MIXING OF DUST IN PROTOPLANETARY DISKS AND THE SOLAR NEBULA
by
ANNA LOUISE HAUGSJAA HUGHES
B.A., University of Montana, 2004
M.S., University of Colorado, 2006

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirement for the degree ofDoctor of Philosophy
Department of Astrophysical and Planetary Sciences
2011Hughes, Anna Louise Haugsjaa (Ph.D., Astrophysical and Planetary Sciences)
Mixing of Dust in Protoplanetary Disks and the Solar Nebula
Thesis directed by Associate Professor Dr. Philip J. Armitage

Understanding the small-dust component of protoplanetary disks is key to understanding theconditions for planet formation. Small dust grains, particularly at large distances, provide ourprimary observational window into the physics of protoplanetary disks, being much more easily
observed than the gas component. Furthermore, the distribution of these grains must ultimately
control the timing and locations for planetesimal formation, the first major step toward planet
formation. For my thesis work, I have used numerical simulations to model the radial distributionof dust grains as they are acted upon by the gas disk, including the evolution of the disk (outward
expansion and inward accretion), radial and azimuthal drag of the gas flow on the particle orbits,
and turbulent mixing of the particle ensemble radially within the disk. I have run simulations using a
range of particle sizes and disk-model parameters and consider primarily two phenomena: the radial
diffusion of hot, inner disk particles outward to large AU, relevant to the compositional makeup of
bodies within our own solar system, and the time evolution of the global dust-to-gas ratio, which
dictates the supply of solid material to the planetesimal- and planet-forming regions. I find that,while the degree out outward mixing depends sensitively on a number of disk-model parameters, thebehavior of the global dust-to-gas distribution is relatively uniform between different disk-model
simulations, suggesting that, while still mysterious, the conditions for planetesimal formation arecommonly met across a range of disk configurations. Observed disk compositions correlate poorly
with most observable disk parameters. However, my simulations suggest compositional properties
are most-strongly controlled by the initial conditions of young disk systems.

iii

DEDICATION
Dedicated to all of my teachers, past, present, and future.

ACKNOWLEDGEMENTS
Learning has long been a central element of my life and so I feel particularly the strong influencethat my teachers have had on my life and on the person I am today. My teachers come in many
forms, and I would like to here directly thank a few who have also been a part of my community
and my sense of self:
Phil Armitage, Elinita Brown, Josh Colwell, David Friend, Diane Friend, Anson Haugsjaa,
Cindy Haugsjaa, Steve Hughes, Jim Jacobs, Jonas Pologe, Hanna Sizemore, Eijiro Uchimoto, Andrew Ware, Alycia Weinberger.
Many people shape who we are and the things we are able to accomplish, and so I have neglecteda great many people from the above list. I appoligize. Know that I have long found myself
surrounded by good people, and that you are all greatly appreciated.

v

Contents
1 Introduction and the Early Solar Nebula

1

2 Building a 1D Disk-Evolution Model
2.1 Observational Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Basic Setup and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Modeling Disk Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Numerical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Parameterized EUV Photoevaporation . . . . . . . . . . . . . . . . . . . . . .
2.4 Evolving the Disk Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Energy-balanced Midplane Temperature . . . . . . . . . . . . . . . . . . . . .
2.4.2 Power-law–Parameterized Temperature . . . . . . . . . . . . . . . . . . . . .
2.5 Observational Comparisons for Model Disks . . . . . . . . . . . . . . . . . . . . . . .
2.6 Calculating Gas Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8
9
12
16
17
20
21
25
26
32
35
38
41

3 Building the Particle Transport Model
3.1 Forces on a Grain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Advection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 The Basics of Gas Drag in Orbit . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Calculating a Steady Radial Velocity . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Fidelity of Steady-Velocity Trajectories . . . . . . . . . . . . . . . . . . . . .
3.3 Turbulent Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Numerical Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Fidelity of Model-Ensemble Diffusion . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Two Theories for Radial Particle Diffusion . . . . . . . . . . . . . . . . . . . .
3.4 A Cartoon Model for Grain-Growth Constraints . . . . . . . . . . . . . . . . . . . .
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43
44
50
50
58
62
65
66
69
80
83
90

4 Outward Mixing: Implications for Stardust91
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.2 Observational Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.3.1 Model Disk Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.3.2 Particle Transport Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.4 Effects of Individual Transport Processes . . . . . . . . . . . . . . . . . . . . . . . . 100
4.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

vi

4.6

4.5.1 Two Radial Gas-Flow Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.5.2 Different Particle Sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.5.3 Varying the Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.5.4 More/Less Compact Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5 Outward Mixing: Extended Parameter Study132
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.2 Some Numerical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.2.1 Randomized Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.2.2 Varying the Inner Disk Edge . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.3 Varying the Time of Particle Initialization . . . . . . . . . . . . . . . . . . . . . . . . 140
5.4 Varying the Initial Disk Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.5 Varying α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.6 The Disk Temperature Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
5.6.1 Varying the Baseline Temperature . . . . . . . . . . . . . . . . . . . . . . . . 151
5.6.2 Varying qT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6 The Necessity of Local Structure and Small-scale Concentration for Planetesimal
Formation
164
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.2.1 Model Disk Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.2.2 Particle Transport Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.2.3 Forming Composites of Multiple Single-grain-size Simulations . . . . . . . . . 172
6.3 Results: The Global Distribution of Solids . . . . . . . . . . . . . . . . . . . . . . . . 176
6.4 Comparison with Required Enhancement Factors . . . . . . . . . . . . . . . . . . . . 183
6.5 Discussion and mm-size Grains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
Bibliography

206

A Analytical Test Cases for Diffusion

216

B Numerical Techniques
222
B.1 Time-Explicit Viscous-Disk Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . 222
B.1.1 Numerical Setup and Fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
B.1.2 Conservation and the Grid-spacing Scheme . . . . . . . . . . . . . . . . . . . 226
B.2 Some Simple Numerical Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
B.2.1 Interpolation: Trajectories and Probabilities . . . . . . . . . . . . . . . . . . . 231
B.2.2 Gradients and q-values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
B.2.3 Large Summing Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
B.3 Itterative Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
B.3.1 Strategies and the First vsrd Itterator . . . . . . . . . . . . . . . . . . . . . . 235
B.3.2 A Map for Finding Steady Radial Drift Velocities . . . . . . . . . . . . . . . . 240
B.3.3 Midplane Disk Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
C Glossary of Constants and Variables

247
vii

List of Tables
2.1

Fiducial Power-law Temperature fitting constants . . . . . . . . . . . . . . . . . . . . 34

6.1
6.2
6.3

Full-list of Power-law Temperature fitting constants . . . . . . . . . . . . . . . . . . 169
Radial Zones Used for Initiating Grain Sizes . . . . . . . . . . . . . . . . . . . . . . . 172
Disk parameters for Youdin & Shu (2002) and my fiducial disk model . . . . . . . . 186

C.1 Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
C.2 Physical Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
C.3 Symbolic Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

viii

List of Figures
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14

Evolving surface densities for fiducial models . . . . . . . . . . . . . . . . . . . . .
Evolving midplane densities for fiducial models . . . . . . . . . . . . . . . . . . . .
EUV photoevaporation gap-opening with my disk-evolution model . . . . . . . . .
Schematic of Bell & Lin (1994) opacity vs. temperature . . . . . . . . . . . . . . .
Bell & Lin (1994) opacities for a range of mean gas densities . . . . . . . . . . . . .
Energy-balanced midplane-temperature distribution . . . . . . . . . . . . . . . . . .
Surface densities from energy-balanced disk temperatures . . . . . . . . . . . . . .
First fit for evolving power-law temperatures . . . . . . . . . . . . . . . . . . . . .
Fits in t for the power-law parameters of evolving disk temperature . . . . . . . . .
Contours of Σg for a disk evolving with different T -models . . . . . . . . . . . . . .
Outer-disk gas radii for model disks . . . . . . . . . . . . . . . . . . . . . . . . . . .
Total masses and mass-loss rates for model disks . . . . . . . . . . . . . . . . . . .
Deviation between vφ,g and vK for several z = xHg . . . . . . . . . . . . . . . . . .
Two models for the radial velocity of disk gas . . . . . . . . . . . . . . . . . . . . .

13
14
25
29
30
31
31
32
34
35
36
37
39
41

3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
3.16
3.17
3.18
3.19
3.20
3.21
3.22
3.23

Disk schematic of particle-interaction zones . . . . . . . . . . . . . . . . . . . . . .
Cartoon for the mechanics of photophoresis versus radiation pressure . . . . . . . .
Particle scale-heights for 2 mm grains . . . . . . . . . . . . . . . . . . . . . . . . . .
Drift velocities for grains above the midplane . . . . . . . . . . . . . . . . . . . . .
Stokes versus Epstein boundary curve . . . . . . . . . . . . . . . . . . . . . . . . .
Integrated trajectories for different-sized particles . . . . . . . . . . . . . . . . . . .
Orbital stability-boundaries with vr,merid and grain-flow cartoon . . . . . . . . . . .
CR,maxv curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Steady mean-radial velocities for different grain sizes . . . . . . . . . . . . . . . . .
Versus various drift-velocity calculations from the literature . . . . . . . . . . . . .
Integrated versus simulated trajectories . . . . . . . . . . . . . . . . . . . . . . . .
Compared trajectories for different-sized particles . . . . . . . . . . . . . . . . . . .
Trajectories of long infall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
∆tturb and pin values in the static disk models . . . . . . . . . . . . . . . . . . . . .
Example diffusivity of transport model versus analytical solution . . . . . . . . . .
Analytic versus numeric diffusion: Several a = b profiles, v0 = 0 . . . . . . . . . . .
Diffusion: Several a = b profiles, v0 = −1.5ν0 . . . . . . . . . . . . . . . . . . . . .
Diffusion: Non-steady cases (a 6= b) . . . . . . . . . . . . . . . . . . . . . . . . . . .
Testing varying the diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Variations in ∆tturb and pin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Testing diffusion time-stepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Testing diffusion time-stepping: a = b = 2, v0 = −1.5ν0 . . . . . . . . . . . . . . . .
Resonant-like diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45
47
48
49
54
55
56
57
59
60
63
63
64
68
68
71
72
73
74
75
76
77
78

ix

3.24
3.25
3.26
3.27
3.28
3.29
3.30

Testing diffusion time-stepping: static-disk models . . . . . . . . . . . . . . . . . .
Two equations for particle diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . .
Ratio of mapped dust distributions for two Dp -relations . . . . . . . . . . . . . . .
Dust-positions results for two Dp -relations . . . . . . . . . . . . . . . . . . . . . . .
Two cartoon grain-growth models . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Grain growth in the Dullemond & Dominik (2005) raindrop model . . . . . . . . .
The cartoon grain-growth models: parameters . . . . . . . . . . . . . . . . . . . . .

4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
4.17
4.18
4.19
4.20
4.21
4.22
4.23
4.24

Model-disk radial profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Basic diffusion: well-coupled grains in a static, still disk . . . . . . . . . . . . . . . 101
Basic transport: including different model parameters . . . . . . . . . . . . . . . . 102
Particle positions versus starting positions . . . . . . . . . . . . . . . . . . . . . . . 104
Particle transport in the baseline model . . . . . . . . . . . . . . . . . . . . . . . . 105
Accretion-flow case: fraction beyond 25 AU . . . . . . . . . . . . . . . . . . . . . . 106
Accretion-flow case: concentration beyond 25 AU . . . . . . . . . . . . . . . . . . . 107
Midplane-flow case: concentration beyond 25 AU . . . . . . . . . . . . . . . . . . . 109
Radial velocities of different-sized particles . . . . . . . . . . . . . . . . . . . . . . . 111
Versus grainsize: maximum concentrations beyond 25 AU . . . . . . . . . . . . . . 112
Versus grainsize: time-windows for peak outer-disk concentrations . . . . . . . . . . 113
Versus grainsize: concentration beyond 25 AU at t = 106 years . . . . . . . . . . . 114
Versus Schmidt number: maximum concentrations beyond 25 AU . . . . . . . . . . 116
Versus Schmidt number: time-windows for peak outer-disk concentrations . . . . . 118
Static disk models: fraction and concentration beyond 25 AU . . . . . . . . . . . . 120
Gas mass distributions and particle source regions . . . . . . . . . . . . . . . . . . 122
Versus Rd : maximum fractions and concentrations beyond 25 AU - 20 µm . . . . . 122
Versus Rd : time-windows for peak outer-disk concentrations . . . . . . . . . . . . . 123
Versus Rd : maximum fractions and concentrations beyond 25 AU - 2 mm . . . . . 124
Mass accretion rates with time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Parameters - Accretion-flow case: fractions beyond 25 AU with time . . . . . . . . 127
Parameters - Midplane-flow case: fractions beyond 25 AU with time . . . . . . . . 128
Parameters - Accretion-flow case: concentrations beyond 25 AU with time . . . . . 129
Parameters - Midplane-flow case: concentrations beyond 25 AU with time . . . . . 130

5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15

Versus seed number: CN curves for 20 µm grains . . . . . . . . . . . . . . . . . . . 135
Versus seed number: peak concentrations beyond 25 AU and grain-size curves . . . 135
Versus seed number: time-windows for 20 µm grains . . . . . . . . . . . . . . . . . 136
Versus Rin : Σg and accretion-velocity profiles varying Rin . . . . . . . . . . . . . . 137
Versus Rin : Peak CN values for 20 µm and 2 mm grains . . . . . . . . . . . . . . . 138
Versus Rin : CN curves and half-maximum time-windows - 20 µm . . . . . . . . . . 139
Versus Rin : peak CN as a function of grain size . . . . . . . . . . . . . . . . . . . . 140
Versus t0 : maximum concentrations beyond 25 AU . . . . . . . . . . . . . . . . . . 141
Versus t0 : CN curves and time-windows for peak concentrations - 20 µm . . . . . . 142
Versus t0 and grain size: maximum CN beyond 25 AU . . . . . . . . . . . . . . . . 143
Versus MD,0 : Peak CN values and time windows - 20 µm . . . . . . . . . . . . . . 145
Versus MD,0 : Peak CN values as a function of grain size . . . . . . . . . . . . . . . 146
Versus α: Peak CN values for 20 µm and 2 mm-sized grains . . . . . . . . . . . . . 148
Versus α: Peak CN values as a function of grain size . . . . . . . . . . . . . . . . . 149
Versus α: Peak CN as a function of time - 20 µm . . . . . . . . . . . . . . . . . . . 150
x

80
82
83
84
85
88
89

5.16
5.17
5.18
5.19
5.20

Versus TAU : Peak CN for 20 µm and 2 mm grains . . . . . . . . . . . . . . . . . . . 152
Versus TAU : CN as a function of time - 20 µm . . . . . . . . . . . . . . . . . . . . . 153
Σg and vacc profiles in disks varying qT . . . . . . . . . . . . . . . . . . . . . . . . . 154
Versus qT : Peak fractions mixed outward and peak CN - 20 µm and 2 mm . . . . . 156
Versus qT : CN (t) (20 µm) and peak CN as a function of grain size . . . . . . . . . 157

6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11
6.12
6.13
6.14
6.15
6.16
6.17
6.18
6.19
6.20

Example enhancement map and size components . . . . . . . . . . . . . . . . . . . 175
Enhancment map for fiducial and energy-balanced disks . . . . . . . . . . . . . . . 177
E at t = 1.5 Myr for grain-size components and truncated distributions . . . . . . 178
Enhancment maps for simulations varying the diffusivity . . . . . . . . . . . . . . . 179
Enhancement maps for simulations in different disk models . . . . . . . . . . . . . 180
Enhancement at t = 1.5 × 106 years for different disk models . . . . . . . . . . . . . 181
Evolving dust-mass distributions for the fiducial simulations . . . . . . . . . . . . . 182
Dust surface-density profiles for different disk models . . . . . . . . . . . . . . . . . 183
Required Enhancements assuming Ric = 1/4 . . . . . . . . . . . . . . . . . . . . . . 185
Required Enhancements assuming Ric = Ri0c from Lee et al. (2010) . . . . . . . . . 188
Eprecip in the fiducial model versus measured E for several qs values . . . . . . . . 189
Eprecip versus measured E in larger disks . . . . . . . . . . . . . . . . . . . . . . . . 190
Eprecip versus measured E just before disk photoevaporation . . . . . . . . . . . . . 191
Enhancement map for particles initiated at t0 = 0 everywhere in the disk . . . . . . 192
Eprecip versus measured E for the t0 = 0, 0.2µm–2 cm case . . . . . . . . . . . . . . 193
Eprecip versus measured E for the t0 = 0, 0.2–20 µm case . . . . . . . . . . . . . . . 193
Dust surface-density and normalized distributions for headwind drag only . . . . . 194
Eprecip versus measured E with a wall at small AU . . . . . . . . . . . . . . . . . . 195
τs = 1 contours overlying simulated grain distributions - 20 µm and 0.2 mm . . . . 197
Πη contours versus τs = 1 grain-size contours for two disk models . . . . . . . . . . 199

B.1
B.2
B.3
B.4
B.5
B.6
B.7
B.8
B.9
B.10
B.11

Comparing numerical disk evolution to the analytic solution . . . . . . . . . . . . . 225
Diagram of Σg dummy evolution across 3 grid cells . . . . . . . . . . . . . . . . . . 227
Plots of mass and angular-momentum conservation in evolving disks . . . . . . . . 229
Comparison of disk evolution on an outwardly expanded grid . . . . . . . . . . . . 230
Diagram of loop/walk iterative solving . . . . . . . . . . . . . . . . . . . . . . . . . 236
Diagram of bisection iterative solving . . . . . . . . . . . . . . . . . . . . . . . . . . 238
Diagram of iterative solving and zero-solution space for Epstein-drag v srd . . . . . 239
Schematic for vsrd solutions with Epstein drag . . . . . . . . . . . . . . . . . . . . . 240
Epstein vsrd map with three possible solutions . . . . . . . . . . . . . . . . . . . . . 242
Map for searching for 0 or 2 vsrd solutions when vφ,g < vK /2 . . . . . . . . . . . . . 243
Schematic for solving for an energy-balanced disk temperature . . . . . . . . . . . . 245

