout | ∝ xout ∝ m1/3 (Eq. (2.25)), or slightly lower due
to the cosine term in Eq. (2.52). The cosine term in Eq. (2.49), cos ϕout , is proportionalto ∼ m−1/3 (Fig. 2.24). Thus, the outflow speed in the x-direction is nearly constantor decreases as the planetary mass increases.

F IGURE 2.24: Angle of the outflow, ϕout (Eq. (2.50)), cos ϕout , and thex-coordinate of the outflow point, xout (Eq. (2.52)). We set Mhw = 0.

For higher-mass planets, m & 0.6, we found that the points where the gas flow
speed in the x-direction has the maximum values are located at the region close
to the planetary orbit (Fig. 2.23b). This leads to the complex behaviour of v+
x,max
and |v−x,max | in Fig. 2.22a. We discuss again the outflow speed in the x-direction forhigher-mass planets, m ≥ 0.6, later in Sect. 2.3.8.
Next, we consider the effect of the headwind of the gas. The outflow speed in thex-direction toward the inside (outside) of the planetary orbit becomes weak (strong)as Mhw increases (Figs. 2.22b–d). This is caused by the second term in the squareroot in the right-hand side of Eq. (2.48), which was originated the global pressure
force due to the sub-Keplerian motion of the gas, Fhw = 2Mhw e x . Since the fluidrecieves the global pressure force in the positive direction in the x-axis, the outflow toward the inside (outside) of the planetary orbit is reduced (enhanced). When
Mhw = 0.1, the strongest Mach number considered in this chapter, our analytic solution, |v−
x,out |, deviates from the results obtained by hydrodynamical simulations,−|v x,max |. We discuss again the outflow speed under the influence of the strong headwind later in Sect. 2.3.8.

2.3.7

Velocity distribution of the outflow

Equation (2.49) provides the maximum flow speed in the x-direction at the midplane
region of a disk. Outflow of the gas, however, has a spread in the x-direction with the2.3. Results

57±
typical width of wout(e.g., Fig. 2.13). Here we aim to model the velocity distributionof the outflow of the gas at the midplane region.
We found that the following Gaussian function can reproduce the numerical result:
"
#
"
#
+ 2
− 2
( x − xout)
( x − xout)+−
v x,out ( x ) = v x,out exp −+ v x,out exp −,(2.53)
2( D + )2
2( D − )2

where we assumed that the full width at half maximum corresponds to the halfwidth of the outflow region√w±
FWHM = 2 2 ln 2 D ± = out .
2(2.54)

Figure 2.25 compares Eq. (2.53) with the flow velocities in the x-direction obtained
from hydrodynamical simulations at the midplane. These results will be used again
when we model the velocity of dust influenced by the planet-induced gas flow (laterin Chap 4).

58

Chapter 2. Gas flow induced by low-mass planets

F IGURE 2.25: Velocity distribution of the outflow in the x-direction
at the midplane region. The blue solid lines correspond to Eq. (2.53).
The yellow dots are obtained from hydrodynamical simulations, corresponding to the maximum values of the flow speed in the xdirection at each radial grid of hydrodynamical simulations within−++
the limited region where wout
≤ x ≤ woutand L−
y,out ≤ y ≤ Ly,out (thegas velocities at the white dots in Fig. 2.23). The vertical dotted lines±
denote the widths of the outflow, wout
. We adopt v x,out ( x ) = 0 where−+| x | ≤ Ratm , x ≤ wout , and x ≥ wout . This implies that we omit the
influences of the envelope rotation and perturbation by the density
waves excited by an embedded planet in this figure.

2.3.8

Caveats

Our analytic solution deviates from the results obtained by hydrodynamical simulations when m & 0.6. As mentioned in Sect. 2.3.4, the width of the outflow for±
the planet with m & 0.3 is constrained by the width of the subsonic region, xsubsonic,which only depends on Mhw . Thus, the outflow angle to the x-axis at the outflow
point is important to estimate the outflow speed in the x-direction. However, inparticular for higher-mass planets, the slope of the curve representing the outline of±the outflow at the outflow point changes significantly with changes in xout
. The fitting formula for the outline of the outflow does not trace exactly the actual recycling
streamline. Thus, our analytic solution will become less accurate for higher-massplanets, m & 0.6.
We would expect that such a high-mass regime may not need to be consideredin the context of influences of the gas flow on the motion of dust or pebbles. When

2.3. Results

59

the planetary mass reaches pebble isolation mass (Lambrechts and Johansen, 2014):
L14
Miso
' 20

 a 3/4M⊕ ,5 au(2.55)a growing planet generates a pressure bump outside of its orbit. A subsequent study
derived a detailed pebble isolation mass (Bitsch et al., 2018):hMiso ' 25
0.05

!3 "

3
0.34
log αacc!4#+ 0.66 M⊕ ,(2.56)

In our dimensionless unit, Eq. (2.56) can be described by (Kuwahara and Kurokawa,
2020b):
miso ' 0.6,

(2.57)We plotted Eq. (2.57) in Fig. 2.22. When m & 0.6, dust or pebbles can be trapped at
the edge of the pressure bump and may not reach the outflow region, | x | < |wout |.
As long as we focus on the influence of the outflow on drifting dust, we would notnecessarily consider the flow speed for higher-mass planets, m & 0.6.
Our analytic solution fails to reproduce the outflow speed toward inside of theplanetary orbit under the influence of the strong headwind, Mhw ∼ 0.1 (Fig. 2.22d).
We speculate that the deviation of the analytic solution from numerical results iscaused by the smooth transition of the flow regime, and thus changes in the inflow
or outflow point. We assumed Pin = (0, 0, zin ), but the inflow point seems to shiftslightly in the negative direction in the x-axis (e.g., 2.17b). The outflow toward insideof the planetary orbit occurs when the planet-induced gas flow is in the flow-shear
regime. This happens when m > 0.03 in the range of the parameters consideredin this chapter (Kuwahara and Kurokawa, 2020b). As discussed in Sect. 2.3.1, thetransition of the gas flow field from the flow-shear to the flow-headwind regime
occurs smoothly. It seems difficult to model the transition of the 3D structure of theflow field.
However, as long as we consider the influence of the outflow on pebble accretion, the importance of the outflow toward inside of the planetary orbit is reduced
compared to the outflow toward outside of the planetary orbit. Dust or pebbles drift
from the outer to the inner region of the disk due to the global pressure gradient.
Thus, the outflow toward inside of the planetary orbit would be less important for
the radial drift of dust or pebbles.

60

Chapter 2. Gas flow induced by low-mass planets

2.4

Discussion of this chapter

2.4.1

Comparison to the previous studies

Atmospheric radius
Assuming the gas chiefly flows in and out of the twice the atmospheric radius, wederived an analytic solution for the outflow speed. Thus, the atmospheric radius iscrucial to estimate the flow speed. In this chapter, we estimated the atmosphericradius based on the definition in Moldenhauer et al., 2022 and obtained the fitting formula (Eq. (2.25)). It should be noted that Eq. (2.25) differs from the scalinglaws obtained by Moldenhauer et al., 2022. Moldenhauer et al., 2022 performed 3D
radiation-hydrodyamic simulations in a local frame for the embedded planets with
Mpl = 1–10 M⊕ at 0.1 au (corresponding to m ∼ 0.3–3 in our disk model; Fig. 2.2a).
Based on Eq. (2.24), the authors estimated the atmospheric radius and introduced
the following fitting formula:
RM22
atm = a



Mpl
M⊕

n
R⊕ ,

(2.58)

where a = 4.7 ± 0.6 and n = 0.67 ± 0.08 when the headwind is included, and a =5.7 ± 0.3 and n = 0.62 ± 0.04 without the headwind. In our dimensionless unit,
Eq. (2.58) can be rewritten asnR̃M22
atm,w/ head ' 0.23a × m ,(2.59)

nR̃M22
atm,w/o head ' 0.26a × m .

(2.60)We plotted Eqs. (2.59) and (2.60) in Fig. 2.26. The atmospheric radius obtained fromour hydrodynamical simulations is larger than Eqs. (2.59) and (2.60). Moldenhauer
et al., 2022 suggest that Ratm increases approximately proportional to ∼ R2Hill .
Several previous studies reported different scaling for the atmospheric radius.
D’Angelo and Bodenheimer, 2013 performed global 3D radiation-hydrodyamic simulations for the embedded planets with Mpl = 5–15 M⊕ at 5–10 au (corresponding
to m ∼ 0.15–0.45 in our disk model; Fig. 2.2a). In D’Angelo and Bodenheimer, 2013,
realistic equation of state was considered and dust opacities were included. Assuming that an envelope is a sphere, the authors estimated the size of the envelopeas ∼ 0.4–0.6RBondi from the shape of density contours obtained from their hydrodynamical simulations. Lambrechts and Lega, 2017 performed 3D global radiationhydrodynamic simulations for the planets with Mpl = 5 M⊕ at 5.2 au (corresponding
to m ∼ 0.26 in our disk model; Fig. 2.2a). The authors considered dust opacities andthe luminosity from accreting solid. From the analysis of the mass flux within the
Hill sphere of the planet, the authors estimated that the envelope size is ∼ 0.3RHill .
Cimerman et al., 2017 performed 3D radiation-hydrodynamic simulations in a local
frame corotating for the planet. The authors defined the planetary envelope as theregion where the azimuthally averaged ratio of radial to azimuthal velocities falls
below a threshold of 10%, h|vr /vφ |iφ ≤ 0.1. They reported that the size of the envelope was 0.25RBondi , 0.2RBondi , and 0.07RBondi for the planets with m = 0.38, 0.75,
and 1.9, respectively.
Figure 2.26 compares the atmospheric radii obtained from radiation-hydrodynamic
simulations of above-mentioned previous studies with those from our study. Weemphasize that the definition of the atmospheric radius differs among the previous studies. Thus, in particular, our results should compare with those obtained by2.4. Discussion of this chapter

61

Moldenhauer et al., 2022. Although many differences among these studies make adirect comparison of results difficult, our results agree with the results of previous
studies within a factor of ∼ 2, except with D’Angelo and Bodenheimer, 2013.
The dependence of the atmospheric radius on the dimensionless cooling time β
is a remaining issue. In this chapter, the radiative cooling is implemented by using
the β cooling model (Gammie, 2001; Kurokawa and Tanigawa, 2018). Kurokawa andTanigawa, 2018 showed that the atmospheric radius is positively correlated to β. The
atmospheric radius would increase (decrease) as β increases (decreases). Following
Kurokawa and Tanigawa, 2018, we set the value of β as β = (m/0.1)2 (Eq. (2.13)).
The dimensionless cooling time β would change the dust opacity. The dust depletion would decrease the relaxation time (Malygin et al., 2017). The influence of thecooling time on our results should be investigated in future studies.

F IGURE 2.26: Comparison of the atmospheric radii obtained in ourstudy and those in previous studies. The square symbols are obtained
from our hydrodynamical simulations. Different colors correspond todifferent Mach numbers, Mhw . The gray- and pink-shaded regions
are given by the fitting formula for the atmospheric radius, Ratm,fit(Eq. (2.25)). The black dashed and dotted lines are given by Moldenhauer et al., 2022. The purple-shaded region corresponds to theresults of D’Angelo and Bodenheimer, 2013. The blue and red diamonds corresponds to the results of Lambrechts and Lega, 2017, andCimerman et al., 2017, respectively.

62

Chapter 2. Gas flow induced by low-mass planets

Isolated envelope
As described in Sect. 2.3.2, an isolated envelope emerges around an embedded
planet due to the buoyancy barrier. This means that the envelope is stable against
convection. The accretion of dust particles onto the planet or the luminosity of theplanetary core would lead to fully convective envelope (Rafikov, 2006; Lambrechtsand Lega, 2017; Popovas et al., 2018), where the envelope has an isentropic profile.
In this case, the complete recycling of the gas is allowed due to the absence of thebuoyancy force.
Even when the envelope becomes convective, once it begins to be cooled, a positive entropy gradient leads to the formation of a radiative layer. The buoyancy stars
to prevent the inflow of the gas from reaching deeper regions in the envelope. Then,the envelope is divided into three layers: convective, radiative, and recycling Lambrechts and Lega, 2017.
However, the fully convective envelope may not be established when we consider the typical value of the accretion rate of dust or the opacity (Ikoma et al., 2000;
Lambrechts et al., 2014; Lambrechts and Lega, 2017; Lee et al., 2018). In this study,
the convective layer does not appear because we did not consider the heat source.
Outflow speed
We plotted the analytic solutions of the outflow derived from previous studies inFig. 2.22a. Ormel, 2013 considered 2D, isothermal, inviscid, and compresible flow
around a low-mass planet, m ≤ 0.03. The author derived an approximate solution
for the flow in terms of the stream function. In Fig. 2.22a, we plotted the flow speed
in the x-direction obtained by Ormel, 2013:
"
#
1
vO13
(2.61)
x,out = m cos ỹ Ci( ỹ ) − sin ỹ ( π − 2 Si( ỹ )) ,2√
where ỹ = y/ 2 (y > 0) and Ci(u), Si(u) are the sine and cosine integral functions:Z ∞cos(t)dt,tu
Z usin(t)
dt.
Si(u) =t0

Ci(u) = −(2.62)(2.63)

Equation (2.61) gives the x-component of the flow speed within the horseshoe region
as a function of y. Owing to the symmetry, we only consider x > 0 and y > 0.
Assuming that the streamline in the horseshoe region can be described by our fitting
function (Eq. (2.32)), we set+y( xout)= q+ 2( xout
)
+ 2
+ 2
(wout
) − ( xout),

(2.64)+
in Eq. (2.61) where xout
is given by Eq. (2.52).
Ormel, 2013 considered the 2D flow, where the recycling streamlines do not appear. Nevertheless, Eq. (2.61) agrees with our results when m . 0.1 (Fig. 2.22a). The
outflow streamlines at the midplane region were closely related to the horseshoe
streamlines, because they were originated from the horseshoe flow at high altitudes

2.4. Discussion of this chapter

63(Fung et al., 2015). The 3D structure of the recycling streamline does not change significantly in isothermal and nonisothermal simulations. Equation (2.61) depends on
the y-coordinate. Thus, it is crucial to estimate the flow speed at which point in thehorseshoe region. In Eq. (2.61), we set the y-coordinate based on the results in thischapter. This may lead to the agreement in the results between Ormel, 2013 and ourstudy.
However, Eq. (2.61) overestimates our results when m & 0.1. We expect that thelinearized approximation used in Ormel, 2013 is no longer valid this regime. In thederivation of Eq. (2.61), the following two approximations were considered:
1. the perturbed flow depends only on y.
2. The limit x  1 and x  y were considered (the linearized approximation).
Since the y-component of the flow is dominated by the background shear flow, thefirst assumption is justified. The second assumption implies that only the horseshoe
region (co-orbital region) was considered. Ormel, 2013 only considered the lowmass regime (m  1), where the following relation is satisfied: Rin < RBondi <+RHill < H. We estimate the flow speed at x = xout
in Eq. (2.61), which increases withthe planetary mass. Thus, the second assumption will break down for a higher-massplanet.
Fung et al., 2015 considered isothermal and viscous flow. The authors performed
3D hydrodynamical simulation for the planet with m = 0.56 and found that theoutflow speed at the midplane region was ∼ 0.2–0.4cs . Fung et al., 2015 applied
Bernoulli’s theorem along with a certain horseshoe streamline and derived the outflow speed as:
v


u
u
1
1
u
F15
.
| = tv2in + 2m 
|vout
(2.65)−qRout2
2
Rin + RoutFung et al., 2015 considered the possibility that gas starts falling from the disk scaleheight and assumed that the distance of the outflow point scales with the half-widthof the horseshoe region. Following Fung et al., 2015, we substituted Rin ∼ H andRout ∼ wHS (Eq. (2.27)) into Eq. (2.65). We here assume that the outflow occurs at anangle of φ ∼ ±π/4 to the x-axis. In Fig. 2.22a, we plotted the following modified
formula for the outflow speed in the x-direction:
v 

u
u
1
1
u 
 × cos(π/4),
vF15−q(2.66)
x,out = t2mwHS2
H 2 + wHS
where the contribution of v2in was neglected. As discussed in Fung et al., 2015,
Eq. (2.66) should be considered an upper limit because the height of the inflow point
assumed in Fung et al., 2015 was higher than ours. Thus, Eq. (2.66) overestimatesour results.
In our previous study, we proposed that the outflow speed can be described by(Kuwahara et al., 2019):
q
|vK19
|
=
3R2grav .
(2.67)out

64

Chapter 2. Gas flow induced by low-mass planets

In Fig. 2.22a, combining the analysis in this chapter with those in Kuwahara et al.,2019, we plotted the following modified formula for the outflow speed in the xdirection:q
vK19=
3R2grav cos ϕout ,
x,out
#
"
 
q
dy
,
(2.68)= 3R2grav cos tan−1
dx x= xK19
out

where dy/dx is given by Eq. (2.51) and we used
K19
xout
= Rgrav × cos(π/4),(2.69)

based on the conclusion in Kuwahara et al., 2019. Equation (2.68) underestimatesour results when m . 0.3. We describe the differences with Kuwahara et al., 2019as below. First, Kuwahara et al., 2019 assumed an isothermal and Keplerian gas
disk (Mhw = 0) and performed 3D hydrodynamical simulations with embedded
planets, m ∈ [0.03, 2]. Second, Eq. (2.67) was derived from Bernoulli’s theorem,
but the authors set the inflow and outflow points on the sphere of radius Rgrav =min( RBondi , RHill ) (Eq. (2.3)). Third, from the analysis of hydrodynamical simulations, Kuwahara et al., 2019 found that the outflow speed has maximum values near
the intersections of a line of y = − x and a circle of radius Rgrav .
In the isothermal cases, there is no clear boundary demarcates the bound atmosphere from the disk gas (Ormel et al., 2015b). The inflow of the gas intrudes deep
inside the gravitational sphere of the planet. Thus, in the isothermal conditions, theinflow and outflow points were located at the points closer to the planetary corethan in the nonisothermal cases, leading to the underestimation of the results in thischapter when m . 0.3. In other words, Equation (2.68) can serve as the lower limitof the outflow speed in the x-direction for the planets with m . 0.3.

2.4.2

Applicability of Bernoulli’s theorem

In Sect. 2.3.1, we estimated the outflow speed using Bernoulli’s theorem. The applicability of Bernoulli’s theorem depends on the thermodynamics of the gas alonga recycling streamline. In Eq. (2.42), we assumed that the enthalpy of the gas canbe described by c2s ln ρg , meaning that we considered the isothermal limit. However,
our hydrodynamical simulations utilized the β cooling model in which the radiative
cooling occurs on a finite timescale. Thus, the Bernoulli’s invariant (Bernoulli’s function), B, is not strictly conserved along the streamline. In this section, we discuss theapplication of Bernoulli’s theorem.
Let us begin with a qualitative discussion. When a parcel of the gas approaches
to the planet, the temperature of the parcel increases due to the adiabatic compression. The temperature of the gas relaxes toward the background temperature withthe dimensionless timescale β = (m/0.1)2 (Eq. (2.13)). Assuming a parcel of the gas
with the x-coordinate x ∼ Rgrav , the crossing time on which the gas crosses the gravitational sphere of the planet at shear velocity, ∼ −3/2Rgrav Ω, can be estimated by
the shear timescale, ∼ Ω−1 (Kuwahara and Kurokawa, 2020b). Thus, the assumption for the thermodynamics of the recycling flow depends on the ratio of the coolingtime to the dynamical timescale.
Figures 2.27–2.29 show the specific streamlines of the gas and the changes of the
Bernoulli’s function along a recycling streamline. The Bernoulli’s function fluctuates2.4. Discussion of this chapter

65

near the planet, which means that B is not strictly conserved along the recycling
streamline.

F IGURE 2.27: Streamlines of 3D planet-induced gas flow around theplanet with m = 0.05 and Mhw = 0.03. The solid lines are the specificstreamlines of the gas. We highlight a recycling streamline with thecolor contour. The color contour represents the Bernoulli’s function.
The black sphere denotes the isolated envelope whose size is 0.04 H.
The arrows represent the direction of the gas flow. The cooling timeis β = 0.25. Panels a and b: Perspective views of the flow field. Panel c:
x-y plane viewed from the +z-direction.

F IGURE 2.28: Same as 2.27, but with the results obtained fromm02-hw003. The cooling time is β = 4.

66

Chapter 2. Gas flow induced by low-mass planets

F IGURE 2.29: Same as 2.27, but with the results obtained fromm05-hw01. The cooling time is β = 25.

Figures 2.30–2.32 give us the more quantitative description of the Bernoulli’s
functions along the recycling streamlines shown in Figs. 2.27–2.29 as a function of
the y-coordinates. The overall trends in these figures are common regardless of theassumed planetary mass and the Mach number of the headwind of the gas.
We first focus on the low-mass regime, where the cooling time is shorter thanthe dynamical timescale (m . 0.1). We consider m = 0.05 and Mhw = 0.03, where
the cooling time is β = 0.25 (Fig. 2.27). The Bernoulli’s function along the recycling
streamline is almost identical to the background (unperturbed) value in the far filed,
B∞ , which can be described by,
B∞ =

v2g,∞

3 2
− x∞
− 2Mhw x∞ ,22

(2.70)

where we obtained x∞ (> 0) from the hydrodynamical simulation. In Eq. (2.70),
we assumed that the enthalpy and the z-component of the tidal potential canceledeach other out in the far field because of the hydrostatic equilibrium (Sect. 2.3.6). The
Bernoulli’s function fluctuates near the gravitational sphere of the planet (|y| . 0.05),
but its amplitude is small, ∆B = 0.036, because of the efficient cooling of the gas.
Since the cooling time depends on the planetary mass, one would expect thatthe amplitude of the fluctuation of the Bernoulli’s function becomes large as theplanetary mass increases. Figures 2.31 and 2.32 show the changes of the Bernoulli’s
function along the recycling streamline for higher-mass planets, m = 0.2 and 0.5.
Same as in Fig. 2.30, the Bernoulli function is almost identical to the unperturbed
value in the far field, but it deviates from the unperturbed value when |y| . Rgrav(Figs. 2.31b and 2.32b). As we expected, the amplitude of the fluctuation is larger
for the large planetary mass, ∆B = 0.25 (when m = 0.2) and 0.63 (when m = 0.5).
This may lead to a mismatch between our analytic formula of the outflow speed andthe numerical results in particular for a higher-mass planet (Fig. 2.22). Although
the Bernoulli’s function is not strictly conserved along the recycling streamline, we
conclude that Bernoulli’s theorem can be applicable to estimate the outflow speed aslong as the cooling time is comparable or shorter than the dynamical timescale.
It should be noted that the colored streamlines in Figs. 2.27–2.29 do not exactly2.4. Discussion of this chapter

67

pass through the inflow point assumed in Eq. (2.44), Pin , due to limitations of thevisualization. The Bernoulli’s invariants have different values for each streamline.
Thus, the values of B in Figs. 2.30–2.32 do not match with those of Bin (Eq. (2.45)).
Finally, we look at the changes in each term in Eq. (2.42) and summarize thetrends as below:- The contribution of the kinetic energy of the gas is negligible near the planet,
which is consistent with the previous studies (Fung et al., 2015; Kuwahara etal., 2019).
- The enthalpy increases near the gravitational sphere of the planet, |y| . Rgrav ,due to an increase in the gas density.
- The acceleration of the gas is caused by the release of the x-component of thetidal potential, −3/2x2 .

68

Chapter 2. Gas flow induced by low-mass planets

F IGURE 2.30: Quantities along the recycling streamline through point( x, y, z) = (0.001, 0.01, 0.02). The black solid and dotted lines represent the Bernoulli’s invariant, B, and that at the far field, B∞ =−0.027. The colored dashed lines correspond to the first to sixth terms
in the Bernoulli’s function (Eq. (2.42)). The purple dashed-dotted line
corresponds to the x-coordinate of the recycling streamline. This figure should be compared to Fig. 2.27. Panel a: Plot in the entire regionof the domain of hydrodynamical simulation. Panel b: Enlarged viewof the range where |y| ≤ RHill .

2.4. Discussion of this chapter

F IGURE 2.31: Same as 2.30, but the results obtained from m02-hw003.
We picked up the recycling streamline through point ( x, y, z) =(0.001, 0.01, 0.44). This figure should be compared to Fig. 2.28.

69

70

Chapter 2. Gas flow induced by low-mass planets

F IGURE 2.32: Same as 2.30, but the results obtained from m05-hw01.
We picked up the recycling streamline through point ( x, y, z) =(0.001, 0.01, 0.35). This figure should be compared to Fig. 2.29.