xiChapter 1

Introduction and the Early Solar
NebulaUnderstanding the small-dust component of protoplanetary disks is vital if we are to understandplanet formation. It is the small-dust component that links two of the biggest mysteries of the
field: How should we model the disk turbulence and viscosity that drive both a disk’s evolution andits structure? And how do planetesimals, the precursors of planets, form, from what, and in whatenvironments? While fluid-dynamic theory is able to broadly sketch the shape and evolution of gas
disks around young stars, observations of such disks, as well as the physical and chemical evidenceof our own solar system primarily trace the solid material, comprising only about 1% of thesesystems. These observations are filled with seeming contradictions: hot-melt minerals embedded incomets along-side pristine ices; large grains orbiting their stars within the tenuous outer reaches of
their disks. In this thesis, I use numerical simulations to trace and explore the aerodynamic radialtransport of dust particles within evolving protoplanetary disks. The questions I have focused on
are these: How is protoplanetary dust distributed? And how is it mixed? Ultimately, the question
is: What is it, really, that builds planetary systems?
Protoplanetary disks are an integral component of the star-formation process. As prestellar
material from a parent molecular cloud begins to collapse, the intrinsic angular momentum of thecloud dictates that a substantial portion of that material falls into a flattened disk structure, rather
than directly onto a central point mass. Mass infalling at later times will tend to fall across wider
radial extents of the disk while viscosity within the disk simultaneously allows for the rapid inward
accretion of most of the disk material as well as the outward spread of some at the outer disk edge.
This disk of material is termed the protoplanetary disk (called the Solar Nebula for that of our own
1Solar system). It persists for some millions of years as gas continues to accrete onto the central starand provides the environment in which dust solids may be processed to larger sizes and perhaps
eventually into planets. Eventually, all the gas of the disk will be stripped away, either throughaccretion or loss mechanisms such as photoevapration, leaving only the large bodies, rocky and icy
material, behind. Such later disks of solid material are often collisional and are termed debris disks.
However, observationally, all disks of material surrounding stars may be called circumstellar disks,
including gas-rich protoplanetary disks and debris disks both.
Circumstellar disks of all types are often first detected as excess in the infrared emission comingfrom a star, resulting from the heating and re-radiation of stellar light by the disk material. Somelarge, nearby debris disks can be resolved in optical images that show starlight reflecting off of thetenuous debris-disk surface, much as we view the rings of Saturn. Young protoplanetary disks,
however, are often optically thick and dense, observed as dark dust lanes obscuring the light of
their central star (if viewed nearly edge on) or as silhouettes against the background illuminationof their parent star-forming cloud.
Despite this dense, robust presence within their star-forming environment, the gas densities andpressures at the midplane of a protoplanetary disk are so low as to be classified as medium to highvacuum by absolute Earth standards, with pressures at the midplane ranging from ∼ 10 −4 –10−13
bars between the innermost and outermost regions of the disk. Furthermore, the dust particles
that I consider in my simulations are mostly microscopic; millimeter-sized particles are consideredquite large in the context of my results, though other studies have called grains up to centimeterscales ”well-coupled” to the gas flow. The disk-evolution timescale is broadly set by the lifetimes of
the gas disks, which range from around 3–6 Myr. Therefore, transport processes or loss of material
that occur on much shorter timescales, ∼50,000 years or less, are often termed ’rapid’ and are ofparticular interest.
The chronology of a planet-forming system falls into three broad stages:
1. The initial collapse of the parent cloud to form a solar-mass star proceeds on roughly 10 5 yrtimescales (Lin & Pringle, 1990; Enoch et al., 2009), and in this time the bulk of the central
mass and disk system is assembled.
2. The lifetime of the gas disk is on the order of 3–6 Myr before it dissipates due to accretion onto2the star as well as via photoevaporation to free space. Recent theory suggests that the earliest
disk accretion proceeds in episodic, high-intensity bursts, as most commonly measured rates
are too low to build solar masses on the above timescales (Kenyon et al., 1990; Evans et al.,2009), but by about 106 years, accretion through the disk drops down to around 10−8 M¯ per
year (Hartmann et al., 1998). In general, gas disks are no longer observed by around 5–10
Myr years (Haisch, Lada, & Lada, 2001) and therefore gas-giant planets, if they form, must
do so within that time period.
3. Terrestrial-planet formation is a longer process, typically requiring around 100 Myr in current
simulations of solar-system formation (Raymond et al., 2009). The Kuiper belt and Oort cloudare also created during this time as the giant planets scatter planetesimals near their orbits
outward. The Nice model for solar-system history (Morbidelli et al., 2009a) predicts thatlarge scale instability in the planet orbits lasted through around 700 Myr at least.
These timescales are in contrast with the 10 Gyr expected lifetime of our sun, the ∼4.5 Gyr age of
our Earth, and the hundreds of millions of years for which complex life has existed on its surface.
For the above planet-formation chronology, the first, collapse stage is dominated by fluids
theory and gas dynamics and is probed primarily by astronomical observations of star-forming
clouds. Within this stage, the dust component exists either as an uncondensed vapor (at hightemperatures near the forming star) or as fine, sub-micron-sized grains like those observed within
the interstellar medium (ISM). As such, the dust is therefore treated as a well-coupled component,
dynamically inseparable from the gas. In the last, terrestrial-planet–building stage, the physics isdominated by large-body orbital dynamics, the gas presumed to either be gone or to have largely
negligible final impact on the outcome of events. In this stage, scientific nuances are drawn largely
from the orbital architecture of both our own and extra-solar planetary systems, and the importantparticle components are believed to be at minimum tens of kilometers in size.
The middle, protoplanetary-disk stage, then, is an interface for both gas dominated and solidbody, celestial-mechanics–dominated disk dynamics, as well as simultaneously the transition fromvery small to very large solid particle sizes, including the process of planetesimal formation, which
addresses the transition from cm-sized pebbles up to hundreds-of-kilometer sized bodies suitable forplanet building. For this interface stage, observations of dust distributions in extra-solar protoplan3

etary disks offer some insight. But, the physics of dust-gas interactions in low-pressure, orbiting
systems is difficult to probe with laboratory experiments. However, physical clues to the processes
and chronologies of our own Solar Nebula do exist in the form of the mineralogies and compositionsof meteorites, asteroids, and comets. Some important clues include:* Calcium-Aluminum-rich Inclusions (CAIs). These (sub)mm-sized grains are formed bycondensation at some of the highest temperatures found within the early Solar-Nebula gases.
They are the oldest dated solids in the solar system, with ages of ∼4,568 Myr (Morris &Desch, 2010), but are found within many meteorites with bulk ages 1–3 Myr younger (Cuzzi,
Davis, & Dobrovolskis, 2003).
* Chondrules. These are submm–cm sized spheres of apparently shock-melted rock that makeup 30–80% of primitive meteorites and appear to have formed at least 1 Myr after CAIs (Cuzzi,
Davis, & Dobrovolskis, 2003). They are one of the primary building-blocks of the earliest large
solar-system bodies and age-dating techniques indicate their formation persisted for at least
2.5 Myr (Amelin et al., 2002).
* Iron-core meteorites. These are meteorites formed from fragments of the core of one orseveral differentiated-asteroid(planetesimal) parent bodies. Recent evidence suggests some
may have ages almost as old as CAIs (Kleine et al., 2005; Bottke et al., 2006), implying theformation of massive solid bodies at the earliest times of Solar-Nebula history.
* Stardust samples. In 2006, the Stardust mission returned to Earth with samples of material
captured from the comet 81P/Wild 2 (Brownlee et al., 2006). This comet is believed to haveformed beyond the orbit of Neptune and was originally thought to be composed of pristine
Solar-Nebula material. However, many of the dust grains recovered are distinctly larger thanthose observed in the ISM. Furthermore, a large fraction of the silicate grains are crystalline,
implying processing at high temperatures almost certainly at disk radii much less than where
the comet formed. One of the grains recovered by Stardust is CAI-like in nature, and otherparticles appear to contain chondrule fragments (Brownlee, Joswiak, & Matrajt, 2011).
The results of the Stardust mission provided the primary motivation for the first part of my thesis
research: to study the viability of outward mixing of Stardust-sized dust grains within a turbulent,
4protoplanetary gas-disk environment. While some studies had already explored turbulent mixing asa means to explain crystalline silicates in comets, most were performed within static disk models andneglected the specific dependence on grain size of aerodynamic-transport properties. For my initialapproach, I considered force-balance trajectories of individual grains orbiting and experiencing gas
drag within a laminar disk flow. I then added a random walk to these trajectories to simulate theturbulent diffusion of a large ensemble of simulation particles. Tracking an ensemble of particles
rather than modeling the dust as a fluid, as is most-commonly done, allows me to examine mixing of
dust grains from multiple source regions within a single simulation. While I model one-dimensional(radial) transport only, this simple approach is fairly computationally inexpensive and has allowed
me to explore transport of dust-grains over a wide range of parameter space, while still including
disk evolution within the computation.
Many studies of grain motion and planetesimal and planet formation employ steady-disk models for simplicity. However, the constraints imposed by observations tell us that disks are notsteady systems. Furthermore, characteristics of evolving-disk models (particularly those important
to grain transport, such as mass-density profiles and gas-flow velocities) can differ substantiallyfrom those with a steady-disk configuration. That the elements of planetary systems form over arange of time (as demonstrated by the age-distributions of chondrules within meteorites), rather
than instantaneously, means that forming systems sample a wide range of protoplanetary-disk conditions and must survive within these changing environments. These factors have been especiallyemphasized by the simulations pertaining to the second focus of my thesis research: using aerodynamic drag and diffusion calculations to consider the global distribution of dust solids within an
evolving disk.
Observations of dust populations in disks around other stars, the chondrule-dense nature ofmost meteorites, and current theories of planetesimal formation all point to large (mm–cm sized)
particles being an important, resilient component of protoplanetary disks and the Solar Nebula.
However, the fairly straight-foward theories of headwind drag predict that these particles will fallin onto the central star on rapid timescales so-far incompatible with the evidence from CAI andchondrule chronology discussed above. This rapid-infall behavior is a robust outcome of virtually
all aerodynamic grain-transport calculations, including those presented in this thesis, and I explore
the universality and ramifications of this effect using my transport simulations.
5This thesis is roughly divided between the physics and methodology used for my particletransport simulations and the investigations performed using those simulations. In Chapter 2, I
describe the model I use to define the evolving disk environment in which my simulation particles
experience transport and include a discussion of the relevant observational parameters used todefine and constrain my disk models. Next, in Chapter 3, I discuss my model for radial transportof dust grains within the disk model. I describe both the relevant physics involved, and include
test cases showing that drag-advection and particle-diffusion within my model conform well tohigh-precision numerical calculations and analytic solutions, respectively.
In Chapter 4, I present the first results of my particle-transport simulations and include an
overview discussion of how the different pieces of my model setup affect the simulation results.
In this chapter, I focus on the outward mixing of inner-disk particles pertinent to the Stardust
results and observations of extra-solar disk crystallinity. I find that the transport of 20 µm-sizedgrains (like those of Stardust) to the comet-forming region is compatible with turbulent mixing
within a Solar-Nebula model disk, though there is a sharp decline in outward mixing for mm-sizedgrains. While observations of disks around other stars show few correlations between crystallinity
and other physical observables of the system, I show that the observations are compatible with aspread in mixing resulting from scatter in the initial compactnesses of disks just after formationof the star-disk systems. In Chapter 5, I explore outward mixing for a number of other diskmodel parameters. Qualitatively, these results also agree disk observations. The degree of outward
mixing is found to be dependent on a few poorly-constrained model parameters, such as the disk
radial-temperature profile, and on disk conditions at times too early to be readily observable within
other disk systems. However, outward mixing is found to be independent of the global baseline
disk temperature, as well as largely independent of the α-scaling of the disk viscosity and the initial
mass of the disk, in agreement with non-correlations between disk mass and crystallinity observed
for disks around other stars.
Finally, in Chapter 6, I focus on the global distribution of dust grains relative to the evolving
gas profile. Several recent theories of planetesimal formation suggest that this process may begreatly aided by local increases in the dust-to-gas surface-density ratio (disk metallicity) above solar
values. Radial drift of grains due to headwind drag has been proposed as a mechanism to produce
such enhancements at small disk radii, and early calculations gave promising enhancement results.
6

In this chapter, I test this radial-drift hypothesis including disk evolution and radial diffusionof the dust population. I find this mechanism non-viable to produce the required enhancement
values, which can be substantial in an evolving-disk scenario. These simulations also highlight thelong-standing problem of the rapid loss of mm-sized grains from a smoothly defined disk model,
incompatible with observations. I therefore suggest that axisymmetric disk models are insufficient
to represent dust transport on a fully global scale and that non-axisymmetric structure and local
enhancements likely play an important role in the evolving solids distribution.
A glossary is included at the end of the manuscript in Appendix C as a reference for the various
symbolic variables used throughout the text. In Appendix A, I derive analytic solutions for the
diffusion of a contaminant within a gas disk, while in Appendix B, I specify some of the particular
numerical methods used within my code.

7

Chapter 2

Building a 1D Disk-Evolution Model
There are two primary models used in this work. The particle-transport model described in Chapter 3 tracks the motions of an ensemble of particles radially within a protoplanetary gas disk. The
model I describe in this chapter defines that gas disk. It is a 1D, vertically isothermal, α-disk modelthat evolves in time due to viscous spreading and accretion, where it’s primary outputs are the disk
surface-density distribution as a function of radius, Σg (R), and the disk temperature distribution,
T (R), which, for the earlier runs, is held fixed to a static R−1/2 profile. From Σg (R) and T (R),the model also calculates bulk properties like the local gas density, ρg , as well as the radial andazimuthal (orbital) gas velocities, vr,g and vφ,g , important for computing gas-drag effects on dust
particles within the disk. A parameterization of EUV photoevaporation is also included, but isimportant only near the end of the disk lifetime when accretion rates fall to a few×10 −10 M¯ yr−1 .
The primary parameters I use to distinguish between different disk models are: M D,0 , the initial
disk mass, Rd , which defines the initial compactness or spatial size of the disk, and α, which scales
the magnitude of the disk viscosity. All three of these parameters have a strong impact on the
lifetime and evolution of the disk structure (§2.5). My fiducial disk model uses MD,0 = 0.03M¯ ,Rd = 20AU, and α = 10−2 .
In this chapter and generally, when I refer to the disk and disk evolution, I am speaking primarily
about the gas component of the disk which, while harder to access observationally than the dusty
component, is easier to model and describe using basic physics and fluid dynamics and forms thebulk (∼99%) of the disk material. The mechanics of protoplanetary-(gas)disk physics are still not
well understood and I use the α model (described in §2.2 and first described by Shakura & Sunyaev(1973)) to parameterize disk turbulence, viscosity and evolution. Nevertheless, many observations

8

exist to constrain the evolutionary timeline of a disk, and I discuss these constraints and observed
properties in §2.1. In §2.2, I outline the properties of the model disks used in this thesis and discuss
the disk structure that results from the vertically-isothermal assumption. In §2.3, I discuss thetheory used to calculate the model-disk evolution. While some of my work uses a disk model witha static disk-temperature profile, other simulations include the evolution of disk temperature due
to energy-balance of stellar flux and heating by accretion. I describe the model for calculating
evolving disk temperatures in §2.4. In §2.5, I present some diagnostics for comparing the model
disks to observed disks and therefore selecting appropriate values for some of the intrinsic model
parameters. Finally, in §2.6, I describe the method for calculating various disk-gas velocities thatprovide aerodynamic drag in my particle-transport simulations. I summarize the main points of
my disk-evolution model in §2.7.

2.1

Observational Understanding and Constraints

While some of our understanding of the Solar Nebula and protoplanetary disks comes from studying
solar-system bodies and debris, most of the constraints on disk physics come from observations ofdisks around other stars. It is from observations that we derive statistics on the sizes of disks, therate at which disk matter is accreted onto host stars, and the average disk lifetimes. I will sketch
the observational constraints on disk characteristics here, and also briefly discuss what is termed
the minimum-mass Solar Nebula, which can be used as a tool for placing our own past disk incontext with those we observe today.
Probably the disk properties that are the most straight-forward to measure are disk lifetimes
and the rates of mass accretion onto disk-host stars. Disk properties are highly variable, even amongsame-age samples born into the same cluster, but an understanding of disk lifetimes can be gathered
via statistics and large surveys of star-forming regions. These surveys tell us that, in general, mostsolar-type stars less than 1 Myr old have disks, and that most lose them by the time they are
3–6 Myr old (Haisch, Lada, & Lada, 2001; Luhman et al., 2010; Manoj, 2010). Furthermore, disk
dispersal appears to be quite rapid, on the order of a few ×105 years, though near-IR signaturesof disk material persist somewhat longer than do accretion signatures (Fedele et al., 2010). There
are several well-known disk examples (e.g., TW Hydra, ∼ 8 Myr old) that are longer-lived thanthe mean (Manoj, 2010), but little to no evidence for active disks persisting beyond 10 Myr (Fedele9et al., 2010). From the fall-off rate of the statistics, some stars must lose their disks on 1 Myrtimescales, and both disks born into either low metallicity environments or into the denser starclusters appear to be shorter lived (Yasui et al., 2009; Luhman et al., 2010). However, my disk
models are aimed at solar-type systems not including external, cluster-environment-type effects.
Therefore, model disks with very short lifetimes are probably not good fits to the observations formy purposes.
Accretion signatures of disk material falling onto host stars can be measured via excess luminosity seen in the stellar spectra, and these measurements fall into a fairly well-established rangeof values, most commonly 10−9 –10−7 M¯ yr−1 (Gullbring et al., 1998; Isella, Carpenter, & Sargent,2009). While any given age bin may see as much as an order-of-magnitude scatter in mass-accretion
rates, statistically, these rates fall off over time, being commonly ∼ 10−8 M¯ yr−1 in 1–2 Myr oldsystems (Hartmann et al., 1998), a few×10−9 M¯ yr−1 in some 4 Myr old systems (Sicilia-Aguilar,
Henning, & Hartmann, 2010), and as low as a few×10−10 M¯ yr−1 in some 5 Myr old systems(Dahm, 2010). These accretion rates do not, however, account for the formation of the host star on
the timescales that the star-disk systems typically become well-established (< 1Myr), and recent
evidence points to accretion rates being time-variable with episodic bursts of high-accretion (up to
10−4 M¯ yr−1 ), at least at early times when the star-forming envelope is still loading mass onto theouter disk (Dunham et al., 2010; Manoj, 2010).
Combining the simple alpha-disk model of disk evolution (discussed in §2.2) with observed disk
lifetimes and accretion-rates has allowed for estimates of the model-viscosity-scaling parameter, α,with typical values found on the order of α = 10−2 (Hartmann et al., 1998). The alpha-model
presumes that disk turbulence is the primary driver of disk viscosity, and while belief has long heldthat turbulence is fundamental to protoplanetary-disk physics, recent measurements are only justreaching the sensitivety necessary to describe the levels of turbulence present in accreting disks
around other stars. Tentative measurements of turbulent linewidths measured are a few to a few
tenths the local sound speed, consistent with α ∼ 10−2 estimates (M. Hughes et al., 2011).
The masses and spatial extents of disks are also important parameters for defining these systems,
but are more difficult to pin down observationally. Most disk observations are in the form ofspectral-energy-distributions (SEDs) and disk dust (only about 1% of the disk mass) emits moreclearly than the gas, both for SED and spatially resolved observations. Power-law fits to dust
10emission tend to show disks extending out to a few hundred AU, while observations of CO emission
in the same systems more than double those disk radii, detecting gas as far out as 1000 AU (M.
Hughes et al., 2008) (though small disks may show gas detection only out to a couple hundred AU(Öberg et al., 2010)). Because CO may be seen to emit at quite low gas densities (Beckwith &Sargent, 1993), this apparent discrepancy in disk-radius measurements has been largely explainedby applying a similarity-solution to the disk-structure model that accounts for the viscous expansionof the disk and an exponential drop-off in surface density at the outer disk edge (M. Hughes et al.,2008). Models of disk structure using similarity solutions report a characteristic radius for the disk,rather than an outer radius. This characteristic radius is typically about half the power-law dust
emission measure of the radius and so on the order of a quarter or less the gas-emission radius.
Characteristic radii of this type reported in the literature show quite a range in disk sizes, withcharacteristic radii of < 20 to ∼200 AU (M. Hughes et al., 2008; Isella, Carpenter, & Sargent,
2009; Andrews et al., 2010). Furthermore, some observations support the idea of disk expansion
over time, noting increases in these radii over the range of disk ages (Isella, Carpenter, & Sargent,2009).
Observational estimates of disk masses typically require some assumptions about the disk dustgas composition as well as some model for the dust-particle size distribution. Nevertheless, many
estimates of disk masses have been made, and these also point to distinct variation between disks
born into similar environments. Typically, the numbers range from ∼ 0.1–40% M ¯ (Jørgensen etal., 2009; Watson et al., 2009; Öberg et al., 2010; Andrews et al., 2010), and again, there is clear
evidence that disk mass falls off with the age of the system. Jørgensen et al. (2009) measured
disk masses of 1.7–46% M¯ for Class 0 sources where the disks are still being fed by the infallingenvelope, but of only 0.81–5.3% M¯ for Class I sources about 1 Myr old, while Dahm (2010)
measured disk masses of 3.7 × 10−6 –6.5 × 10−3 M¯ for a set of 5 Myr old sources. The range oftypical, young disk masses seems to be somewhere around 1–10% M¯ . Interestingly, this is also therange given by Weidenschilling (1977b) for his estimate of the minimum-mass Solar Nebula. Thisis an estimate of the minimum mass needed in our solar-system-forming disk to be able to form ourplanets. It is based on taking the masses of the planets (and the asteroid belt) and then adding
extra volatile mass to reach solar composition. The estimate has such a wide range due mostly
to uncertainties in the compositions of the giant planets. Another commonly cited minimum-mass
11

model is the Hayashi-1981 model (see Armitage (2010) pp 4–5), which contains a mass interior to
30 AU of 0.01 M¯ .

2.2

Basic Setup and Properties

While some studies use static, steady disk models, where the disk surface-density profile is in a
steady-state balance with the viscosity and the inward flow of accretion (defined in §2.3.1), one ofthe primary uses of the disk model that I use is to explore the effects of the evolution of the disk
surface-density. I do use a steady disk as a comparison case in Chapter 4, but otherwise, the model
disks that I use begin with a t = 0 surface-density profileṀ0Σg (R, t = 0) =
3πν

Ã

1−

r

Rin
R

!

µ
¶
R
exp −
,
Rd

(2.1)

where Σg is the surface density, R is radial distance from the central star, t is time, Ṁ0 is the t = 0
accretion rate onto the central star, ν is the local disk viscosity, Rin is the inner-disk boundary(set equal to the inner-grid boundary, R1/2 = 0.099 AU), and Rd is a variable controlling the´³
pcompactness of the t = 0 profile. The ν −1 1 − Rin /R terms do reference the steady-disksolution to the primary disk-evolution equation (with irrevocable mass loss, and therefore Σ g →0, past Rin ). However, the exp (−R/Rd ) term references the similarity solution for a viscously
expanding disk and allows for a finite disk-mass distribution. The calculation for the initial disk
mass for this distribution is given in §2.5, and I primarily define the disk models that I use with
the initial total disk mass, MD,0 , and the initial compactness, Rd (as well as α).
This surface-density profile is evolved numerically due to viscous spreading and photoevaporation, as described in §2.3. This evolution is shown in Figure 2.1 for the fiducial disk models ofChapters 4, 5, & 6 where MD,0 = 0.03M¯ , and Rd = 20 AU (and α = 10−2 ). Chapters 4 & 5 disks
use a static temperature profile with T ∝ R−1/2 , appropriate for slowly accreting, passive, flareddisks, while the Chapter 6 disks evolve from initially steeper, hotter profiles. However, both setsscale to a passive-disk temperature at 1 AU of 278.9 K. As each disk evolves, the surface density
drops (due to accretion onto the central star), and the disk spreads outward before dissipation via
photoevaporation occurs after ∼5.5 Myr. I define Σg (R) and T (R) on a radial grid of 600 cells
spaced logarithmically in R. Innermost and outermost grid points are placed at 0.1 and 15,000 AU,
12

Static T

102

t=0
t=105 yr
t=106 yrt=4x106 yr
t=5.3x106 yr
t=5.34x106 yr

104

Sigmag (g cm−2)

Sigmag (g cm−2)

10

4

100
10−2
10−4
10−60.1

1.0

Evolving Power−law T

102
100
10−2
10−4
10−60.1

10.0 100.0 1000.0
R (AU)

1.0

10.0 100.0 1000.0
R (AU)

Figure 2.1: Surface-density profiles at different times for the two fiducial disk models of Chapters 4,
5, & 6. Both use α = 10−2 , and t = 0 values of Rd = 20 AU and MD,0 = 0.03 M¯ . However,
the Chapters 4 & 5 fiducial models use a static power-law disk-temperature profile, whereas the
Chapter 6 fiducial model uses an evolving power-law temperature profile that is initially hotter andsteeper. Nevertheless, the evolution of the two models is very similar except that the evolvingtemperature model evolves slightly more quickly near t = 0 and has a slightly longer lifetime before
inside-out clearing via EUV photoevaporation, with gap-opening at t = 5.47 × 10 6 yr.
respectively.
The vertical structure of the disk comes from basic hydrostatic force-balance:

−

GM? z
2 3/2

(R2 + z )

−

1 ∂P
= 0,
ρg ∂z

(2.2)

where G is the gravitational constant, M? is the mass of the central star, z is the height above thedisk midplane, and ρg and P are the local gas-density and pressure, respectively. The first term on
the left-hand side corresponds to the vertical force of gravity from the central star (for simplicity
and because my disks are usually not massive enough for it to matter, my models do not include
the self-gravity of the disk), and the second term corresponds to the gaseous pressure support. For
an ideal gas, P = nkB T = ρg kB T /µmH , where kB is Boltzmann’s constant, T is temperature, andµ is the average mass of a gas particle in proton masses (mH ). For a vertically isothermal disk inthe thin-disk limit, (z/R)2 ¿ 1, the local gas density is given byρg = ρg,0 exp

µ

·
¸¶
µ
¶
R2
1 z2
z2
1−
+ ...
≈ ρg,mid exp −
Hg2
2 R2
2Hg2

(2.3)

where ρg,mid is the gas density at the mid-plane, and the local disk-gas scale height is now defined13

Static T

10−8
10−10
10−12

t=0
t=105 yr
t=106 yrt=4x106 yr
t=5.3x106 yr
t=5.34x106 yr

10−6rhog,mid (g cm−3)rhog,mid (g cm−3)

10−6

10−14
10−16
10−18
10−200.1

1.0

Evolving Power−law T

10−8
10−10
10−12
10−14
10−16
10−18
10−200.1

10.0 100.0 1000.0
R (AU)

1.0

10.0 100.0 1000.0
R (AU)

Figure 2.2: Midplane gas-density profiles for the two fiducial disk models of Chapters 4, 5, & 6.
Companion plot to Figure 2.1.
here asHg =skB T R 3
cs=,µmH GM?
ΩK,mid(2.4)

where cs is the sound speed of the gas, and ΩK,mid is the Keplerian angular velocity at the disk
midplane.
Next, in order to derive the midplane density from the surface density, I use the definition of
the surface density:z2
ρg,mid exp −ρg dz ≈
Σg =
2Hg2
−∞
−∞
Z ∞

Z ∞

As Hg does not depend on z, and

Z ∞

e

−ax2

dx =

−∞

ρg,mid =ρg (z) =

µ

r

¶

dz .

(2.5)

π
, we have
a

Σg
√
;
2πHg
µ
¶
Σg
z2
√
.
exp −
2Hg2
2πHg

(2.6)
(2.7)

Unless the disk temperature falls off very steeply in R (T ∝ R−3 or steeper), the disk scale-height
will increase with distance from the star and ρg,mid will fall off more steeply than Σg . Figure 2.2
plots the evolving midplane-density profiles for the fiducial disk models shown in Figure 2.1.
Equation (2.4) shows a clear relationship between temperature and scale-height for a vertically
isothermal disk. Next is to relate these quantities to the disk viscosity.

14

The concept of α-viscosity, put forth by Shakura & Sunyaev (1973), assumes that the viscosity inthe disk is driven by turbulence and is approximately governed by the size and turnover velocity of
the largest eddies. It assumes that “the scale of the eddies is less than the disc thickness (certainlytrue) and that the turbulence is subsonic (probably true)” (Pringle (1981); parenthetical comments
original). This then leads toν = αcs Hg(2.8)

where ν is the viscosity, and α ≤ 1 is a dimensionless scaling parameter. Using Equation (2.4), thisis often re-writtenν = αΩK,mid Hg2 .

(2.9)

For a passive, flared disk temperature of T ∝ R−1/2 , Equation (2.8) leads to ν ∝ R and therefore
steady-disk models with Σg ∝ 1/R. For my fiducial models, I adopt α = 10−2 . This value falls
within the range of values estimated observationally (Hartmann et al., 1998; Hueso & Guillot, 2005;
King, Pringle, & Livio, 2007) and allows dissipation of my model disks in less than 10 Myr, as
discussed in §2.5.
Other models of disk viscosity used in the literature include, e.g., the beta-prescription for
viscosity, which assumes that viscosity is dependent only on local orbital parameters within the disk;
viscosity based on gravitational-instability criterion within the disk; and dead-zone disk models,
based on phenomena predicted for magnetorotational models of disk turbulence. Like the alphaprescription, a beta-prescription is most-often used as a simple power-law-in-radius prescription fordisk viscosity. Viscositity models based on gravitational instability are often used in conjunction
with an alpha-prescription, evaluating local disk conditions for Q-parameter gravitational instability
and assigning generally a higher viscosity parameter to those regions based on gravitational overturnproperties. Finally, a dead-zone disk model is often described in terms of an alpha-prescription,
but with a radial jump in the assumed alpha-value. A deadzone model assumes that ionization
rates in the densest inner regions of a disk are incompatible with MRI-driven viscosity, creating an
inner-disk dead-zone region where the total-disk viscosity is lower and mass is subsequently piled
up relative to the higher-visocisity disk beyond the dead-zone. In a 2D model, the dead-zone isconfined to near the midplane of the disk, with active-viscosity zones in the disk surface-layers
where ionization remains efficient.
15

Returning to the alpha-prescription and disk model used in this work, for my verticallyisothermal disk-evolution model, the disk temperature is the fundamental parameter, so that the
primary parameters of interest may be summed up withHg =1
ΩK,mid

s

kB T
;
µmH

ν = αΩK,mid Hg2 ;
µ
¶
Σg
z2
exp −
.
and ρg ≈ √
2Hg2
2πHg
If the temperature is defined by a power law, then T , Hg , and ν can be written interchangeably as
T (r) = T◦ rqT ;Hg (r) = H◦ rqH ;ν (r) = ν◦ rqν ,(2.10)

where
qT = 2qH − 3 = qν −3.
2

(2.11)For the temperature-static disk models that I run, qT = −1/2, qH = 5/4, and qν = 1, appropriate
for a passively heated, flared disk.

2.3

Modeling Disk Evolution

In this section, I discuss the mechanics of modeling 1D gas-disk evolution, including the basic theory
and numerical approach for viscous spreading of mass and angular momentum in a cylindrical,
gravitationally bound system, as well as the parameterization used to include loss of disk mass dueto photoevaporation. This model for disk evolution assumes that the star and disk are alreadyformed at t = 0, with a fixed, central, gravitational mass (and a fixed stellar luminosity), as initialinfall and system formation are beyond the scope of the studies of this thesis. The model also
assumes that disk evolution occurs in isolation, neglecting radiative or gravitational interactions
that might take place with members of the system’s birth cluster.

16

2.3.1

TheoryThe derivation for the most basic form of the 1D (radial) disk evolution equation can be found in
Pringle (1981) and results in an equation for the evolution of the disk surface density, which is:
·
³
´¸
1 ∂
1
∂Σg
3 ∂
2
2
=
R
νΣg R
∂t
R ∂R
∂R

(2.12)

Equation (2.12) is derived by invoking conservation of mass and conservation of angular momentum
within the disk and assuming azimuthal symmetry. Conservation of mass gives:

R

∂Σg
∂
+
(RvR Σg ) = 0 ,∂t∂R(2.13)

where vR is the radial velocity of the disk material. Conservation of angular momentum gives:

R

¢
¢
∂ ¡
∂ ¡
1 ∂Gν
Σg R 2 Ωg +RvR Σg R2 Ωg =
,
∂t
∂R
2π ∂R

(2.14)

where Ωg is the orbital angular velocity of the disk gas, and Gν is the local torque of an outerannulus of material acting on the annulus of material immediately interior. It is given byGν (R, t) = 2πR2 dFν = 2πR2 νΣg A = 2πR2 νΣg R

dΩg
,
dR

(2.15)

where dFν is the viscous force per unit length, and A is the rate of the shearing.
Assuming that the disk is slowly varying and ∂Ωg /∂t = 0 for simplicity, expanding the second
term in Equation (2.14), and substituting in Equation (2.13), eliminates the ∂Σ g /∂t term andprovides an equation for RvR Σg . Plugging this equation back into Equation (2.13) eliminates theunknown vR and yields the disk-evolution equation (Pringle, 1981) (2.9):
∂Σg
1 ∂
=−
∂t
R ∂R

"µ

¶
µ
¶#
∂Ω
∂ ¡ 2 ¢ −1 ∂gR Ωg
νΣg R3.
∂R∂R∂R(2.16)

Equation (2.16) is a more general form of the 1D disk evolution equation, and yields the more
tractable, standard-caseform of the equation given in Equation (2.12) for the assumption thatrGM?
Ωg = ΩK,mid =. Therefore, the standard disk-evolution equation (which is used to evolve
R3
17

the model-disk surface-density profile in my disk evolution model) assumes a) Keplerian velocities
for the disk gas, b) zero gravitational contribution from the disk gas, and c) a constant stellar-massvalue.
Equation (2.12) can be made more compact by using a change of variables: X =√R (Pringle,

Verbunt & Wade, 1986). This compact form is useful for numerical applications (§2.3.2 & Appendix B.1) and has:∂Σg
3 1 ∂2=(νΣg X) .
∂t
4 X 3 ∂X 2

(2.17)The disk-evolution equation (2.12) is best physically represented by instead expanding thesecond derivative in R, to produce
∂Σg
∂2
91 ∂
= 3 2 (νΣg ) +(νΣg ) .
∂t∂R
2 R ∂R(2.18)

Here, the change in disk surface density is broken into a diffusive and an advective component,
showing that the disk as a whole will tend to spread out, but with a preferred direction of flux forthe mass and angular momentum (mass inward and angular momentum outward). Importantly, it
is clear that for the special case when the product νΣg equals a constant in R, that the surfacedensity will remain unchanged. Σg ∝ 1/ν is then the steady-state solution that (for a fixed viscosity
profile) an evolving disk profile will approach over time.
One modifier to the simple νΣg = C steady-disk profile comes from the inner edge of the disk,
where a zero-torque boundary condition at Rin provides a pinch-off of the disk surface density. The
steady-disk profile derived by Pringle (1981) (Eq. 3.9) is:
Ṁ
νΣg =
3π

Ã

1−

r

Rin
R

!

(2.19)

where Ṁ is the uniform rate of mass accretion through the disk at all R.
Pringle (1981) derives an analytic solution for disk evolution from the standard equation (2.12)
for the case of a constant, uniform viscosity. However, the derivation is easily expanded for any
radial power-law relation for the viscosity, ν = ν0 Rb . The derivation is performed for the simplest-18

case initial condition of a delta function in Σg :

Σg (R, t) = Σg,0 δ (R − R0 ) .

(2.20)

Because this derivation follows much of the same form as the derivation for the diffusion of acontaminant within a steady accretion disk that is outlined in Appendix A, I will not detail it here.
A summary of the derivation is as follows
1. Changing variables with X =√R produces the compact form of the equation (2.17).

2. Because the viscosity has been defined as independent of time and a power-law in R, theequation is now separable in R and t.
3. The time component yields exponential decay in time scaled by λ2 , the square of the constantof separation.
4. The spatial component requires another change of variables to produce a Bessel equation with
Bessel-function solutions, Jn (x), dependent on λ.
5. Using the δ-function initial condition, integrating over all λ, and using a Hankel transform
finally switches things over to a modified Bessel function, Iβ (x).
The final, general solution, assuming initial-condition (2.20) and power-law viscosity (b 6= 2, as
b = 2 will not produce a Bessel equation), is given by

Σg (R, t)δ(R−R◦ )·¸µ ¶1 1Ṁ
4β 4β R−b/2 R◦ 4 + 2 b
√
√
=
t
R
4π 3ν◦
3ν◦
"
#
·
¸
αc /2
α
α
c
c
2 R◦ + R
2 (R◦ R)∗ exp −4β
Iβ 8β
3ν◦ t
3ν◦ t

(2.21)

αc = 2 − bβ =1
2 (2 − b)where Iβ is the modified Bessel function of the first kind of order β. In the case where the viscosity
is uniform everywhere in the disk (b = 0) the final solution reduces to:ṀΣg (R, t) =12πν◦ t

µ

R◦
R

¶1/4

µ
¶
µ
¶
R◦2 + R2
R◦ R
exp −
I1/4
,
12ν◦ t6ν◦ t
19(2.22)

which is precisely equivalent to Pringle’s expression (Pringle, 1981) (2.13) if one perfoms the coordinate transforms into his notation:

χ=

R
;
R◦

τ=

12νt
R◦2

In Appendix B.1, I detail my numerical method for disk evolution and include test comparisons
between the output of my code and this simple-case analytic solution.

2.3.2

Numerical Approach

In this subsection, I give an overview of the numeric setup for disk evolution in my code. The
detailed discussion of numerical disk evolution is presented in Appendix B.1, along with tests
for convergence, fidelity to the analytic test-case solution, and conservation of mass and angularmomentum. This is a fairly standard setup for 1D disk evolution and grid-spacing (see, e.g., Lin& Pringle (1990); Hueso & Guillot (2005)).
The disk-evolution model uses the most compact form of the basic disk-evolution equation,∂Σg
3 1 ∂2=(νΣg X) ,∂t
4 X 3 ∂X 2
where X =√R. This form is both the most tractable numerically and leads to the best convergence,since the addition of multiple terms is a large source of error in numerics. Because of taking thisform of the equation, some studies (e.g., Pringle, Verbunt & Wade (1986); Hueso & Guillot (2005))
have therefore used constant ∆X grid spacing. However, a logarithmic grid spacing provides thebest conservation of mass and angular momentum and that is what I use in my disk evolution
model (this also provides the grid spacing for the total disk-evolution + particle-transport model
as a whole).
µRi = Rmin
p
Xi =
Ri ,

Rmax
Rmin

¶(i−1)/(ngrid −1)(2.23)

where ngrid is the total number of points on the grid, Rmin = R1 is the innermost grid point, andRmax = Rngrid is the outermost grid point. In my fiducial disk models I use ngrid = 600, Rmin = 0.1
20

AU, and Rmax = 15000 AU (a large radial extent is best for preventing loss of disk mass past theouter grid boundary).
Disk evolution is evaluated using a standard time-explicit finite-difference scheme, which, forthe basic disk-evolution equation (with uneven grid spacing), looks like this (Press et al. (1992)§19.2):jΣj+1
g,i − Σg,i

∆t

3 1 1
=
4 Xi3 ∆Xi"

(νΣg X)ji+1 − (νΣg X)ji(νΣg X)ji − (νΣg X)ji−1+∆Xi+1/2∆Xi−1/2#,(2.24)

where i and j are the spatial and temporal indices, repsectively, ∆Xi = Xi+1/2 − Xi−1/2 is thewidth of grid cell i and ∆Xi+1/2 = Xi+1 − Xi is the distance from grid point i to i + 1. Numericalstability for this first-order scheme is given by the condition,2∆t ≤ min
3

Ã

!

,

∆Xi ∆Xi−1/2 Xi2νi!

∆Xi ∆Xi−1/2 Xi2
νi(2.25)

and for my disk-evolution model I use
2∆t = ζ∆t × min3Ã,(2.26)

where ζ∆t = 1/2. This time-stepping constraint is the primary draw-back to using logarithmic grid
spacing; the very small grid cells in the inner-disk regions require the evolution be calculated witha very small global time-step (∆t ∼ 0.05 years in my fiducial models). The effect is counter-acted
a bit by the fact that the viscosity values tend to be lowest in the inner disk (the turbulence issmaller-scale). However, since the time-stepping does go as 1/ν, simulations using larger values of
α for the disk-viscosity parameterization are proportionally more expensive to run.