2.4.3

Angular momentum of the gas

Figures 2.30–2.32 show the changes of the x-coordinates of the gas along the recycling streamline, where the x-coordinates of the gas at the edges of the calculation
domain of the hydrodynamical simulation do not match each other. This implies
that the angular momentum of the gas increases after the gravitational interaction
with the planet.
To understand this phenomenon, we focus on the changes of the enthalpy along
the recycling streamline. The gas flows in at high latitudes of the gravitational sphere2.4. Discussion of this chapter

71of the planet at its horseshoe turn and exits through the midplane region of the disk.
The transient flow is vertically compressed. Thus, the enthalpy of the gas along therecycling streamline has a higher value than the unperturbed value at the midplane
region (c2s ln ρg (0) → 0). If B is strictly conserved along the streamline, an increase in
the enthalpy along the streamline must be offset by contributions from other terms
in Eq. (2.42). Here, this is accomplished by an increase in the x-component of thetidal potential, −3/2x2 , leading to an increase in the x-coordinate of the recycling
streamline (e.g., Fig. 2.31b).
We note that our hydrodynamical simulations were performed in the local domain. Thus, our results may need to be compared with the results obtained by globalhydrodynamical simulations in a future study.

2.4.4

Implications for the dust motion under the influence of the outflow

Simulations in previous studies and in this chapter have shown that a planet embedded in a protoplanetary disk induces outflow in the midplane region. The dust
scale height is given by (Youdin and Lithwick, 2007):

Hd =St 1 + 2St
1+
αdiff 1 + St

−1/2
H,

(2.71)

where St is the Stokes number (dimensionless stopping time of a particle) and αdiff
is the dimensionless turbulent parameter, which describes turbulent diffusion in ourmodel. Assuming St ∼ 10−3 –10−1 and αdiff ∼ 10−4 –10−2 , Eq. (2.71) gives Hd /H ∼10−2 –100 . Because the vertical scale of the outflow is estimated to be a few tensof percent of the Bondi or Hill radius of the planet, planets with a dimensionlessmass m = RBondi /H > 10−2 –100 have the potential to affect the motion of the dust
particles.
We compared the outflow speed to the terminal velocity of particles within theatmospheric radius to discuss the influence of the outflow on the motion of the dust
particles. Given the force balance between the gas drag and the gravity of the planet
acting on the particle,
GMpl
∆v
,
∼
tstop
R2

(2.72)

where ∆v is the terminal speed of the particle relative to the gas and tstop is the
stopping time expressed by tstop = St/Ω. We substituted R = 2Ratm,fit into Eq. (2.72)
and obtained
∆v =

m St
cs .
4 R2atm,fit(2.73)We plotted the changes of the relative velocity of the small particles as a function of
the dimensionless planetary mass for each Stokes number, St = 10−3 , 10−2 , and 10−1
in Fig. 2.33. As shown in this figure, the gas flow around a planet may affect the incoming small dust particles when the outflow speed exceeds the relative velocity ofthe particles.

72

Chapter 2. Gas flow induced by low-mass planets

F IGURE 2.33: Same as 2.22, but with the terminal velocity of pebbles(Eq. (2.73); blue dashed-dotted lines). Different colors correspond todifferent Stokes numbers. The vertical axis is on a log scale.

Here we derive the upper limit of the Stokes number that affected by the outflow by the order-of-magnitude estimate. For simplicity, we neglect the effect of theheadwind of the gas and only consider x ≥ 0. From Eq. (2.73), the outflow affects
the motion of dust particles when√
8 3 R3atm,fit
2Ratm,fit+
St .
.
(2.74)
cos ϕout cos+mwout
Since the cosine term in Eq. (2.74) has on the order of O(10−1 ), when m . 0.1 ( Ratm,fit =C1 × RBomdi ), we obtain√
St . 8 3 C13 m2 × [cos term] ∼ m2 ,(2.75)
and when m & 0.1 ( Ratm,fit = C2 × RHill ),
√
8 3 3
St .
C2 × [cos term] ∼ 0.1,
3(2.76)

where C1 = 0.7–1, C2 = 0.3–0.4. The outflow could affect the motion of dust particles when St . min(m2 , 0.1).
The above findings suggest that the outflow could inhibit the accretion of dust
particles onto a planet and may√ delay the planetary growth. The growth of protoplanets may halt when m ∼ St due to the outflow barrier, which gives m ∼ 0.1

2.4. Discussion of this chapter

73

when we assume St ∼ 10−2 . This value could be comparable or smaller than the pebble isolation mass, miso & 0.6 (Eq. (2.57)). The dense and hot envelope in the vicinityof the planet may also induce the disruption and vapourization of solid particles,
which would further prevent accretion onto the planet (Alibert, 2017), though theinner part of the envelope is isolated from the recycling flow (Kurokawa and Tanigawa, 2018). Furthermore, even the particles with a relatively large Stokes number—for instance St > min(m2 , 0.1)—may also be affected by the outflow within the gravitational sphere of the planet. Since the Stokes number is inversely proportional to
the gas density both in the Epstein and Quadratic regimes (e.g., Ormel and Klahr,2010), the effective Stokes number is considered to decrease inside the envelope
where gas density is much higher than that of the background. In such cases, theparticles become more susceptible to the outflow barrier.
Our results also suggest that the outflow of the gas would inhibit the radial driftof dust in a disk. The dust particles drift from the outer to the inner region of thedisk (Whipple, 1972; Weidenschilling, 1977a). These particles could be trapped by
the outflow of the gas toward the outside of the planetary orbit
In this chapter, we restrict our discussion to the order-of-magnitude estimate. In
reality, the growth rate of the planet depends on the accretion rate of particles onto
the planet, and hence depends on the motion of particles influenced by the planetinduced gas flow. How the radial distribution of dust particles in a disk change
under the influence of the planet-induced gas flow remain unclear. Further studiesof the interaction between the planet-induced flow and dust particles are needed.
Thus, in Chapter 3, we investigate the influence of the planet-induced gas flow on
the motion of dust particles. In Chapter 4, we investigate the influence of the planetinduced gas flow on the spatial distribution of dust in a disk.

75

Chapter 3

Dust motion influenced by theplanet-induced gas flow
This chapter is based on
Ayumu Kuwahara & Hiroyuki Kurokawa
Astronomy and Astrophysics 633, A81 (2020a)and
Ayumu Kuwahara & Hiroyuki Kurokawa
Astronomy and Astrophysics 643, A21 (2020b)

3.1

Introduction of this chapter

Planets form in protoplanetary disks. In the last decade, the pebble accretion model
has been studied by many authors, where protocores accrete mm- to cm-sized dust
particles (pebbles; Ormel and Klahr, 2010; Lambrechts and Johansen, 2012; Ormel,
2017; Johansen and Lambrechts, 2017). Accreting pebbles onto the protocores canform cores massive enough to trigger gas giant formation within the lifetime of gas
disks (Lambrechts and Johansen, 2012). A key parameter to control the outcome ofthe planet formation models is the accretion rate of particles onto a planet, which
can be estimated by investigating the motion of the particles relative to the planet(e.g., Ormel and Klahr, 2010; Visser and Ormel, 2016). Small dust particles not only
experience the gravitational interaction with the growing planet but also a gas dragforce from the surrounding disk gas. Since smaller dust particles are more sensitive
to the gas flow, it is crucial to investigate the influence of the gas flow on the motionof mm–cm-sized small particles.
So far the conventional pebble accretion model has only been discussed in theunperturbed gas flow (e.g., Ormel and Klahr, 2010; Lambrechts and Johansen, 2012;
Ormel and Kobayashi, 2012; Lambrechts and Johansen, 2014; Lambrechts et al., 2014;
Guillot et al., 2014; Sato et al., 2016; Ida et al., 2016; Ormel, 2017; Johansen and Lambrechts, 2017; Ormel and Liu, 2018; Liu and Ormel, 2018). Recent hydrodynamical
simulations have revealed that a planet embedded in a protoplanetary disk induces
gas flow with a complex three-dimensional (3D) structure (Chapter 2; Ormel et al.,2015b; Fung et al., 2015; Fung et al., 2019; Lambrechts and Lega, 2017; Cimerman
et al., 2017; Kurokawa and Tanigawa, 2018; Kuwahara et al., 2019; Béthune andRafikov, 2019; Moldenhauer et al., 2021; Moldenhauer et al., 2022). The anteriorposterior horseshoe flows extending in the orbital direction of the planet have acharacteristic vertical structure that resembles a column. A substantial amount ofgas from the disk enters the gravitational sphere of the planet (inflow) and exits(outflow), causing atmospheric recycling.

76

Chapter 3. Dust motion influenced by the planet-induced gas flow

The planet-induced gas flow could affect the motion of the dust particles andmay alter the accretion rate of them onto a planet. The accretion rate of small particles (∼ 0.1–1 mm) is reduced in the 2D planet-induced flow (Ormel, 2013). The motion of particles in the 3D planet-induced gas flow is shown to be more complicated.
Popovas et al., 2018 and Popovas et al., 2019 incorporated particles into their hydrodynamical simulations and found that small particles (0.01 mm–1 cm) move awayfrom the planet in the horseshoe flow, effectively avoiding accretion onto Earth- andMars-mass planets. In Chapter 2, we showed that the speed of midplane outflow
could exceed the terminal speed of incoming dust. This suggests that the outflow
may inhibit accretion.
Though these studies suggest the importance of considering the gas flow induced by a planet, a comprehensive study investigating a range of planetary masses
and pebble sizes is missing. In this chapter, to investigate the influence of the planetinduced gas flow on the motion of dust particles, we perform hydrodynamical simulations of the gas flow around an embedded planet and compute the dust trajectories.
In Sect. 3.2 we describe our numerical methods. In Sect. 3.3, we show the results
obtained from a series of orbital calculations. In Sect. 3.4, we discuss the accretion
rate of dust onto a planet.

3.2. Method

3.2

Method3.2.1

Model overview

77

In this section, we provide an overview of our model. Our approach is summarized in Fig. 3.1. First, assuming a compressible, inviscid sub-Keplerian gas disk
with the vertical stratification due to the vertical component of the stellar gravity,
we performed 3D nonisothermal hydrodynamical simulations (see Chapter 2). Hydrodynamical simulations were performed in the frame co-rotating with the planet(the black sphere in Fig. 2.1). Next, we calculated the 3D trajectories of dust particles
influenced by the planet-induced gas flow. We numerically integrated the equationof motion of dust using the hydrodynamical simulation data. 3D orbital calculations
were performed in the frame co-rotating with the planet (Fig. 3.1). Throughout allof our simulations, the length, times, velocities, and densities are normalized by thedisk scale height, H, reciprocal of the orbital frequency, Ω−1 , sound speed cs , andunperturbed gas density at the location of the planet, ρdisk , respectively (see Sect.
2.2.2; Chapter 2).

78

Chapter 3. Dust motion influenced by the planet-induced gas flow

F IGURE 3.1: Schematic picture of an orbital calculation of dust particles. We first performed 3D hydrodynamical simulations on thespherical polar grid (Chapter 2). We then calculated the trajectories
of particles influenced by the planet-induced gas flow in the frame
co-rotating with the planet. The dashed circle represents the limited part of the calculation domain of hydrodynamical simulations,
Rhydro . The starting point of orbital calculation is beyond Rhydro . The
gas velocity is assumed to be the sub-Keplerian shear both inside andoutside of Rhydro in the unperturbed flow case, but is switched to thegas velocity obtained from the hydrodynamical simulations within
Rhydro in the planet-induced gas flow case.

3.2.2

Three-dimensional hydrodynamical simulations

We performed nonisothermal, inviscid 3D hydrodynamical simulations of the gasof the protoplanetary disk around an embedded planet (described in Sect. 2.2). The
obtained gas flow data were used to simulate the orbits of dust particles influenced
by the planet-induced gas flow. We listed our parameter sets used in this Chapter(simulations with an asterisk in Table 2.1).

3.2. Method

3.2.3

79

Three-dimensional orbital calculation of dust particles

We calculated the trajectories of dust particles influenced by the planet-induced gas
flow in the frame co-rotating with the planet (Fig. 3.1). The x-, y- and z-directions
are similar to those used in the hydrodynamical simulations. The dimensionlessequation of motion of a particle is described by 2vy,d + 3x
x
dvd
m  


=
− 3 y + Fdrag ,(3.1)−2v x,ddtRz0where vd = (v x,d , vy,d , vz,d ) is velocity of the dust and ( x, y, z) is the coordinates ofthe dust (Ormel and Klahr, 2010; Visser and Ormel, 2016). The first and second termon the right-hand side of Eq. (3.1) are the Coriolis and tidal forces and the two-bodyinteraction force with the embedded planet. The third term is the gas drag force
acting on the dust expressed by
Fdrag = −vd − vg
,
St

(3.2)

where St is the dimensionless stopping time of a dust particle, called the Stokes
number,
St = tstop Ω.

(3.3)

We assume St = 10−4 –100 . The relation between the Stokes numbers and dimensional particle sizes for a steady viscous-accretion disk model is shown in Fig. 3.2(see also Sect. 2.2.7). The parameter range in this chapter is St ∼ 10−4 –100 , whichcorresponds to the dust particles of ∼ 0.1 mm–1 m at ∼ 1–10 au for the steadystate viscous-accretion disk (Fig. 3.2). The gas velocity is assumed to be the speedof the sub-Keplerian shear outside the limited part of the calculation domain of hydrodynamical simulations, Rhydro (which is defined later), but is switched to the gas
velocity obtained from the hydrodynamical simulations within Rhydro in the planetinduced gas flow case.

80

Chapter 3. Dust motion influenced by the planet-induced gas flow

F IGURE 3.2: Relation between the Stokes number and particle size.
Different colors correspond to different Stokes numbers. The grayshaded region corresponds to the Stokes gas drag regime.

We omitted the dimensionless vertical component of the tidal force, −zez in theequation of motion of dust (Eq. (3.1)). Through the test integrations with this term,
we found that most dust particles settled in the midplane region of the disk before
they enter the calculation domain of hydrodynamical simulations when St & 0.1. In
reality, turbulent stirring would balance the vertical component of the tide on statistical average. The sources of this turbulence will be discussed in Sect. 5.1.1. Though
our hydrodynamical simulations do not resolve the turbulence, we assumed that the
turbulent and tidal forces in the vertical direction canceled each other out. Considering the effect of random motion due to the turbulence (Xu et al., 2017; Picogna et al.,2018) is beyond the scope of this chapter.
In a steady accretion disk, the gas surface density is given by
Σg =

Ṁ∗
.
3παacc r2 h2 Ω

(3.4)

The stopping time of the dust particle depends on the thermal structure of the disk.
Thus, the Stokes number is given by (Fig. 3.2; Ida et al., 2016)√
2πρ• s9,(Epstein regime : s ≤ λmfp )
(3.5)
 Σ
4
g
St =

9
 4ρ• σmol s2


(3.6)
 9µm h , (Stokes regime : s ≥ 4 λmfp ),H

3.2. Method

81

In the region where viscous heating dominates, the Stokes number is described by(Ida et al., 2016):

St '







!−1/5
!4/5
!−3/5



αacc
Ṁ∗
M∗

−
3


 1.3 × 10

1M
10−3
10−8 M /yr










!
!3/5
!



s
r
ρ•



×
,


1 au
1cm
1 g cm−3






(Epstein regime),






(3.7)






!1/10
!−1/5
!7/20



M∗
Ṁ ∗
αacc

−4

5.6 × 10



1M
10−3
10−8 M /yr










!
!−21/20
!2



r
s
ρ•



×
,


1 au
1cm
1 g cm−3






(Stokes regime).






In the region where stellar irradiation dominates, the Stokes number is described by

St '



































0.93 × 10−3




















6.3 × 10−4













L∗
1L

×

!2/7

M∗
1M

r
1 au

!15/14!−9/14

αacc
10−3

ρ•
1 g cm−3

!

!

Ṁ ∗
10−8 M /yr

! −1

!
s
,
1cm
(3.8)

(Epstein regime),

L∗
1L

!−1/7

M∗
1M

!4/7

r
1 au

!−9/7

ρ•
1 g cm−3

!

s
1cm

!2
,

(Stokes regime).

Since the mean free path of the gas is proportional to the reciprocal of the gas density,
λmfp ∝ ρg−1 , the stopping time in the Stokes regime is independent of the gas density.
In our model, the Stokes number is independent of the local gas density in the Stokes
regime, whereas it is proportional to ρg−1 in the Epstein regime. The gas density
increases significantly in the vicinity of the planet, then the effective Stokes number
decreases as the gas density increases. The threshold value of St for the transition
varies as a function of the distance from the star.
In our co-rotating frame, the x- and y-components of the initial velocity of dust
particles are given by the drift equations (Weidenschilling, 1977b; Nakagawa et al.,82

Chapter 3. Dust motion influenced by the planet-induced gas flow

1986, Eqs. (1.15) and (1.16)):2St
Mhw ≡ vdrift ,1 + St2
Mhw
3
v∞
− x,
y,d = −
2
2
1 + Stv∞
x,d = −

(3.9)
(3.10)

We integrated Eq. (3.1) using a fifth-order Runge-Kutta-Fehlberg method (Fehlberg,
1969; Shampine et al., 1976; Es-Hagh, 2005). In this scheme each time step is controlled automatically depending on the relative acceleration of the dust to the planet.
The maximum relative error tolerance was set to 10−8 , which ensures numerical convergence (Ormel and Klahr, 2010; Visser and Ormel, 2016). We fixed the y-coordinateof the starting point of dust at |ys | = 40RHill (Ida and Nakazawa, 1989). The x- andz-coordinates of the starting point of dust, xs and zs , are the parameters. We terminated orbital calculation when either one of the following conditions was met: (1)−1the calculation time reached the sufficiently long time of computation, 104 ΩK, (2)a particle reached the edge of the calculation domain, |y| ≥ 40RHill , or (3) accreted
onto the planet, R ≤ 2Rinn .
When m = 0.03 and 0.1, we found that the horseshoe flow formed unexpected
vortices, which influences the dust trajectories (e.g., Fig. 2.8; Chapter 2). The originof these vortices is unknown, but it is likely to be a numerical artifact due to thespherical polar coordinates centered at the planet, in which the resolution becomes
too low to resolve the horseshoe flow far from the planet when the assumed planet
mass is small. Only in these cases we use the limited part of the calculation domain,
Rhydro = 0.6Rout , to avoid the effects of the vortices (Table 2.1).

3.2.4

Calculation of accretion rate of dust

Width of accretion window and accretion cross section
We defined the width of the accretion window as:
wacc (z) = ∑ ( xmax,i (z) − xmin,i (z)) ,(3.11)iwhere xmax,i (z) and xmin,i (z) are the maximum and minimum values of the x-componentof the starting point of accreted dust particles at a certain height. The subscript, i,denotes the number of the accretion window. When we include the headwind of thegas, the accretion of dust occurs asymmetrically with respect to the corotation radius
for the dust particle. In the unperturbed flow, the width of the accretion window is
identical to the maximum impact parameter of accreted dust particles, which is expressed by (Ormel and Klahr, 2010; Lambrechts and Johansen, 2012; Guillot et al.,2014; Ida et al., 2016; Sato et al., 2016):"   #
St 0.65b∞ = b0 exp −,(3.12)
2wheresb0 ' min 2

mSt, 2St1/3 RHill
Mhw

!
.

(3.13)

3.2. Method

83

Using this definition, we defined the accretion cross-section as
Aacc = ∑

Z z

max,i

Z xmax,i (z)zmin,ii

xmin,i (z)dxdz ,(3.14)

where zmax,i and zmin,i are the maximum and the minimum value of the z-componentof the starting point of accreted dust particles. We reduced the spatial intervals stepwise near the edge of the accretion window and determined xmax,i (z) and xmin,i (z)with sufficient accuracy. The initial spatial intervals are 10−2 × b∞ in the x- and zdirections.
Accretion rate
The accretion rate of dust, the sum of dust mass flux through the accretion crosssection, is calculated by: Z z
Z xmax,i (z)max,iṀacc = ∑ 2
ρd (z)vy,∞ dxdz ,izmin,ixmin,i (z)(3.15)

where ρd is the density distribution of dust
Σd

"

1
exp −
ρd ( z ) = √
2
2πHd



z
Hd

2 #
.

(3.16)

Owing to the symmetry of the system, we only consider the region where z ≥ 0 andmultiply Eq. (3.15) by two. The dust surface density, Σd , is given by
Σd =

Ṁd,∞
,
2πr |vdrift |(3.17)

where Ṁd,∞ is the radial inward mass flux of dust in the gas disk. We fixed theinward dust mass flux as Ṁd,∞ = 102 M⊕ /Myr, which is consistent with the typical value of the dust flux used in a previous study (Lambrechts et al., 2019). The
existence of massive dust reservoirs (∼ 102 M⊕ ) in the outer regions of the young
disks has been suggested by (sub)millimeter dust observations (Ricci et al., 2010;
Andrews et al., 2013; Ansdell et al., 2016). The dust mass is correlated with theaccretion rate onto young stars. Given a typical T Tauri star with a disk accretion
rate 10−8 M /Myr, the dust mass is ∼ 102 M⊕ (Manara et al., 2016). The dust massflux can be analytically estimated as ∼ 100–300 M⊕ /Myr when we assume the disk
lifetime is ∼ 1–3 Myr (Lambrechts and Johansen, 2014).

84

Chapter 3. Dust motion influenced by the planet-induced gas flow

3.3

Results

3.3.1

Results overview

The main subject of this chapter is to clarify the influence of the planet-induced gas
flow on the motion of dust particles. In Sect. 3.3.2, we show the results of orbital
calculations.

3.3.2

Orbital calculations

Trajectories of dust particles in 2D
We first focus on the dust trajectories at the midplane region of the disk. Let us consider an endmember case, Mhw = 0, and the Stokes gas drag regime. In this case,the planet-induced gas flow is in the flow-shear regime and the Stokes number ofdust does not depend on the gas density. Figure 3.3 shows the trajectories of dustat the midplane. We assumed m = 0.3 in this figure. The width of the accretion
window is narrower in the planet-induced gas flow than in the unperturbed flow, inparticular for the smaller Stokes numbers, St . 10−1 (Figs. 3.3d and e). In the unperturbed flow, the dust particles that successfully accrete onto the planet originatedin the vicinity of the planetary orbit, x ∼ 0 (e.g., Figs. 3.3a and b). In the planetinduced gas flow, however, most dust particles coming from this region move awayfrom the planet along the horseshoe flows (Figs. 3.3d and e). The dust coming froma window between the horseshoe and the shear regions can accrete onto the planet.
In Chapter 2, we predicted that the motion of dust could be affected by the outflow of the gas when St . min(m2 , 10−1 ) (Eq. (2.74)). As shown in Fig. 3.3, theoutflow of the gas deflects the trajectories of dust significantly. Thus, even when
the dust particles enter the gravitational sphere of the planet, the accretion of dust
particles onto a planet is suppressed due to the outflow barrier.
We found voids in the upper left and lower right regions of the planet when dustis small (St . 10−2 –10−1 ; Figs. 3.3d and e). The midplane outflow and resulting
voids in dust trajectories are not found in 2D simulations (Ormel, 2013, their Fig.
12), but appear in the 3D ones. This is because the recycling streamlines do notappear in 2D simulation. The outflow of the gas occurs in a local region close to theplanet, but it could affect the global distribution of dust in disks. The outflow of
the gas occurs in the radial direction to the disk (x-direction in Fig. 3.3). Once theoutflow disrupts the motion of the small dust particles in disks, the inward drift ofdust could be inhibited, leading to the changes in the dust surface density, Σd . Thismechanism is studied later in Chapter 4.
For the large Stokes number, St = 1, we obtained similar results both in the unperturbed and in the planet-induced gas flow (Figs. 3.3c and f). When the Stokes
number is large, dust is loosely coupled with the gas. Though the disk gas is perturbed by the planet, its influence on the larger dust is weak.
We confirmed that the shape of trajectories of dust particles did not differ significantly from that in the Epstein gas drag regime. However, the accretion rate in the
Epstein gas drag regime did not match with those in the Stokes gas drag regime. We
discuss the influence of the gas drag regime later in Sect. 3.4.1.

3.3. Results

85

F IGURE 3.3: Trajectories of dust particles at the midplane in the unperturbed gas flow (upper right) and in the planet-induced gas flow
(lower right) with different Stokes numbers. The orange and blue
solid lines correspond to the trajectories of dust particles which accreted and did not accrete onto the planet, respectively. The twoheaded orange arrow indicates the width of the accretion window at
the midplane, wacc (0). The planet-induced gas flow around a planet
is obtained from m03-hw0 (lower left). The color contour in the lower
left panel represents the flow speed in the x-direction. We set zs = 0
for the orbital calculation. The dashed-dotted and dashed lines correspond to the Bondi and Hill radii of the planet, respectively. Upper left panel shows the schematic illustration of the structure of theplanet-induced gas flow and the relative motion of dust.