2.3.3

Parameterized EUV Photoevaporation

To include EUV photoevaporation in my disk-evolution model, I use the parameterization presented
in Alexander & Armitage (2007) of the work of Font et al. (2004). Because I am interested in only
the surface-density profile as the disk evolves, photoevaporation is treated as simply a mass-loss

21

term, so that the basic equation is
·
³
´¸
∂Σg
3 ∂
1/2 ∂
1/2
=
R
νΣg R
− Σ̇wind (R, t) ,
∂t
R ∂R
∂R

(2.27)

where Σ̇wind is the rate of surface-density loss due to photoevaporation. Numerically, the viscous andphotoevaporative disk evolution are performed separately, so that the surface-density profile is firstviscously evolved as outlined in §2.3.2 to produce a Σvisc (R)j+1 profile. Then the photoevaporation
component is subtracted following
³
´j
j+1
Σj+1
=
(Σ
)
−
∆t
Σ̇
.
visc iwind
i(2.28)i

If the calculated mass loss is greater than the available mass remaining in a grid cell, the surfacedensity in that cell is set equal to zero.
Following Alexander & Armitage (2007), Σ̇wind (R, t) is broken into two components, which
essentially define a first and second stage of EUV photoevaporation. The two components arecalled the diffuse wind, Σ̇diffuse , and the direct wind, Σ̇direct , referring to the respective sources ofionizing radiation.
The diffuse wind dominates in the beginning and is caused by radiation that scatters off thedisk atmosphere and ionizes the surface layers of the disk. It is easy to see how a wind will belaunched at and beyond a disk radius where the high temperature of the ionized layer allows it tobe gravitationally unbound. This radius is given by

Rg =

GM?
,
c2s,ionz(2.29)

where cs,ionz = 10 km s−1 is the sound speed of the ionized gas. However, interior to this point,
there is also a subsonic flow of material that, for this parameterization, may cause loss of disk gas asR
far inward as= χR = 0.1. Therefore, the diffuse wind is set equal to zero interior to χR = 0.1.
Rg
Exterior to this boundary, the equation used for the diffuse wind is given byΣ̇diffuse (R, t) = KdiffpΦdiff (t)

22

Ã

2
15/2

χR

25/2

+ χR

!1/5

∗ (χR − 0.1)Dph exp [Bph (χR − 0.1)] ,(2.30)

where Kdiff is a constant containing numerical fit parameters for the diffuse wind density andwind launch velosity, Φdiff is the ionization flux, and Dph = 0.2457 and Bph = −0.3612 are fitting
constants for the the launch velocity of the diffuse wind. Kdif f is given by

Kdiff =√

3C1,ph Aph µionz mH cs,ionzq,π αB Rg3(2.31)

where µionz = 1.35 is the average mass of an ionized wind particle in proton masses, m H , αB =2.6 × 10−13 cm3 s−1 is a recombination coefficient for atomic Hydrogen, and C1,ph ' 0.14 andAph = 0.3423 are, once again, numerical fitting factors. Aph scales the wind launch velocity relative
to cs,ionz , and C1,ph scales the base density profile.
In the first stage of EUV photoevaporation, when the wind is diffuse-wind dominated, Φ diff =Φ = 1042 photons s−1 (Alexander, Clarke, & Pringle, 2005; Alexander & Armitage, 2007). However, once the viscous accretion rate drops low enough that photoevaporated material is no longer
replaced by inward-flowing gas, a gap opens in the disk at R = Rgap 1 , and photoevaporation transitions from the diffuse-wind regime, to the direct-wind regime. During the transition from diffuse
to direct wind,Φdiff (t) = ΦµRthin (t)Redge (t)¶2,(2.32)

where Rthin (t) < Rgap < Redge (t), Rthin is the transition radius from the inner-disk material
to the optically thin gap, and Redge is the inner edge of the outer disk outside the gap. EUV
photoevaporation is driven by ionization of hydrogen, and the optically-thin limit used to define
the edges of the gap, Rthin and Redge is given by Σg < mH /σ13.6eV , where σ13.6eV = 6.3 × 10−18
cm2 is the absorption cross-section for ionizing photons. Once the inner disk has viscously drained
onto the central star, Rthin = 0 and the diffuse wind is completely shut off.
The direct wind is due to ionizing radiation that directly strikes the flared inner edge of theNote that in Alexander & Armitage (2007), Rgap is defined as Rcrit ≈ 1.4 (M? /M¯ ) AU, technically independent of where the gap actually opens (usually somewhat interior to 1.4 AU). While
there seem to be other photoevaporative physics involved in defining Rcrit , because of this discrepancy, my disk-evolution code simply waits to define Rgap until a gap is opened and sets it equal tothat initial gap location.
1

23

disk, just beyond the gap. Interior to the gap there is zero direct-wind photoevaporation. For
R > Rgap , the direct wind is given bypaph −3/2
Σ̇direct (R, t) = fph (R, t) Kdir Redge(t) Φdir (t)sR1−2aph,Hg (R)(2.33)

where fph (R, t) is a smoothing function, used to prevent the direct-wind component from diverging(unphysically) at small R, Kdir is made up of constants from the Σ̇direct -fit relation, Φdir is the
direct-wind ionization flux, and aph = 2.42 is a direct-wind power-law scaling in R with-respect-toRedge . The fit for the direct wind presented in Alexander & Armitage (2007) is based on the resultsof Alexander, Clarke, & Pringle (2006). The smoothing function is given by·µ

R − Redge (t)f (R, t) = 1 + exp −Hg (R = Redge )

¶¸−1

.

(2.34)

And,
Kdir =

C2,ph µionz mH cs,ionz
,
√
π αB

(2.35)

where C2 = 0.235 is a scaling constant (suitable for Hg /R ' 0.05).
Once the inner disk drains and the direct wind dominates, Φdir = Φ. However, during thetransition from diffuse to direct wind, the direct-wind ionizing flux is filtered by the optical depthof the inner disk, and
"

!Z
Ã
#
Redge (t)χ2phΣg
σ13.6eV√Φdir (t) = Φ exp −
exp −
dR ,
mH
2
2πHg
0

(2.36)

where χph = 2.5 accounts for the inclusion of gas-density effects χph scaleheights above the disk
midplane. The initial gap-opening is somewhat sensitive to the evaluation of the integral in Equation (2.36), as well as to the grid-space resolution of the model in the inner disk, so logarithmic
grid-spacing is a plus for this.
Figure 2.3 shows a replica the gas-disk evolution in Figures 2 & 3 in Alexander & Armitage(2007) using my disk-evolution code. Compared to their figures, the inner disk drains faster than it
should. However my simulations are not concerned with the dynamics of particles orbiting withina rapidly clearing inner disk, and so I consider this to be a sufficiently accurate match for my24alpha = 10

100

t=0
t=2x104 yr
t=3.3x104 yr
t=3.6x104 yr
t=4x104 yr
t=5x104 yr

10−2
10−4
10−6
0.1

1.0

10.0 100.0 1000.0
R (AU)

Surface Density (g cm −2)

Surface Density (g cm −2)

102

−2

102

alpha = 10

−3

100

t=0
t=2x105 yr
t=3.3x105 yr
t=3.6x105 yr
t=4x105 yr
t=5x105 yr

10−2
10−4
10−6
0.1

1.0

10.0 100.0 1000.0
R (AU)

Figure 2.3: Disk evolution and EUV-photoevaporation gap opening for disk models mimicking
those presented in Alexander & Armitage (2007): Ṁ0 = 10−10 M¯ yr−1 , Rd = 500 AU. The innerdisk drains slightly too fast with my disk model compared with the Alexander & Armitage (2007)
results, but they are otherwise a good match.
purposes.

2.4

Evolving the Disk Temperature

The simulations that I run to examine the outward mixing of inner-disk grains for Chapters 4 &5 use a static disk-temperature profile for the model-disk structure and evolution. However, atearly times, during disk formation via infall and the initial period of energetic accretion, disks
are believed to be much hotter (Cassen, 2001), particularly in the inner, planet-forming regions.
Therefore, before running simulations of the global dust distribution for Chapter 6, I upgraded the
disk-evolution model to include evolution of the disk-temperature profile. Interestingly, although
in the α model for disk viscosity ν ∝ T , the end result is not a strong effect on the disk evolution
or lifetime. While hot inner disks do drain more quickly, most of the disk mass is held at large
distances from the star where the change in computed disk temperature is small and slightly morerapid expansion at early times actually results in a slightly longer disk lifetime.
My disk-evolution code currently allows for two methods of modeling disk-temperature evolution, the second method being a parameterization of the first and therefore less computationallyexpensive. For the first method, I solve for the temperature at the disk midplane according toenergy-balance between radiative cooling and the incident starlight at the disk surface, and heatgenerated at the midplane by the dissipation of energy via accretion. For the second method, I fit
25

a power-law distribution, T = T0 RqT , (truncated at T = 10 K in the outer disk) to the modeled
distribution and allow T0 and qT to change with time, following a prescribed decay toward a passivedisk profile. Because my simulations do not closely examine the chemical or mineralogical effects
associated with the range of possible disk temperatures, this second, faster method for calculating
disk-temperature evolution is sufficient for most of my applications. However, one initial run ofdisk evolution accounting for the full energy-balanced temperature is first required for a given setof disk parameters in order to choose an appropriate evolving–power-law approximation.

2.4.1

Calculating an Energy-balanced Temperature at the Disk Midplane

For the energy-balanced model, I solve for the gas temperature at the disk midplane for each radialgrid point at each disk-evolution time step. I assume that the disk is everywhere in a steadystate so that the energy-flux from radiative cooling is balanced exactly by the heating fluxes fromabsorption of external radiation and from viscous dissipation within the disk. I assume that thedisk is optically thick in the radial direction so that all energy transport is vertical relative tothe disk midplane and can then solve for the temperature at each radial grid point in isolation.
For simplicity, I assume that all energy loss is radiative and so do not account for some portion
that may be converted into mechanical energy, e.g., in a disk wind. Therefore, the equation forenergy-balance is
Fcool = Fvisc + Fstar + Fcloud ,(2.37)

where Fcool is the rate of energy loss per unit area due to radiative cooling, Fvisc is the rate ofheating by viscous dissipation, Fstar is the energy flux of radiation from the central star, and Fcloudis the energy flux from the background radiation of the cloud in which the disk has formed.
Radiative heating and cooling are straight-forward. Remembering that the disk has two sides,
Fcool = 2σB ²Te4 ,(2.38)

where σB = 5.6704 × 10−5 ergs s−1 cm−2 K−4 is the Stefan-Boltzman constant, ² = 1 in the
optically-thick regime is the radiative efficiency of the gas, and Te is the effective temperature at26

the disk surface. Following the approximation used in Ciesla (2009),
4
Fstar = 2σB ²Tirr
=

L? φ
,
2πR2

(2.39)

where Tirr is the effective local temperature of irradiation from the central star, L ? is the luminosityof the central star, and φ is the local angle of incidence between the disk and the starlight. For mostof the science simulations of this thesis, I set L? = 5.0175L¯ , in order to match T (1AU) = 278.9K
in the fiducial, passive-disk temperature models of Chapters 4 & 5. I follow Ciesla (2009) in setting
φ = 0.05 everywhere. For the background radiation
4,
Fcloud = 2σB ²Tcloud(2.40)

where Tcloud = 10K is the background temperature of the cloud environment.
For heating due to the viscous dissipation of energy, one returns to the equations for viscous
torque and disk evolution discussed in Pringle (1981) and in §2.3.1 above. On an annulus of gas ofarea 2πR∆R, work is done by the viscous torque, Gν , at a rate of:
·
¸
∂Ωg
∂Gν
∂
Ωg
∆R .
∆R ≡
(Ωg Gν ) − Gν
∂R
∂R
∂R

(2.41)

The first term of the equivalence represents the energy-transport rate, while the second term represents the rate of energy dissipation as heat within the disk. Using the relation for G ν given in
Equation (2.15), per unit areaGν Ω0g ∆R
Fvisc =
= R2 νΣg
2πR∆R

µ

∂Ωg
∂R

¶2

.

(2.42)

Inserting these energy-flux definitions into the energy-balance Equation (2.37) gives
2σB Te4 = R2 νΣ

µ

∂Ωg
∂R

¶2

¡ 4
¢
4
+ 2σB Tirr
+ Tcloud.

(2.43)

However, that Equation (2.43) is correct only in the optically thick regime, while the outer, opticallythin regions of the disks are also important for my studies of long-distance particle transport. To

27

include optically thin solutions and to solve for the temperature at the disk midplane rather thanthe effective, radiative temperature at the surface, I use the approximation derived by Nakamoto& Nakagawa (1994):
1
4
σB Tmid= R2 νΣg
2

µ

∂Ωg
∂R

¶2 µ

3
1
τR +
8
2τP

¶

¢
¡ 4
4
,
+ σB Tirr+ Tcloud(2.44)

where Tmid is the temperature at the disk midplane, τR is the Rosseland mean optical depth, andτP is the Planck mean optical depth. Some notes on this deriviation/approximation:* Nakamoto & Nakagawa (1994) note that in the optically thick regime, the radiative-diffusion
3τR
4= Te4 +
approximation, in terms of the viscous heating, gives Tmid
Fvisc .
16σB
* In terms of heating by external radiation, re-radiation of that energy is assumed to be inefficient so that a vertically isothermal approximation can be made for those contributions.
The Rosseland mean optical depth is defined in terms of the disk surface density and the Rosselandmean opacity, κR .
τR = κ R Σ g .

(2.45)

Note that untangling optical depth and opacity in the literature is a bit tricky due to different
methods used to set up the problem. Some papers use Equation (2.45) as I have shown it (Nakamoto& Nakagawa, 1994; Stepinski, 1998), whereas others use τR = κR Σg /2 (Ruden & Pollack, 1991;
Hueso & Guillot, 2005; Ciesla, 2009; Rice & Armitage, 2009). However, true contradictions intheory likely appear in few, if any of these derivations. Finally, Nakamoto & Nakagawa (1994) useτP /τR = 2.39, citing it as more traditional but with some observational results maybe pointing
to τP /τR ∼ 1 being more accurate — though it is probably also temperature dependent. For
simplicity, in this temperature-evolution model, I use
τP
= 1.
τR

(2.46)

The choice of τP /τR appears to have a negligible impact on temperatures calculated for my model
disks, as shown in Figure 2.6.
Next, solving Equation (2.44) for the midplane disk temperature requires solving two other28

Figure 2.4: Schematic of Bell & Lin (1994) opacity vs. temperature (at ρ g = 1 × 10−7 g cm−3 ).
The opacity is divided into eight temperature domains with four of those domains dependent on
the mean gas density. Grey’ed lines indicate the trends toward lower and lower density.

equations simultaneously. First, the opacity, κR , depends on both the temperature and the meandensity of the gas, the later of which I approximate with the disk-midplane density given in Equation (2.6) for a vertically isothermal disk, ρg,mid ∝ 1/T . Second, in the α-approximation given in
Equation (2.9), the viscosity is also temperature-dependent, following ν = αΩ K Hg2 ∝ T . Therefore,
deriving an energy-balanced disk-temperature requires solving for Tmid , κR (and ρg,mid ), and ν
simultaneously. The numerical methodology for this is detailed in Appendix B.3.3.
By far, the most troublesome piece of the solution comes from the Rosseland mean opacity,
for which I use the parameterization of Bell & Lin (1994). In this parameterization, the opacity
is devided into eight regimes dependent on temperature. In each regime, a different mechanism
dominates andκ = κi ρag i T bi ,(2.47)

where κ is defined in units of cm2 g−1 . The boundaries between the different regimes are definedat κi ρai T bi = κi+1 ρai+1 T bi+1 .

Tbound (i → i + 1) =

·µ

κi
κi+1

¶

ρ

(ai −ai+1 )¸1/(bi+1 −bi ).

Bell & Lin (1994) also use a smoothing technique to transition between each regime, but for abasic, 1D disk model, such refinement is not necessary, and my model simply divides the opacity
29

opacity (cm2 g−1)

105
100
10−5
10−10
10−15
10−20

Density: −3(g cm )
10−7
10−9
10−12
10−15
10−20
10−25
10−30
10−41

10−25 0
10

101

102

103
T (K)

104

105

Figure 2.5: Bell & Lin (1994) opacities for a range of mean gas densities. Opacities divided into
discreet temperature domains, rather than smoothed as in Bell & Lin (1994).

into discreet temperature domains. These domains are dominated, as defined by Bell & Lin (1994),
by: ice grains; the evaporation of ice grains; metal grains; the evaporation of metal grains; molecules;
H-scattering; bound-free and free-free processees; and electron scattering. Note that above T ∼2000K, the Bell & Lin (1994) are known to be consistently low. However, such high temperatures
are almost always above the regime of those seen in my model disks.
In fact, not all of the opacity domains are dependent on the mean density, and Figure 2.4 plotsa schematic of the opacity versus temperature, with grey’ed lines indicating the trends toward
lower and lower gas density. As the trend lines shift, the transition points between some domains
cross other domain boundaries. Where this occurs, the middle-most domain is discarded and a
new domain boundary calculated using trend lines from the domains to either side. Figure 2.5
plots the Bell & Lin (1994) opacities vs. temperature for a range of mean-density values. Belowρ ∼ 10−10 g cm−3 , the bound-free–free-free domain is lost, as are the molecules domain below
ρ ∼ 10−18 g cm−3 , the metal-grains domain below ρ ∼ 10−25 g cm−3 , and the evaporation-of-icesdomain below ρ ∼ 10−26 g cm−3 .
The results of solving for the energy-balanced temperature at t = 0 in the fiducial disk model are
shown in Figure 2.6. Note that in the inner regions the temperature is dominated by accretion (withlower temperatures for lower disk viscosity), while the temperature in the outer disk is dominated30Midplane Temperature (K)1000

E.B. Fiducial
tauP/tauR = 2.39
L* = 1Lsun
alpha = 10−3
Static T Fiducial

100

10
0.1

1.0

10.0 100.0 1000.010000.0
R (AU)

Surface Density (g cm −2)

Figure 2.6: Midplane-temperature distribution at t = 0 calculated with the energy-balance–
temperature code for the fiducial disk parameters (MD,0 = 0.03M¯ , Rd = 20, α = 10−2 , L? ' 5L¯ )unless otherwise specified. The static temperature used for the fiducial disks of Chapters 4 & 5
is included for reference. Note that the test run with τP /τR = 2.39 is virtually identical to theτP /τR = 1 case.

104
t=0
t=2000 yr
t=105 yr
t=106 yrt=4x106 yr

103
102
101
100
10−1
10−2
10−30.1

1.0

10.0
100.0
R (AU)

1000.0Figure 2.7: Surface-density profiles from running the disk-evolution code with an energy-balanced
temperature distribution and the fiducial-disk parameters listed in Figure 2.6.

31

by stellar and background radiation. Such a non-smooth temperature distribution does have anobservable effect on the evolving surface-density profile at early times, as shown in Figure 2.7 forthe fiducial-disk model. Since (in the α-viscosity model) ν ∝ T , a greater inward flow of material
occurs where the temperature is highest, and the inner disk quickly settles into a quasi-steady statewith lower Σg in regions of high T and a mass flux that is roughly constant in R.

2.4.2

Calculating an Evolving Disk Temperature using a Power-law Parameterization

Converting the temperature evolution derived using the energy-balance method to a fitted, evolving,power-law distribution is a two step (two fit) process. First is to fit the temperature distribution
at each time step with a power-law distribution:

T (R, t) = TAU (t)

µ

R
1AU

¶qT (t).

To make this fit, I have chosen to fix this first power-law fit to the modeled temperature distribution

Midplane Temperature (K)at the grid points for R ∼ 0.2 & ∼ 200 AU, as shown in Figure 2.8. Therefore,

t=0
t=105 yr
t=106 yr

1000

100

10
0.1

1.0

10.0
100.0
R (AU)

1000.0Figure 2.8: Plots energy-balanced disk temperatures (thick lines) compared to the first fit to an
evolving power-law temperature (darker, thin lines) where the two are matched at R ' 0.2 and 200
AU at each time step.

32

µ

¶
µ
¶
T0.2
R0.2
qT (t) = ln
/ ln
,
T200
R200
(T0.2 − T200 ).
TAU (t) =
qT
qT
(R0.2
− R200)(2.48)(2.49)

Because of the complex structure of the energy-balanced temperatures, the q T and TAU functions
derived this way are not entirely smoothly varying in time. In order to run simulations using an
evolving power-law temperature distribution with my code, I next fit qT (t) and TAU (t) in t using aquasi-exponential decay toward a passive-disk temperature profile. For each power-law parameter(qT , TAU → x)

" µ ¶ #
t bx
x (t) = (x0 − x∞ ) exp −+ x∞ .
tx(2.50)

The fit is made separately for qT (t) and TAU (t) for each disk-evolution model. x0 and x∞ are takenfrom the initial and final conditions provided by Equations (2.48) & (2.49) and the disk-evolution
model, discounting sudden discontinuities near t = 0 when the disk surface-density at small R is
first adjusting into a quasi-steady state. This fit is also made by matching the curves at two points(in t this time). However, these two points, (t1 , x1 ) and (t2 , x2 ), are chosen by inspection separately
for each disk-evolution model to provide a nice fit. Using these points,

A1 = lnµx1 − x ∞x0 − x ∞

¶

,

bx
tx

µ

x2 − x ∞x0 − x ∞ln (A1 /A2 )=,ln (t1 /t2 )!1/bx
Ã
tb2x − tb1x.
=
A1 − A 2
A2 = ln¶,(2.51)(2.52)

The fits for qT and TAU for the fiducial-disk model of Chapter 6 are shown in Figure 2.9, plotted
beside the Equations (2.48)&(2.49)-derived values. The power-law fit parameters thus chosen forthis disk-evolution model are shown in Table 2.1. Figure 2.10 plots and compares Σ g contours
for the fiducial-disk model using the original static temperature distribution, the energy-balanced
temperature distribution, and the power-law fitted temperature distribution. Interestingly, despite
the marked difference in temperatures (and therefore viscosities) in the inner disk at early times, thelarge-scale and long-term disk evolution are all quite similar (and the power-law–fitted temperatureparameterization is a decent match for the energy-balanced model). This suggests that the outer

33

TAU fits

qT fits−0.45

1st fit (temp matched)
2nd fit (time smoothed)

500

−0.50

qT(t)

TAU(t) (K)

600

400
300
200

−0.55
−0.60

0

−0.65

5.0×105 1.0×106 1.5×106 2.0×106
Time (years)0

5.0×105 1.0×106 1.5×106 2.0×106
Time (years)

Figure 2.9: Plots the quasi-exponential fit in t (Equation (2.50) ) against the initial fit at R ' 0.2
and 200 AU for the fiducial disk model.

MD,0 = 0.03M¯ , Rd = 20AU, α = 10−2 :TAU,0 = 500K
qT,0 = −0.61
TAU,∞ = 279.94K
qT,∞ = −0.497683
tTAU = 3.494091 × 105 yr
tqT = 3.9992166 × 105 yr
bTAU = 0.924903
bqT = 0.965539
Table 2.1: Fitting constants for the evolving, power-law temperature distribution of the fiducial
disk model. For fitting constants of the full set of disk models run with evolving temperature, see
Table 6.1 in Chapter 6.

34

6×106Time (years)

5×106

1 g cm−2

0.1

4×106
3×106

10 g cm−2

2×106
10−4

1×106
0
−1

100

0

1
2
log10 R(AU)34

Figure 2.10: Surface-density contours for an evolving disks using the fiducial-disk-model parameters.
Green contours plot the temperature-static disk (fiducial model for Chapters 4 & 5. Red contours
plot the disk evolving with an energy-balanced temperature distribution. Yellow contours plot thedisk with a power-law-fitted evolving temperature distribution (fiducial disk model for Chapter 6.

disk, housing most of the disk mass and where the temperature variations are smallest, holds
primary control over the global, viscous disk evolution.

2.5

Observational Comparisons for Model Disks

Here I include a brief comparison of the model disks used in this thesis to the observational constraints laid out in §2.1, considering specifically: disk mass, mass-accretion rate, outer gas radius,and lifetime. The fiducial disk model uses MD,0 = 0.03M¯ , Rd = 20 AU (similar to a t = 0
characteristic radius of 20 AU), and α = 10−2 . In Figures 2.11 & 2.12, I plot Rout , the outerdisk radius estimated based on CO emission, the total disk masses, and the total mass-loss rates
as a function of time for model disks varying Rd , MD,0 , and α, as well as for the fiducial model
evolved using each of the three disk-temperature models. Note that I have included an R d = 80 AU
case, and an α = 10−3 case, neither of which are used in my simulations, due to their longeivity.

35

800

800
b)

a)

Rout (AU)

600

600

400
200
0

400

Rd (AU)
5
10
20
40
80
0

MD0 (Msun)
1%
3%
9%

200
0

2×106 4×106 6×106 8×106 1×107

800

0

800
c)

Rout (AU)600d)
StaticEnergy−balanced
Power−law evolved
temperature

600

400

400

alpha
10−3
5x10−310−2
2x10−2200
0

2×106 4×106 6×106 8×106 1×107

0

200
0

2×106 4×106 6×106 8×106 1×107
Time (years)

0

2×106 4×106 6×106 8×106 1×107
Time (years)

Figure 2.11: Rout for model disks using a range of disk parameters. The baseline model uses
MD,0 = 0.03M¯ , Rd = 20 AU, and α = 10−2 .
The MD,0 = 0.09M¯ also has an over-long lifespan, but has been included in Chapters 5 & 6 asa test of the effect of the disk mass on the mixing and global distribution of dust solids in mysimulations. The rapid loss of disk mass at the end of the disk lifetimes is due to photoevaporative
clearing by EUV radiation, rather than accretion onto the central star, and occurs on a timescaleconsistent with observations (as these models do not allow for disk clearing by planet formation).
The disk masses are generally consistent with medium-sized disks, and the mass-accretion rates fall
comfortably within the range observed across the progression of disk ages.
To define Rout for Figure 2.11, I follow M. Hughes et al. (2008), who note that the outer radius
as measured by CO emission appears to occur when the midplane surface density drops below thecritical density for the rotational transition. At 20 K, the critical density is ∼ 4.4 × 10 4 CO cm−3 .
This is a CO mass density of 2.06×10−22 g cm−3 and, using their interstellar value for the CO-to-H2
mass ratio of 10−4 , this comes to a total gas density of 2.06 × 10−18 g cm−3 . Therefore, I have
set the Rout plotted in Figure 2.11 at the boundary in the model disks where ρg,mid ≥ 3 × 10−18
36

0.00

0

2×106 4×106 6×106 8×106 1×107

0.03
MD0 (Msun)
1%
3%
9%

0.02
0.01
0.00

0

2×106 4×106 6×106 8×106 1×107

0.03
alpha
10−3 −3
5x10
10−2 −2
2x10

0.02
0.01
0.00

0

2×106 4×106 6×106 8×106 1×107

0.03
temperature
StaticEnergy−balanced
Power−law evolved

0.02
0.01
0.00

0

2×106 4×106 6×106 8×106 1×107
Time (years)−dMD/dt (Msun yr−1)

0.01−dMD/dt (Msun yr−1) −dMD/dt (Msun yr=−1)

Rd (AU)
5
10
20
40
80

0.02

−dMD/dt (Msun yr−1)

Disk Mass (Msun)
Disk Mass (Msun)
Disk Mass (Msun)
Disk Mass (Msun)

0.03

10−6
10−7
10−8
10−9
10−10

0

2×106 4×106 6×106 8×106 1×107

10−6
10−7
10−8
10−9
10−10

0

2×106 4×106 6×106 8×106 1×107

0

2×106 4×106 6×106 8×106 1×107

0

2×106 4×106 6×106 8×106 1×107
Time (years)

10−6
10−7
10−8
10−9
10−10
10−6
10−7
10−8
10−9
10−10

Figure 2.12: Total disk masses and mass-loss rates for models disks using a range of disk parameters.
The baseline model uses MD,0 = 0.03M¯ , Rd = 20 AU, and α = 10−2 .
g cm−3 , using Equation 2.6 for a vertically-isothermal disk structure. In all of my model disks, I
find an outer gas radius of several hundred AU measured for the majority of the disk lifetime, withsome fall-off with time as the disk accretes onto the star and thins.

37

2.6

Calculating Gas Velocities

In this section, I overview three calculations for the disk-gas velocities, vφ,g , vr,acc , and vr,merid .
While the particle-transport simulations presented in this thesis are restricted to the midplane,
I include the vertical dependence of these velocities (assuming a vertically isothermal disk) forcompleteness.
One of the most important velocities for studying small-particle transport in protoplanetarydisks is the azimuthal (orbital) velocity of the gas. This velocity deviates from Keplerian due tothe radial pressure gradient within the gas, which acts to somewhat offset the gravity of the central
star. The gas orbital-angular velocity, Ωg , is easily derived using radial force balance (Takeuchi &Lin, 2002) whereRΩ2g −

GM? R(R2 + z 2 )3/2

−

1 ∂P
= 0.
ρg ∂R

(2.53)At the midplane, the pressure gradient is outward, causing the gas to orbit at slower-than-Keplerian
speeds, but in a flared disk, high above the midplane, or just outside a gap in the gas, the pressure
gradient is inward and the gas rotation super-Keplerian.
Using P = ρg kB T /µmH and the definition for ρg in a vertically-isothermal disk (Equation (2.7) ),one can solve for Ωg for any general disk surface-density and temperature structure:
µ
¶
µ
¶
¸
·
Hg2 R ∂ΣgR ∂T3
Ω2g (R, z) = Ω2K (z) + Ω2K,mid 2+−
R Σg ∂R
2T ∂R
2
µ
¶¸
·
2
z
3
R ∂T
,
+ Ω2K,mid 2+
R 2 2T ∂R↓Ω2g (R, z) = Ω2K (z) + Ω2K,mid

Ã

·
¸
·
¸!
Hg2
1
3
z2 3 1qΣ + qT −
+ 2
+ qT
,
R2
2
2
R 2 2

(2.54)

where qΣ and qT are the local radial power-law slopes for the gas surface-densitys and temperature,GM?
respectively (see Appendix B.2.2 for the numerical calculation), ΩK (z) =is the(R2 + z 2 )3/2
general-Keplerian angular velocity, and ΩK (z = 0) = ΩK,mid .
The actual orbital speed of the gas is given by vφ,g = RΩg . Figure 2.13 plots the deviation
between vφ,g and vK , the Keplerian speed, for several different vertical scale-height positions within
the fiducial disk model of Chapter 6 at t = 0. Note that this variation is extreme near the outer38(vphi,g−vK)/vK

0.10

0.05

0.00

−0.05

−0.10
0.1

1.0

10.0
R (AU)

100.0

1000.0Figure 2.13: Plot of the difference between vφ,g and vK for several different heights above the disk
midplane at t = 0 in the fiducial disk model of Chapter 6. Curves at half-scale-height incrementsfrom z = 0 (black) to z = 4Hg (orange). My transport simulations take place at the midplane,
where the gas almost always orbits at sub-Keplerian velocities.

edge of the disk where the surface density drops toward zero, and also that the gas velocity isgreater than Keplerian at the very inner disk edge.
Because the radial-flow velocity of the gas depends on aspects of disk turbulence that cannot
yet be predicted from first principles, my simulations consider two end-case calculations for v r,g .
The first is the 1D, vertically-averaged gas velocity, vr,acc , referred to as the accretion-flow velocity.
This is the bulk radial flow required to maintain mass conservation for 1D viscous disk evolution. It
is derived from the same equations as the basic form of the 1D disk-evolution equation (beginningof §2.3.1) using the same simplifying assumptions of Ωg = ΩK,mid and ∂Ωg /∂t = 0. Combining
Equations (2.13), (2.14), & (2.15) yields

vr,acc = −

´
3
∂ ³
1/2
νΣ
R
,
g
Σg R1/2 ∂R(2.55)

which produces a gas flow that is predominantly inward. Importantly, however, there is a regionof outward-flowing gas in the outer disk where the disk is expanding. The boundary of this region
moves outward as the disk evolves and spreads.

39

Almost certainly, an accurate 2D (R, z) depiction of radial gas flow within a disk requires
variation in that flow with height above the disk midplane. One model to calculate such variation
assumes that the viscosity is constant in z and that there is no vertical mixing of angular momentum;
all transfers take place in the radial direction only. The basic property of such a model in a flareddisk is that there is rapid inflow of material in the surface layers of the disk, while at the midplane,
for most disk radii, the flow is actually directed outward (Urpin, 1984; Rózyczka, Bodenheimer &Bell, 1994). To calculate this flow (the meridional gas velocity, vr,merid ), I follow Takeuchi & Lin(2002) and use the azimuthally symmetric Navier-Stokes equation,∂ ¡ 2 ¢
∂ ¡ 2 ¢
vr,g Rρg
R Ωg + vz,g Rρg
R Ωg =
∂R
∂z
µ
¶
µ
¶
∂Ωg
∂Ωg
∂
∂
∂ ¡ 2 ¢
R3 νρg+R3 νρg− Rρg
R Ωg ,
∂R
∂R
∂z
∂z
∂t

(2.56)

which at the midplane yields
¸ ·
µ
¶
µ
¶
¸
∂Ωg
∂Ωg
∂ ¡ 2 ¢ −1 1 ∂R2 ν ∂
32 ∂Ωgvr,merid =
R νρg
+
ρg
−R
.
R Ωg
∂R
Rρg ∂R
∂R
ρg ∂z
∂z
∂t
·

(2.57)

Note that while Takeuchi & Lin (2002) assume a power law for the gas distribution and use thatto derive a simplified expression for the flow structure of the gas, I use the input of the disk
surface density from my disk-evolution model to solve Equation (2.57) numerically by assuming
vertically uniform temperature and viscosity, as in the expansion for Ωg above. I discuss my numericcalculations for the R- and t-derivatives in Appendix B.2.2.
Figure 2.14 shows the accretion-flow velocity and the midplane-flow velocity at the midplaneat t = 0 and at t = 1 Myr in the fiducial-disk model of Chapters 4 & 5. Note that the outward
expansion of the disk dominates both velocity models at large disk radii.
In order to derive an expression of vr,merid valid at z > 0, one must solve Equation (2.56)
including some expression for vz,g , the gas vertical velocity. Here I show one simple approach. If
I assume that a given parcel of gas will tend to remain at it’s original scale-height position withinzg
the disk such that
= K = a constant, then I haveHgvz,g =dd(zg ) = K (Hg ) = K
dt
dt
40

µ

dRg
dt

¶µ

dHg
dR

¶

,

Radial Velocity (cm s −1)

400
300

t=0; fiducial disk
t=106 yr; fiducial
steady disk

200
100

midplane
flow

0
−100
−200
0.1

accretion flow

1.0

10.0
R (AU)

100.0

1000.0Figure 2.14: Radial velocities of the disk gas in the fiducial disk model of Chapter 4, as well as in
the steady disk discussed in §4.5.4. Red curves plot the flow for the accretion-flow model. Blue
curves plot the flow at the midplane in the meridional-flow model.
andzvz,g = vr,g
Hg

µ

dHg
dR

¶

.

(2.58)

Using the same vertically-isothermal expansion for ρg (and the z-derivatives of ρg and Ωg ), I canthen write
µ
¶µ
¶¸
∂Ωg
∂ ¡ 2 ¢ R2 z ∂Hgvr,g=
R Ωg +
∂R
Hg
∂R
∂z
µ
µ
µ
¶
µ
¶·
¶
¶µ
¶¸
∂Hg
∂Ωg
∂Ωg
1 ∂
1 ∂Σg
1
z2
R3 ν
+ R2 ν
−
1+ 2
+
R ∂R
∂R
∂R
Σg ∂R
Hg ∂R
Hg
µ 2 ¶
µ
¶
µ
¶
∂ Ωg
∂Ωg
z
2
2 ∂Ωg
−
R
ν
−
R
.
R2 ν
∂z 2
Hg2 ∂z∂t·

2.7(2.59)

Summary

In this chapter I have described the model I use for protoplanetary-gas-disk evolution, structure,and environment properties. This is a 1D (radial) model ultimately defined by radially gridded
values for the disk surface density and temperature. My primary use for this model is to define theaerodynamic environment experienced by orbiting small grains in a typical Solar Nebula-analog
41

disk, and to explore the effects of disk evolution on aerodynamic particle transport.
These disks do not, unfortunately, include the very early stages of star formation and the buildup of the disk, but do allow for exploration of changes to the initial disk conditions. I use the
Shakura & Sunyaev (1973) alpha-prescription to describe the disk viscosity and viscous evolution
and use a somewhat narrow range for α and the base disk parameters, with a fiducial model of
MD,0 = 0.03M¯ , Rd = 20 AU, and α = 10−2 . This provides a model disk both of sufficient mass
to act as a credible stand-in for our own Solar Nebula, and having a lifetime and mass-accretion
rate consistent with observations of disks around other stars. While other disk-evolution/viscosity
models do exist applicable within a 1D treatment, these models are often described in terms ofalpha-parameterizations. Using an alpha-prescription, therefore, allows the greatest ease of comparison between my simulations and scenarios using others’ disk models.
It is important to note that disks do change dramatically over the course of their lifetimes,
having initially more compact configurations and high rates of accretion that also allow for massive
outflow at the expanding outer edges. This early high accretion means that a substantial fractionof mass is lost early, and later disks are considerably more tenuous and spread out. While surfacedensity profiles in an alpha-model disk tend to approach an inverse with the viscosity profile, onlyin the very inner disk (inner few AU) do profiles match well with those of steady-disk models.

42

Chapter 3

Building the Particle Transport
Model
The particle-transport model is built separately from the evolving gas-disk model described in
Chapter 2, which is used to define the environment in which the simulation dust particles reside.
While some studies have treated the dust as a fluid (Gail, 2001; Bockelée-Morvan et al., 2002;
Dullemond, Apai, & Walch, 2006; Ciesla, 2009), I have chosen to treat it as an ensemble of discreet
particles and particle trajectories. This choice allows me to examine not only variations in the
distribution of disk solids, but also in the mixing between solids that originate in vastly different
regions of the disk. My particle-transport model tracks an ensemble of particles within the 1D(radial) gas-disk environment and subjects the particle orbits to two non-gravitational forces: aerodynamic drag against the mean gas flow, and turbulent diffusion of the ensemble within the diskgas. Coagulation and other grain-grain interactions are neglected, as are back-ractions of the solids
on the disk-gas flow or collective effects in the grain motion. However, for some later simulations, I
employ a simple model of early grain-growth processes to place rough constraints on the timescales
for the appearance of larger particles (micron–centimeter sizes) within the disk.
The particle trajectories therefore consist of two components: mean radial motion assuming
drag against the mean disk flow, and a random-walk component due to turbulent diffusion. These
two physical effects are represented numerically by two components to the dust radial velocity:
vsrd , the mean radial velocity induced by gas drag, and vturb , the turbulent random-walk velocity.
These velocities are calculated at each evolutionary time step of the gas-disk model (defined in§2.3.2) for each grid point of the model disk; values between grid points are linearly interpolated

43

as necessary. Together, these velocities are used to integrate the dust-grain trajectory as:rd (t + ∆t) = rd (t) + ∆t × (vsrd ± vturb ) ,(3.1)

where rd is the dust-particle radial position in the disk, and ∆t is the time step. The time step ischosen locally, to insure accurate integration of particle trajectories, but is capped globally at thedisk model’s disk-evolution time step (or at the output time if the disk model is static).
In the following section, I outline the range of forces and effects a dust grain might experience
within the environment of a protoplanetary disk, with special attention to the aerodynamic forces
modeled by my particle-tranport model. In §3.2, I discuss the basics of gas drag and the advective
component of the model, and in §3.3, I focus on the random-walk diffusive component. In §3.4, I
discuss a set of simple models used to place rough constraints on grain growth in various regionsof the disk, and in §3.5, I summarize the workings of this model.