We continue to focus on the dust trajectories at the midplane region of the disk,
but include the headwind of the gas. Let us consider an endmember case, Mhw =0.1. In this case, the planet-induced gas flow is in the flow-headwind regime. Figure
3.4 shows the trajectories of dust at the midplane. We assumed m = 0.03 in thisfigure.
When the Stokes number is smaller than St ≤ 10−2 , the trajectories of dust
near the planetary orbit are deflected by the recycling flow (Figs. 3.4d–f). Our
previous prediction—the outflow of the gas would affect the motion of dust when
St . min(m2 , 10−1 ) (Eq. (2.74))—is also confirmed in the flow-headwind regime.
Similar to the flow-shear regime, the deflection of the trajectories of dust in the flowheadwind regime could affect the dust surface density, which is investigated later in
Chapter 4.
We found that the width of the accretion window is wider in the planet-induced
gas flow than in the unperturbed flow when St = 10−4 , the smallest Stokes number
considered here (Fig. 3.4d). This is in contrast to the trend in Fig. 3.3, where the widthof the accretion window in the planet-induced gas flow in the flow-shear regime is

86

Chapter 3. Dust motion influenced by the planet-induced gas flow

narrower than those of the unperturbed shear flow for the smaller Stokes numbers.
The difference is caused by the headwind of the gas. Some of the dust from inside
the planetary orbit move away from the planet along the horseshoe flow. In theflow-shear regime, the horseshoe flow that lies near the planetary orbit reduces thewidth of the accretion window (Fig. 3.3). On the other hand, in the flow headwind
regime, since the horseshoe flow shifts significantly to the negative direction in thex-axis, it does not suppress the accretion of dust particles. The outflow occurs onlyin the fourth quadrant of the x-y plane. The accretion of dust particles coming fromthe region where y > 0 is not inhibited. Dust is susceptible to becoming entangled in
the recycling flow. This leads to an increase in the encounter time of dust particles,
and thus an increase in the width of the accretion window.
We confirmed that the trajectories of dust and the width of the accretion window are identical to those in the unperturbed flow when St ≥ 10−1 . The shapeof trajectories of dust particles did not differ significantly from that in the Epsteingas drag regime. However, the accretion rate in the Epstein gas drag regime differs
significantly from that in the Stokes regime and unperturbed flow. We discuss theinfluence of the gas drag regime later in Sect. 3.4.1.

F IGURE 3.4: Same as Fig. 3.3, but with the flow structure obtained
from m003-hw01.

Trajectories of dust particles in 3D
Next we focus on the 3D behavior of dust trajectories. Let us consider an endmember
case, Mhw = 0, and the Stokes gas drag regime. Figure 3.5 shows the 3D trajectoriesof dust with St = 10−2 influenced by the planet-induced gas flow. Similar to the 2D
case, dust coming from the vicinity of planetary orbit moves away from the planet
and do not accrete onto it, because the horseshoe streamlines extend in the verticaldirection keeping its configurations. In the 3D case, the dust coming from the narrow3.3. Results

87

region between the horseshoe and the shear regions sharply descends toward theplanet with terminal velocity when they approach the planet from the polar region.
This downward motion is due to the gravity of the planet and the advection due
to the gas inflow (e.g., Fig. 2.17). Disk gas enters the gravitational sphere from thepolar region. The inflow has the velocity component of −z-direction, which causes
rapid changes in the motion of the particle. The outflow of the gas also deflects the
trajectories of dust and makes the voids in the second and fourth quadrants of x-y
plane because the outflows extend in the vertical direction (e.g., Fig. 2.13; Chapter
2).
Similar to the 2D case, the shape of trajectories of dust did not differ significantly
from that in the Epstein gas drag regime.

88

Chapter 3. Dust motion influenced by the planet-induced gas flow

F IGURE 3.5: Trajectories of dust with St = 10−2 influenced by theplanet-induced gas flow. We set zs = 0.2 H for the orbital calculation.
We used the results obtained from m03-hw0 to calculate dust trajectories. Different colors correspond to different xs . The dashed-dottedand dashed lines correspond to the Bondi and Hill radii of the planet,
respectively. Panel a: x-y plane viewed from the +z direction. Panel b:
x-z plane viewed from the y direction. Panel c: y-z plane viewed fromthe + x direction.

3.3. Results

89

We continue to focus on 3D behavior of dust trajectories, but include the headwind of the gas (Mhw = 0.1). Figure 3.6 shows the 3D trajectories of dust with
St = 10−4 influenced by the planet-induced gas flow, where the planet-induced gas
flow is in the flow-headwind regime. In this case, the inflow of the gas originatedfrom the horseshoe flow at high altitudes no longer occur. However, even when dust
particles come from high altitudes, zs ∼ RBondi , they accrete onto the planet (Fig. 3.6).
This is caused by the recycling flow in the vicinity of the planet. The typical scaleof the recycling flow is the size of the atmosphere (Ratm ∼ RBondi for the assumed
parameter set in Fig. 3.6). In the flow-shear regime, since the horseshoe flow has avertical structure like a column, dust coming from high altitudes moves away fromthe planet near the planetary orbit. In the flow-headwind regime, the horseshoe flow
does not inhibit the accretion of dust particles. When dust that comes from high altitudes reach the vicinity of the planet, a fraction of those that reside in z . RBondi
are entangled in the recycling flow. This causes an increase in the encounter time fordust to cross the Bondi radius. Thus, dust particles can accrete onto the planet evenwhen they come from high altitudes.

90

Chapter 3. Dust motion influenced by the planet-induced gas flow

F IGURE 3.6: Same as Fig. 3.5, but with the flow structure obtained
from m003-hw01 and with St = 10−4 . We set zs = 0.03 H for theorbital calculation.

3.4. Discussion of this chapter

3.4

Discussion of this chapter

3.4.1

Accretion rate of dust onto a planet

91

So far we only considered the motion of dust particles influenced by the planetinduced gas flow. We found that the trajectories of dust in the planet-induced gas
flow differ significantly from those in the unperturbed flow for a wide range of investigated dust sizes (Sect. 3.3). The changes in the trajectories would lead to thechanges in the accretion rate of dust particles onto a planet, and thus affect growthof the planet.
Figure 3.7 summarizes the accretion rate of dust particles onto a planet for a
fixed turbulent parameter, αdiff = 10−3 , obtained from a series of orbital calculations
in our study. We first consider the accretion rate in the flow-shear regime and focus
on the Stokes gas drag regime (solid lines in Fig. 3.7a). Since the influence of theplanet-induced gas flow is weaker for the large Stokes number, the accretion rate ofdust particles in the planet-induced gas flow is identical to that in the unperturbed
flow when St & 10−1 . When St . 10−1 , the accretion rate influenced by the planetinduced gas flow deviates from that in the unperturbed flow.
For a higher-mass planet (m = 0.3; the blue solid line in Fig. 3.7a), the obtained
result is counterintuitive. The width of the accretion window decreases significantlyin the planet-induced gas flow when St . 10−1 (Fig. 3.3d–f), but the accretion rate in
the planet-induced gas flow either match or is slightly larger than that in the unperturbed flow. This phenomenon can be understood as follows; in the unperturbed gas
flow, the accretion window lies near the planetary orbit, where the relative velocity∞ | ∝ |x|
between dust and the planet is low because the velocity is proportional to |vy,d(Eq. (3.10)). In the planet-induced gas flow, the accretion window outside (inside)
the planetary orbit shifted to the positive (negative) direction in the x-axis (Fig. 3.3),
where the relative velocity is large. The reduction of the accretion cross section andthe increase of relative velocity cancel each other out. This leads to the accretion ratein the planet-induced gas flow case being almost equal to that in the unperturbed
flow.
The above explanations shall not apply to a low-mass planet (m = 0.03; the redsolid line in Fig. 3.7a), where the accretion rate for the smaller dust influenced by theplanet-induced gas flow is smaller than that in the unperturbed flow. Because thehorseshoe width is proportional to the planetary mass (Eq. (2.27); Masset et al., 2006;
Paardekooper and Papaloizou, 2009; Masset and Benítez-Llambay, 2016; Jiménez
and Masset, 2017), the shifting rate of the accretion window decreases as the planetary mass decreases. As a consequence, when the planetary mass is small, m = 0.03,
the accretion window outside (inside) the planetary orbit does not shift to the positive (negative) direction in the x-axis enough to cancel its reduction.
When we adopt the Epstein gas drag regime, the reduction of the width of theaccretion window is more significant than that in the Stokes gas drag case for thesmaller dust (the dashed lines in Fig. 3.7). The gas density increases significantlyin the vicinity of the planet, R . Rgrav , then the effective Stokes number decreases
as the gas density increases, leading to a significant reduction in the width of theaccretion window. The increase of the relative velocities of dust does not fully offset
the significant reduction of the width of the accretion window. Thus, the accretion
rate in the Epstein gas drag regime is smaller than that in the unperturbed and Stokes
gas drag cases regardless of assumed Stokes numbers and planetary masses (thedashed lines in Fig. 3.7).

92

Chapter 3. Dust motion influenced by the planet-induced gas flow

Next, we consider the accretion rate in the flow-headwind regime (Fig. 3.7b).
Similar to the flow-shear regime, the accretion rate of dust in the planet-induced gas
flow is identical to that in the unperturbed flow when the Stokes number is large,
St & 10−2 , regardless of the assumed gas drag regime, because the influences of theplanet-induced gas flow is weaker for the large Stokes number. When we adopt the
Stokes gas drag regime, we found that the accretion rate influenced by the planetinduced gas flow exceeds that in the unperturbed flow when St . 10−2 (the solidline in Fig. 3.7b). The explanation in Figs. 3.4 and 3.6 can be applied to this figure.
The location of the accretion window does not change in the planet-induced gas flowin the flow-headwind regime, but the width of the accretion window is larger in theplanet-induced gas flow than that in the unperturbed flow (e.g., Fig. 3.4). Thus,
Fig. 3.7b reflects the results of Fig. 3.4.
When we adopt the Epstein gas drag regime, the effective Stokes number decreases significantly in the vicinity of the planet, R < Rgrav . The dust particles tend
to follow the recycling flow and past the planet. Thus, the accretion rate in the Epstein gas drag regime is smaller than that in the unperturbed and Stokes gas dragcases.

3.4. Discussion of this chapter

93

F IGURE 3.7: Accretion rate as a function of the Stokes number inthe unperturbed flow (dotted lines), planet-induced gas flow (Stokes
gas drag regime; solid lines), and the planet-induced gas flow (Epstein gas drag regime; dashed lines), respectively. Different colors
correspond to different planetary masses and Mach numbers. Panela: the planet-induced gas flow is in the flow-shear regime. Panel b:
the planet-induced gas flow is in the flow-headwind regime. We set
αdiff = 10−3 . Panel a: we did not calculate the accretion rate for
the Stokes number with St < 10−3 , thus each line is truncated at
St = 10−3 .

Figure 3.8 shows the dependence on the turbulent parameter for fixed planetary
mass and Mach number, m = 0.03 and Mhw = 0. The dust scale height is smaller
for weak turbulence for a fixed Stokes number (Eq. (3.16)), leading to an increasein the accretion rate of dust. On the other hand, the influence of the outflow of the

94

Chapter 3. Dust motion influenced by the planet-induced gas flow

gas on the accretion rate is stronger for a small αdiff , in which case the dust settles in
the midplane of the disk. This is because the outflow has the maximum speed near
the midplane region of the disk (e.g., Fig. 2.13). Suppression of the accretion of dust
particles due to the planet-induced gas flow becomes prominent for smaller Stokes
numbers and smaller turbulence parameters.

F IGURE 3.8: Same as Fig. 3.7, but this figure shows the dependenceof the accretion rate influenced by the planet-induced gas flow on
the turbulent parameter, αdiff . Different colors correspond to different
turbulent parameter. We assume m = 0.03 and Mhw = 0.

3.4.2

Comparison to previous studies

Assuming a case with 2D and inviscid fluid, Ormel, 2013 derived the steady state solution of 2D flow around an embedded planet. The author calculated the trajectories
of small particles using the derived flow pattern. Ormel, 2013 showed the trajectories of dust with St = 10−4 , 10−3 , and 10−2 around the planet, with m = 0.01 (Fig.12of Ormel, 2013). Two cases are shown: Mhw = 0 and Mhw = 0.05.
The former case corresponds to the flow-shear regime. Though the recycling
flow does not appear in 2D case, the author reported that the width of the accretion
window is smaller for the small Stokes number. The dust particles with small Stokes
numbers tend to follow the gas streamlines. The dust comes from the narrow region
between the horseshoe and the shear regions. These results are in agreement withours. The latter case corresponds to the flow-headwid regime. Thus, the accretion of3.4. Discussion of this chapter

95

dust is expected to be promoted based on our results. However, accretion of small
dust particles is suppressed in Ormel, 2013. A plausible reason is that the size of theatmosphere is different in 2D and 3D. Ormel, 2013 found that the averaged size ofthe atmosphere in 2D is ∼ 2.5 RBondi when m = 0.01 and Mhw = 0.05. The small
dust particles follow the gas streamlines outside the planetary atmosphere, and theypassed the planet without breaking through the atmosphere. In our study, the sizeof an isolated envelope is ∼ RBondi (Fig. 2.6). In 3D, the small size of an isolated
envelope allows dust to approach close to the planet. The extension of the Bondi
crossing time of dust due to the recycling flow further promotes the accretion ofdust particles.
Popovas et al., 2018 investigated the accretion of dust particles onto the Mars- to
Earth-mass planets (m = 0.008, 0.04, and 0.08) in the planet-induced gas flow in the
Epstein regime. They performed 3D nonisothermal simulations in the local frame
co-rotating with the planet. After the sufficient relaxation of the fluid motion anddensity distribution of the gas, they injected millions of tracer particles with different
sizes (St = 3 × 10−5 –3). The initial stratification of particles is the same as that of thegas. They measured the accretion rates of a sub-population of particles that initiallyresided within the Hill sphere. They have reported that the accretion rate increases
as the planetary mass increases, and it scales linearly with dust size regardless of
the assumed planetary mass when St = 3 × 10−5 –3 × 10−1 . In their subsequent
study, they concluded that the convective motion within the envelope caused by
the accretion heating does not significantly affect the accretion rate (Popovas et al.,2019). Similar to their results, the accretion rates increase with the planetary mass
(Fig. 3.7). In our study, however, the slope of the accretion rate changes significantly
for the assumed planetary mass and turbulent parameter. The accretion rate does notscale linearly over the same dust size range as Popovas et al., 2018. The maximum
accretion rate in Popovas et al., 2018 is Ṁacc ∼ 102 M⊕ /Myr when m = 0.08 andSt ∼ 1–3, but the achieved accretion rate in our study is smaller by one or two ordersof magnitude even for a larger planet, m = 0.3 ( Ṁacc ∼ 3–30 M⊕ /Myr; Fig. 3.7).
The higher accretion rate obtained in Popovas et al., 2018 may be the result of theabsence of turbulent stirring. In contrast to our study, Popovas et al., 2018 consideredthe vertical component of the tidal force, −zΩez , so that the particles settle to the
midplane with the settling timescale, (StΩ)−1 , which may lead to a high accretion
rate for the larger dust.

3.4.3

Caveat: the uniform distribution of dust particles

In this chapter, assuming an uniform dust surface density, we discussed the accretion rate of dust particles onto the planet. However, this assumption would not bevalid under the influence of the planet-induced gas flow. In Sect. 3.3.2, we foundvoids in dust trajectories, which were caused by the outflow of the gas. The outflow toward the outside (inside) of the planetary orbit of the gas deflects the dust
trajectories radially outward (inward) to the disk. In this chapter, we only follow
trajectories within the local domain of orbital calculations. The existence of voids indust trajectories would indicate that the global dust distribution could be affected by
the planet-induced gas flow. Thus, the planet-induced gas flow may alter the dust
surface density around the planetary orbit. To obtain a quantitative description of
the accretion rate of dust onto the planet, further studies of the interaction between
the planet-induced flow and dust distributions in disks are needed. These studies
could provide a possible formation scenario of observed dust substructures in disks(Chapter 1). We investigate the influence of the planet-induced gas flow on the dust

96

Chapter 3. Dust motion influenced by the planet-induced gas flow

surface density in Chapter 4 and discuss the implications for planet formation theory
and observations of protoplanetary disks in Chapter 5.

97Chapter 4

Dust ring and gap formation by theplanet-induced gas flow
Sections 4.2–4.4.1 are based on
Ayumu Kuwahara, Hiroyuki Kurokawa, Takayuki Tanigawa, and Shigeru Ida
Astronomy and Astrophysics 665, A122 (2022)

4.1

Introduction of this chapter

Recent high-spatial-resolution observations have revealed substructures in dust profiles (hereafter dust substructures), such as rings and gaps in both young and evolved
protoplanetary disks (e.g., ALMA Partnership et al., 2015; Andrews et al., 2018; Longet al., 2018; Long et al., 2020; Marel et al., 2019). Several mechanisms — planet-freeand planet-dependent — have been proposed to explain the dust rings and gaps indisks.
The planet-free mechanisms are related to perturbations to the disk gas pressureand dust motion in the radial direction of the disk. Those include dust accumulation at the edge of the dead zone (Flock et al., 2015), dust growth at snow lines(Zhang et al., 2015), dust sintering (Okuzumi et al., 2016), self-induced dust pileup(Gonzalez et al., 2015; Gonzalez et al., 2017), secular gravitational instability (SGI;
Takahashi and Inutsuka, 2014; Takahashi and Inutsuka, 2016; Tominaga et al., 2018;
Tominaga et al., 2019; Tominaga et al., 2020), magnetic disk winds (Suriano et al.,2017; Suriano et al., 2018), a self-organization process due to nonideal magnetohydrodynamic effects (Béthune et al., 2017), a hypothetical axisymmetric radial pressure bump (Dullemond et al., 2018), and thermal wave instability (Watanabe andLin, 2008; Ueda et al., 2019; Ueda et al., 2021; Wu and Lithwick, 2021).
Disk-planet interaction is the other mechanism for interpreting the dust ringsand gaps in disks. Paardekooper and Mellema, 2004 were the first to discover thatplanets can open dust gaps in gas disks. A planet embedded in a disk perturbs thesurrounding gas. The tidal torque by a giant planet can remove gas from the vicinityof the planetary orbit (Goldreich and Tremaine, 1979; Goldreich and Tremaine, 1980;
Lin and Papaloizou, 1979), leading to the formation of a gas gap in a disk (e.g., Lin
and Papaloizou, 1986; Kley, 1999; Nelson et al., 2000; Crida et al., 2006; Duffell and
MacFadyen, 2013; Fung et al., 2014; Kanagawa et al., 2015). The young protoplanetary disks contain abundant mm- to cm-sized dust particles (e.g., Testi et al., 2003).
These particles drift from the outer to the inner region of the disk due to the globalpressure gradient (the radial drift of dust; Whipple, 1972; Weidenschilling, 1977a).
Dust particles can be trapped at the edge of the gas gap, the so-called dust filtration
mechanism, forming a dust ring and a gap (e.g., Rice et al., 2006; Paardekooper andMellema, 2006; Pinilla et al., 2012; Dong et al., 2015; Kanagawa et al., 2018). Thus, in98

Chapter 4. Dust ring and gap formation by the planet-induced gas flow

the case of the dust substructure created by a gas gap, one would expect the spatial
distribution of dust substructures to be correlated with that of gas.
However, this is not always the case (e.g., Zhang et al., 2021; Jiang et al., 2022),
and it poses a challenge for the dust substructure formation model predictions by
gas-gap-opening planets for several systems. Moreover, rings and gaps seem to beubiquitous in the outer region of large, bright protoplanetary disks (& 10 au; e.g.,Andrews et al., 2018), but the occurrence rate of giant planets per star is limited to∼ 10% (Johnson et al., 2010; Fernandes et al., 2019). The period-mass distribution of
putative giant planets inferred from the observed dust gap widths is not reconciledwith the current observed period-mass distribution of giant planets (Bae et al., 2018;
Lodato et al., 2019; Disk Dynamics Collaboration et al., 2020). Although this discrepancy might be solved by considering large-scale migration and planetary growth(Lodato et al., 2019), it is not conclusive that most dust substructures are derived
from interactions between disks and gas-gap-opening planets.
The ambiguous relationship between the spatial distribution of gas and dust
could be solved by considering low-mass planets that do not carve a deep gas gap
in a disk. Here, we focus on the potential of gas flow induced by low-mass, non-gasgap-opening planets to sculpt dust substructures in protoplanetary disks. Recent
hydrodynamical simulations have revealed that a low-mass planet embedded in adisk induces gas flow with a complex three-dimensional structure (Chapter 2; Ormelet al., 2015b; Fung et al., 2015; Fung et al., 2019; Lambrechts and Lega, 2017; Cimerman et al., 2017; Kurokawa and Tanigawa, 2018; Kuwahara et al., 2019; Béthune
and Rafikov, 2019; Moldenhauer et al., 2021). The key point is the outflow of thegas, which occurs in the radial direction to the disk. The outflow of the gas deflects
the trajectories of dust particles significantly, even in the case of low-mass planets(∼ 1 M⊕ Kuwahara and Kurokawa, 2020a; Kuwahara and Kurokawa, 2020b, Chapter 3). Once the radial outward gas flow disrupts the motion of the particles, theinward drift of particles could be inhibited, leading to the formation of a dust substructure. This mechanism is studied in this chapter.
Since planet-induced gas flow is a local phenomenon, a high-resolution simulation is required to capture the detailed 3D structure of the gas flow within the
Bondi and Hill spheres. However, the influence of the 3D planet-induced gas flow
is missing in previous studies on the effects of planets on the global dust distribution in disks. So far, the influence of planet-induced gas flow on dust particles has
been investigated in terms of the accretion rate of particles onto the growing planet(Popovas et al., 2018; Popovas et al., 2019; Kuwahara and Kurokawa, 2020a; Kuwahara and Kurokawa, 2020b; Okamura and Kobayashi, 2021, Chapter 3).
The structure of this chapter is as follows. In Sect. 4.2 we describe our numericalmethods. In Sect. 4.3 we show the results obtained from a series of numerical calculations. The main result of this chapter is presented in Sect. 4.3.6, where we focus
on the steady-state dust surface density under the influence of the planet-induced
gas flow. In Sect. 4.4, we discuss the time evolution of the dust surface density
influenced by the planet-induced gas flow and provide an analytical argument.
Sections 4.2–4.4.1 are modified from Kuwahara et al., 2022.

4.2. Methods

4.2

Methods

4.2.1

Model overview

99

In this section, we provide an overview of our model. Our approach is summarized in Fig. 4.1. First, assuming a compressible, inviscid sub-Keplerian gas disk
with the vertical stratification due to the vertical component of the stellar gravity,
we performed 3D nonisothermal hydrodynamical simulations (see Chapter 2). Radiative cooling is implemented by using the β cooling model (Gammie, 2001). The
self-gravity of gas is neglected. Hydrodynamical simulations were performed in theframe co-rotating with the planet (the black sphere in Fig. 4.1). These simulations
resolve the region’s local gas flow, including the planet’s Hill and Bondi spheres.
Second, we calculated the 3D trajectories of dust particles influenced by theplanet-induced gas flow (see Chpter 3). We numerically integrated the equation of
motion of dust particles using the hydrodynamical simulation data. 3D orbital calculations were performed in the frame co-rotating with the planet (the local box in
Fig. 4.1). Through each 3D orbital calculation, we sampled the positions and velocities of particles at specific time intervals. We then obtained the spatial distributionof dust influenced by the planet-induced gas flow in the local domain (Sect. 4.2.3).
Although our orbital calculations were performed in the local domain, in reality,
the dust particles distribute outside the domain as well (i.e., in the full azimuthal direction of a disk). Thus, third, we assumed the uniform and Gaussian distributionsof dust in the azimuthal and vertical directions outside the local domain of orbital
calculations (Sect. 4.2.4), and then calculated the azimuthally and vertically averaged drift velocities of dust (Sect. 4.2.4) based on the obtained spatial distributionsof dust both inside and outside the local domain of orbital calculations.
Finally, we computed the dust surface density by incorporating the obtained azimuthally and vertically averaged drift velocity into a one-dimensional (1D) advectiondiffusion equation (Sect. 4.2.5).
Throughout all of our simulations, the length, times, velocities, and densities
are normalized by the disk scale height, H, reciprocal of the orbital frequency, Ω−1 ,sound speed cs , and unperturbed gas density at the location of the planet, ρdisk ,respectively (see Sect. 2.2.2; Chapter 2).

100

Chapter 4. Dust ring and gap formation by the planet-induced gas flow

F IGURE 4.1: Schematic illustration of our model. We performed 3Dhydrodynamical simulations on the spherical polar grid (the black
sphere), which hosts a planet located at its center (the brown-filled
circle; Sect. 4.2.2). We consider the planet on a fixed circular orbit with the orbital radius, a. We adopt unperturbed sub-Keplerian
shear flow at the outer boundary of the hydrodynamical simulations(the green arrows). We then calculated the trajectories of particles
influenced by the planet-induced gas flow in the frame co-rotatingwith the planet (the local box; Sect. 4.2.3). We used the limited partof the computational domain of hydrodynamical simulation, Rhydro ,in orbital calculations of dust particles (Sect. 4.2.3). The x and z
coordinates of the starting point of particles, xs and zs , are the parameters. The starting point of orbital calculation is beyond Rhydro .
The spatial interval of orbital calculation in the x-direction is δx (Sect.
4.2.3). The x-coordinate of the particle at the edge of the domain oforbital calculation is xe . The x-coordinate of the particle after one
synodical orbit, xs0 , is given by Eq. (4.13) (Sect. 4.3.3). The schematic
trajectories of particles are shown by the dashed blue lines (without
the influence of planet-induced gas flow) and solid orange lines (perturbed by the planet-induced gas flow). The open circles on the trajectory correspond to the position of a particle at specific time intervals, ∆t(v∞y,d , zs ) (Eq. (4.1)). The black arrows on the trajectories are
the velocities in the x-direction of a particle at each position, v x,d . Itshould be noted that we do not follow trajectories outside the calculation domain of orbital calculations. Instead, we assumed the uniform and Gaussian distributions of dust particles in the azimuthaland vertical directions outside the local domain of orbital calculation, respectively, where the particles have a fixed velocity in the xdirection, vdrift . Based on the obtained spatial distribution of dust in
the global domain, we calculated the azimuthally and vertically averaged drift velocity, hv x,d iy,z (Eq. (4.7)). Finally, we calculated thedust surface density by incorporating hv x,d iy,z into the 1D advectiondiffusion equation (Eq. (4.12)).

4.2. Methods

4.2.2

101

Three-dimensional hydrodynamical simulations (the local domain)

We performed nonisothermal, inviscid 3D hydrodynamical simulations of the gasof the protoplanetary disk around an embedded planet (described in Sect. 2.2). The
inviscid assumption is justified in Sect. 4.2.3. The obtained gas flow data were used
to simulate the orbits of dust particles influenced by the planet-induced gas flow
(described later in Sect. 4.2.3). We listed our parameter sets used in this chapter(simulations with an asterisk in Table 2.1).

4.2.3

Dust trajectory simulations (the local domain)

To clarify the influence of the planet-induced gas flow on the radial drift of dust
particles in a disk, we calculated dust trajectories in gas flow obtained by hydrodynamical simulations (Sect. 4.2.3). We generated the spatial distribution of dust
influenced by the planet-induced gas flow in the local domain (Sect. 4.2.3).
Three-dimensional orbital calculation of dust particles
We calculated the trajectories of dust particles influenced by the planet-induced gas
flow in the frame co-rotating with the planet (Fig. 4.1). The x, y and z-directions are
similar to those used in the hydrodynamical simulations. A majority of our methods
for orbital calculations are the same as described in detail Sect. 3.2, where we studied
the dust motion influenced by the planet-induced gas flow. We describe the differences in the numerical setting of orbital calculations with that described in Sect. 3.2
in Sect. 4.2.6.
We numerically integrated the equation of motion of dust particles (3.1). We used
the gas velocity obtained from the hydrodynamical simulation to calculate the gas
drag force acting on the dust within the limited part of the computational domain,
Rhydro (3.2.3 and Table 2.1). Throughout this chapter, we assumed the Stokes gas
drag regime. The choice of the gas drag regime did not affect our results. The gas
drag force is divided into two regimes, the Epstein and the Stokes regimes, depending on the relationship between the size of the particle and mean free path of the gas(Eqs. (3.5) and (3.6)), which differ in terms of the dependence on the gas density. In
this chapter, we assumed that the gas drag force is independent of the gas density,
which is strictly valid only in the Stokes regime but not in the Epstein regime. However, the region of particular interest in this chapter is typically beyond the Bondi
region, where the gas density is almost constant and identical to the initial (unperturbed) value. We confirmed that the choice of the gas drag regime does not affectthe results and thus, the results can be applicable for dust particles both in Epsteinand Stokes regimes.
The y-coordinate of the starting point of particles is fixed at |ys | = 40RHill (Ida
and Nakazawa, 1989). The x and z coordinates of the starting point of particles, xs
and zs , are the parameters. The ranges of xs and zs are xs ∈ [−5RHill , 5RHill ] andzs ∈ [0, 1]. Owing to the symmetry of the system, we only consider z ≥ 0. We
set the upper limit as z = 1. At high altitudes, z > 1, the dust density decreases
significantly (described later in Eq. (4.2)). The ranges of xs and zs fully cover the sizeof the perturbed region, which can be scaled by the Bondi radius in our parameter
sets (m ∈ [0.03, 0.3]).

102

Chapter 4. Dust ring and gap formation by the planet-induced gas flow

Spatial distribution of dust in the local domain
Through each orbital calculation, we sampled the positions and velocities of the particles at fixed small time intervals. We then obtained the spatial distribution of particles in the local calculation domain as schematically illustrated in the upper rightcorner of Fig. 4.1.
We first describe the spatial intervals of the starting points of the orbital calculations, which determine the radial and vertical distributions of the incoming dust
(Fig. 4.1). We integrated the equation of motion for dust particles with their initial
spatial intervals of δx = 0.01b∞ in the x-direction and of δz = 0.01 in the z-direction.
The maximum impact parameter of accreted particles in the unperturbed flow, b∞ ,is expressed by Eq. (3.12)The spatial interval in the x-direction, δx = 0.01b∞ , has on
the order of ∼ 10−4 –10−3 H.
Next, we describe the time interval for sampling the position and velocity ofa particle to generate the dust spatial distribution within a local domain of orbital
calculations. The time interval is given by
∆t =

L

,
|v∞
y,d | Nloc ( zs )(4.1)

where L = 80RHill (the size of the calculation domain) and Nloc (zs ) is the numberof dust particles on an unperturbed orbit (without the influences of gravity of theplanet and the planet-induced gas flow) in the local calculation domain (describedlater in Eq. (4.2)). The dependence of ∆t on |v∞
y,d | means that the dust mass flux isconstant in the x-direction of the local calculation domain. To take into account thevertical distribution of incoming dust, we assumed the Gaussian distribution:" #
1 zs 2
Nloc (zs ) = N0 exp −
,
(4.2)
2 Hd
where N0 is the number of dust particles at the midplane. We adopt N0 = 500. In
Eq. (4.2), the scale height of dust particles, Hd , is given by (Eq. (2.71); Dubrulle et al.,
1995; Cuzzi et al., 1993; Youdin and Lithwick, 2007):

Hd =St 1 + 2St
1+
αdiff 1 + St

−1/2
H,

(4.3)

where αdiff is the dimensionless turbulent parameter, which describes turbulent diffusion in our model. In this chapter, we assume αdiff = 10−5 , 10−4 , and 10−3 . Wenote that the obtained spatial distribution of dust in the local domain deviates fromthe Gaussian distribution in the vertical direction because of the gas flow in the vertical direction (e.g., Fig. 2.17).
We assumed the inviscid fluid of an ideal gas and did not consider the effect ofrandom motion of dust due to the turbulence in a series of hydrodynamical simulations and orbital calculations (Kuwahara and Kurokawa, 2020a; Kuwahara andKurokawa, 2020b). The dimensionless turbulent parameter, αdiff , is used when we
consider the vertical distribution of dust (Sects. 4.2.3 and 4.2.4) and the turbulent
diffusion of dust in the radial direction (described later in Sect. 4.2.5). As long as we
focus on the range of the turbulence strength considered in this chapter, the turbulence likely affects the dust only. The viscosity suppresses the 3D flow only whenits α-viscous coefficient is large, αdiff & 10−2 (Fung et al., 2015; Lambrechts andLega, 2017). We note that several previous studies reported another transition at4.2. Methods

103

even lower αdiff . Lambrechts and Lega, 2017 found the oscillational flow pattern at−1
αdiff . 10−3 . In our inviscid simulations, we also found the fluctuation after 10 ΩK,
but its amplitude is small: <10% in the x-component of the outflow velocity. McNally et al., 2019 considered the planets with m & 0.3 and found the growth of a setof Rossby wave instability vortices outside the planetary orbits when αdiff . 10−4 intheir 2D global hydrodynamical simulations. Though these vortices do not appear
in our local hydrodynamical simulations, their influences on dust drift may need to
be investigated in a future study.

4.2.4

Model for global dust distribution

So far, we only considered the distribution of dust particles in the local domain oforbital calculation because we did not follow trajectories of dust outside the domain (Sect. 4.2.3). In reality, the dust particles distribute outside the domain as well(namely, in the full azimuthal direction of the disk). We then considered the spatial distribution of dust outside the local domain of orbital calculation (Sect. 4.2.4).
Based on the obtained spatial distribution of dust both inside and outside the local
domain of orbital calculation, we calculated the azimuthally and vertically averaged
drift velocity of dust (Sect. 4.2.4).
Spatial distribution of dust in the global domain
We assumed the uniform distribution of dust particles in the azimuthal direction
outside the local domain of orbital calculation, where the particles have the fixedvelocity in the x-direction, vdrift . We confirmed that the velocities of dust particles
were almost identical to that of the initial values at the edge of the local domain oforbital calculation (Eqs. (3.9) and (3.10)). Thus, this assumption was justified.
To maintain consistency with orbital calculations, outside the domain, the number of dust particles on a hypothetical orbit (the orange solid line outside the local
domain of orbital calculation, which is schematically illustrated in the upper left
corner of Fig. 4.1) is given by!
2πaNout (zs ) '− 1 Nloc (zs ),(4.4)
Lwhere we assume a  x. As with the orbital calculations, we consider the rangesof x ∈ [−5RHill , 5RHill ] and z ∈ [0, 1]. The spatial intervals of hypothetical orbits
are δx and δz in the x- and z-directions, respectively. Thus, the "global box" of size
10RHill × (2πa − L) × 1 H 3 contains ∑100
i =0 Nout (i/δz ) × 10RHill /δx particles.
Azimuthally and vertically averaged drift velocity of dust
Based on the obtained spatial distribution of dust both inside and outside the local domain of orbital calculation (Sect. 4.2.4), we calculated the azimuthally andvertically averaged drift velocity of dust. We divided the obtained spatial distribution of dust in the global domain into grids in the x-direction with equal spacing,∆x = 10 × δx. The total number of sampling particles in each grid is given by
Ntot,i = Nloc,i + Nout,i ,(4.5)

where Nloc,i is obtained from orbital calculations and Nout,i is given by Nout,i =Nout × (∆x/δx ). The subscript i denotes the grid number. Each grid contains Ntot,i ∼104

Chapter 4. Dust ring and gap formation by the planet-induced gas flow

106 sampling particles. We confirmed that the sample number is large enough toachieve numerical convergence.
The azimuthally and vertically averaged x-component of the velocity on the ith
grid is calculated by

R 1 R 2πhvix,d iy,z =ρd vix,d dydz
0,
R 1 R 2π
ρd dydz
0 0

0

(4.6)

where ρd is the density distribution of dust (Eq. (3.16)):" #1 zs 2ρd = ρd,0 exp −,2 Hdwith the midplane density, ρd,0 = 1. We numerically calculated Eq. (4.6) asNhvix,d iy,z = loc,iNtot,iNloc,iij∑ j=1 v x,dNloc,i+Nout,iv ,Ntot,i drift(4.7)

where the subscript j denotes the number of particles in the ith grid. The azimuthally
and vertically averaged drift velocity of dust is schematically illustrated in the lowerright corner of Fig. 4.1.

4.2.5

Simulation of dust surface density

We computed the dust surface density by incorporating the azimuthally and vertically averaged drift velocity into the 1D advection-diffusion equation as schematically illustrated in the lower-left corner of Fig. 4.1. Instead of calculating the dust
surface density directly from the obtained spatial distribution of dust, we solve theadvection-diffusion equation for the following reason. The obtained spatial distribution of dust is a snapshot (Sect. 4.2.4). Though we considered the effect of turbulent
diffusion in the vertical direction through Eq. (2.71), we did not consider it in theradial direction so far.
The dust surface density, Σd , can be obtained by solving the following 1D advectiondiffusion equation:∂Σd∂+( F + Fdiff ) = 0,∂t∂x adv(4.8)

Fadv = vd,adv Σd ,∂ΣFdiff = −Ddiff d ,
∂x

(4.9)

where

(4.10)

where vd,adv is the advective velocity of dust and Ddiff is the diffusion coefficient forthe dust (Youdin and Lithwick, 2007):

Ddiff =

αdiff.
1 + St2(4.11)

Because we focus on a radial range sufficiently narrow compared to the orbital radius of the planet, we assumed that the gas surface density is constant in these dust
surface density simulations. Thus, the gas surface density can be omitted in Eq. (4.8).

4.2. Methods

105

Steady state
In Sect. 4.3.6, where we focus on a steady state, we integrated Eq. (4.8) once, andthen obtained
Fadv + Fdiff = vdrift Σd,0 ,(4.12)

where Σd,0 is the initial dust surface density. We assumed vd,adv = hv x,d iy,z in theabove equation, where hv x,d iy,z is given by Eq. (4.7). The size of the calculation domain for solving Eq. (4.12) is x ∈ [−100RHill , 100RHill ], which is much larger thanthat of the orbital calculations. Outside the domain of orbital calculations in the radial direction, | x | > 5RHill , we set hv x,d iy,z = vdrift (see also 4.2.7). As a boundary
condition, we assumed Σd = Σd,0 = 1.
Time evolution
In Sect. 4.4.2, where we discuss the time evolution of the dust surface density, we
integrated Eq. (4.8) using the finite-volume method. We assumed vd,adv = hv x,d iy,z .
The size of the calculation for solving Eq. (4.8) is x ∈ [−5, 5], which is divided into
103 cells with equal spacing, 0.01 H. We describe the boundary conditions for theadvective and diffusive flux as below. We allow the advective flux to across the innerboundary. We set Fadv = vdrift Σd,0 at the outer boundary. We use a zero-diffusiveflux boundary condition for the inner boundary, Fdiff = 0 at x = −5, while we allow
the diffusive flux to across the outer boundary.
Analytical estimation
In Sect. 4.4.3, we provide an analytical argument for the drift velocity of dust influenced by the planet-induced gas flow and compute the dust surface density. When
we solve the 1D advection-diffusion equation using the analytic formula for the dust
velocity, we assume vd,adv = vdust ( x ), where vdust ( x ) is given later by Eq. (4.33).

4.2.6

Numerical settings: Dust trajectory simulations

Most of our methods of orbital calculations are the same as described in Sect. 3.3.2,
apart from the configuration of termination conditions. In Chapter 3, where we
studied the motion of dust particles influenced by the planet-induced gas flow, we
terminated orbital calculation when either one of the following conditions was met:
−1(1) the calculation time reached the sufficiently long time of computation, 104 ΩK,(2) a dust particle reached the edge of the calculation domain, |y| ≥ 40RHill (Fig. 4.1),or (3) accreted onto the planet, R ≤ 2Rinn . However, we found that this treatment
causes several problems when studying the influence of gas flow on dust drift (Sect.
4.2.5).
First, we found that the x-component of the gas velocity, v x,gas , is significant in
the vicinity of the planet, R . 0.3RBondi (Figs. 4.2 and 4.3). The particles that enter the planetary atmosphere circulate close to the planet and then accrete onto theplanet (Figs. 4.2b and 4.3b). We found that the azimuthally and vertically averaged
drift velocity of dust calculated by Eq. (4.7) fluctuated significantly in the vicinity ofthe planetary orbit, | x | . 0.3RBondi , due to the large v x,gas . To eliminate the fluctuation of the azimuthally and vertically averaged drift velocity, we assumed that theparticles accreted onto the planet when R ≤ 0.3RBondi in all cases.

106

Chapter 4. Dust ring and gap formation by the planet-induced gas flow

Second, dust particles are trapped and circulate many times in the horseshoe region when the drift timescale of particles, tdrift ∼ wHS /|vdrift |,is comparable to thehorseshoe libration timescale, tlib ∼ 8πa/(3ΩwHS ). We terminated orbital calculations to reduce the computational time when the horseshoe turn was detected twice.

F IGURE 4.2: Streamlines of gas (left) and trajectories of dust particles
with St = 10−2 (right) at the midplane of the disk. The flow structurearound a planet is obtained from m03-hw001. With this parameter
set, the planet-induced gas flow is in the flow-shear regime (see Sect.
2.3.1). The color contour represents the flow speed in the x-direction,
v x,gas . We set zs = 0 for the orbital calculations. The dotted anddashed circles are the Bondi and Hill radii of the planet, respectively.
The thick blue lines in the right panel correspond to the trajectories ofthe dust particles that satisfy xs0 ≥ xs (see Fig. 4.1 and Sect. 4.3.3 fordetails).

4.2. Methods

107

F IGURE 4.3: Same as Fig. 4.2, but with the flow structure obtained
from m003-hw01 and St = 10−4 set for the trajectories. In this figure,the planet-induced gas flow is in the flow-headwind regime (see Sect.
2.3.1).

4.2.7

Numerical settings: Azimuthally and vertically averaged drift velocity of dust

When we calculated the azimuthally and vertically averaged drift velocity of dust,
hv x,d iy,z , we found unexpected peaks in hv x,d iy,z at the positions away from the planetary orbit (outside the region influenced by the planet-induced outward gas flow,| x | & 1). These peaks are caused by the density waves excited by an embedded
planet. While our hydrodynamical simulations can resolve the local gas flow pattern in detail, they do not handle the global structure in the azimuthal direction in a
disk. Since the density waves extend in the azimuthal direction, they are interrupted
at the edge of the computational domain of our hydrodynamical simulations. We
found that the locations of the unexpected peaks coincide with those where the density waves are interrupted. Figure 4.4 shows the locations of the outflows and thedensity waves. An embedded planet excites the density waves as shown in Fig. 4.4.
Since the density waves extend in the azimuthal direction, they are interrupted at theedge of the computational domain of our hydrodynamical simulations, | x | ∼ 3 H.
Figure 4.4 also shows that the outflow of the gas appears closer to the planetary orbit
than the density waves.
Previous studies have reported that, in global disk simulations, the effects ofdensity waves on dust drift were minor (Paardekooper and Mellema, 2006; Muto
and Inutsuka, 2009; Rosotti et al., 2016; Dong et al., 2017a). Therefore, we assumed
that the dust particles have a fixed velocity, vdrift , outside the region influenced by

108

Chapter 4. Dust ring and gap formation by the planet-induced gas flow

the planet-induced outward gas flow. In this region, the artificial effect of the interrupted density waves could be seen (schematically illustrated in the lower-rightcorner of Fig. 4.1).

F IGURE 4.4: Flow structure at the midplane of the disk around aplanet obtained from m03-hw001. The color contours represent theflow speed in the x-direction (top) and the gas density (bottom).

4.3. Results

4.3

Results

4.3.1

Results overview

109

The main subject of this chapter is to clarify the influence of the planet-induced gas
flow on the radial distribution of dust particles in a disk. Section 4.3.2 summarizes
the characteristic 3D structure of the planet-induced gas flow obtained by 3D hydrodynamical simulations. In Sects. 4.3.3–4.3.5, we show the results of our orbital
calculations. Section 4.3.6 presents the results obtained by solving the 1D advectiondiffusion equation for dust surface density.

4.3.2

Three-dimensional planet-induced gas flow

The planet-induced gas flow has a complex 3D structure. A substantial amount ofgas from the disk enters the Bondi and Hill spheres at high latitudes (inflow), andexits through the midplane region of the disk (outflow). The 3D flow structure depends on Mhw (Chapter 2; Ormel et al., 2015b; Kurokawa and Tanigawa, 2018). In
Chapter 2, we classified the planet-induced gas flow into the flow-shear and theflow-headwind regimes (e.g., Figs. 4.2 and 4.3; Kuwahara and Kurokawa, 2020b).
The flow transition mass was originally derived to distinguish these hydrodynamical regimes. The transition of the gas flow field from the flow-shear to the flowheadwind regime occurs smoothly (e.g., Fig. 2.10). We found that the flow transition
mass is not applicable for the classification of the effects on dust drift — which is thefocus of this chapter — but it is still useful to understand the tendency of our results.
In this chapter, the gas flow regime should be considered only as a qualitative guideto whether it is "shear-dominated" (two outflows occur both inside and outside theplanetary orbit) or "headwind-dominated" (outflow occurs dominantly outside theplanetary orbit).

4.3.3

Existence of the forbidden region

We first show that the gas flow induced by low-mass planets affect the radial driftof the dust particles. Given a dust particle launched at ( xs , ys ), the x-coordinate of aparticle after one synodical orbit at y = ys can be described by v
xs0 ' xe − 2πa − L × drift,(4.13)
v∞y,dwhere xe is the x-coordinate of a particle at the edge of the orbital calculation domain(upper right corner of Fig. 4.1). Equation (4.13) is valid because the velocity of dust
is identical to that of the unperturbed value at the edge of the domain of orbital
calculation (Sect. 4.2.3). We then set the following simple criteria:
1. when xs0 < xs , the dust particles continue the radial drift.
2. When xs0 ≥ xs , the planet-induced gas flow inhibits the radial drift. We only
use the starting and ending points of an orbital calculation at this stage.
We found that a forbidden region exists outside the planetary orbit, where theradial drift of dust particles is inhibited (Fig. 4.5). Figure 4.5 shows the vector fieldof dust velocities influenced by the planet-induced gas flow in the x-z plane. The
mass of the planet and Mach number of the headwind are m = 0.3 and Mhw = 0.01,
respectively. We considered the Stokes numbers of St = 10−4 , 10−2 , and 3 × 10−2 .

110Chapter 4. Dust ring and gap formation by the planet-induced gas flow

In these cases, the planet-induced gas flow is in the flow-shear regime. A significant
outward drift of dust particles can be seen near the midplane (dark green arrows in
Fig. 4.5), because the outflow has the maximum speed near the midplane region of
the disk. We discuss the motion of dust within the forbidden region in detail in Sect.
4.3.4.
Similar to the flow-shear regime, we confirmed that a forbidden region exists
outside the planetary orbit in the flow-headwind regime (see Sect. 4.3.5).

F IGURE 4.5: Vector field of dust velocities in the cases with St = 10−4
(left panel), 10−2 (middle panel), and 3 × 10−2 (right panel) under thecommon gas flow obtained from m03-hw001. The color contour from
red to black represents the x component of the gas flow velocity averaged in the y-direction within the calculation domain of hydrodynamical simulation around a planet, hv x,gas iy . Arrows represent thedirection of movement of dust particles. The color contour of the arrows from green to purple represents the distance of drift of a dust
particle after one synodical orbit. The hatched region enclosed byblack curves corresponds to the forbidden region. The two-headed
blue arrow in panel b can be compared to the region shown by the
thick blue lines in Fig. 4.2b. The dotted and dashed lines are the Bondiand Hill radii of the planet, respectively. Thanks to the symmetry of
the system, we only focus on the region where z ≥ 0. We discuss thedust motion in the x-z plane in detail in Sect. 4.3.4 motion within theforbidden region.

4.3.4

Dust motion within the forbidden region

We first describe the motion of dust particles in the x-direction. When the Stokes
number is small (St = 10−4 ), the forbidden region lies in a wide range of Fig. 4.5a.
Counterintuitively, we found that the forbidden region exists even above the outflow
whose vertical scale is ∼ 0.5RBondi . To understand this phenomenon, we show the
trajectories of dust particles with St = 10−4 at high altitudes in Fig. 4.6. Figure 4.6
compares the dust trajectories in the planet-induced gas flow (solid lines) with thosein the unperturbed gas flow (dashed lines). In the distant region far from the planet,y > −4, the orbital radii of dust in the planet-induced gas flow are larger than thosein the unperturbed gas flow (Fig. 4.6a). Figure 4.6b shows xe > xs in the planetinduced gas flow. This is caused by the gas flow structure at high altitudes. We notethat the changes in the orbital radii of dust should be carefully considered. Since ourhydrodynamical simulations were local, we did not trace the gas flow structure in
the full azimuthal direction. Thus, our results may need to be compared with theresults obtained by global hydrodynamical simulations in a future study.

4.3. Results

111

When the Stokes number is large, the width of the forbidden region decreases
significantly above the outflow (Fig. 4.5b). The width of the forbidden region for
St = 10−2 (Fig. 4.5b) is narrower than that for St = 10−4 (Fig. 4.5a). When St =3 × 10−2 (Fig. 4.5c), the width of the forbidden region is quite narrow (. 0.1 H). Thisis because, when the Stokes number is large, the dust particles are less affected by
the gas flow.
Next, we describe the motion of dust particles in the z-direction. The dust particles influenced by the planet-induced gas flow ascend when they pass through near
the planet location (solid lines in Figs. 4.6b and c). The upward motion of dust
particles is caused by the meridional gas flow induced by the planet. In the case of
low-mass planets, the meridional gas flow occurs as follows: the outflow streamlines connect to the horseshoe streamlines (called "transient horseshoe flow"; Funget al., 2015). In the flow-shear regime, gas flows in at high latitudes of the gravitational sphere of the planet at its horseshoe turn. The transient flow is verticallycompressed, then the upward gas flow occurs due to the decompression of the gas(see, e.g., Fig. 11 of Fung et al., 2015).

112

Chapter 4. Dust ring and gap formation by the planet-induced gas flow

F IGURE 4.6: Trajectories of dust particles with St = 10−4 in theplanet-induced gas flow (solid lines) and in the unperturbed subKeplerian shear flow (dashed lines). We used the result obtained fromm03-hw001 to calculate dust trajectories. We set zs = 0.5 for the orbital
calculations. Different colors correspond to different xs . The filled
circles and cross symbols in panel b correspond to xs and xe , respectively. The dotted and dashed circles are the Bondi and Hill radii ofthe planet, respectively. Panel a: x-y plane viewed from the +z direction. Panel b: x-z plane viewed from the −y direction. Panel c: y-z
plane viewed from the + x direction. This figure should be compared
to Fig. 4.5a.