3.1

Forces on a Dust Grain in a Gas Disk

While I focus in this thesis on aerodynamic effects to drive grain motions within protoplanetarydisks, there are a number of other effects one may consider for their influence over grain transport.
Figure 3.1 shows a schematic of a disk and some of the different regions and regimes where varioustransport processes might occur, and in this section, I briefly outline the range of forces acting on
small grains in protoplanetary-disk environments.
Basics aerodynamics interactions, like advection within the bulk flow of the accreting disk andheadwind drag, are familiar and tend to predominate throughout the dense-gas regions of the disk
into which particles are intermixed, such as region E in Figure 3.1. However, gas-flow transportof grains is not necessarily restricted to the main disk. Some disks are known to exhibit jets or
to experience photoevaporation of their bulk material. A well known model, termed the X-windmodel, proposes that grains very near the central star may become swept up in a magnetocentrifugal
outflow and deposited further out in the disk (Shu, Shang, & Lee, 1996), as depicted for region Aof Figure 3.1.
As indicated by the X-wind model, most disks are believed to be threaded by large-scale magnetic fields. Indeed, one of the leading theories to explain disk accretion argues for instabilities in44

Figure 3.1: Disk schematic of a variety of different particle-interaction zones: A) within an outflow,B) at the edge of an inner cleared region, C) in the optically thin disk-surface layers, D) for settled
particles at the midplane, E) within the main disk flow, and F) within disk-gas turbulent eddies.
the gas flow created by interaction with such a magnetic field (Balbus & Hawley, 1991). This thenraises the question of whether grain charging and interactions with magnetic or electric fields play
an important role in the transport of protoplanetary dust grains. This is known to be the casein the disk of ring material around Saturn. There, space-craft have captured images of featurestermed spokes, short-lived, dark smudges forming radial patterns of considerable extent over thesurface of the B ring (Grün et al., 1983). It is believed that these features form when collisional
events produce a population of small particles that are then lofted above the surface of the rings
within a photoelectrically created surface electric field (Mitchell et al., 2006). One of the keys tomaking such a process efficient, however, is the extreme-vacuum environment. Grain charging andlifting is dependent on the particle’s surface-area-to-mass ratio, which is highest for the smallest
grains, which are therefore electrically transported the most efficiently (e.g., Hughes, Colwell, &DeWolfe (2008)). However, within a gas disk, it is this same high surface-area-to-mass ratio thatmost efficiently couples particles to the gas flow, as discussed in §3.2.1 below. Therefore, trans45

port of grains via electric or magnetic-field effects is likely potentially important only in the most
tenuous regions of a disk.
In the tenuous surface layers of a disk (C of Figure 3.1), radiation effects become importantbecause the gas here is optically thin to stellar illumination. This means that fine dust grains at
the disk surface are efficiently heated (Chiang & Goldreich, 1997) and also that forces like radiationpressure become important. The force of radiation pressure goes as

Frad =

πs2d I,c(3.2)(where sd is the grain radius, and I is the intensity of the incident illumination) and is therefore
most efficient at transporting small grains. Combining the radially outward radiation pressure fromthe central star with a buoyant force due infrared radiation from the heated disk, Vinković (2009)
proposes that grains a few microns in size may be transported large distances outward along thedisk surface due to radiation-pressure forces.
Equally restricted to low-optical-depth regions but also dependent on non-zero gas pressure, isphotophoresis, which produces outward motion of a grain due to an illumination-induced temperature gradient over the grain surface. With photophoresis, gas particles absorbed and re-emitted
from the hot side of a grain leave with a faster velocity than those from the cold side, thus imparting momentum away from the light source (Krauss & Wurm, 2005). This is depicted in Figure 3.2
panel-a opposite the standard picture for acceleration due to radiation pressure in panel-b. In therelatively cold environment of a protoplanetary disk, the force due to photophoresis goes asFph =πs3d IJ1 P
,
kth T

(3.3)

where J1 is an assymetry parameter, P is the local gas pressure, kth is the thermal conductivityof the grain, and T is the local temperature. This force can be orders-of-magnitude greater thanradiation pressure and is more efficient for large grains as well as for porous agglomerates, whichhave lower kth values, making them better at establishing strong thermal gradients over theirsurfaces. While the illumination requirement means that this effect will not operate throughout
the bulk of a radially optically-thick disk, Krauss & Wurm (2005) predict that it may be responsible

46

Figure 3.2: Diagram of the mechanics of the force of (a) photophoresis and (b) radiation pressure.
Photophoresis depends on light heating a grain that then interacts with a pressurized gas, whereas
radiation pressure is an interaction between the grain and incoming light directly.
for clearing out optically thin regions of some inner disks. Pushing back all the dust grains fromsome inner region would drastically lower the optical depth of that region, allowing photophoresis
to operate at greater radius and push the grains still further out. This could lead to the pile-up ofa sharp-edged dust ring at the radius where photophoresis balances inward drift due to headwind
drag on the particles, as indicated for region B of Figure 3.1.
A somewhat less exotic effect is the settling, usually of larger grains, toward the disk midplane (Dof Figure 3.1). If grains are allowed to settle out, then there is a whole host possible interactions that
may occur, both collective and aerodynamic. Large dust-to-gas density ratios or collective motionsof grains may lead to a slower infall of those grains than is otherwise experienced by individual
particles (Weidenschilling, 2003; Bai & Stone, 2010a). Particles large enough to be only marginallycoupled to the gas may excite streaming instabilities that can lead to particle clumping (Johansen,& Youdin, 2007; Bai & Stone, 2010a). Furthermore, increasing the density of dust grains will leadto more frequent collisions, accelerating grain coagulation and fragmentation processes (Dullemond& Dominik, 2005). While these processes are all interesting, none are explored directly in thetransport simulations of this thesis. This is in part because they each represent separate disciplines
into themselves, and in part because I have focused on the transport of small grains, which tend tobe well-mixed vertically with the disk gas.
Turbulence is expected to be very efficient at mixing small grains to height above the disk
47

2 mm
1.0

Hp/Hg

0.8
t=0
t=0.5x106 yr
t=106 yr
t=1.5x106 yr

0.6
0.4
0.2
0.0
0.1

1.0

10.0
100.0 1000.0
R (AU)

Figure 3.3: Particle scale-heights for 2 mm grains in the fiducial disk model of Chapters 4 & 5.
Curves are truncated at 1.0 because the particle scale-height cannot be greater than the scale-heightof the disk gas.
midplane. Using an alpha-model for disk turbulence (as described in §2.2), the scale-height ofparticles, Hp , in a disk is expected to go as (Cuzzi et al., 2005)

Hp ' H grα,ΩK tstop(3.4)

where Hg is the gas scale-height, and tstop is the gas-drag stopping time on the particle (discussed in§3.2.1). In Figure 3.3, I plot the predicted scale-height of mm-sized grains in the fiducial disk modelof Chapters 4 & 5, which uses an alpha-scaling of α = 10−2 . This is more than sufficient turbulence
to produce efficient vertical mixing and even such relatively large dust particles are expected to
be well-mixed throughout the disk structure for a substantial portion of the disk lifetime. Notethat a wide spread in particle heights may have an appreciable effect on the net drag-induced flow
experienced by particles. This effect is discussed in detail in, e.g., Takeuchi & Lin (2002) and Ciesla(2009). In Figure 3.4, I plot drift velocities for particles at different heights (in units of local diskgas scale-height) above the midplane. In a flared disk model, gas actually orbits at super-Keplerianvelocities near the disk surface, causing a tailwind drag that exerts a radially outward pull on
particle orbits. The effect is strongest for slightly larger particles that are marginally coupled tothe disk gas. While an exponential fall-off in the particle distribution with Hp means that few
48Radial Velocity (cm s −1)
Radial Velocity (cm s −1)

2 mm

20 microns
3000

3000
2500

t=0

2500

2000

2000

1500

1500

1000

1000

500

500

0

0

−500
0.1

1.0

10.0

100.0 1000.0

3000

−500
0.1

t=1.5x10 yr

2500

2000

gas
z=1 Hgz=2 Hgz=3 Hgz=4 Hg

2000

1000

10.0

100.0 1000.0

1000
500

0

0
1.0

t=1.5x106yr

1500

500

−500
0.1

1.0

3000
6

2500

1500

t=0

10.0
100.0 1000.0
R (AU)

−500
0.1

1.0

10.0
100.0 1000.0
R (AU)

Figure 3.4: Drift velocities for grains at height z above the disk midplane in the fiducial disk modelof Chapter 6 for two different times (top and bottom) and two different grain sizes (left and right).
particles will experience this effect, it does act as a caveat to my particle-transport simulations,which consider gas drag and particle transport only at the disk midplane.
While there are many varied and interesting effects operating to affect grain transport within
protoplanetary disks, those that I am most interested in for the work of this thesis operateubiquitously on the bulk of small-dust material to effect large-scale radial transport. Both gasdrag/advection and turbulent diffusion (just as, if not more, efficient radially as vertically) should
be in effect throughout the bulk of the disk and provide the basic framework for dust-transport
processes. Furthermore, these processes are equally important in the inner, planet-forming regions,as in the outer-regions best viewed by observations of disks around other stars.

49

3.2

The Advection of Particles Due to Gas DragIn the particle-transport model, the advection of a dust grain is calculated dependent on the size ofthe grain and on the local temperature, density, and flow conditions of the gas disk. These conditions
are used in the model to calculate a steady mean-radial velocity for the dust particle at each disk
grid point, which is then used in Equation (3.1) to calculate the individual trajectories of eachgrain in the model ensemble. In this section, I first outline the physics of the radial and azimuthalgas drag acting on the particle orbit (§3.2.1), then discuss how those physics are translated intoa steady mean-radial dust velocity at each grid point (§3.2.2). Finally (§3.2.3), I compare theparticle-transport model trajectories to precision force-balance integrations of the particle motion
to demonstrate the fidelity of my gridded velocities in capturing the radial-transport dynamics ofparticles advected within a laminar disk gas flow.

3.2.1

The Basics of Gas Drag in Orbit

The most familiar phenomenon concerning gas drag in a protoplanetary disk is headwind drag.
Because the disk gas is radially pressure-supported, it orbits at a slightly lower velocity than thedust, which orbits closer to the Kepler velocity. This causes drag against the forward motion of theorbiting particle, sapping its angular momentum, so that it spirals inward onto the central star,sometimes quite rapidly (e.g., within ∼ 6 × 104 years for a 2 cm grain initiated at 1 AU in a typicaldisk and in < 5 × 103 years if the inward accretion flow of the gas is included). There are instanceswhere the gas pressure support may be reversed radially (such as just outside a gap in the disk
surface-density profile caused by a massive planet or by photoevaporation), so that the gas orbits
faster and causes a tailwind drag that boosts particles to wider orbits. However, this is essentiallythe same phenomenon as headwind drag, and from here on I use ’headwind drag’ to refer to both.
In many discussions of drag-induced radial migration [e.g., (Youdin & Shu, 2002)] the difference
between the gas and dust azimuthal (orbital) velocities is considered the ∆v of primary importance.
However, as I will show, the difference in the gas and dust radial velocities can have a strong impact
on the particle dynamics. Furthermore, in Chapters 4 & 5 I will highlight potential variations in
the radial gas-flow velocity as a parameter-of-significance for the results of my outward-mixing
simulations.

50

When the force of gas drag is linearly proportional to ∆v (Epstein drag), the equations of
motion for a particle in a laminar disk-gas flow, accounting for both the gas-drag force and theparticle orbit, may be written as
dvr,d
dt
dvφ,d
dt

2 − v2vφ,dK(vr,d − vr,g )
,
R
tstop
vφ,d vr,d (vφ,d − vφ,g )
= −
−
,
R
tstop

=

−

(3.5)
(3.6)

where vr,d and vφ,d are the radial and azimuthal velocities of the dust particle; vr,g and vφ,g are thelocal radial and azimuthal velocities of the disk gas; vK is the local Keplerian velocity; and tstop
is the exponential-stopping time, the time over which a constant drag force brings v d a factor of ecloser to vg . Comparing tstop to the local Kepler time (1/ΩK , where ΩK is the Keplerian orbital
angular velocity) provides an assessment of the relative importance between the gas motions andthe Keplerian orbital motions on the particle dynamics, indicating whether a particle is well- or
loosely-coupled to the gas motions. However, the more general form of the particle-motion equations
is
dvr,d
dt
dvφ,d
dt

2 − v2
vφ,d
K

FD,r
,
R
md
vφ,d vr,d FD,φ
= −
+
,
R
md
=

+

(3.7)
(3.8)

where FD,r and FD,φ are the radial and azimuthal components of the force of gas drag; and m d isthe mass of the dust particle. Note: all of these are the equations in cylindrical coordinates andneglect motion in the vertical plane (normal to the disk and the particle orbit).
In general, for fine grains in a protoplanetary disk (up to a few centimeters at 1 AU or tens ofmeters at 10 AU), the force FD is in the Epstein-drag regime. Unlike the more familiar ∆v 2 draglaws, Epstein-drag is linear in ∆v, because it applies when the radius of the dust grain, s d , is lessthan the mean-free-path of the gas, λmf (and when ∆v ¿ vtherm , the thermal velocity of the gas).
In the Epstein-drag regime, the drag force is given by (Weidenschilling, 1977a)

FD = −mdCRρg vtherm (~vd − ~vg )
3

51

(3.9)

where CR is the particle surface-area-to-mass ratio; ρg is the local-gas volume density, and vtherm is
the gas thermal velocity. Then the equations of motion for the particle may be written (Takeuchi& Lin, 2002)
dvr,d
dt
dvφ,d
dt

2 − v2vφ,dK

1− CR ρg vtherm (vr,d − vr,g ) ,R
3vφ,d vr,d 1= −− CR ρg vtherm (vφ,d − vφ,g ) .
R
3=(3.10)(3.11)For larger particles or higher gas densities (closer to the star), Stokes drag applies, which
resembles Epstein drag at low Reynolds number (Re) but transitions to the familiar ∆v 2 drag force
at Re ≥ 800. Here (Weidenschilling, 1977a)Re =2sd ρg |~vd − ~vg |,η(3.12)

where η is the gas kinematic viscosity (equal to νm /ρg , the molecular viscosity over the gas density).
For gas particles modeled by hard spheres, η is given by (Clarke & Carswell, 2007)
5
η=
16

√
πµmH kB T,
σcross(3.13)

where µ is the average mass of a gas particle in proton masses (m H ); kB is the Boltzmann constant;
T is the local disk temperature; and σcross is the collisional cross section of a gas particle. For mysimulations, I use µ = 2.34 (Nakagawa, Sekiya, & Hayashi, 1986), and σcross ≈ σH2 = 2.4 × 10−15
cm2 (Chapman & Cowling, 1970). σH2 is the cross-section of molecular hydrogen, the dominantgas molecule in the disk, and, while slightly greater than the cross-section of a more-massive gas
particle, is still sufficiently representative.
From Weidenschilling (1977a), in the Stokes-drag regime
FD = −CD πs2d ρg|~vd − ~vg |2,
2

and the drag coefficient is given by

CD ' 24Re−1 =12η
sd ρg ∆v
52

for Re < 1 ,(3.14)

CD ' 24Re−0.6 =CD '24η 0.6
0.6
0.6
20.6 s0.6
d ρg ∆v

24
8000.6for 1 < Re < 800 ,for Re > 800 .

Plugging these equations for Stokes drag into Equations (3.7) & (3.8) and using

πs2d
CRyields=md
4

the following full set for the Stokes-drag equations of particle motion:

Re < 1 :
dvr,d
dt
dvφ,d
dt

2 − v2
vφ,d
K

3CR η
(vr,d − vr,g ) ,R
2sd
vφ,d vr,d 3CR η= −−(vφ,d − vφ,g ) ;
R
2sd

=

−

(3.15)(3.16)

1 < Re < 800 :2 − v2
vφ,d
K

dvr,d
dt

=

dvφ,d
dt

= −

3CR η 0.6 ρ0.4
g|vr,d − vr,g |0.4 (vr,d − vr,g ) ,
0.6
0.6
2 sd

(3.17)

vφ,d vr,d 3CR η 0.6 ρ0.4
g−|vφ,d − vφ,g |0.4 (vφ,d − vφ,g ) ;R
20.6 s0.6
d(3.18)

R−

Re > 800 :
dvr,d
dt
dvφ,d
dt

2 − v2vφ,dK

3CR ρg|vr,d − vr,g | (vr,d − vr,g ) ,8000.6vφ,d vr,d 3CR ρg= −−|vφ,d − vφ,g | (vφ,d − vφ,g ) .
R
8000.6=R−(3.19)(3.20)

Note that while these equations for the acceleration in the radial and azimuthal directions require
the respective r and φ velocity components as input into the drag, the Re value and regime-ofrelevance must be calculated using the full magnitude of ~vd − ~vg .
The Stokes-drag and Epstein-drag forces are equivalent at λmf /sd = 64/45π ≈ 4/9 and Figure 3.5 plots the particle-size boundary where this occurs as a function of R for t = 0 of thefiducial-disk models used in Chapter 6. From this figure it is clear that Epstein drag usually applies. The exceptions occur for particles of about centimeter size and larger in the innermost regionsof the disk. The simulations run for this thesis all assume Epstein-regime drag forces.
To investigate gas-drag induced transport of particles, I can consider the simplest case oflaminar-gas flow with no turbulent forcing of the particle motions, and numerically solve for eachparticle trajectory directly. To do this, I use a Runge-Kutta-style integrator, where, in the Epsteindrag regime, Equations (3.10) & (3.11) are coupled with radial acceleration and the full-set of53

Stokes−drag vs. Epstein−drag boundary103

sd (cm)

102
101

20 cm

100

2 cm

10−1

2 mm

10−2

Static Temp.
Energy−balanced Temp.
Power−law−fitted Temp.

10−3
10−4 0.1

1.0

10.0
R (AU)

100.0

1000.0Figure 3.5: For the fiducial models of Chapter 6 (t = 0), a plot of the sizes at which the Epsteindrag and Stokes-drag forces are equivalent. Stokes(Epstein) drag for sizes above(below) the curve.
Note that lower local temperature (like in the Static-temperature model) mean denser disk gas.
integrated equations is
drd
dt
dvr,d
dt
dvφ,d
dt

= vr,d ,´³
2 − v2vφ,dK
4π s2d=−ρg vtherm (vr,d − vr,g ) ,rd
3 md
vφ,d vr,d 4π s2d= −−ρg vtherm (vφ,d − vφ,g ) ,rd
3 md(3.21)

defining the particle motion radially within the 1D disk, where rd is the dust-grain radial location.
The integrator uses step-size adjustment to reach a relative accuracy of 10 −5 –10−6 in rd , vr,d ,and vφ,d . Note, however, that the integrator becomes numerically expensive when the individual
terms in Equations (3.10) & (3.11) are large but sum to values that are small. Therefore, if theinitially computed acceleration (dvr,d /dt or dvφ,d /dt or both) is very small (< 10−14 cm s−2 ), theintegrator sets the small acceleration equal to zero. Also, if vφ,d ∼ vK (10−10 relative difference)vK
2 − v 2 -term in Equation (3.10) is set equal to zero, but only if 1 C ρ vthen the vφ,d.
R g therm ≥K
3R
2 − v 2 -term is still important andIf vφ,d ∼ vK but the second condition is not met, then the vφ,dK
the integrator instead calculates the particle trajectory to the lower relative accuracy of 10 −5 . To

54

Radial Position (AU)

(a)

2.5

0.2 microns
2 microns
20 microns
0.2 mmRadial Position (AU)from 1 AU

3.0

2.0
1.5
1.0
0.5
0.0

0

1×104 2×104 3×104 4×104 5×104
Time (years)

from 500 AU

600

(b)

500
400
300
200
100
0

2×104 4×104 6×104 8×104
Time (years)

0

Figure 3.6: Particle trajectories within a tenuous, photoevaporating disk in the midplane-flow gasvelocity case for four different particle sizes. (Alexander & Armitage (2007) α = 10 −2 disk; ρd = 3
g cm−3 .)maintain stability in this dust-trajectory integration, I scale the called integration time-steps to1≤ 0.1 ×, which provides well-converged and stable output.
ΩK,midFigure 3.6 plots gas-drag trajectories for the case of outwardly flowing gas at the disk midplanein the α = 10−2 photoevaporating disk of Alexander & Armitage (2007) (Ṁ0 = 10−10 M¯ , Rd = 500
AU) discussed in §2.3.3. The trajectories include two different starting locations and a range ofparticle sizes, and illustrate two important effects. First, for certain disk locations and particle sizes(changing in time) particle motion in radially outward, rather than inward. Second, depending on
local conditions, the fastest inward radial-drift migration occurs for a particle of some intermediatesize.
To examine the first effect, consider the case when the particle orbit is stable (dv r,d /dt =dvφ,d /dt = vr,d = 0). Plugging this condition into the Epstein-drag equations of motion (3.10) &(3.11) yields
CR,steady =³´
2 − v23 vK
φ,gvtherm ρg vr,g R.