4.3.5

Width of the forbidden region

There is a clear dependence of the width of the forbidden region on the Stokes number. As shown in Fig. 4.5, the width of the forbidden region is wider for small Stokes
number.

4.3. Results

113

Figure 4.7 shows the changes of the vertically averaged width of the forbidden region weighted by the dust number density, hwf iz , as a function of the Stokes number
for different planetary masses and Mach numbers. The vertically averaged width of
the forbidden region was calculated by

R1
h wf i z =0ρd wf (z) dz
,
R1
ρ
dz
0 d

(4.14)

where wf (z) is the width of the forbidden region as a function of z. Same as Eq. (4.6),
we only consider 0 ≤ z ≤ 1. Overall trends indicate that hwf iz is a decreasing
function of the Stokes number, because the influences of gas flow are weaker for thelarge Stokes number. We note that the forbidden region does not exist in the limit of
St → 0. The inflow to the Bondi sphere of the planet balances the outflow from it.
This replenishment occurs through the meridional circulation of the gas (e.g., Funget al., 2015). When St → 0, dust particles cycle along with the replenishment of thegas and, consequently with the outflow, meaning that the forbidden region does notexist in this limit. On the other hand, if the upper layer of the disk is dust-poor, we
expect that the forbidden region does exist. This occurs when the dust scale height issmaller than the disk scale height, Hd < H. We discuss again the required conditions
in our model later in Sect. 4.3.6 on turbulence strength.
For the larger Stokes numbers, St > 3 × 10−2 , the forbidden region does notexist in our parameter spaces (m ∈ [0.03, 0.3] and Mhw ∈ [0.01, 0.1]). The following
sections show only the results for St ≤ 3 × 10−2 .

114Chapter 4. Dust ring and gap formation by the planet-induced gas flow

F IGURE 4.7: Vertically averaged width of the forbidden region, hwf iz .
Different colors correspond to different Mhw . We adopt αdiff = 10−4 .

4.3.6

Influence of planet-induced gas flow on the dust surface density

The essential points of the previous sections are summarized in Fig. 4.1. So far wehave discussed the influence of the planet-induced gas flow on the radial drift ofdust particles without the turbulent diffusion in the radial direction (Sects. 4.3.3 and4.3. Results

115

4.3.5, where we only used the starting and ending points of an orbital calculation).
This section shows the steady-state dust surface density obtained by solving the
1D advection-diffusion equation (Eq. (4.12)). We found that the dust surface density profiles are different in the flow-shear (shear-dominated) and flow-headwind(headwind-dominated) regimes.1 Hereafter, we refer to the regions where dust is
depleted and accumulated as "dust gap" and "dust ring," respectively.
Dust ring and gap formation by planet-induced gas flow: Flow-shear regime
When the outflow occurs dominantly both outside and inside the planetary orbit,that is, when the planet-induced gas flow is in the flow-shear regime, the dust is accumulated outside the planetary orbit (dust ring) and depleted around the planetary
orbit (dust gap; Fig. 4.8d). Panels a–d of Fig. 4.8 show the gas flow structure averaged in the y-direction within the calculation domain of hydrodynamical simulation(m01-hw001; Fig. 4.8a), the difference of the azimuthally and vertically averaged drift
velocity of dust particles from the unperturbed steady-state drift velocity (Eq. (4.7);
Fig. 4.8b), advective and diffusive flux of dust (Fig. 4.8c), and the steady-state dust
surface density (Fig. 4.8d), respectively. We set αdiff = 10−5 in Figs. 4.8b-d. We
overplotted the forbidden regions for different Stokes numbers in Fig. 4.8a.
We found that the azimuthally and vertically averaged drift velocity of dust
particles has positive and negative peaks outside and inside the planetary orbit(Fig. 4.8b). The positive peak is located at the position overlapping the forbidden
region. The negative peak is caused by the outflow toward the inside of the planetary orbit, where the inward drift of dust particles is enhanced.
The advective flux dominates in the distant region far from the planet, | x |  1(Fig. 4.8c). The dust particles drift inward to the central star with the steady-state
velocity, vdrift , and thus the dust surface density has the fixed value, Σd = Σd,0 .
The dust particles accumulate at the location where hv x,d iy,z switches from negative to positive due to the outflow toward the outside of the planetary orbit, which isshown by the solid lines of St = 10−2 and 10−3 in Fig. 4.8d. The gradient of the dust
surface density caused by dust accumulation results in an outward (inward) diffusive flux on the right (left) side across the peak of the dust surface density (Fig. 4.8c).
The tail of the dust ring extends beyond the width of the forbidden region.
The dust particles pass through the forbidden region by turbulence diffusion andare transported inside the planetary orbit (Fig. 4.8c). Since the outflow occurs inside
the planetary orbit in the flow-shear regime, the inward advective flux is enhancedby the outflow. As a result, the dust surface density decreases around the planetary
orbit, followed by the formation of the dust gap.
We found that only the dust gap formation occurs in a few cases (e.g., St = 10−4
in Fig. 4.8d), such as when the amplitude of the positive peak in the azimuthally andvertically averaged drift velocity of dust is smaller than that of the negative peak.
The width and depth of the dust gap are larger for smaller Stokes numbers. The
typical width of the dust gap is ∼ 1 H in most cases, which corresponds to the widthof the region influenced by the planet-induced outflow (e.g., Fig. 4.2). In Fig. 4.8, thewidth of the dust ring has on the order of ∼ 0.1–1 H, but it strongly depends on the
Stokes number and the turbulence strength (shown later in Figs. 4.11 and 4.12, seeSect. 4.3.6 on turbulence strength).
1 Again, the following results cannot be directly classified based on the hydrodynamical regimes

introduced in Chapter 2 because the transition of the gas flow field from the flow-shear to the flowheadwind regime occurs smoothly. We use the gas flow regimes as a qualitative guide. The important
point is whether the dominant outflow occurs inside or outside the planetary orbit.

116

Chapter 4. Dust ring and gap formation by the planet-induced gas flow

F IGURE 4.8: Gas flow structure averaged in the y-direction within thecalculation domain of hydrodynamical simulation around a planet(panel a), the difference of the azimuthally and vertically averaged
velocity of dust particles from the unperturbed steady-state drift velocity, hv x,d iy,z − vdrift (panel b), advective (solid lines) and diffusive(dashed lines) flux of dust with St = 10−3 in the steady state (panel c),
and the dust surface density (panel d). Panels a, b, and d: Different colors correspond to different Stokes numbers. Panel a: Result obtained
from m01-hw001. The solid gray lines correspond to the streamlines.
The dashed circle is the Hill radius. The regions enclosed by blue
curves correspond to the forbidden regions. The horizontal dashedlines in panels b and d correspond to hv x,d iy,z = vdrift and Σd /Σd,0 = 1.
The figure displayed at the lower right of panel b shows hv x,d iy,z . Panels b–d: We adopt αdiff = 10−5 . To improve legibility, we only plot theregion where | x | ≤ 2.

4.3. Results

117

Dust ring formation by planet-induced gas flow: Flow-headwind regime
When the outflow occurs dominantly outside the planetary orbit (flow-headwind
regime), the dust is accumulated outside the planetary orbit without the depletionof dust around the planetary orbit, as shown in Fig. 4.9d. Similar to Fig. 4.8, panels
a–d of Fig. 4.9 show the gas flow structure averaged in the y-direction within thecalculation domain of hydrodynamical simulation (m01-hw01; Fig. 4.9a), the difference between the azimuthally and vertically averaged drift velocity of dust particles
and the unperturbed steady-state drift velocity (Fig. 4.9b), advective and diffusive
flux of dust (Fig. 4.9c), and the dust surface density (Fig. 4.9d), respectively. We set
αdiff = 10−4 in Figs. 4.9b-d.
In the flow-headwind regime, the azimuthally and vertically averaged drift velocity of dust particles has only a positive peak, which leads to dust accumulation
outside the planetary orbit (Figs. 4.9b and d). We note that hv x,d iy,z for St = 10−2 in
Fig. 4.9 is always negative (see the small panel displayed in Fig. 4.9b). Even when
the drift speed of dust is always negative, the dust particles accumulate slightly outside the planetary orbit (St = 10−2 ; Fig. 4.9d). This is caused by the traffic jam of dustthat results from changes in hv x,d iy,z . The width of the dust ring and the concentration of the dust into a ring are larger for the small Stokes number. The concentrationstrongly depends on the turbulence strength (shown later in Figs. 4.11 and 4.12, seealso Sect. 4.3.6 on turbulence strength).
Similar to the flow-shear regime, the advective flux dominates in the distant region far from the planet, | x |  1, (Fig. 4.9c). The outward (inward) diffusive flux
occurs on the right (left) side across the peak of the dust surface density (Fig. 4.9c).
Inside the planetary orbit, the advective flux is not disturbed because the influenceof the outflow toward the inside of the planetary orbit is weak. Thus, the dust gap
does not form in a steady state.

118

Chapter 4. Dust ring and gap formation by the planet-induced gas flow

F IGURE 4.9: Same as Fig. 4.8, but with the result obtained fromm01-hw01 in panel a and αdiff = 10−4 adopted in panels b–d. Since
we only plot the region where | x | ≤ 2, the outer part of the ring isinterrupted.

Dependence on planetary mass
Figures 4.10–4.12 (see also Figs. 4.13 and 4.14) show the dependence of dust surface
density influenced by the planet-induced gas flow on the planetary mass. We found4.3. Results

119

the following trends: (1) an apparent dust ring and (or) gap formation by the planetinduced gas flow occurs whenm ≥ 0.1(4.15)and (2) dust gap formation is susceptible to occur for a higher-mass planet.
Regarding the former trend, we recall that the width of the forbidden region increases with the planetary mass, except when the Stokes number is extremely small(St . 10−3 ; Fig. 4.7). When m = 0.03, the smallest planetary mass considered inthis chapter, the forbidden region does exist, but its width is small (hwf iz . 0.2;
Fig. 4.7a). The influence of the planet-induced gas flow on the radial drift of dust is
weak. The turbulence diffusion can smooth the dust surface density, except when
the turbulence strength is weak (αdiff . 10−5 ; Figs. 4.10g–i).
The latter trend is related to the response of the topologies of the planet-induced
gas flow to an increase in the Mach number. A larger Mhw is needed for a highermass planet to change the gas flow regime from the flow-shear to the flow-headwind
regime (Eq. (2.23)). Within a range of the Mach numbers considered in this chapter,when m = 0.3, the influence of the outflow toward inside the planetary orbit remains
and the dust gap forms. We found the following empirical relation between theplanetary mass and the Mach number based on our results. The dust gap forms
when (Fig. 4.13):
m/Mhw & 3.

F IGURE 4.10: Dust surface density with m = 0.03 for the different
Stokes numbers, the Mach numbers, and the turbulent parameters.
Different colors correspond to different Stokes numbers. The minimum and the maximum values of the dust surface density are summarized in Figs. 4.13 and 4.14.

(4.16)

120

Chapter 4. Dust ring and gap formation by the planet-induced gas flow

F IGURE 4.11: Same as Fig. 4.10, but with m = 0.1. The grayshaded region represents an excessive accumulation of dust particles,
where the dust-to-gas ratio exceeds unity, assuming the initial valueof Σd,0 /Σg = 0.01. Since we only plot the region where −1 ≤ x ≤ 4,
the outer part of the ring is interrupted. The dashed black line corresponds to Σd /Σd,0 = 1.

4.3. Results

121

F IGURE 4.12: Same as Fig. 4.10, but with m = 0.3. The grayshaded region represents an excessive accumulation of dust particles,
where the dust-to-gas ratio exceeds unity, assuming the initial valueof Σd,0 /Σg = 0.01. Since we only plot the region where −1 ≤ x ≤ 4,
the outer part of the ring is interrupted. The dashed black line corresponds to the initial value.

Dependence on turbulence strength
Figures 4.13 and 4.14 show the dependence of dust surface density influenced by theplanet-induced gas flow on the turbulence strength, αdiff . In common with Figs. 4.13
and 4.14, the influence of the planet-induced gas flow on the dust surface density
is stronger for a small αdiff , in which case the dust settles in the midplane of thedisk, and the velocity fluctuations caused by the planet-induced gas flow are large.
This is because the outflow significantly influences the radial drift of dust near the
midplane (Fig. 4.5). Then the large surface density gradient needs to achieve a steadystate, which results in a significant accumulation or depletion of dust.
Within a range of the turbulence strengths considered in this chapter, the apparent dust ring and gap structures are found only when αdiff ≤ 10−4 (Figs. 4.13 and4.14). When αdiff is very small, αdiff = 10−5 , we found an excessive accumulation ofdust outside the planetary orbit in some cases (Figs. 4.14g–i). We discuss an excessive dust concentration outside the planetary orbit in Sect. 5.3.

122

Chapter 4. Dust ring and gap formation by the planet-induced gas flow

F IGURE 4.13: Minimum dust surface density, Σd,min , as a functionof the planetary mass and the Stokes number. Different rows correspond to different turbulent parameters, αdiff . Different columns
correspond to different Mach numbers, Mhw .

F IGURE 4.14: Same as Fig. 4.13, but with panels showing the maximum dust surface density, Σd,max , as a function of the planetary massand the Stokes number. We note that the color contours are on a linear
scale in panels a–f but on a log scale in panels g–i.

We assumed that a planet did not perturb the gas surface density (Sect. 4.2.5),

4.3. Results

123

and found that the influences of the planet-induced gas flow on the dust surface
density were more pronounced for the small Stokes number (Figs. 4.13 and 4.14).
However, one might expect that the spatial distribution of dust with the very small
Stokes number (St  1) would match that of the gas.
We consider an endmember case where St = 10−4 and assume that the planetinduced gas flow is in the flow-shear regime. First, we only focus on the dust motion
at the midplane region of the disk. Even when St = 10−4 , the voids in the dust
distribution are created by the outflow of the gas (e.g., Fig. 4.2b), because the outflow
streamlines originated from the horseshoe streamlines at high altitudes (e.g., Fig. 3.5:
Fung et al., 2015).
Next, we consider the vertical distribution of dust. The transient horseshoe gas
flow transports the dust particles at high altitudes to the midplane region (orange
arrows in Fig. 4.15).2 After transportation, the dust trajectories at the midplane region should match those shown in Fig. 4.2b. Thus, the voids remain in the dust
distribution.
The downward motion of dust particles is identified even at z ∼ H (Fig. 4.15).
When the dust scale height is smaller than the disk scale height, Hd < H, dust-poor
gas is supplied to the forbidden region. Then the surface density of dust deviatesfrom that of the gas even when St  1. Thus, from Eq. (2.71), our model for the dust
substructure formation without the perturbation of the gas surface density would
require the following condition: αdiff < St  1.
2 The dust is stirred up by the meridional gas flow, but its magnitude is small (Fig. 4.15). Recent
two-fluid (dust + gas) 3D global simulations also suggest that the meridional gas flow induced by alow-mass planet does not effectively stir dust particles (Krapp et al., 2022). We discussed the upward
motion of dust in Sect. 4.3.4.

124

Chapter 4. Dust ring and gap formation by the planet-induced gas flow

F IGURE 4.15: Same as Fig. 4.5a, but the color contour from blue to red
represents the z component of the gas flow velocity averaged in they-direction within the calculation domain of hydrodynamical simulation around a planet, hvz, gas iy . The color contour of the arrows frompurple to orange represents the difference in the z coordinates between the ending and starting points of orbital calculations.

4.4. Discussion of this chapter

125

4.4

Discussion of this chapter

4.4.1

Caveat: the steady-state solution

In Sect. 4.3.6, we only provided the steady-state solution of the 1D advection-diffusion
equation (Eq. (4.12)). The timescale to reach the steady state depends on an inward
advective flux of dust from the outer region of the disk. The drift speed of dust is
low when the Stokes number is small. Thus, the advective flux of dust is small for a
fixed initial dust surface density. It might take a long time to reach a steady state.
The timescale to reach the steady state can be estimated by the timescale for thedust concentration in a ring. The dust accumulates outside the planetary orbit withthe timescale taccum :taccum ∼

Mring
Ṁd

,

(4.17)

where Mring is the total mass of the mass of the ring and Ṁd is the radial inward
mass flux of dust. The ring mass can be described by
Mring = M̃ring H 2 Σd,0 ,(4.18)

where M̃ring is the dimensionless mass of the ring that is obtained from our numerical results (Figs. 4.10–4.12). The initial dust surface density can be described by
1 + St2StṀd
ṀdΣd,0 '=
2πr |vdrift |2πr2 ΩK h2!

d ln pd ln r! −1.

(4.19)

From Eq. (3.4), the gas surface density in a steady accretion disk can be rewritten as(Ida et al., 2016)
10−5
Σg '
6π 2

Ṁ∗
10−8 M /yr

!

M∗
1M

!−1/2

r
1 au

!1/2

αacc
10−3

! −1
h−2 M /au2 .

(4.20)

From Eqs. (4.19) and (4.20), Σd,0 can be expressed as
10−5
Σd,0 'π2

1 + St2
St!

Ṁd102 M⊕ /Myr

!

M∗
1M

!−1/2

r
1 au

!1/2
h−2 M⊕ /au2 .

(4.21)

Thus, the timescale for the dust accumulation can be estimated by
10−7 × M̃ring
taccum 'π2

1 + St2
St

!

M∗
1M

!−1/2

r
1 au

!5/2
Myr.

(4.22)

Here we consider the planet with m = 0.1, the Stokes number St = 10−4 , andthe turbulent parameter αdiff = 10−4 . When Mhw = 0.03 (∼ 1 au; Fig. 4.11e),Eq. (4.22) gives taccum ∼ 6000 yr (corresponding to ∼ 3.8 × 104 Ω−1 ). When Mhw =0.1 (∼ 50 au; Fig. 4.11f), Eq. (4.22) gives taccum ∼ 133 Myr (corresponding to ∼2.3 × 106 Ω−1 ).
From these estimations, in the inner region of the disk (. 1 au), the dust surface
density can reach the steady state within the typical disk lifetime of a few megayears(Haisch et al., 2001), but it seems difficult to reach the steady state in the outer region(& 10 au).

126

4.4.2

Chapter 4. Dust ring and gap formation by the planet-induced gas flow

Time evolution of the dust surface density

In this section, we investigate the time evolution of the dust surface density influenced by the planet-induced gas flow. Here we only consider the changes in Σdfor a fixed Stokes number. We neglect complex processes such as dust growth andfragmentation. As described in Sect. 4.2.5, we integrated the 1D advection-diffusion
equation (Eq. (4.8)) using a finite-volume method. We assumed Σd,0 = 1 in the initialstate. In the previous section, we mentioned that the timescale to reach the steadystate depends on an inward advective flux of dust from the outer region of the disk.
The drift timescale is longer for small Stokes number. Moreover, we found that an
apparent dust ring and (or) gap formation in the distribution of small dust grains(St . 10−3 ) by the planet-induced gas flow occurs when m ≥ 0.1 and αdiff . 10−4
(Sect. 4.3.6). Thus, in this section, we restrict our attention to the higher-mass planets with m ≥ 0.1, weak turbulence, αdiff ≤ 10−4 , and the small Stokes numbers,
St ≤ 10−3 .
Figure 4.16 shows the time evolution of the dust surface density, where we assumed m = 0.3, St = 10−3 , and αdiff = 10−4 . In this figure, the dust surface density
almost reaches the steady state in t ∼ 105 Ω−1 . The depth of the dust gap around theplanetary orbit increases as the time increases. We found that the dust accumulate
inside the planetary orbit in an early phase of time evolution, t . 104 Ω−1 . The dust
ring inside the planetary orbit was not found in the steady-state profile (e.g., Figs.
4.10–4.12) because the turbulence diffusion smooth the dust surface density. For theparameter set in Fig. 4.16, the planet-induced gas flow is in the flow-shear regime(two outflows occur both inside and outside the planetary orbit). The outflow toward the inside of the planetary orbit enhances the inward drift of dust particles.
Since we assumed the uniform dust surface density in the initial state, in the earlystage of the time evolution, the dust accumulates temporarily inside the planetary
orbit due to the outflow toward the inside of the planetary orbit.

4.4. Discussion of this chapter

F IGURE 4.16: Time evolution of the dust surface density for different
Mach numbers, Mhw = 0.01 (panel a), Mhw = 0.03 (panel a), and
Mhw = 0.1 (panel c). We set m = 0.3, St = 10−3 , and αdiff = 10−4 .
The dashed lines represent the steady-state solutions obtained in Sect.
4.3.6. The color contour represents the time. We used the azimuthally
and vertically averaged drift velocity of dust obtained from numerical simulations, vd,adv = hv x,d iy,z , to solve the 1D advection-diffusion
equation.

127

128

Chapter 4. Dust ring and gap formation by the planet-induced gas flow

Figure 4.17 shows the time evolution of the dust surface density for the smallest
Stokes number considered in this chapter. In this figure, we assumed m = 0.3,
St = 10−4 , and αdiff = 10−4 . The dust surface densities reach the steady state in
Figs. 4.17a (Mhw = 0.01) and b (Mhw = 0.03), but in panel c (Mhw = 0.1), there is
still deviation between the dust surface density at t = 105 Ω−1 and the steady-state
profile.
In Fig. 4.17c, where we adopt the largest Mach number considered in this chapter(M = 0.1), the influence of the outflow toward the outside of the planetary orbit
is stronger than that of the outflow toward the inside of the planetary orbit. The
steady-state solution predicted that the apparent dust ring could form outside theplanetary orbit. However, since the drift speed of dust is low, the timescale for thedust accumulation due to the outflow toward the outside the planetary orbit is long.
Thus, the dust surface density does not reach the steady state within the calculation
time, t = 105 Ω−1 .
We found the difference between the depth of the dust gap at t = 105 Ω−1 andthat in the steady-state, where the depth of the dust gap becomes deeper than theprediction of the steady-state solution. This trend can be understood as follows: (1)the outflow toward the outside (inside) of the planetary orbit inhibits (enhances) the
inward drift of dust around the planetary orbit, leading to the formation of deep gap
in the early stage of the time evolution. (2) Since we assumed constant advective flux
at the outer boundary, the dust continues to be supplied near the planetary orbit. (3)The dust gap is gradually filled due to the turbulence diffusion, and then the dust
surface density reaches the steady state.

4.4. Discussion of this chapter

129

F IGURE 4.17: Same as Fig. 4.16, but with m = 0.3, St = 10−4 , andαdiff = 10−4 .

A significant deviation between the dust surface density at the end of calculation,
105 Ω−1 , and that in the steady-state can be seen in Fig. 4.18, where we assumed

130

Chapter 4. Dust ring and gap formation by the planet-induced gas flowm = 0.1, St = 10−4 , and αdiff = 10−5 . In Fig. 4.18a (Mhw = 0.01), the dust ringremains inside the planetary orbit. In Figs. 4.18b (Mhw = 0.03) and c (Mhw =0.1), the concentration of the dust into a ring at t = 105 Ω−1 differs significantly
from that in the steady state. The steady-state solution predicted that an excessive
dust concentration outside the planetary orbit would appear. However, similar to
Fig. 4.17c, since the drift speed of dust is low, the dust surface density does not reachthe steady state within the calculation time. On the other hand, we found the deep
gap around the planetary orbit (Figs. 4.18b and c), which is not found in the steadystate. This is caused by temporary dust filtration due to the outflow toward theoutside of the planetary orbit as mentioned in the preceding paragraph.
The above findings imply that both the dust ring and gap could form even in
the flow-headwind regime (headwind-dominated regime) within the typical disk
lifetime in aerodynamically small dust grain distribution under weak turbulence. In
this section, we considered the time evolution of the dust surface density for a fixed
Stokes number. In reality, the time evolution of dust surface density is determinedby complex processes such as dust growth and fragmentation. These additional
processes should be included in further studies.

4.4. Discussion of this chapter

F IGURE 4.18: Same as Fig. 4.16, but with m = 0.1, St = 10−4 , andαdiff = 10−5 .