(3.22)

CR,steady then defines a particle size (surface-area-to-mass ratio), as a function of R and the local
disk conditions, for which the particle orbit is steady. Note that there are two important end cases:
1. If vr,g → 0 then CR,steady → ∞ (“inifinitely small” particles) because only massless particles
will orbit at any forced non-Keplerian gas velocity without experiencing radial acceleration.
2. If vφ,g → vK then CR,steady → 0 (“infinitely large” particles) because only the most massive
55

1010
(a)
2x10−4 microns

CR (cm g−1)

108
106

t=0
t=2x104 yr
t=3.3x104 yr
t=3.6x104 yr
t=4x104 yr
t=5x104 yr

104
102

0.2 mm

0
100.1

1.0

10.0 100.0 1000.0
R (AU)

106(b)

CR (cm g−1)

10

5

104
103
102
101
100

0.1

1.0
R (AU)

10.0

Figure 3.7: (a) Orbital stability boundaries in a photoevaporating disk (Alexander & Armitage(2007) α = 10−2 ) with outward-flowing gas, vr,merid . (See reference-Σg profiles in Figure 2.3.) (b)CR,steady curve at t = 0 and a schematic of radial particle motions. Black-dashed line denotes
vφ,g = vK . Black-dotted line denotes vr,g = 0 (outward-flowing exterior, inward-flowing interior).
Note, larger particle sizes = smaller values of CR .

particles are impervious to radial gas drag, orbiting at vK no matter what.
Figure 3.7 plots CR,steady at the midplane for the photoevaporating disk of Figure 3.6, with
vr,g = vr,merid , at different times (panel-a). The curves are broken where CR,steady < 0 (negativelymassed particles). Including the boundaries where vr,g = 0 and vφ,g = vK in panel-b for t = 0,
one sees that the steady-orbit curves divide the disk–particle-size parameter space into different
transport zones. We know that very small particles (large CR ) can move outward with outward-

56

CR (cm g−1)

1010

CR,maxv
CR,steady108

t=0
t=3.6x104 yr

106
104
102
100
0.1

1.0

10.0 100.0 1000.0
R (AU)

Figure 3.8: CR,maxv curves (with CR,steady curves for comparison) before and after gap opening by
EUV photoevaporation in the Alexander & Armitage (2007) α = 10 −2 disk. Note that CR,maxv
mirrors the disk surface-density profile.

flowing gas, and that super-Keplerian gas at the inner edge of the disk produces tailwind-drag thatcan move particles outward. Large particles are most influenced by the difference in v φ,g and vK(in more tenuous regions of the disk, the definition of large moves to smaller and smaller particle
sizes), and small particles are more likely to be advected radially with the radial gas flow.
Next, to consider the second effect of particle motions demonstrated in Figure 3.6. Weidentstop
schilling (1977a) notes that the greatest (azimuthal) drag-induced migration occurs when≈torbit1
3. For Epstein drag, tstop =. Assuming a Keplerian orbit for simplicity, then,2π
CR ρg vtherm
CR,maxv ≈3ΩK,ρg vtherm(3.23)

where CR,maxv is the particle surface-area-to-mass ratio at which the maximum radial drift will
occur. This particle-size is linked to the well-known radial-drift barrier in grain-growth and particletransport studies where particles roughly centimeters to decimeters in size in a model Solar Nebula
drift inward onto the parent star so rapidly that they quickly deplete the disk of solids. Figure 3.8
plots the CR,maxv values for the disk of Figure 3.6 at the midplane at two times, compared with
the CR,steady values. Note that strong-drift forcing for the inner-disk region where the gas is super-57

Keplerian allows particles of certain sizes to follow a photoevaporating disk edge outward (at least
for a while) as is seen in Figure 3.6 at later times. Also, this strong-headwind-drag effect, as it
depends on the local gas density, comes into play for smaller and smaller particles in more tenuous
regions of the disk.

3.2.2

Calculating a Steady Radial Velocity

For my transport simulations, I solve for the mean radial velocity of the dust grains, v srd , subject to√
Epstein drag against the mean gas flow. In my simulations, I assume that ρg = ρg,mid = Σg / 2πHg ,the gas density at the disk midplane. This means that the particle trajectories are calculated forthe maximum coupling between gas and dust motions. While Equations (3.10) & (3.11) represent
the precise equations of particle motion, including both orbital dynamics and gas drag, I follow
Takeuchi & Lin (2002) and simplify these forces by assuming that the radial acceleration of thegrain is zero, dvr,d /dt ≈ 0, and that the azimuthal acceleration corresponds to a change in the
Keplerian velocity with R, dvφ,d /dt ≈ −vK vr,d /2R. The equations for the radial and azimuthal
motions of a grain now produce two equations for the steady mean radial velocity, v srd , as a functionof its steady azimuthal velocity, vsφd ,vsrd = vr,g +
vsrd

´
³
2 − v2
3 vsφdK

CR Rρg vtherm(vsφd − vφ,g )1= − CR Rρg vtherm.
3(vsφd − vK /2)(3.24)

I solve these equations iteratively for vsrd using the gas-disk bulk flow from my disk model at eachradial grid point. (See Appendix B.3 for details.)
Figure 3.9 shows the mean radial velocities of the dust compared to the radial velocity of the gas.
Very small grains (. 1µm) are well coupled to the gas flow, but for larger grains, at large distancesfrom the star (tens of AU) and at late times in the disk evolution, the dust motion significantly
departs from the gas flow.
There are a number of methods used in the literature to calculate or estimate the radial-drift
velocities of particles in a disk. The simplest consider the difference between the gas and dust
azimuthal velocities only, sometimes simply adding this drift velocity to the gas radial velocity
assuming direct radial advection. This method can be effective in the inner, dense regions of a disk,
58

Radial Velocity (cm s −1)

Midplane Flow Case400

2 micronsgas
2 microns
20 microns
0.2 mm
2 mm

300
200
100

2 mm

0
−100
−200
0.1

1.0

10.0
R (AU)

100.0

1000.0Figure 3.9: Steady mean-radial velocities for different grain sizes at t = 0 in the fiducial disk modelof Chapters 4 & 5. Assumes gas flow at the disk midplane that is radially outward (see §2.6) andρd = 3 g cm−3 .

but is much less so where the disk gas is tenuous. In Figure 3.10 panels-a1 and -a2, I plot radialdust velocities for two cases according to my methods versus the azimuthal-only–drift limiting cases
presented in Weidenschilling (1977a). The first limit is for tstop ¿ torbit (well-coupled particles),and is given by (for Epstein drag)vr,drift =µρd s dρg vtherm

¶µ

1 dP
ρg dR

¶

,

(3.25)

where ρd is the internal density of the grain and P is the local gas pressure. This is one of the mostcommonly cited formulations for estimating particle drift lifetimes within a disk, the other being
the same, but substituting cs , the sound speed, for vtherm , often preferred for order-of-magnitude
estimates. The other limit shown is for tstop À torbit (large, decoupled particles), and is given by1
vr,drift = 2ΩKµρg vthermρd s d

¶µ

1 dP
ρg dR

¶

.

(3.26)

The above limits can be combined into a continuous equation for vr,drift across the range ofparticle stopping times, as given in, e.g. Nakagawa, Sekiya, & Hayashi (1986) and Bai & Stone

59

Radial Velocity (cm s −1) Radial Velocity (cm s −1) Radial Velocity (cm s −1)

t = 2x106 years

t = 0

2000
400

a2)

a1)

0

200
0

−2000

−200

−4000

−400

−6000

gas
ts << torb
ts >> torb
transport model

−600

−8000

−800
−1000
10

100

1000 −10000

100

1000

2000
400

b2)

b1)

0

200
0

−2000

−200

−4000

−400

gas
BS10: Z0
BS10: Z0x2
BS10: Z0x10transport model

−6000

−600
−8000

−800
−1000
10

400

100

1000

−10000

100

1000

2000
c1)

c2)

200

0

0

−2000

−200

−4000

−400
−6000

−600

−8000

−800
−1000
10

100
R (AU)

1000

gas
TL02
transport model

100

1000
R (AU)

Figure 3.10: Plots of the particle-transport model calculations for radial drift of a 20 µm grain versus
others’ calculations for the drift velocity. Dashed curves denote the calculated velocity + the radial
gas velocity for calculations that neglect the radial gas velocity. Curves shown for two times in
the fiducial disk model of Chapters 4 & 5 for the case of outward-flowing gas at the midplane.
Panels-a1 and -a2 consider the limits to drift velocity presented in Weidenschilling (1977a). -b1
and -b2 consider the velocity from Bai & Stone (2010a) (BS10) for three local dust-to-gas densities(Z0 = 0.015. -c1 and c2 consider the velocity from Takeuchi & Lin (2002) (TL02).

60

(2010a). This combination produces

vr,drift = −2τsηδφ vK .
(1 + ²ρ )2 + τs2(3.27)

Note that the above equation also includes the effect that regions with high dust-to-gas ratios will
experience slower particle drift (when ²ρ = ρp /ρg is large, where ρp is the local volume densityof solids), as shown in panels-b1 and -b2 of Figure 3.10. In Equation (3.27), τ s = ΩK tstop is thenormalized stopping time, and ηδφ references the radial shearing within the disk, usually referenced
as (approximations) ηδφ vK = vK − vφ,g or ηδφ = − (∂P/∂R) /2ρg RΩ2K . However, perhaps becauseof the predominance of the order-of-magnitude limit for Equation (3.25), Bai & Stone (2010a) havechosen to specify τs of Epstein drag using cs in place of vtherm : τs = ΩK ρd sd /ρg cs . This has theeffect of off-setting the vr,drift curve from that calculated by my method, as shown in panels-b1 and-b2 of Figure 3.10, and calculating a somewhat larger inward drift for particles in the main part ofthe disk than necessary.
Finally, Takeuchi & Lin (2002) derive a widely used equation for the radial-drag velocity of particles that includes drag from the radial gas flow explicitly. It continues the simplifying assumptions
used to arrive at Equations-set (3.24) to produce

vr,drag =vr,g /τs − ηδφ vK,
τs + 1/τs(3.28)

where Takeuchi & Lin (2002) derive their ηδφ directly from radial orbital-pressure balance: ηδφ =− (∂P/∂R) /Rρg Ω2K . Plots of the Takeuchi & Lin (2002) radial-velocity values compared to my own
are shown in panels-c1 and -c2 of Figure 3.10. Equation 3.28 provides radial-velocity values very
similar to my own, if somewhat slower, along the range of disk-verus-gas conditions. However, myparticle-transport model continues to use the itterative method to solve for dust radial velocities
rather than this analytical expression largely because, as shown in the next subsection, my method
has been tested and proven against direct-integration trajectories of gas-dragged, orbiting particles.

61

3.2.3

Fidelity of Transport Trajectories Compared to Force-Balance Numeric
Integrations

The particle-transport model is constructed to use the gridded, steady-radial dust velocities discussed above, so that the trajectories calculated are a first-order approximation of the full gas-dragdynamics. Here I compare these trajectories to their corresponding precision-numerical integrations calculated as discussed in §3.2.1. Most of the comparison simulations presented below are runusing the midplane-flow-velocity case, as defined in §2.6. I begin by presenting particle trajectories
calculated within an initially tenuous disk (Rd = 500 AU, Σg (1AU) ∼ 1 g cm−2 at t = 0) that is
photoevaporatively cleared within .105 years. Such tenuous disks provide loose particle-gas coupling. Combined with an outward-flowing gas velocity, the radial trajectories of the dust particles
can be complex and so provide good tests of my particle-transport model. The dust-grain sizes
quoted assume a rocky internal density of 3 g cm−3 .
Figure 3.11 plots, for both numeric integrations and particle-transport simulations, the trajectories and radial velocities of 0.2 mm-sized particles initiated at three different radial locations
within this tenuous disk. Here, such large particles are only loosely coupled to the gas flow, withexponential-stopping times (tstop ) at t = 0 of a few to greater than ten Kepler times (and of no
shorter than 0.1 Kepler times throughout the simulations). Nevertheless, there is good agreement
between my particle-transport simulations and the more precise numeric integrations.
However, the particle-transport model trajectories do lack certain details; as shown in the lower
panels of Figure 3.11, they do not account for orbital eccentricity. Thus, the particle-transport
simulations are limited in their depiction of complex gas-particle interactions. In panels-a2 and b2, these interactions pertain to a particle disengaging from an outward-sweeping photoevaporation
front. However, the results of interest to the studies of this thesis (as explored in Chapters 4, 5,& 6) are not sensitive to such a high level of detail. Most important, instead, are the broad
effects pertaining to variations in particle size and the evolving disk surface density. As a test ofthese considerations, I ran simulations for particles of sizes 0.2 µm, 2 µm, 20 µm, and 0.2 mm
in the tenuous disk, from 20 starting locations, 0.5–500 AU. Figure 3.12 compares the numericalintegrations versus particle-transport simulations for those runs initiated at 1 AU and 500 AU. For
the total run set, the exponential-stopping times ranged from less than 10 −4 to greater than 10
Kepler times, and in general, the relative radial positions of the simulated trajectories agreed with62

r0=242 AU
r0=168 AU
r0=117 AU

250
200
150
100
50
0

2×104 4×104 6×104
Time (years)

0

124

8×104

(a2)
from r0=242 AU

122
dust transport model
numeric integration

120
118
116
5×104

6×104 7×104 8×104
Time (years)

9×104

Radial Velocity (cm s −1)(a1)

Radial Velocity (cm s −1)

Radial Position (AU)Radial Position (AU)

300

2000
(b1)

1000
r0=117 AU

0−1000

242 AU−2000
168 AU

−3000
−4000

2×104 4×104 6×104
Time (years)

0

8×104

1500
1000

(b2)

500
0
−500
−1000
−1500 4
5×10

6×104 7×104 8×104
Time (years)

9×104

0.2 mm20 microns
2 microns
0.2 microns

from 1 AU

3.0

(a)

2.5

Radial Position (AU)Radial Position (AU)Figure 3.11: Radial position (a1, a2) and radial velocity (b1, b2) of simulated particle trajectories.
Comparison between precision numerical integrations (heavy, dark lines) and the particle-transportmodel trajectories (thin, pale lines). In this figure, 2 mm-sized particles were initiated at threedifferent radial positions within a tenuous, evolving disk and were subject to the midplane-flow
gas-velocity case. Lower panels show a zoom-in of the trajectory initiated at r 0 = 242 AU.

2.0
1.5
1.0
0.5
0.0

0

4

4

4

4

1×10 2×10 3×10 4×10 5×10
Time (years)

4

from 500 AU

600

(b)

500
400
300
200
100
0

2×104 4×104 6×104 8×104
Time (years)

Figure 3.12: Particle trajectories within a tenuous, evolving disk in the midplane-flow gas-velocity
case for four different particle sizes. Comparison between high-precision numerical integrations(heavy, dark lines) and particle-transport model trajectories (light lines). (a) Particles initiated at
R = 1 AU. (b) Particles initiated at R = 500 AU.
63

2 microns
20 microns
0.2 mm

600
(a1)
500

Relative Delta r d

Radial Position (AU)

Large, Compact Disk − Midplane−Flow Case400
2 microns

300
200

20 microns

100
0.2 mm

0

105

0.05
0.00
−0.05
−0.10
−0.15
2 microns
20 microns
0.2 mm

−0.20
−0.25

106
Time (years)

(b1)

105

106
Time (years)

0.2 microns
2 microns
20 microns
0.2 mm20(a2)
15

Relative Delta r d

Radial Position (AU)Nominal Disk − Accretion−Flow Case

0.2 microns

1050

0.2 mm105
Time (years)

0.005
(b2)

−0.005−0.010−0.015−0.020106

0.2 microns

0.000
0.2 mm

0.2 microns
2 microns
20 microns
0.2 mm105
Time (years)

106

Figure 3.13: (a1, b1) Particles initiated at 500 AU within a massive, but initially compact disk
for three particle sizes in the midplane-flow gas-velocity case. (a2, b2) Particles initiated at 15
AU within the fiducial-disk for four particle sizes in the accretion-flow gas-velocity case. Note thatthe two smallest particle sizes follow nearly identical trajectories. In the left panels, trajectories
compared between high-precision numerical integrations (heavy, dark lines) and particle-transportmodel trajectories (light lines). In the right panels, the relative difference in the radial positions
over time between the numerical integrations and the particle-transport simulations. [(simulated integrated) / integrated] Note that the time-axes are in log-scale.
the numeric integrations to within 2% or better.
Where my particle-transport simulations deviate the most from the numeric integrations is forparticles experiencing a long period of steady infall. This is true regardless of how well-coupled
the particles are to the gas motions, and is an effect of the particle-transport model repeatedly
overshooting the estimate of the infalling particle trajectory. Panel-a1 of Figure 3.13 plots trajectory comparisons for particles initiated at 500 AU in a massive, but initially compact disk
(MD,0 = 0.05M¯ and Rd = 10 AU). In this scenario, the particles begin very decoupled from the64

gas (tstop > 103 Kepler times), and after their initial infall toward the main disk they become
moderately well-coupled (tstop ∼ 10−2 and smaller). In panel-b1 I plot the relative difference inthe particle radial positions between the particle-transport model trajectories and the numericalintegrations. Here one can see that for each infall (including the final infall of the largest particle) these two trajectory calculations diverge as the simulated trajectories outpace those of thenumeric integrations. Therefore, exaggerated, rapid infall of particles may be a drawback of myparticle-transport model. However, inspection of all of my simulated trajectories suggests that thisexaggerated infall should have a negligible effect on my general understanding of the timing andtrends of particle transport. As a check I have also run comparison integrations using the accretionflow case in the baseline-disk model of Chapter 4. I ran four particle sizes (0.2 µm–0.2 mm) initiatedat 15 AU and the results (trajectory comparisons and relative differences in radial positions) areplotted in Figure 3.13 panels-a2 and -b2. Initially the particles move outward with the outwardly
expanding disk, but soon they are swept inward as the usual, steady, inward-accretion flow spreads
throughout the main disk. There is good agreement between the simulations and the numericalintegrations, with positional differences of less than 2% for all but the very final stages of infall.
Therefore, in the limit of zero turbulent diffusion, the simulated trajectories produced by myparticle-transport model appear to be a sufficiently accurate representation of radial particle transport in an evolving disk under the influence of gas drag, due to both the pressure-supported–azimuthal and the radial gas flows.

3.3

Turbulent Diffusion of the Particle Ensemble

Turbulent diffusion of the particle ensemble in my particle-transport model is represented by arandom walk added to the individual grain trajectories. The magnitude of this random walk is
directed by the local radial diffusivity for a particle of a given size, while the inward/outward
statistics of the walk also depend on the local disk-mass distribution and on the time stepping used
in a given simulation. In this section, I first define the physical and numerical setup for effecting
diffusion of the particle ensemble (§3.3.1). Next, I demonstrate the fidelity of my particle-transportmodel in reproducing analytical test-case solutions of radial diffusion within a gas disk (§3.3.2).
Finally, I consider two different theories for the scaling of the dust to the gas diffusivities andcompare the basic results of the standard-case simulations (of Chapters 4 & 6) using each (§3.3.3).
65

3.3.1

Numerical Calculation of Stocastic Integrations

To model the turbulent diffusion of the particle ensemble, I add a random walk to the individual
grain motion via the addition of a turbulent velocity component, vturb , used in Equation 3.1. Thisturbulent velocity is set by the dust-particle diffusivity, Dp , and by the time step of the random
walk motion; it is given by
vturb = ±

r

2Dp
.
∆t

(3.29)In my particle-transport model, the time step of the random walk is limited by the time step of
the gas-disk-evolution model, but is also allowed to be no greater than the local Kepler time, 1/Ω K(ΩK is the Keplerian angular velocity), and no greater than either ∆R5 /vsrd or ∆R5 /vturb , where∆R5 is the radial distance across five grid cells in the direction of particle motion.
For the bulk of my simulations, I follow Youdin & Lithwick (2007) in calculating the radial
diffusion coefficient of the dust grains via

Dp = D g

1 + 4τs2(1 + τs2 )2,(3.30)

where Dg is the gas diffusion coefficient, and τs = ΩK tstop . (See §3.3.3 for a possible alternative toEquation (3.30).) In the Epstein-drag regime, tstop = 3/ (CR ρg vtherm ). The dust and gas diffusion
coefficients are equal throughout most of the disk, except in the outermost regions where the gas
surface density is very low, and the dust diffusion coefficient drops to zero. In my fiducial models,I assume that because the disk viscosity is derived from turbulence within the disk, Dg = ν.
While Equation (3.29) provides an accurate first-order representation of the particle diffusivity,
it is not sufficient for producing accurate diffusion of the particle ensemble in my model gas disk.
Among other things, it does not account for the mass distribution of the disk gas. In a real random
walk, the time and distance of each step vary according to the local properties of the gas environment. In my model, however, the time step is fixed by other, mostly numerical, considerations.
Therefore, to allow my particle ensemble to diffuse according to the actual gas distribution, I must
calculate an appropriate probability for whether a given particle in the ensemble will step radiallyinward (pin ) or outward (pout ).
The proper weighting between pin and pout is a function of the imposed time step. Instanta-

66neously (∆t → 0), pin /pout = 1, because ∆t = 0 represents only a single random walk encounter.
However, for ∆t > 0, the probability that multiple encounters will occur within that time step
becomes nonzero. Furthermore, additional encounters will occur not at the initial position being
considered, but either inward or outward of that location depending on the outcome of past encounters. Therefore, the properties of the diffusive medium (the gas) outside the local point of interest
play a role in determining pin /pout for ∆t > 0. The spatial extent of this region of interest is set by
how far particles may travel for a given ∆t. For the particle ensemble, this is the root-mean-squareof the displacement of the diffusing particles, or∆Rrms =rDEp(∆x)2 = vturb ∆t = 2Dp ∆t .