131

132

Chapter 4. Dust ring and gap formation by the planet-induced gas flow

4.4.3

Analytical estimation of the drift velocity of dust influenced by theplanet-induced gas flow

High-resolution 3D hydrodynamical simulations are required to investigate the influence of the planet-induced gas flow on dust dynamics, which are computationallyexpensive. The nature of the flow field depends on the planetary mass and the headwind of the gas. The planetary mass and the headwind of the gas at the location of
the planet change in the growth process of the planet. Here we aim to analyticallymodel the azimuthally and vertically averaged drift velocity of dust influenced by
the planet-induced gas flow (e.g., Figs. 4.8b and 4.9b). Similar to Chapter 3, weconsider the motion of dust particles in the frame co-rotating with the planet (e.g.,3.1), but in this section we only consider the midplane region of the disk. The x- andy-directions are similar to those used in the dust trajectory simulations (Chapter 3).
We first assume that a particle is initially on a circular orbit in the far field (|y| →∞). We define the difference between the orbital radii of the planet and the particle
as bd . The change of the radii of the particle after the encounter with the planet isgiven by (Muto and Inutsuka, 2009):
∆bd =

St
1 + St2Z ∞−∞Fx (t)dt +2St2
1 + St2Z ∞−∞Fy (t)dt,(4.23)

where Fx (t) and Fy (t) are the forces acting on the particle in the x- and y-directions.
The integration with respect to t in Eq. (4.23) must be performed along the particle’s
trajectory. Muto and Inutsuka, 2009 considered the gas drag caused by the deviationof the gas velocity from Keplerian rotation, planet’s gravity, steady accretion flowof the disk, and the influence of the spiral density wave excited by the planet in
Eq. (4.23). In Sect. 4.3.6, we found that the dust substructure formation by gas flow
occurs only when St . 10−2 . Thus, in this section, we restrict our attention to thesmall Stokes numbers, St . 10−2 . In such cases, the change of the orbital radius of
the particle is determined by the gas drag force.
For the y-direction, we consider the dimensionless drag force due to the subKeplerian rotation of the disk
Fy (t) = −

Mhw
ey .
St

(4.24)For the x-direction, we consider the influence of the outflow of the gas
Fx (t) =v x,out ( x )ex ,St(4.25)

where v x,out ( x ) is given by 2.53 (Chapter 2). We assumed that the outflow speed
in the x-direction is independent of time. We did not consider the influence of thesteady accretion flow caused by angular momentum transport of the disk, which isincluded in Muto and Inutsuka, 2009. Though the influence of the steady accretion
flow is important for small particles, the influence of the outflow is much stronger
by orders of magnitude.
We consider the changes of bd during the encounter in a box of finite size, Ly ,and set the integration range from 0 to T for the y-direction and from 0 to Tout forthe x-direction, respectively, where T and Tout are the times taken for the particle
to cross the box and the outflow region. We substituted Eqs. (4.24) and (4.25) into

4.4. Discussion of this chapter

133

Eq. (4.23) and obtained
∆bd =

2St
1
v
( x ) Tout −Mhw T.
2 x,out
1 + St
1 + St2(4.26)The y-component of the velocity of the particle is dominated by the backgroundshear speed, even when the dust passes the outflow region (Kuwahara and Kurokawa,
2020b). Thus, we set
T'

Ly
,
v∞
y,d

(4.27)

Tout '

Ly,out,v∞y,d(4.28)

where Ly,out is given by Eq. (2.33) (Chapter 2). In Sect. 4.2.4, we considered the global
distribution of dust (in the full azimuthal direction of the disk), and calculated theazimuthally and vertically averaged drift velocity. Thus, considering one synodical
orbit, we set Ly = 2π ( a/H ), where we assumed a  bd . From Eqs. 4.26–4.28, we
obtain∆bd,T v x,out ( x ) Ly,out H
2St
=
Mhw .
−
2
2π
a
1 + St
1 + St2

vdust ( x ) ≡(4.29)

The first and second terms in the right-hand side of Eq. (4.29) represent the fluctuation of the drift velocity of dust due to the outflow of the gas and the unperturbed
drift speed, vdrift , respectively. Equation (4.29) describes the dust velocity in the xdirection at the midplane influenced by the planet-induced gas flow.
So far, we restrict our analysis to the midplane region. However, as shown in
Sect. 2.3.5, the outflow extends in the vertical direction (Figs. 2.13 and 2.14). The
dust scale height increases as the turbulence strength increases for a fixed Stokes
number (Eq. (2.71)). To include these 3D effects, we derive the vertically averaged
outflow speed in the x-direction weighted by the dust density distribution. Based on
Sect. 2.3.5, the velocity distribution of the outflow in the z-direction can be expressedby the exponential function (Eq. (2.35)). Thus, we introduced
" 
2 #
z
v x,out ( x ) → v x,out ( x, z) ≡ v x,out ( x ) × exp −,(4.30)
Ratm,fit
for the outflow velocity. The vertically averaged outflow velocity weighted by thedust density distribution can be described byR∞ρd v x,out ( x, z)dz
hv x,out ( x )iz = 0 R ∞ρd dz
0Ratm,fit=qv x,out ( x ).
(4.31)
2
Ratm,fit + 2Hd2
Equation (4.31) represents the effective velocity of the outflow in the x-direction fordrifting dust. It depends only on the x-coordinate. In Eq. (4.25), we assumed thatthe outflow speed is constant with time. In Eq. (4.31), the x-coordinate must be134

Chapter 4. Dust ring and gap formation by the planet-induced gas flowcorrelated with the x-coordinate of the starting point of dust in the far field, x = xs .
However, as shown in Fig. 4.2, only the dust particles coming from the outside
the horseshoe flow reach the outflow region. Thus, a correction must be performed
for the x-coordinate in Eq. (4.31) by the half-width of the horseshoe region. In otherwords, we correct the positions of positive and negative peaks in the velocity distribution of the gas, v x,out ( x ), by the half-width of the horseshoe region. The xcoordinates of the edges of the horseshoe region for the incoming dust (the dust
particles coming from the first and third quadrants of the x-y plane) are given by
wHS − | xcor,g | and wHS + | xcor,g | for the dust outside and inside the planetary orbit,
respectively. The effective velocity of the outflow in the x-direction for drifting dust
can be described byhvHS
x,out ( x )iz = q

Ratm,fit

R2atm,fit + 2Hd2#"+
( x − xout
− (wHS − | xcor,g |))2+
× v x,out exp −2( D + )2
"
#!
−
2
+
(
w
+
|
x
|))
(
x
−
x
cor,g
HS
out
+v−
,
x,out exp −22( D − )(4.32)±
where xout
and wHS are given by Eqs. (2.52) and (2.27), respectively. We substituted
Eq. (4.32) into Eq. (4.29) , and obtained the drift velocity of dust influenced by theoutflow as
 
hvHS
2St
H
x,out ( x )iz Ly,out
vdust ( x ) =−Mhw .
(4.33)
22π
a
1 + St
1 + St2

Figures 4.19–4.21 compare our analytic formula (Eq. (4.33)) with the results obtained from numerical simulations. Equation (4.33) reproduces the results obtained
from numerical simulations for a wide range of planetary mass, Mach number of theheadwind of the gas, Stokes number, and the turbulent parameter. The additional
results are shown in Appendix B.
We first describe overall trends of Figs. 4.19–4.21. In this section, we only consider the small dust particles, St . 10−2 . The strong gas drag force dominates the
motion of these small particles. Thus, we would expect that the velocity distributionof dust influenced by the planet-induced gas flow to be correlated with that of gas.
The amplitude and width of peaks in Eq. (4.33) are determined by the maximum
speed of the outflow and width of the outflow region in the x-direction. In Sect.
2.3.6 we derived an analytic formula for the gas velocity distribution, which shows
excellent agreement with the results of hydrodynamical simulations. Thus, in Figs.
4.19–4.21, the amplitude and width of the positive and negative peaks in vdust ( x )match with those in numerical results, hv x,d iy,z .
We found that the x-coordinate of the position of peaks in our analytic formula,vdust ( x ), deviates slightly from those in numerical results, hv x,d iy,z . This is due tothe correction of the peak position made in Eq. (4.32), where we shifted the peak positions by the half-width of the horseshoe region. While our hydrodynamical simulations were performed in the local frame, the half-width of the horseshoe region,
Eq. (2.27), was introduced based on the results of global 3D hydrodynamical simulations (Paardekooper and Papaloizou, 2009; Masset and Benítez-Llambay, 2016;
Jiménez and Masset, 2017). This leads to a mismatch between our analytic formula4.4. Discussion of this chapter

135

and numerical results.
Equation (4.33) has only one or two peaks inside or (and) outside the planetary
orbit and has a smooth distribution within the horseshoe region, | x | . wHS . On theother hand, the numerical results show the fluctuation within the horseshoe region,
which is not included in our analytic formula. The velocity fluctuation within thehorseshoe region affects the shape of the bottom of the dust gap, but does not affectthe other properties of the dust ring and gap, such as width and amplitude (as shown
later in Sect. 4.4.4).
A second large peak may appear near the planetary orbit in the numerical results
under strong headwind condition (dashed lines in Fig. 4.20g,h, and i). In the vicinityof the planet, the recycling flow swirled from − x to + x direction due to the Coriolis
force may affect the motion of dust particles (e.g., Fig. 2.8). Since we do not consider
the velocity of the gas within the region where | x | ≤ Ratm (Eq. (2.53); Sect. 2.3.7),
such effect is not included in Eq. (4.33), leading to a mismatch between our analytic
formula and numerical results.

F IGURE 4.19: Difference of the azimuthally and vertically averaged
velocity of dust particles from the unperturbed steady-state drift velocity. We assumed m = 0.03 and αdiff = 10−4 . The solid anddashed lines are obtained from Eq. (4.33), vdust ( x ), and numericalsimulations, hv x,d iy,z , respectively. Different rows and columns correspond to different Stokes numbers and different Mach numbers, respectively.We set H/a = 0.03 for this plot.

136

Chapter 4. Dust ring and gap formation by the planet-induced gas flow

F IGURE 4.20: Same as Fig. 4.19, but we assumed m = 0.1 and αdiff =10−5 .

F IGURE 4.21: Same as Fig. 4.19, but we assumed m = 0.3 and αdiff =10−4 .

4.4. Discussion of this chapter

4.4.4

137

Simulation of dust surface density using the analytic formula

We calculate the dust surface density using an analytic formula of the drift velocityof dust influenced by the planet-induced gas flow, vdust ( x ) (Eq. (4.33)). Both thesteady-state solution and the time evolution of the dust surface density are obtainedby solving the 1D advection-diffusion equation (Eq. (4.8) and Eq. (4.12)), where we
adopt vd,adv = vdust ( x ).
Steady-state solution
Figures 4.22–4.24 show the steady-state dust surface density. We compare the results
obtained by solving the 1D advection-diffusion equation where the advective velocity of dust was set to the analytic formula, vd,adv = vdust ( x ), with the results where
vd,adv = hv x,d iy,z .
Overall trend indicates that the width of the dust gap and positions of dust ringand gap in the steady-state solution obtained from Eq. (4.12) with the advective velocity vd,adv = vdust ( x ) are in agreement with those with vd,adv = hv x,d iy,z . This isbecause the x-coordinate of the position peaks in our analytic formula for the dust
velocity influenced by the planet-induced gas flow is in agreement with that obtained from numerical result (Figs. 4.19–4.21). However, the depth of the dust gap
and the amplitude of the dust ring do not always match each other.
In Fig. 4.22, where we assumed m = 0.03, the steady-state solution obtained
from Eq. (4.12) with the advective velocity vd,adv = vdust ( x ) are in agreement withthose with vd,adv = hv x,d iy,z . For the parameter set in this figure, the outflow speed
is low and the influences of the outflow on the dust surface density is small. Thus,
the discrepancy between the steady-state solutions obtained from Eq. (4.12) withvd,adv = vdust ( x ) and vd,adv = hv x,d iy,z is also small.
In Figs. 4.23 and 4.24, however, the steady-state solutions obtained from Eq. (4.12)
with vd,adv = vdust ( x ) differs significantly from those with vd,adv = hv x,d iy,z . For instance, an order of magnitude difference in the steady-state solutions obtained from
Eq. (4.12) with vd,adv = vdust ( x ) and that with vd,adv = hv x,d iy,z can be seen in thedepth of the dust gap in Fig. 4.23g (m = 0.1 and Mhw = 0.01) and Figs. 4.24d–f(m = 0.3 and Mhw = 0.01–0.1). The orders of magnitude difference in the concentration of the dust into a ring between the steady-state solutions obtained from
Eq. (4.12) with vd,adv = vdust ( x ) and that with vd,adv = hv x,d iy,z can be seen in Figs.
4.23d–f (m = 0.1 and Mhw = 0.01–0.1) and 4.24d–f (m = 0.3 and Mhw = 0.01–0.1).
These significant deviations suggest that the simulations of the dust surface density
are highly sensitive to the advective velocity of dust when the assumed Stokes number and turbulence strength are small. Under weak turbulence, the large surface
density gradient needs to achieve a steady state. A factor of difference of the amplitude of peaks in vd,adv could result in the orders of magnitude difference of the dust
surface density.

138

Chapter 4. Dust ring and gap formation by the planet-induced gas flow

F IGURE 4.22: Steady-state dust surface density for different Stokes
numbers and Mach numbers. The advective velocities of dust were
set to the analytic formula, vd,adv = vdust ( x ) (solid lines), and thedrift velocity of dust that calculated numerically, vd,adv = hv x,d iy,z(dashed lines). We set m = 0.03 and αdiff = 10−5 . Different rows
correspond to different Stokes numbers, St. Different columns correspond to different Mach numbers, Mhw . This figure should be compared to Fig. 4.19.

4.4. Discussion of this chapter

F IGURE 4.23: Same as Fig. 4.22, but we set m = 0.1 and αdiff = 10−5 .
This figure should be compared to Fig. 4.20.

F IGURE 4.24: Same as Fig. 4.22, but we set m = 0.3 and αdiff = 10−4 .
This figure should be compared to Fig. 4.21.

139

140

Chapter 4. Dust ring and gap formation by the planet-induced gas flow

Time evolution of the dust surface density
In this section, similar to Sect. 4.4.2, we compute the time evolution of the dust
surface density using our analytic formula for the drift velocity of dust influenced
by the planet-induced gas flow.
Figures 4.25–4.26 show the time evolution of the dust surface density. We compare the results obtained by solving the 1D advection-diffusion equation where theadvective velocity of dust was set to the analytic formula, vd,adv = vdust ( x ), with theresults where vd,adv = hv x,d iy,z .
Similar to the steady-state solution, overall trend indicates that the width of thedust gap and positions of dust ring and gap in the dust surface density obtained
from simulations with the advective velocity vd,adv = vdust ( x ) are in agreement withthose with vd,adv = hv x,d iy,z . This is because the x-coordinate of the position peaks in
our analytic formula for the dust velocity influenced by the planet-induced gas flow
is in agreement with that obtained from numerical result (Figs. 4.19–4.21). Furthermore, the depletion of dust around the planetary orbit over time and the smoothingof the dust ring (further concentration of the dust into a ring) inside (outside) theplanetary orbit over time are reproduced by the simulations with the advective velocity vd,adv = vdust ( x ) (e.g., Fig. 4.25).
However, the dust surface density obtained from simulations with vd,adv = vdust ( x )and that with vd,adv = hv x,d iy,z starts to deviate in some parameter sets at t &104 –105 Ω−1 , as shown in Figs. 4.26e and f (m = 0.1, St = 10−4 , and αdiff = 10−5 ) ,and Fig. 4.27 (m = 0.3, St = 10−3 , and αdiff = 10−4 ).

4.4. Discussion of this chapter

F IGURE 4.25: Time evolution of the dust surface density for different
Mach numbers.. The advective velocities of dust were set to the analytic formula, vd,adv = vdust ( x ) (left column), and the drift velocityof dust that calculated numerically, vd,adv = hv x,d iy,z (right column).
We set m = 0.3, St = 10−4 , and αdiff = 10−4 . The color contour
represents the time.

141

142

Chapter 4. Dust ring and gap formation by the planet-induced gas flow

F IGURE 4.26: Same as Fig. 4.25, but we set m = 0.1, St = 10−4 , andαdiff = 10−5 .

4.4. Discussion of this chapter

143

F IGURE 4.27: Same as Fig. 4.25, but we set m = 0.3, St = 10−3 , andαdiff = 10−4 .

4.4.5

Comparison to previous studies

Many authors have investigated the planet-driven dust ring and gap formation in a
disk. One of the major differences between the conventional models and our modelis that we focus on non-gas-gap-opening, low-mass planets. Consequently, the proposed mechanism works well for the particles with small Stokes numbers.
Gas-gap-opening planets have long been a plausible candidate to explain the
dust rings and gaps in disks. A giant planet carves a density gap in a gas disk (e.g.,
Lin and Papaloizou, 1986), and then the particles are trapped at the edge of the gas
gap. The depth of the dust gap increases with the Stokes number (Zhu et al., 2012;
Weber et al., 2018). When particles are tightly coupled with the gas, they can passthrough the gas gap due to the turbulent diffusion or the viscous accretion flow (Riceet al., 2006; Zhu et al., 2012; Pinilla et al., 2016; Weber et al., 2018; Bitsch et al., 2018;
Drażkowska et al., 2019).
Several studies have shown that a low-mass planet that does not carve a significant gas gap in a disk can form dust substructures. Paardekooper and Mellema, 2006
found that ∼ 15 M⊕ planet perturbs the dust surface density and then forms the dust
gap in the distribution of the dust particles larger than 150 µm. From the analyticalpoint of view, Muto and Inutsuka, 2009 showed that ∼ 2 M⊕ can open the dust gap
when St & 0.1 in the absence of the global pressure gradient. The axisymmetric dust

144

Chapter 4. Dust ring and gap formation by the planet-induced gas flow

depletion can be generated in an inviscid disk due to steepening of nonlinear density waves launched by ∼ 8 M⊕ planet when St & 2 × 10−3 (Zhu et al., 2014). The
annular enhancements in the dust disk are caused by the pressure perturbation by alow-mass embedded planet (∼ 15 M⊕ ) when St & 2 × 10−2 (Rosotti et al., 2016). The
depth of the dust gap or the concentration of the dust into a ring increases with the
Stokes number (Zhu et al., 2014; Rosotti et al., 2016). Dust gap forms due to the tidal
torque from a planet (∼ 15 M⊕ ) when the Stokes number is large, St & 1 (Dipierro
et al., 2016; Dipierro and Laibe, 2017). These previous studies did not consider the
influences of the radial outward gas flow induced by an embedded planet, which is
the mechanism we study in this chapter.
Compared with the studies mentioned above, our study reveals the opposite
trend: the dust ring and gap are likely to form when the Stokes number is small.
The properties of the dust ring and gap, such as width, depth, and amplitude, aremore pronounced for the small Stokes number (Figs. 4.10–4.12), because smaller
dust particles are more sensitive to the gas flow. Our local approach cannot explicitly handle the global effects that investigated in the previous studies (e.g., shallow
gas-gap opening by a low-mass planet), but a local model is needed to reveal the
influence of the outflow of the gas.
Here we consider how our model can coexist with the other dust substructureformation mechanisms. In an inviscid disk, low-mass planets could open shallow
gas gaps through nonlinear density wave steepening (Goodman and Rafikov, 2001;
Rafikov, 2002; Zhu et al., 2014; Dong et al., 2017a; Dong et al., 2018). Linear waves excited by a low-mass planet steepen to shocks after they propagate a distance (Goodman and Rafikov, 2001):
xsh ≈ 0.93

γ+1
m
12/5

!−2/5
H,

(4.34)

where γ = 1.4 is the adiabatic index. Equation (4.34) gives xsh ∼ 1.5–3.8 H withina range of the planetary masses considered in this chapter, m = 0.03–0.3. These
values are larger than the widths of the outflow of the gas, |wout | < 2/3 H. We
would expect that the large dust particles accumulate at | x | & xsh due to the shallow
gas-gap opening. Small particles that are tightly coupled with the gas, on the other
hand, can diffuse or pass through the gas gap due to the turbulence, the viscousaccretion flow, or both (e.g., Rice et al., 2006). These small dust particles are trappedby the outflow of the gas. Thus, when we consider the shallow gas-gap opening in
an inviscid disk, we would expect that the locations of the dust rings and the widthsof the dust gaps in the populations of the large and small dust grains do not match
each other. The dust ring in the large dust population may appear ∼ 1 H farther
away than that in the small dust population. The width of the dust gap in the large
dust population may be wider than that in the small dust.

4.4.6

Caveat: the spatial distribution of dust in the global domain

The distribution of dust particles outside the local computational domain of orbital
calculation is nontrivial. The planet-induced gas flow deflects the trajectories of particles and creates the voids in the x-y plane (Kuwahara and Kurokawa, 2020a; Kuwahara and Kurokawa, 2020b, see also Fig. 4.2b). Outside the local domain, however,these voids might be filled by the drift of particles in the x-direction, and turbulent
diffusion. Our local simulations could not handle the void closing. In estimating
dust drift velocity, we assumed that the effects of the planet-induced gas flow are4.4. Discussion of this chapter

145

negligible outside the domain of the orbital calculations (namely, the void is closed
outside the domain). This assumption is a conservative approach to give a minimum
effect of the planet-induced gas flow on dust drift. In other words, if a planet can
open a gap in our 1D model under this assumption, it means that gap could form
even if we treat the global dust motion explicitly.

4.4.7

Caveat: non-migrating planet

Throughout this thesis, we considered non-migrating planets. Low-mass planets
would undergo the fast inward migration (the so-called type I migration; Ward,
1986; Tanaka et al., 2002). Timescale of the type I migration can be described byM∗ M∗
τtype1 ∼Mpl Σg a2Ha!3Ω−1 = 104

m
0.1

! −1

Σg a2 /M∗
10−3

! −1
Ω −1 ,(4.35)

which is comparable or shorter than the timescale for the dust ring and gap formation due to the planet-induced gas flow (e.g., Sect. 4.4.2). When the migration
timescale is shorter than the timescale for planet-driven dust ring and gap formation, the position of dust ring and gap does not necessarily coincide with the planetary orbit (Kanagawa et al., 2021), and may lead to a deviation of the dust surface
density profile from the steady-state solution.
The migration timescale could be affected by the planet-induced gas flow (Funget al., 2015). Fung et al., 2015 reported that under the influence of the planet-induced
gas flow the embedded planet with m = 0.56 migrates three times slower than predicted by linear calculations. The influence of the planet-induced gas flow on thetype I migration is not fully understood. Further simulations for a wide range ofplanetary masses and Mach numbers of the headwind of the gas are required.

147

Chapter 5

Discussion and outlook
5.1

Implications for the origins of the architecture of a planetary system

In this section, we propose a formation scenario of planetary systems based on thepebble accretion model. We first summarize two parameters: the turbulent parameter and the size distribution of the solid materials in a disk, and then give the global
structure of the disk in terms of these parameters. Finally we introduce our scenario
based on our results.

5.1.1

Turbulence in a protoplanetary disk

Protoplanetary disks are dissipated with the timescale of few million years due toangular momentum transport in disks. One of the possible origin of the mechanismsof angular momentum transport is gas turbulence. In this chapter, we consider thegas turbulence stirs dust in the vertical direction to the disk and also causes theturbulent diffusion of dust in the radial direction. Assuming an isotropic turbulence,
we use the notation of αdiff for the turbulence strength of the gas.
The conventional model considered that the disk was originated by the magnetorotational instability (MRI; Balbus and Hawley, 1991). The MRI can develop in the
innermost region of the disk (. 0.1 au), where the temperature of the disk gas is highenough (& 1000 K) and thus the ionization degree of the gas is high. The turbulence
strength produced by the MRI has on the order of αdiff ∼ 10−3 –10−2 (e.g., Davis et al.,2010; Simon et al., 2012). However, the MRI would not operate at larger radii, & 0.1
au. The MRI could be damped by Ohmic and ambipolar diffusion, which create an
MRI-inactive region (dead-zone; Gammie, 1996). The dead-zone lies between a few
times 0.1 au and about a few tens of au in a disk, in which the value of the turbulent
parameter should be smaller, αdiff . 10−4 .
Hydrodynamic instabilities are other mechanisms that generate turbulence indisks. The three instabilities are known to operate in the disk: vertical shear instability (VSI, Urpin and Brandenburg, 1998; Urpin, 2003; Nelson et al., 2013; Barker
and Latter, 2015; Lin and Youdin, 2015), convective overstability (COV, Klahr and
Bodenheimer, 2003; Klahr and Hubbard, 2014; Lyra, 2014; Latter, 2016), subcriticalbaroclinic instability (SBI, Klahr and Bodenheimer, 2003; Petersen et al., 2007; Lesurand Papaloizou, 2010), and zombie vortex instability (ZVI, Marcus et al., 2013; Marcus et al., 2015). These instabilities develop in different regions of the disk, namely, inregions of different opacities. The disk opacities determine the thermal cooling timesof disk gas and determine which hydrodynamic instability is dominant. Assuming
a MMSN disk model, Malygin et al., 2017 introduced a mapping of hydrodynamic
instabilities. In the adiabatic case (corresponding to . 1 au), the ZVI dominates andgenerates αdiff ∼ 10−3 ; in the isothermal case (corresponding to & 10 au) , the VSI

148

Chapter 5. Discussion and outlook

dominates and generates αdiff ∼ 10−4 ; in between these regions (corresponding to∼ 1–10 au), the COV dominates and generates αdiff ∼ 10−4 –10−2 . Self-regulation
may limit the turbulence strength produced by ZVI to as little as αdiff ∼ 10−5 . The
VSI-driven turbulence may be suppressed by the toroidal magnetic fields in a disk
(Cui and Bai, 2020; Cui and Lin, 2021), or dust evolution (Fukuhara et al., 2021).
When the dust grows to ∼ 0.1–1 mm at > 10 au, the VSI may be stabilized completely (Fukuhara et al., 2021).
The turbulence strength beyond a few tens of au can be constrained observationally. Recent observations suggest low levels of turbulence, αdiff . 10−5 –10−3 , fromdust settling (Pinte et al., 2015; Stephens et al., 2017; Dullemond et al., 2018; Villenave et al., 2022), or direct molecular emission line measurements (Flaherty et al.,2015; Flaherty et al., 2017; Flaherty et al., 2018). Dullemond et al., 2018 considered
dust trapping in radial pressure bumps as a mechanism to produce dust rings in fivedifferent systems observed by the ALMA large program DSHARP campaign. The
constrained αdiff /St differs in each ring but it is larger than ∼ 10−2 . The relatively
weak level of turbulence suggested from the observations is also preferable for theformation of ring structures by secular gravitational instability (Takahashi and Inutsuka, 2014; Takahashi and Inutsuka, 2016; Tominaga et al., 2019).

5.1.2

Size distribution of dust

The planet-induced gas flow strongly influences on the distribution of aerodynamically small dust, St . 10−2 –10−1 . Such a small Stokes number corresponds to thedust particles of ∼ 0.1–100 mm in the outer region of the disk, > 10 au, for the steadyviscous-accretion disk model assumed in this thesis (Fig. 3.2). The size of the dust
particles in disks can be constrained observationally and theoretically. Recent observations constrain the maximum dust size in the HL Tau disk: ∼ 0.1mm (polarization
observations at millimeter wavelengths; Kataoka et al., 2016) or ∼ 1 mm (spectralenergy distribution of the millimeter emission; Carrasco-González et al., 2019). Dust
coagulation models can also constrain the size of the dust in a disk. Okuzumi and
Tazaki, 2019 has shown that the dust particles covered with the H2 O-ice mantle cangrow up to ∼ 50 mm (corresponding to St ∼ 5 × 10−2 in the HL Tau disk) between
the H2 O and CO2 snow lines. Inside the H2 O snow line, where the dust grains exist
as nonsticky silicate grains, the maximum dust size in their model is ∼ 1 mm (corresponding to St ∼ 10−3 in the HL Tau disk). Outside the CO2 snow line, since the CO2
ice is as nonsticky as silicate grains (Musiolik et al., 2016a; Musiolik et al., 2016b), theupper limit of the size of the dust in their model is ∼ 0.1–1 mm (corresponding toSt ∼ 10−3 –10−2 in the HL Tau disk).
Given such a size distribution, the growth of the planets inside the H2 O snowline
may be expected to be slower than that between the H2 O and CO2 snowlines if
we consider the reduction of accretion probability of small dust due to the planetinduced flow.

5.1.3

Planet formation via pebble accretion in the planet-induced gas flow

Now we introduce the formation scenario of planetary systems based on our results
and above discussions. Dust particles with St ∼ 10−3 –100 can be referred to as
pebbles. We adopted three hydrodynamic instabilities as the possible origins of the
turbulence: ZVI, COV, and VSI. We divided the disk into three sections according
to the previous studies and assumed turbulence strength in each section as: αdiff ∼10−5 (. 1 au), αdiff ∼ 10−3 (∼ 1–10 au), and αdiff ∼ 10−4 (& 10 au; Malygin et al.,5.1. Implications for the origins of the architecture of a planetary system

149

2017; Lyra and Umurhan, 2019). The outermost region (> 100 au) may be MRIactive, but here we set αdiff ∼ 10−4 from the standpoint of observational constraints.
Given the size distribution of the solid materials in a disk (Okuzumi and Tazaki,2019), we assumed that the pebbles have St ∼ 10−3 (. 1 au), St ∼ 3 × 10−2 (∼ 1–10au), and St ∼ 3 × 10−3 (& 10 au).
In Chapter 3, we estimated the accretion rate of dust for the fixed planetary massand Mach number based on the results of orbital calculations (Fig. 3.7). In this section, we aim to obtain the growth timescale of the planet for the various planetary
masses and orbital distances under the influence of the planet-induced gas flow. Assuming an uniform dust surface density and the Stokes gas drag regime, we use thefollowing formula for the accretion rate of dust onto the planet influenced by theplanet-induced gas flow (Okamura and Kobayashi, 2021):"

bOK21
Pcol,3D = ( Pcol,2D )−2 + Pcol,2D ∞0.65Hd

−2 #−1/2,(5.1)

where Pcol,2D is the collision rate between the dust and the planet in the 2D limit andOK21 is the maximum impact parameter of the accreted dust particles influenced byb∞
the planet-induced gas flow. These are given by the following formulae:
Pcol,2D = min( Pcol,ss , Pcol,set , Pcol,ho ),

(5.2)

where
!1/4
ρ• cs λmfp
Pcol,ss = 2 6C
St
R2Hill Ω,ρg RHill Ω RHill2/3
Pcol,set = 3 2COK21 St
R2Hill Ω,
s
"
#
3St
3
RBondi−ξ
Pcol,ho = 2R2 Ω,RHill ( RBondi /RHill )2( RBondi /RHill ) Hill

√

and

OK21


!1/4
!1/8

p


cs λmfp8
OK21 ρ•
St
RHill (when Pcol,2D = Pcol,ss ),Cρg RHill Ω RHill
 3
OK21 =
b∞

1/3


2COK21 St
RHill (when Pcol,2D = Pcol,set ),





2m (when P=P).
col,2Dcol,ho

(5.3)
(5.4)
(5.5)

(5.6)
(5.7)
(5.8)

Following Okamura and Kobayashi, 2021, we set COK21 = 1.5 and ξ = 0.1m. We
adopt ρ• = 1 g cm−3 . The mean free path of the disk gas is given by (Ida et al.,2016):√µmH
2πµmH Hλmfp ='= min(λmfp,vis , λmfp,irr ),(5.9)
σmol ρg
σmol Σg

150

Chapter 5. Discussion and outlookwhere
λmfp,vis = 1

M∗
1M!−11/20L∗
1L

λmfp,irr = 0.65

!3/7!−7/10!−2/5!33/20
Ṁ∗
r
αacc
cm,
1 au
10−3
10−8 M /yr!
!−17/14!33/14
r
M∗
αacc
cm.
1M
1 au
10−3

(5.10)(5.11)The gas surface density is given by (Ida et al., 2016):
Σg = min(Σg,vis , Σg,irr ),(5.12)

where3

Σg,vis = 2.1 × 10

M∗
1M

3

L∗
1L

Σg,irr = 2.7 × 10

!1/5

!−2/7

αacc
10−3
M∗
1M

!−4/5

!9/14

!3/5
!−3/5
Ṁ∗
r
cm,
1 au
10−8 M /yr! −1!−15/14
r
αacc
cm.
1 au
10−3

(5.13)(5.14)We define the growth timescale of the planet as
τgrowth ≡Mp
Pcol,3D Σd.

(5.15)

To maintain consistency with Chap. 3, we fixed the inward dust mass flux as Ṁd,∞ =102 M⊕ /Myr (e.g., Lambrechts et al., 2019). Thus, the dust-to-gas ratio varies as afunction of the Stokes number and the orbital radius (Ida et al., 2016; Kuwahara andKurokawa, 2020a):Σd= 2 × 10−5 f d
Σg

1 + St2
St

!

αacc
10−3

!

Ṁ∗
−
8
10 M /yr

! −1

!
Ṁd,∞
, (5.16)100 M⊕ /Myrwhere f d represents a factor for enhancement or depletion of the dust surface density due to the planet-induced gas flow. Although f d depends on the assumed planetary mass, the Stokes number, the Mach number of the headwind of the gas (corresponading to the orbital distance), and the turbulence strength, we assumed
0.1 (m ≥ 0.1 and r < 1 au),f d = 1 (m ≥ 0.1 and r = 1–10 au),
2 (m ≥ 0.1 and r > 10 au),

(5.17)(5.18)(5.19)

based on the results in Figs. 4.13 and 4.14. For a larger-mass planet, the magnitudeof f d would increase. Thus, Eq. (5.15) gives the upper (lower) limit of the growthtimescale of the planet in the inner (outer) region of the disk (< 1 au or > 10 au).
Figure 5.1 shows the growth timescale of the planet influenced by the planetinduced gas flow as a function of the planetary mass and orbital distance. It should
be noted that we fixed the Stokes number and the turbulence strength in each sectionof the disk (Sects. 5.1.1 and 5.1.2). The dashed white line in Fig. 5.1 marks where thegrowth timescale reaches 1 Myr. We consider this line to be the limit at which theplanet cannot grow any further within the typical lifetime of the disk.

5.1. Implications for the origins of the architecture of a planetary system

151

F IGURE 5.1: Growth timescale of the planet influenced by the planetinduced gas flow. The white dashed line represents the contour of 1
Myr. The horizontal black dotted line corresponds to the pebble isolation mass (Eq. (2.57)). We note that we divided the disk into threesections, < 1 au, 1–10 au, and > 10 au, and assumed a fixed Stokes
number and turbulence strength in each section.

- In the inner region of the diks (. 1 au), where we assume αdiff = 10−5 andSt = 10−3 , the planetary growth via pebble accretion would be inefficientdue to a reduction in the width of the accretion window and dust depletionaround the planetary orbit when m & 0.1. In general, under weak turbulence,
the accretion rate increases as the turbulence strength decreases because pebble accretion occurs two dimensions. On the other hand, the influence of theplanet-induced gas flow on the dust surface density is stronger for a small αdiff .
Thus, the net accretion rate would strongly depend on the dust surface density
around the planetary orbit. In the inner region of the disk, the planet-induced
gas flow tends to be the flow-shear regime and the dust gap forms aroundthe planetary orbit. Under the influence of the planet-induced gas flow, thepebbles coming from a narrow window between the horseshoe and the shear
regions can accrete onto the planet (e.g., Fig. 3.3). The accretion cross sectionof pebbles lies within the dust gap formed by the planet-induced gas flow,
because the outflow deflects the trajectories of pebbles outside the horseshoe
region (Fig. 4.2). The subsequent growth of the protoplanets is highly suppressed by the reduction in the width of the accretion window and dust surface density due to the planet-induced gas flow. The growth of protoplanets
via pebble accretion may halt when m ∼ 0.1, corresponding to a protoplanetwith ∼ 0.7 M⊕ at ∼ 1 au (Fig. 5.2). These protoplanets can avoid runaway
gas accretion within the lifetime of the gas disk, which results in the formationof rocky terrestrial planets. The small rocky protoplanets may also experience
giant impacts. These events lead to the formation of super-Earths.
- In the middle region of the disk (∼ 1–10 au), where we assume αdiff = 10−3
and St = 3 × 10−2 , the influence of the planet-induced gas flow on the dust
surface density is negligible (Fig. 4.11), because of the strong turbulence and152

Chapter 5. Discussion and outlook
large Stokes number. Thus, the accretion rate of dust onto the planet is determined by the influence of the planet-induced gas flow on the width of theaccretion window. The growth of planets via pebble accretion may halt whenm ∼ 0.3–0.5, corresponding to ∼ 3–10 M⊕ at ∼ 1–10 au (Fig. 5.2).
- In the outer region of the disk (& 10 au), where we assume αdiff = 10−4 andSt = 3 × 10−3 ,the planet-induced gas flow tends to be the flow-headwind
regime. Only a dust ring is likely to be formed when the planet-induced gas
flow is in the flow headwind regime Fig. 4.11). In this case, the accretion rateof pebbles could increase due to an increase in the pebble surface density near
the planetary orbit. The pebble surface density would be increased by a factorof ∼ 2 due to the dust ring. In the outer region of the disk, the VSI-driven turbulence could be suppressed (Fukuhara et al., 2021, see also Sect. 5.1.1). This
leads to a significant accumulation of pebbles (e.g., Fig. 4.14i) and could resultin more efficient growth of the planet via pebble accretion (Morbidelli, 2020;
Cummins et al., 2022; Jiang and Ormel, 2022). The growth of planets via pebble accretion may halt when m ∼ 0.1, corresponding to ∼ 3–15 M⊕ at ∼ 10–60
au (Fig. 5.2).

F IGURE 5.2: Final mass of the planet at which the planet cannot growany further within the typical lifetime of the disk, 1 Myr. The vertical dashed lines divided the disk into three sections. Same as inFig. 5.1, we fixed the Stokes number and the turbulence strength ineach section. We assumed the dimensionless final mass of the planet
as m = 0.1 (< 1 au), 0.5 (1–10 au), and 0.1 (> 10 au) based on Fig. 5.1.

Though the formation scenario of a planetary system discussed above involves
many simplification—for instance, we did not consider dust growth and planet
migration—Fig. 5.1 suggests that there is a clear difference in the growth rate of theplanets between the inner and the outer region of the disk. This may help explain
the distribution of exoplanets, where low-mass planets are more frequent than giant

5.2. Implications for protoplanetary disk observations

153

planets (Sect. 1.1; Cumming et al., 2008; Howard et al., 2010; Mayor et al., 2011;
Wright et al., 2012; Dong and Zhu, 2013; Petigura et al., 2013; Zhu and Wu, 2018;
Fernandes et al., 2019).
Even in the middle region of the disk (∼ 10 au), where the dust substructures
are observed, the planetary growth may halt at super-Earth mass. This suggests thatthe existence of long-period low-mass planets in this region. These planets could
be found by the microlensing survey that would be performed by the Nancy Grace
Roman Space Telescope (formerly known as WFIRST Penny et al., 2019; Zhu andDong, 2021).
The suppression of dust accretion onto a planet due to the planet-induced gas
flow in the inner region of the disk may favor the formation of terrestrial planets in
the Solar System. Recently, the early formation of proto-Jupiter has been proposed,
where the proto-Jupiter generates a pressure bump and acts as a barrier against thedrifting dust (Kruijer et al., 2017). This could reduce the dust mass flux in the terrestrial planet-forming region. However, debate over the efficiency of the Jupiterbarrier is still ongoing (Brasser and Mojzsis, 2020; Johansen et al., 2021). Hydrodynamical simulations suggest that small dust grains can pass through the orbit of the
proto-Jupiter (Lyra et al., 2009; Haugbølle et al., 2019; Drażkowska et al., 2019). Even
when the proto-Jupiter does not inhibit the inward drift of dust, the outflow of thegas behaves as a barrier against the accretion of dust particles onto protoplanets in
the inner Solar System, leading to the formation of small rocky terrestrial planets.

5.2

Implications for protoplanetary disk observations

The gas flow induced by low-mass planets can be considered one of the possible origins of the observed dust substructures in disks. The characteristics of the dust ringand gap formed by the planet-induced gas flow are summarized as follows: (1) The
planet-induced gas flow forms the ring and (or) gap in aerodynamically small dust
grain distribution, St . 10−2 . (2) The gas flow induced by a low-mass embedded
planet can generate the dust ring and (or) gap in a radial extent of ∼ 1–10 H aroundthe planetary orbit without creating a gas gap or pressure bump (Fig. 5.3).
Our model for the dust ring and gap formation by low-mass planets can be tested
when the following conditions are satisfied: (1) the dust substructures appear only
for the distribution of small dust grains (. 1 cm) and (2) there is no correlation
between the spatial distribution of gas and dust in a disk. This information can beobtained from observations of dust emission and scattering in different wavelengths
to constrain the dust surface densities for different sizes and of spectral lines that
trace the gas surface density
In the following sections we discuss how our model can be distinguished fromthe other dust substructure formation mechanisms.

154

Chapter 5. Discussion and outlook

F IGURE 5.3: Schematic illustrations of our model for the dust ringand gap formation. The planet-induced gas flow is in the flow-shear
regime (panel a) and in the flow-headwind regime (panel b to form
dust substructures.

5.2.1

Correlation between the spatial distribution of gas and dust

Our model focuses on the low-mass planets that neither open gas gaps nor generatepressure bumps. Thus, correlations between the spatial distribution of gas and dust
would not be observed.
When the dust substructures form due to the gas-gap-opening planet, however,there would be a correlation between the spatial distribution of gas and dust in disks,such as transition disks where the inner disks are depleted in dust relative to theouter disk (Francis and Marel, 2020). In many transition disks, correlations between
emission lines of gas and continuum rings are observed at the outer edge of thecavity, which suggests the existence of a massive planet within the cavity (Marel etal., 2016; Marel et al., 2018; Dong et al., 2017b; Boehler et al., 2017; Fedele et al., 2017).

5.2. Implications for protoplanetary disk observations

155

We note that the observed signature of the gas gap may be caused by planetfree mechanisms. For instance, CO, one of the main tracers of protoplanetary disks,
is chemically processed into other molecules, leading to the changes in the CO gas
abundance (Aikawa et al., 1998; Bergin et al., 2014; Schwarz et al., 2018; Bosman etal., 2018). CO depletion would be caused by the dust evolution in the disk (Xu et al.,2017; Krijt et al., 2018; Booth and Ilee, 2019; Krijt et al., 2020). The CO abundance canbe affected by freeze-out and sublimation of CO on the surface of the dust, which
depends on the radial and vertical motion of dust in the disk. Thus, we need to becareful when constraining planetary masses based on the spatial distribution of thegas obtained by CO observation alone. Observations with multiple chemical species
are needed to constrain the gas distribution in disks and understand the origin of
the gas gap in the disk.

5.2.2

Kinematic features in a protoplanetary disk

When a massive planet is embedded in a disk, kinematic features appear in the threevelocity components of gas in the upper disk layers due to the large-scale meridionalgas flow (Morbidelli et al., 2014; Szulágyi et al., 2014; Fung and Chiang, 2016). The
vertical flows toward the midplane with speeds of ∼ 0.1cs at a region ∼ 2–4 H above
the midplane of the disk around HD 163296 have been detected by observations of
12
CO emissions, which can be interpreted as a sign of the large-scale meridional gas
flow that driven by Jupiter-mass planets (Teague et al., 2019).
Low-mass planets that do not carve a gap can induce the gas flow such as thesmall-scale meridional gas flow (e.g., Fig. 2.14; Fung et al., 2015) or the subsonicpolar inflow and radial outflow (e.g., Fig. 2.17; Ormel et al., 2015b). These flows
occur within the typical scale of the planetary atmosphere (z < H; Fung et al., 2015;
Lambrechts and Lega, 2017; Kuwahara et al., 2019; Béthune and Rafikov, 2019; Funget al., 2019; Krapp et al., 2022). Thus, as long as we focus on the low-mass planets, thekinematic features of gas that can be traceable in upper layers of disks by molecularemissions, for example 12CO (J = 2–1), would not be expected to be observable.
Meanwhile, the vertical location of the molecular emission surface depends onthe molecular isotopologues and species. For instance, since the relative abundancesof CO isotopologues are different, the emission surface approaches the midplane ofthe disk in the order of 12CO, 13CO, C 18O, and C 17O. Other molecules, such as HCN
and C2 H, can be used as tracers. The C2 H lines come from a lower height than
CO isotopologue (Law et al., 2021; Bosman et al., 2021). When observations withmultiple chemical species constrain the velocity structure of the disk (Yu et al., 2021)
and the kinematic features of the gas can only be seen in the region close to the
midplane (z . H) at the orbital location of the dust substructure, such features ofgas and dust support our model.
Furthermore, it is important to investigate the velocity distribution of the disk
gas and compare it to the locations of dust substructures. Recent observation havefound the nonaxsymmetric velocity deviation (the Doppler flip) and it closely alignedwith the dust ring in the disk around HD 100546 (Casassus et al., 2022). Casassus
et al., 2022 argued that the observed Doppler flip may be caused the outflow of thegas induced by a super-Earth mass planet. Our results should be compared withdetailed observations of the gas in disks.

156

5.2.3

Chapter 5. Discussion and outlook

Comparison to observed widths of dust rings and gaps

In this section, we compare the widths of the dust rings and gaps obtained fromour simulations with the protoplanetary disk observations. The planet-induced gas
flow generates the dust gap with a radial extent of ∼ 1 H. The width of the dust gap
formed by the planet-induced gas flow is determined by the width of the outflow
and do not significantly depend on the turbulent parameter and the Stokes number(Figs. 4.11 and 4.12). The width of the dust ring formed by the planet-induced
gas flow has on the order of ∼ 1–10 H, which strongly depends on the turbulent
parameter and the Stokes number (Figs. 4.11 and 4.12). The disk gas scale height fora steady viscous-accretion disk model can be described as (Fig. 5.4):

21/209/7 !
rr
H = max 0.027, 0.024
au.
(5.20)1 au
1 au

F IGURE 5.4: Gas scale height for a steady viscous-accretion disk
model as a function of the orbital distance.

We first focus on the outer region of the disk (& 10 au), where the dust substructures are frequently observed (e.g., Andrews, 2020). Based on our results, the dust
ring is likely to form in this region. Here we consider the following parameter set
to maintain consistency with the formation scenario of the planetary systems (Sect.
5.1): m = 0.1 (corresponding to ∼ 8.5 M⊕ at ∼ 30 au), αdiff = 10−4 , St = 3 × 10−3
(corresponding to ∼ 0.1–1 mm-sized dust), and Mhw = 0.1. We adopted a planetary
mass as the final mass that a planet could reach within a typical disk lifetime, ∼ 1
Myr (Fig. 5.1). For this parameter set, the width of the dust ring in a steady state is∼ 1 H (Fig. 4.11), corresponding to ∼ 0.5–10 au (Fig. 5.4). Given that the distance toa disk is ∼ 100 pc, the capability to resolve the disk at ∼ 0.005–0.1 arcsec is required.
Since ALMA has the angular resolution of & 0.01 arcsec, the dust ring with a widthof < 10 au generated by the planet-induced gas flow could be detectable by ALMA

5.2. Implications for protoplanetary disk observations

157

observations. Rings with the width of a few au have been found by ALMA. For instance, the width of the dust ring in the DSHARP sample ranges from ∼ 3 to ∼ 7
au.
On the other hand, the dust ring with a width of > 10 au have also been commonly observed. Although these wide rings might be composed of multiple narrowrings, a weak turbulence (αdiff < 10−4 ) and small Stokes number (St < 10−3 ) areneeded to form such wide rings by the planet-induced gas flow (e.g., Figs. 4.11f andi).
Even in the outer region of the disk, the dust gap can be formed around the orbitof a higher-mass planet (e.g., Fig. 4.12f, where we assumed m = 0.3 and Mhw = 0.1).
The width of the dust gap in a steady state is ∼ 1 H, corresponding to ∼ 0.5–10 au(Fig. 5.4). Narrow gaps that are smaller than the beam size are frequently observed(Long et al., 2018, , where the beam size is ∼ 16 au). The distance between the ringpeak and gap center normalized to the gap location ranges typically ∼ 0.2–0.3 (Longet al., 2018), meaning that the ring peak is located at ∼ 1.2–1.3 H from the gap center.
This is consistent with our results (e.g., Fig. 4.12). When m & 0.3, the mass of anembedded planet might be inferred from the observed dust gap width. However,to form such a higher-mass planet, a lager Stokes number or smaller turbulence
strength than those assumed in Fig. 5.1 would be required.
Next, we focus on the middle region of the disk (∼ 1–10 au), where the planet
would grow up to m ∼ 0.3–0.5 (Fig. 5.1). Following Sect. 5.1.3, we consider αdiff =10−3 and St = 3 × 10−2 (corresponding to ∼ 1–10 cm-sized dust). In this region,
neither the dust ring and gap formation by the planet-induced gas flow would not
be expected to occur because of the large Stokes number and turbulence strength.
Instead, since the final mass that a planet could reach within a typical disk lifetime
is comparable to the pebble isolation mass (Fig. 5.1), the inward drift of the large
dust particles could be inhibited by a pressure bump (e.g., Lambrechts et al., 2014),
leading to the formation of the gap in the spatial distribution of these dust particles.
Finally, we focus on the innermost region of the disk (. 1 au). We consider thefollowing parameter set: m = 0.1 (corresponding to ∼ 0.7 M⊕ at ∼ 1 au), αdiff =10−5 , St = 10−3 (corresponding to ∼ 1–10 mm-sized dust), and Mhw = 0.01. In
this case, the dust ring and gap form by the planet-induced gas flow. The widths ofring and gap in a steady state are ∼ 1 H (e.g., Fig. 4.11), corresponding to ∼ 0.03–0.5
au (Fig. 5.4). Given that the distance to a disk is ∼ 100 pc, an angular resolution of< 0.001 arcsec is required to resolve this scale. It is difficult to capture the dust ringand gap formed by the planet-induced gas flow in the inner region of the disk evenwith ALMA.
Future observations will provide the capability to probe the inner few au regions.
Future square kilometer array (SKA) observations in cm-wavelengths would resolve
inner few au scale (Braun et al., 2015; Ilee et al., 2020), but the maximum angularresolution is ∼ 0.04 arcsec (Braun et al., 2019). Moreover, since the cm-sized dust
particles (St & 10−2 at . 1 au; Fig. 3.2) are less affected by the gas flow, the dust
ring and gap formed by the planet-induced gas flow would not be expected to beobservable by SKA.
The ngVLA is expected to capture dust thermal emission at ∼mm-cm wavelengths, with the best angular resolution of ∼ 0.001 arcsec (Selina et al., 2018). Simulations of ngVLA observations suggest that the dust gaps with the widths of ∼ 2–3
au are detectable at ∼ 5 au under weak turbulence (αdiff . 10−5 ; Ricci et al., 2018;
Harter et al., 2020). A possible future extension would improve the angular resolution of the current ALMA by several times, which could lead to further dectectionsof dust substructure in the inner few au region (Carpenter et al., 2020; Burrill et al.,

158

Chapter 5. Discussion and outlook2022). The ngVLA and an upgraded ALMA observations may directly detect theenvelopes of embedded planets (Zhu et al., 2022).
To investigate quantitatively the detectability of dust rings and gaps formed by
the planet-induced gas flow, we plan to produce synthetic images for the dust continuum emission at multiple wavelengths based on our results in future studies.

5.2.4

Comments on individual protoplanetary disks

In this section, we address the constraints on planetary masses and disk parameters
based on the results of this thesis. Assuming that the thermal emission of the dust is
optically thin, the dust surface density can be described byΣd (r ) =Iν (r ),
κν Bν ( Td (r ))(5.21)

where Iν is the radial intensity profile, Bν is the Planck function, Td is the temperature of the dust, and τν is the optical depth. If we ignore the changes in the opacity
and the temperature within the dust substructures, the dust surface density is proportional to the observed intensity profile, Iν . Same as in Sect. 2.2.7, we consider thesteady-accretion disks when we convert the dimensionless quantities into dimensional ones.
HD 143006
The disk around HD 143006 (spectral type: G7, Pecaut and Mamajek, 2016; distance:
165 pc, Gaia et al., 2018) has two bright rings at 41 and 65 au (e.g., Andrews et al.,2018). These rings are referred to as B41 and B65, respectively (Huang et al., 2018a).
The width of each ring is 3.9 au and 7.91 au (Dullemond et al., 2018). The dark annuli(the dust gaps referred to as D22 and D51) are observed around ∼ 22 au and ∼ 51
au.
We first focus on the outermost ring, B65. The observed radial intensity profileshows that B65 has a contrast of ∼ 2 with respect to D51, suggesting that the dust
surface density around 65 au is higher than that around 51 au (Huang et al., 2018a).
Assuming a steady viscous-accretion disk model (Sect. 2.2.7), the Mach number ofthe headwind of the gas is Mhw ' 0.1 at 65 au of HD 143006. The width of theoutermost ring B65 corresponds to ∼ 1.8 H. Based on the results of this thesis, thefollowing parameter set could explain B65 (Fig. 4.11f):- m ∼ 0.1 (corresponding to ∼ 16 M⊕ ),- St ∼ 3 × 10−3 (corresponding to ∼ 0.1–1 mm-sized dust),- and αdiff ∼ 10−4 .
The protoplanet can grow up to m ∼ 0.1 under the influence of the planet-induced
gas flow within the typical disk lifetime (Fig. 5.1). Furthermore, in the outer region of
HD 143006, there is no prominent substructure of the spatial distribution of CO gas.
Thus, the ring formation scenario due to the gas flow induced by low-mass planets
could be applicable to B65. The mass of the putative embedded planet inferred fromour study is an order of magnitude smaller than that inferred by previous study,∼ 0.3 MJup , where MJup is the Jupiter mass (Zhang et al., 2018).
On the other hand, HD 143006 has the inner dust gap, D22, whose width is ∼ 22
au ( 1 H; Huang et al., 2018a). Although no depletion is observed in CO emission

5.2. Implications for protoplanetary disk observations

159

between ∼ 15–40 au in HD 143006, the wide dust gap of D22 would not be explainedby single low-mass planet. As long as we consider that the dust surface density
profile has reached the steady state, a dozen of higher-mass planets (m > 0.1) mustbe aligned with a certain interval (∼ 1 H) to form the wide dust gap of D22.
Oph 163131
The disk around Oph 163131 (spectral type: K4, Flores et al., 2021; distance: 147 pc,
Ortiz-León et al., 2017; Ortiz-León et al., 2018; and stellar mass: 1.2 M , Flores et al.,2021) has two rings referred to as ring 1 and ring 2 (Villenave et al., 2022). Ring 1 and2 lie in regions between 60 and 87 au and 98 and 150 au, respectively. These rings are
separated by a clear gap (referred to as gap 1; Villenave et al., 2022). The disk aroundOph 163131 is highly inclined. Recent observations suggest that the efficient settlingof mm-sized dust grains in the disk (Villenave et al., 2020; Flores et al., 2021; Wolff
et al., 2021; Villenave et al., 2022). Villenave et al., 2022 estimated the dust surface
density in Oph 163131 by a radiative transfer model and found that the dust surface
density is increased by a factor of 2 in ring 1 compared to the inside of ring 1. In gap
1, the mm-sized dust grains would be depleted by a factor of 10 or more than in ring2.
We first focus on ring 1, where the Mach number of the headwind of the gas is
Mhw ' 0.1. The width of ring 1 corresponds to ∼ 4–7 H. The following parameterset could explain the properties of ring 1 (Fig. 4.11f):- m ∼ 0.1 (corresponding to ∼ 15 M⊕ at 60 au),
- St ∼ 3 × 10−4 (corresponding to ∼ 0.1 mm-sized dust at 60 au),- and αdiff ∼ 10−4 .
In Oph 163131, a very weak turbulence would be needed to reproduce the observed
mm continuum emission (αdiff . 10−5 at least in the outer region of the disk, ∼ 100
au Villenave et al., 2022). Thus, lower-mass planets could also form the observed
dust substructure (m ∼ 0.03, Fig. 4.10i). From observations, the lower limit of the
Stokes number is constrained as St & 2 × 10−3 at the outer region of the disk, ∼ 100
au, which is consistent with our estimation.
For gap 1, at ∼ 90 au where Mhw ' 0.1, whose width is ∼ 1 H (Villenave et al.,2022), the following parameter set might be applicable (see Figs. 4.12f and i):- m ∼ 0.3 (corresponding to ∼ 65 M⊕ at 90 au),
- St ∼ 10−4 –3 × 10−4 (corresponding to ∼ 0.01 mm-sized dust at 90 au),- and αdiff . 10−4 .
Our estimation for the turbulence strength is consistent with the previous study, but
the inferred mass of the planet is higher than the final mass of the planet obtained inFig. 5.1. Villenave et al., 2022 suggested that the small dust scale height at the outerregion of the disk leads to an efficient accretion of dust onto the planetary seed, andthen the planet with ∼ 10 M⊕ could be formed within < 10 Myr. The ring formation
during the mid-stage of planetary growth may lead to an efficient growth of planet
and form higher-mass planets as we expected in the outer region of Oph 163131(> 10 M⊕ ).

160

Chapter 5. Discussion and outlook

HD 100546
The disk around HD 100546 (distance: 116 pc, Gaia et al., 2018; stellar mass: 2.2 M ,Wichittanakom et al., 2020), classified as the transition disk, has a ring with a peakaround ∼ 26 au and a width of ∼ 21 au. One interesting point unique to this disk isthat a nonaxisymmetric velocity deviations (the so-called Doppler flip) is observed,
which is closely aligned with the peak of the continuum ring. Doppler flips havebeen thought to be the imprint of interactions between disks and giant planets (e.g.,
Pinte et al., 2018). This is counterintuitive, because when the Doppler flip is induced
by a giant planet the dust gap would appear in a position overlapping the Doppler
flip.
Recently, Casassus et al., 2022 suggested that the Doppler flip in the disk aroundHD 100546 could be caused by the gas flow induced by a super-Earth mass planet.
Motivated by the discussion in Casassus et al., 2022 , here we consider the possibilityof the dust ring formation by the planet-induced gas flow. The observed continuum
emission shows that the contrast of the intensity within the ring varies by a factorof ∼ 2 (Pérez et al., 2020). The width of the ring corresponds to ∼ 10 H. Thus, weconsider that the following parameter set would explain the ring in the disk aroundHD 100546 (where Mhw ' 0.1; Fig. 4.11f):- m ∼ 0.1 (corresponding to ∼ 7.5 M⊕ at 26 au),
- St . 10−3 (corresponding to . 1 mm-sized dust at 26 au),- and αdiff ∼ 10−4 .
As suggested by Casassus et al., 2022, the super-Earth mass planet might be embedded in the disk around HD 100546.
However, our scenario for the dust ring formation by low-mass planets cannot
explain the inner cavity of the disk around HD 100546, unless we consider the additional mechanisms such as photoevaporation or rapid growth of dust in the inner
region of the disk, < 20 au.

5.3

Dust growth and sequential formation of planetesimals

From Figs. 4.11 and 4.12, we found an excessive accumulation of small dust outside
the planetary orbit in some cases. An increase in the dust-to-gas ratio may trigger
planetesimal formation via direct growth and streaming instability (SI; Kretke andLin, 2007; Youdin and Goodman, 2005; Youdin and Johansen, 2007; Johansen andYoudin, 2007). The growth rate of SI depends on the size of dust, which has the
maximum value when St ∼ 10−1 –100 (Chen and Lin, 2020; Umurhan et al., 2020).
Thus, SI does not work for small dust grains.
In this study, we did not consider dust growth, but the accumulation of dust
leads to subsequent dust growth due to an increase in the collision frequency ofdust, and then the Stokes number increases. Assuming perfect sticking via collisions without collisional fragmentation and bouncing, the growth timescale can−1
be described by tgrow ∼ (Σg /Σd )ΩK(Takeuchi and Lin, 2005). Comparing the
growth timescale to the drift timescale, when tgrow  tdrift , the dust grains enter
the "growth-dominated phase" (Taki et al., 2021). Taki et al., 2021 derived the Stokes
number in the equilibrium state as follows:√π ΣdSteq ≈,(5.22)
8η Σg

5.3. Dust growth and sequential formation of planetesimals

161

where η is given by (Ida et al., 2016)

η'






0.93 × 10−3




M∗
1M





−3


0.80 × 10

L∗
1L

!−1/5
!2/5
Ṁ∗
αacc
10−3
10−8 M /yr
!−8/7
!4/7
M∗
r
.
1M
1 au

!−7/10

!2/7

r
1 au

!2/20
,

(5.23)(5.24)

Equations (5.23) and (5.24) are valid in the region where viscous heating and stellar
irradiation dominate, respectively. The dimensionless value of η has on the orderof 10−3 and does not change significantly across the entire region of the disk (. 100au). Thus, Steq strongly depends on the local dust-to-gas ratio. When Steq & 1, thedust particles rapidly grow to planetesimal through the positive feedback between
the reduced radial drift and the accelerated collisional growth (Taki et al., 2021).
From Figs. 4.11 and 4.12, the above condition is easily satisfied outside the planetary orbit under weak turbulence. As a result, the dust particles grow rapidly andbegin to decouple from the gas. The planetesimals could form outside the planetary
orbit, where dust accumulates due to the planet-induced gas flow. This may lead tothe sequential formation of planets next to the embedded planet. These additional
processes should be included in further studies.

163

Chapter 6

Summary
In this thesis, we investigated gas and dust dynamics around a planet embedded ina protoplanetary disk. First, we performed 3D hydrodynamical simulations, which
resolve the local gas flow in the region, including the Hill and Bondi spheres of theplanet. We considered the radiative cooling and the sub-Keplerian motion of the gas.
Radiative cooling was implemented by using the β cooling model (Gammie, 2001;
Kurokawa and Tanigawa, 2018). Chapter 2 summarized the results obtained froma series of hydrodynamical simulations. In Chapter 2, we clarified the dependenceof the planet-induced gas flow on the planetary mass and the Mach number of theheadwind of the gas and suggested that the planet-induced gas flow would affectdust dynamics. We describe the main findings of Chapter 2 as follows.
1. An isolated envelope (atmosphere) emerges within the gravitational sphere of
the planet, whose size scales with the Bondi or Hill radius of the planet. Gas
from the disk chiefly flows in and out of the twice the atmospheric radius.
2. The planet-induced gas flow is divided into two regimes, the flow-shear andflow-headwind regimes, depending on the relationship between the dimensionless planetary mass and the Mach number of the headwind of the gas.
Qualitatively, two outflows occur both inside and outside the planetary orbit
in the flow-shear regime, while outflow occurs dominantly outside the planetary orbit in the flow-headwind regime.
3. We found a fitting formula for the outline of the outflow at the midplane region, which could serve a quantitative description of the properties of theplanet-induced gas flow such as the radial and azimuthal extent of the outflow.
4. For a fixed Mach number of the headwind, outflow speed in the radial direction to the disk increases with the planetary mass and has a peak at m ∼ 0.3(corresponding to ∼ 2 M⊕ at 1 au). The outflow toward the outside (inside) of
the planetary orbit increases (decreases) as the Mach number of the headwindof the gas increases (decreases). We derived an analytical solution for the outflow speed from Bernoulli’s theorem, which was in agreement with the resultsof hydrodynamical simulations for a wide range of planetary mass and Mach
number of the headwind of the gas.
5. The outflow at the midplane region has the speed on the order of sound speed,which suggests that the outflow could affect dust dynamics.
Next, we calculated the trajectories of dust influenced by the planet-induced gas
flow. We used the obtained hydrodynamical simulation data to calculate the gas
drag force acting on the dust particle. Chapter 3, which is based on Kuwahara and164

Chapter 6. Summary

Kurokawa, 2020a and Kuwahara and Kurokawa, 2020b, summarized the results obtained from a series of orbital calculations. In Chapter 3, we clarified how local
dust dynamics around the planetary orbit changes under the influence of the planetinduced gas flow and discussed how the planet-induced gas flow affect the global
distribution of dust in disks. We describe the main findings of Chapter 3 as follows.
1. As we expected in Chapter 2, the trajectories of dust in the planet-induced
gas flow differ significantly from those in the unperturbed flow. In the planetinduced gas flow, most dust particles coming from the horseshoe region moveaway from the planet along the horseshoe flows. The dust coming from awindow between the horseshoe and the shear regions accretes onto the planet.
The width of the accretion window is narrower in the planet-induced gas flow
than in the unperturbed flow, in particular for the smaller Stokes numbers.
2. In the Epstein gas drag regime, the accretion of small dust grains (St . 10−3 –10−2 )onto the planet is suppressed by the planet-induced gas flow regardless of theassumed planetary mass in Chapter 3. The suppression of dust accretion onto
the planet is more prominent for a small Stokes number.
3. The outflow toward the outside (inside) of the planetary orbit deflects the dust
trajectories outside (inside) the planetary orbit and creates the voids in a frame
co-rotating with the planet, implying that the radial drift of dust could be inhibited by the planet-induced gas flow. The outflow may affect the global dust
distribution.
Finally, we computed the steady-state dust surface density by incorporating theinfluences of the planet-induced gas flow into the 1D advection-diffusion equation.
Chapter 4 summarized the results of dust surface density simulations. Sections 4.2–
4.4.1 are based on Kuwahara et al., 2022. In Chapter 4, we clarified the influence ofthe planet-induced gas flow on the radial distribution of dust particles in a disk. We
describe the main findings of Chapter 4 as follows.
1. When St . 3 × 10−2 (. 1 cm at ∼ 1–10 au), a forbidden region exists outside
the planetary orbit due to the outflow of the gas, where the planet-induced gas
flow inhibits the radial drift of dust particles as suggested in Chapters 2 and 3.
The width of the forbidden region is wider for smaller Stokes numbers.
2. Under weak turbulence (αdiff . 10−4 ), the gas flow induced by a non-gas-gapopening planet with m & 0.1 (& 0.7 M⊕ at 1 au, or & 3.3 M⊕ at 10 au) generates
dust accumulation (the dust ring) and depletion (the dust gap) with a radialextent of ∼ 1–10 H around the planetary orbit in the distribution of small dust
grains, St . 10−2 (. 1 cm at ∼ 1–10 au). When the outflow occurs dominantly
both outside and inside the planetary orbit (m/Mhw & 3), a dust ring and/or
gap form(s). When the outflow occurs dominantly outside the planetary orbit(m/Mhw . 3), only the dust ring forms outside the planetary orbit in thesteady state.
3. The properties of the dust ring and gap, such as width, depth, and amplitude,
are more pronounced for a small Stokes number, because smaller dust particles
are more sensitive to the gas flow.
Based on the results obtained in Chapters 2–4, and assuming the global structureof the disk in terms of the distribution of the turbulence parameter and the size distribution of dust in a disk inferred from previous studies (Malygin et al., 2017; Lyra

Chapter 6. Summary

165

and Umurhan, 2019; Okuzumi and Tazaki, 2019), we proposed a formation scenario
of planetary systems (Fig. 5.1). In the inner region of the disk (. 1 au), the planetinduced gas flow tends to be the shear-dominated (flow-shear) regime. The dust depletion is likely to occur around the planetary orbit. The growth of the protoplanets
via pebble accretion is suppressed due to the reduction in the accretion rate of dust
onto the planet and dust depletion around the planetary orbit cased by the planetinduced gas flow. In the outer region of the disk (& 1–10 au), the planet-induced gas
flow tends to be the headwind-dominated (flow-headwind) regime. The dust accumulation outside the planetary orbit is likely to occur due to the planet-induced gas
flow, which could result in efficient growth of the planet via pebble accretion, andmay trigger planetesimal formation via rapid growth of the dust particles. Thesefindings may be helpful in explaining the current observed period-mass distribution of exoplanets, where the low-mass planets are more frequent than giant planetsat < 1 au.
The gas flow induced by low-mass, non-gas-gap-opening planets can be considered a possible origin of the observed dust substructures in disks (Fig. 5.3). When(1) the dust ring and gap appear only for the distribution of small dust grains (. 1cm), and (2) there is no correlation between the spatial distribution of gas and dust
in a disk, such dust substructures may indicate the existence of a low-mass planet
embedded in a protoplanetary disk (& 1 M⊕ ). The ngVLA and a possible future extension to the ALMA may provide opportunities for validation of our model for thedust ring and gap formation.
In this thesis, we demonstrate the importance of the planet-induced gas flow inthe context of both planet formation and ptoroplanetary disk observations. To obtain more quantitative descriptions of the influence of the planet-induced gas flowon the planet formation process, we plan to include additional processes such as
dust growth and fragmentation and planet migration in our future work. Our future study will also include a direct comparison to protoplanetary disk observations
using synthetic images from our dust surface density simulations.

167

Appendix A

Additional hydrodynamical
simulations
A.1

Numerical convergence

The atmospheric radius is one of the important physical quantities in this thesis. To
confirm the numerical convergence, we plot the time evolution of the atmosphericradius, based on the definition in Moldenhauer et al., 2022 (Eq. (2.24)). As shownin Figs. A.1–A.4, the atmospheric radius increases in the early stage of the time
evolution. After t & 50–100 Ω−1 , the atmospheric radius is nearly constant withtime. Thus, we concluded that the atmospheric radius obtained our hydrodynamical
simulations with Rsm = 0.1Rgrav reached a steady state.
However, the atmospheric radius depends on the choice of the smoothing length,
Rsm , and thus the smoothing length affect the numerical results. In Sect. A.2, weinvestigate the influence of the smoothing length.

168

Appendix A. Additional hydrodynamical simulations

F IGURE A.1: Time evolution of the atmospheric radius for different
Mach numbers. We adopt m = 0.05 and Rsm = 0.1Rgrav . Different colors correspond to different Mach numbers. The gray-shaded
region is given by the fitting formula for the atmospheric radius,
Ratm,fit .

A.1. Numerical convergence

F IGURE A.2: Same as Fig. A.1, but we adopt m = 0.1.

169

170

Appendix A. Additional hydrodynamical simulations

F IGURE A.3: Same as Fig. A.1, but we adopt m = 0.2. The pinkshaded region is given by the fitting formula for the atmospheric radius, Ratm,fit .

A.1. Numerical convergence

F IGURE A.4: Same as Fig. A.1, but we adopt m = 0.6.

171

172

A.2

Appendix A. Additional hydrodynamical simulations

Dependence on smoothing length

Throughout all of our hydrodynamical simulations in Chapters 2–5, the smoothinglength was set to Rsm = 0.1Rgrav (Table 2.1). To investigate the influence of thesmoothing length, we performed additional hydrodynamical simulations for different smoothing lengths, Rsm = 0 and 3Rin . Figure A.5 shows the finite smoothinglength as a function of the planetary mass.

F IGURE A.5: Smoothing length as a function of the dimensionlessplanetary mass.

Through the test simulations, we found unsteady and unphysical flow pattern,
and finally, the calculations stop. These numerical errors occurred when we performed long-term simulations, t > 500 Ω−1 , in particular for higher-mass planets,
m & 0.3. To investigate this problem, we plot the shell averaged radial velocity in
Figs. A.6 and A.7. These figures show that the small but nonzero inward radial gas
flow occurs at the region close to the inner boundary, which means that the hydrostatic equilibrium is not established in this region. The density of the gas increases
with time due to the continuous accumulation of the gas in the inner region of theenvelope, leading to the numerical problems.
There is a clear correlation between the radial gas velocity in the region close
to the inner boundary and the smoothing length (Figs. A.6 and A.7). The larger
the smoothing length, the smaller the radial gas velocity at the region close to theinner boundary. This can be seen in previous study (e.g., Ormel et al., 2015b). Ormelet al., 2015b suggested that the nonzero inward radial gas flow would disappear ina higher-resolution simulation. However, our 3D hydrodynamical simulations arecomputationally expensive. Thus, we adopt the nonzero smoothing length to avoid
the numerical problem in Chapters 2–5.

A.2. Dependence on smoothing length

F IGURE A.6: Shell averaged radial velocity. We assumed m = 0.1 and
Mhw = 0. Different lines correspond to different smoothing lengths.

173

174

Appendix A. Additional hydrodynamical simulations

F IGURE A.7: Same as Fig. A.6, but we set m = 0.5 and Mhw = 0.

Figures A.8 and A.9 show the time evolution of the atmospheric radius for different smoothing lengths. In Fig. A.8, where we assumed m = 0.1, we found thefluctuation of the atmospheric radius when we adopt a relatively large smoothinglength, Rsm = 3Rin (corresponding to ∼ 40% of the Bondi radius). The fluctuation
does not occur when Rsm = 0.1Rgrav . For a higher-mass planet, m = 0.5, only when
we adopt the nonzero smoothing length, the results seem to reach the steady state(the blue and yellow lines in Fig. A.9).
It should be noted that even when we adopt the nonzero smoothing length, theinward radial gas flow remains in the region close to the inner boundary (Figs. A.6
and A.7). Thus, the unsteady and unphysical flow pattern would eventually appear
when t > 103 Ω−1 .

A.2. Dependence on smoothing length

F IGURE A.8: Time evolution of the atmospheric radius. We assumed
m = 0.1 and Mhw = 0. Different lines correspond to different
smoothing lengths.

175

176

Appendix A. Additional hydrodynamical simulations

F IGURE A.9: Same as Fig. A.8, but we set m = 0.5 and Mhw = 0.
Since we adopt the different length of the calculation time, the redsolid line is truncated at t = 500.

The choice of the smoothing length does not affect significantly the outflow speed(Fig. A.10), in particular for low-mass planets, m . 0.3. In this figure, we adopt
Rsm = 0 and Rsm = 0.1Rsm . For higher-mass planets, m & 0.3, since the unsteadyand unphysical flow pattern appear at t = tend when we adopt Rsm = 0, the outflow
speed obtained from our hydrodynamical simulations with Rsm = 0 deviates fromthat with Rsm = 0.1Rgrav .

A.2. Dependence on smoothing length

F IGURE A.10: Outflow speed at the midplane region of the disk as afunction of the dimensionless planetary mass and the Mach numberof the headwind of the gas. The star and cross symbols are obtained
from hydrodynamical simulations at t = tend with Rsm = 0.1Rgrav
and Rsm = 0, respectively. The red and black solid lines correspondto the analytic solutions of the x-component of the outflow velocity derived from Eq. (2.49). The gray-shaded region represents thedimensionless pebble isolation mass, miso (Eq. (2.57)). The valuesof assumed fitting parameters C1 and C2 are shown in the lowerright corner of each panel, which determine the atmospheric radius(Eq. (2.25)). It should be noted that length of the calculation time is
not consistent for each planetary mass (Table 2.1).

177

179

Appendix B

Additional figures
B.1

Comparison between the analytical estimations of thedrift velocity of dust influenced by the planet-induced
gas flow and numerical results

F IGURE B.1: Difference of the azimuthally and vertically averaged
velocity of dust particles from the unperturbed steady-state drift velocity. We assumed m = 0.03 and αdiff = 10−3 . The solid anddashed lines are obtained from Eq. (4.33), vdust ( x ), and numericalsimulations, hv x,d iy,z , respectively. Different rows and columns correspond to different Stokes numbers and different Mach numbers, respectively.We set H/a = 0.03 for this plot.

180

Appendix B. Additional figures

F IGURE B.2: Same as Fig. B.1, but we assumed m = 0.03 and αdiff =10−5 .

F IGURE B.3: Same as Fig. B.1, but we assumed m = 0.1 and αdiff =10−3 .

B.1. Comparison between the analytical estimations of the drift velocity of dust
181
influenced by the planet-induced gas flow and numerical results

F IGURE B.4: Same as Fig. B.1, but we assumed m = 0.1 and αdiff =10−4 .

F IGURE B.5: Same as Fig. B.1, but we assumed m = 0.3 and αdiff =10−3 .

182

Appendix B. Additional figures

F IGURE B.6: Same as Fig. B.1, but we assumed m = 0.3 and αdiff =10−5 .

183

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