(3.31)

The weighting for pin is set by the gas properties between R − ∆Rrms and R; and for pout by
the properties between R and R + ∆Rrms . Asymptotically, the diffusion must yield everywhere auniform particles(dust)-to-gas ratio. Therefore, after an infinite time, the region with more gas mass
must also have proportionally more dust particles. However, because regions of higher diffusivity
reach the steady state of uniform concentration more quickly, for a finite time step, proportionallymore particles will also mix into a region of higher diffusivity than into one of lower diffusivity.
Therefore,

Z R

Dp Σg R dR
(Mg Dp )in
pin
R−∆R
.
=
= Z R+∆Rpout(Mg Dp )out
Dp Σg R dR

(3.32)

R

To evaluate Equation (3.32) within my model gas disk, I assume uniform, mean values of
the gas surface density and particle diffusivity between each grid point. Because I calculate p inonce at each disk-evolution time step, but ∆t of the individual particle motions may vary due tolocal constraints, I calculate actually pin at each grid point for both ∆tevolve and 21 ∆tevolve , theninterpolate parabolically for other time-step sizes as needed, using pin (∆t = 0) = 0.5.
Figure 3.14 plots the diffusion–time-step caps and corresponding pin values for the static-disk
models used to study mixing in Chapter 4. Both models have a maximum time-cap for data output
of 5000 years, and the static0 disk model has an exponential fall-off of gas mass in the outer disk.
Otherwise both have a roughly uniform mass distribution in radius. From this figure it is clear

67

time caps:
static disk
steady disk

0.54
0.52

102

0.50

100
10

−2

10

−4

stepping (pin):
static disk
steady disk
50−50 stepping

0.48
0.46

10−2 10−1 100 101 102 103 104 105
R (AU)

inward−stepping probability

time−step cap (yr)104

fraction of particles per bin

Figure 3.14: Curves of the maximum turbulent (random-walk) time step for particles in the static0
and steady disk models and the corresponding probablities for a particle to step radially inward.
The global time step here is 5,000 years, setting the cap in the outer disk. Inward of the cap, theturbulent time-stepping in these models is primarily controlled by the local Kepler time (1/Ω K,mid ).

0.012
0.010

t=5x103 yr
t=104 yr

5x103 yr

0.008
0.006

104 yr

0.004
0.002
0.000
0.1

1.0

10.0
R (AU)

100.0

1000.0Figure 3.15: The fraction of particles per radial bin for my particle-transport simulations (palelines) versus the analytical-expected values (dark lines). This figure corresponds to the Clarke
2& Pringle (1988) example where ν ∝ Σ−1
g ∝ R in a static disk with an accretion-flow velocity,
142−1
vacc = −1.5ν/R, and ν = 4.941 × 10 cm s at 1 AU. The simulation was initiated at t = 0 with
104 particles at 40 AU.
68how, in general, the inward/outward-walking probabilities are close to equal, but that the particle
ensemble will probabilistically avoid regions of very low mass density.
Figure 3.15 presents a comparison of the distribution of particles using my particle-transportmodel compared to the expected distribution for an analytical example derived by Clarke & Pringle(1988). While the discreet nature of my particle-transport simulations result in scatter on the
distribution, the overall result is a good match to the analytical theory. In the following subsection,
I compare simulated particle diffusion to other analytical test cases derived in Appendix A.

3.3.2

Fidelity of Model-Ensemble Diffusion

In this section, I present simulations of static-disk test cases of particle diffusion and compare
these simulations to the analytical solutions obtained in Appendix A. I also discuss the effects ofmy choice of simulation time-step on the results. Analytical solutions can be found for a mass ofcontaminants (particles) given by a delta-function at t = 0 and for cases using fixed power-laws
for the disk-gas surface density, Σg , the disk viscosity, ν, and the radial (advection) velocity of the
diffusing contaminant, vR , when those power-laws follow:
Σg = Σ0 R−a ,ν = ν0 Rb → Dkg = ζν ,vR = v0 Rb−1 .

Σ0 , ν0 , v0 , ζ, a, and b are all constants and Dkg is the local diffusivity.
The analytic cases are solved for C (R, t), the concentration of the contaminant, which is equivalent to σ/Σg , where σ is the local surface density of the contaminant. However, the particletransport simulations output the number of particles per radial grid space, so I convert C (R, t)
into an expected mass distribution, mi /mtot , where mi is the contaminant mass in grid cell i, andmtot is the total mass of contaminant in the disk at t = 0. This can then be directly compared to the
distribution of particles in my simulations. Analytical solutions are given for three separate cases:
simplest radial diffusion, with a constant disk-mass distribution (a = 1) and uniform diffusivity(b = 0); a steady disk with a logarithmic solution space (a = b = 2); and other random values of aand b and b 6= 2. Mass-distribution solutions for each of these are listed in Appendix A.
69

To compare my particle-transport model to the derived analytic test-case solutions, I have run
simulations of 10,000 particles initiated at 40 AU. I consider cases for both v 0 = 0 and v0 = −3/2ν0 .
The latter corresponds to the 1D gas-accretion velocity in the steady-disk cases (a = b), but is an
artificially imposed velocity structure in the nonsteady disks. I vary the values of both a andb (the slopes of the disk surface-density (inverse) and disk-viscosity functions, respectively) totest my model in both steady and nonsteady-disk environments. All simulations use ν (1AU) =4.941 × 1014 cm2 s−1 and (unless otherwise stated) ζ = 1. In these test-case simulations, thevalues of Σ0 and the particle size are arbitrary, since the background velocity of the particles, v R ,is prescribed.
Panel-a1 of Figure 3.16 plots a comparison between my numerical simulations of particle transport and the analytically expected contaminant-mass distribution for the case of a = b = 1.5 andzero background velocity (v0 = 0) at two different times. It shows that the overall, evolving distribution of the particle ensemble is in good agreement with the analytical solution. However, thescatter produced by the discreet nature of the particle simulations makes it difficult to judge theaccuracy with which the simulations reproduce this analytical test case. Therefore, panel-b1 plotsan alternate comparison of the simulated and analytical mass distributions. Here the values plottedcorrespond to a summing of the fractional-mass per grid space either from R = 0 outward, or from
R = R0 = 40 AU inward (for R < R0 ) and outward (for R > R0 ). Again, the simulations and theanalytical solutions appear to be in good agreement.
I have simulated particle diffusion with zero background velocity in a steady disk for a range
of a = b-values, and panels-a2 and -b2 of Figure 3.16 display the results at t = 2 × 10 4 years. In
general, my simulations agree well with the analytical solutions. The exceptions occur for very
steep (a = b = 3), or very shallow (a = b = 1) disk profiles and are caused by the finite nature of
my simulation space. The divergence is easy to see in the steep a = b = 3 case. A steep viscosity(diffusivity) profile that goes as R3 leads to rapid transport of particles over very large distancesin the outer disk. Therefore, the outer edge of my simulation space acts as a sink for particles. At
early times, before a significant fraction of particles have been lost, the simulations and analyticalsolutions are in good agreement, as shown in the lower panels, (-a3 and -b3); they deviate fromeach other only at later times. In the shallow a = b = 1 case, it is loss past the inner boundary of
the simulation space that causes the simulations to deviate from the analytical solutions (at later70

vR = 0a = b = 1.5
1.0
(a1)

t=104 yrt=4x104 yr

0.015
0.010

Summed fraction

Fraction per bin

0.020
104 yr4x104 yr

0.005
0.000
0.1

1.0

10.0

(b1)

0.8
104 yr

0.6
0.4
0.2
0.0
0.1

100.0 1000.0

1.0

10.0

100.0 1000.0

10.0

100.0 1000.0

1.0

0.020
(a2)

Summed Fraction

Fraction per bin

t = 2x104 years
a=b=1

a=b=1
a=b=1.5
a=b=2
a=b=3

0.015
0.010

a=b=3

0.005
0.000
0.1

1.0

(b2)
a=b=3

0.8
0.6
0.4
0.2
0.0
0.1

10.0 100.0 1000.0

1.0

a = b = 3
1.0
(a3)

0.015
0.010

t=100 yr
t=500 yr
t=2000 yr

Summed Fraction

Fraction per bin

0.020
100 yr2000 yr

0.005
0.000
0.1

1.0

10.0 100.0 1000.0
R (AU)

(b3)

0.8
0.6
100 yr

0.4
0.2
0.0
0.1

1.0

10.0 100.0 1000.0
R (AU)

Figure 3.16: Steady (a = b disks with zero background velocity. Left panels: comparison of themass distributions of the particles (contaminant) between the particle-transport simulations (palelines) and the analytical solutions (dark lines). The vertical axis is the fraction of total particles(total contaminant mass) per radial grid space. Right panels: comparison between simulations (palelines) and analytical solutions (dark lines) of the summed-mass distributions. Solid lines represent
the distributions summed from R = 0, and dashed lines represent the distributions summed from
R = R0 = 40 AU. Simulations run using 10,000 particles. For the cases of a = b = 1, 1.5, 2, and3, simulations were run using global time-step sizes of ∆t = 2 × 104 , 2000, 1000, and 2000 years,
respectively. The a = b = 3 simulation for the lower panels-a3 and -b3 was run using ∆t = 100
years. In this simulation at t = 100 years, 100% of the particles remain on the grid, but at t = 500
and 2000 years only 93% and 87% remain, respectively.
71

Fraction per bin

0.020
0.015

t = 104 yr

vo = −1.5 visco
1.0

a=b=1 (a)Summed fraction

a=b=1
a=b=1.5
a=b=1.9a=b=2
a=b=2.1
a=b=3
a=b=3

0.010
0.005
0.000
0.1

1.0

10.0 100.0 1000.0
R (AU)

(b)

0.8
0.6
0.4

a=b=1
a=b=3

0.2
0.0
0.1

1.0

10.0 100.0 1000.0
R (AU)

Figure 3.17: Mass-distribution profiles (a) and summed profiles (b) at t = 10 4 years for several
a = b cases with v0 = −1.5ν0 . Comparison between simulations (pale lines) and analytical solutions(dark lines). In (b) solid lines represent summing from R = 0, dashed lines from R = R0 . For thecase of a = b = 1, the simulation was run using a global time-step size of ∆t = 10 4 years. Fora = b = 1.5 and 3, ∆t = 1000 years. For a = b = 1.9, 2, and 2.1, ∆t = 500 years.

times than the t = 2 × 104 years shown in the Figure, due to the slower diffusive evolution).
Loss of particles past the simulation-space boundaries does not cause significant deviation fromthe analytical solution when I include a background-accretion velocity. In Figure 3.17, I plot
the mass-distribution profiles at t = 104 years for a range of a = b values (steady disks) with
v0 = −3ν0 /2. Because the background velocity is inward, it keeps particles away from the outerboundary, and loss past the inner boundary is a part of the analytical solution as well as thesimulations. All of these cases show good agreement between the simulations and the analyticalsolutions.
There is also good agreement for simulations run in non-steady disks (a 6= b). As examples,
panels-a1 and -b1 of Figure 3.18 plot the mass-distribution profiles for the case of a = 1 (a uniform
mass distribution in R), b = 3, and v0 = −3ν0 /2, which has a rapidly evolving distribution. And
panels-a2 and -b2 plot the distributions for a = 3, b = 0 (uniform diffusivity), and v◦ = −3ν◦ /2,
which is slowly evolving. Simulations were also run for these a 6= b cases with v◦ = 0; they also
agree well with their corresponding analytical solutions.
Next, as a check of the scaling between the diffusivity and the background particle velocity, I
present simulations varying ζ (where Dkg = ζν). The results are plotted in Figure 3.19 for three
disk scenarios, including zero and non-zero background velocity, and steady and non-steady disk72

Non−Steady Cases
a=1, b=3,

v0 = −1.5 visc0

1.0
(a1)

t=100 yr
t=200 yr

0.010

100 yr

200 yr

0.005

0.000

Summed fraction

Fraction per bin

0.015

1

10

100
R (AU)

0.2
1

10

100
R (AU)

1000

1.0
(a2)

t=106 yr

106 yr0.020

Summed fraction

Fraction per bin

0.4

v0 = −1.5 visc0

0.025
6

t=2x10 yr

0.015
0.010

2x106 yr

0.005
0.000

100 yr

0.6

0.0

1000

a=3, b=0,

(b1)

0.8

0

20

40
60
R (AU)

80

100

(b2)

0.8

106 yr

0.6
0.4
0.2
0.0

0

20

40
60
R (AU)

80

100

Figure 3.18: Mass-distribution profiles (a1, a2) and summed profiles (b1, b2) at two times for two
non-steady-disk cases. Comparison between simulations (pale lines) and analytical solutions (darklines). In (b1, b2) solid lines represent summing from R = 0, dashed lines from R = R0 . The
simulation for (a1, b1) was run using a global time-step size of ∆t = 10 years, and that for (a2,b2) was run using ∆t = 105 years.
profiles. For all, I see good agreement between my simulations and the analytical solutions. The
only noticable deviation occurs for the shallow disk case — panels-a1 and -a2 — with high relative
diffusivity where there is excess loss of particles past the inner grid boundary.
Finally, I would like to be sure that my particle-transport model correctly simulates diffusionof the particle ensemble for the global time steps used in the static-disk simulations of Chapter 4(∆t = 2000 years), as well as for the time steps required by the disk-evolution model in the evolvingdisk simulations (∆t ∼ 0.05 years). As was described in §3.3.1, maximum limits are placed on theparticle time stepping to ensure that disk conditions local to each particle are obeyed. Panel-a ofFigure 3.20 plots these time-step restrictions for a variety of disk scenarios and scaled diffusivities.
The time-cap due to the simulation-output timestep is here set very high (10 4 years) in order to73zeta=0.5

0.010
zeta=2

0.005

0.000

1

10

100

summed fraction

0.008
0.006
0.004
0.002
0.000
0.1

1.0

10.0

0.2
1

10

0.015
zeta=0.5

(c1)

0.010

0.005
zeta=2

1

10100
R (AU)

100

zeta=0.5

1000

(b2)

0.4
0.2
1.0

10.0

100.0 1000.0− t=200 yr
1.0

summed fraction

fraction per bin

0.4

0.0
0.1

100.0 1000.0

a=1, b=3 − v0=−1.5 visc00.000

zeta=0.5

0.6

0.0

1000

(a2)

a=b=2 − v0=−1.5 visc0 − t=104 yr
1.0(b1)
zeta=0.5
0.8
zeta=1zeta=20.6

0.010

fraction per bin

summed fraction

fraction per bin

0.015

a=b=1 − v0=0 − t=2x105 yr
1.0
(a1)
0.8

1000

zeta=0.5

0.8

(c2)

0.6
0.4
0.2
0.0

1

10

100
R (AU)

1000

Figure 3.19: Mass-distribution profiles (a1, b1, c1) and summed profiles (a2, b2, c2) varying thecontaminant diffusivity by a factor of ζ. The top panels are for the steady-disk case of a = b = 1
with v0 = 0 at t = 2 × 105 years run with a global time-step size of ∆t = 2 × 104 years for ζ = 1
and ∆t = 104 years for ζ = 0.5 and 2. The middle panels are for the Clarke & Pringle (1988)
standard steady-disk case of a = b = 2 with v0 = − 32 ν0 at t = 104 years run with ∆t = 500 years.
The bottom panels are for the non-steady disk case of a = 1, b = 3, with v0 = − 32 ν0 at t = 200
years run with ∆t = 10 years.

74

time caps
time−step cap (yr)

104

(a)
a=1,b=3, v0=−1.5 visc0
a=b=2, v0=−1.5 visc0
a=b=1, v0=0102
OrbitalCap

10

5 Grid SpacesCaps

0zeta=2zeta=1zeta=0.5

10−2

10−4 −1
10

100

101

102
R (AU)

103

104

105

pin values

0.52

in−stepping probability

Output Cap

(b)

0.50
a=b=1, v0=0

0.48
a=b=2, v0=−1.5 visc0

0.46
0.44
0.42 −110

a=1,b=3, v0=−1.5 visc0

100

101

102
R (AU)

103

104

105

Figure 3.20: Turbulent (random-walk) maximum time steps (a) and the corresponding inwardstepping probabilities (b) for particles in a variety of disk models and for different scaled diffusivities.
Time caps include all three forms of time-stepping restrictions: capping by the global time step of
the simulation (∆t = 104 years here), capping by the local Kepler time (1/ΩK ), and the restrictionthat a particle may not diffuse (or advect) more than 5 grid spaces in a single random walk.
The inward-stepping probability is set by the integrated disk properties accessible within a single
random-walk step, and changes for alterations in the random-walk time-step size.

75

show how the other constraints vary with R. A shorter output time or a cap due to an evolving
disk scenario would truncate these curves above that time-step threshold and alter the requisite
pin values where that cap applied. The pin values corresponding to the time-step curves shown areplotted in panel-b.
Smaller time steps mean more steps to simulate a given amount of time in the disk; therefore,
the probabilities of a particle stepping inward versus outward grow ever closer to 50/50. Cumulatively, However, all time-step sizes used must evolve the distribution of the particle ensemble
equally. Therefore, I have run a few test-case simulations using capped time-step sizes ranging
from 10−3 years to 100 years. The resultant distributions versus the analytical solutions are plota=b=1, v0=0,
dt=100 yr
dt=10 yr
dt=1 yr
dt=10−1 yr
dt=10−2 yr
dt=10−3 yr

0.005

0.000

1

10

100

0.8
0.6
0.4
0.2
1

v0 = −1.5 visc0,

10

100

1000

t=200 yr

1.0

0.010
(a2)

0.008
0.006
0.004
0.002
0.000

(b1)

0.0

1000

a=1, b=3,

Fraction per bin

Summed fraction

0.010

1.0
(a1)

Summed fraction

Fraction per bin

0.015

t=2x105 yr

1

10

100
R (AU)

1000

(b2)

0.8
0.6
0.4
0.2
0.0

1

10

100
R (AU)

1000

Figure 3.21: Mass-distribution profiles (a1, a2) and summed-profiles (b1, b2) for two disk cases: asteady case of a = b = 1 with v0 = 0 shown at t = 2 × 105 years, and a non-steady case, a = 1,
b = 3, with v◦ = − 32 ν◦ , shown at t = 200 years. Simulations were run for a range of global time-stepsizes (colored lines) and are compared to the analytical solutions (black/grey lines). In (b1, b2) the
dark lines represent summing from R = 0, and the pale lines summing from R = R0 . The coloredlines of the simulations are largely obscured by the analytical solutions (black/grey) plotted over
top.

76

ted in Figure 3.21 for the case of a = b = 1 and v0 = 0 (panels a1, b1), and for the case of a = 1,
b = 3, and v◦ = −3ν◦ /2 (panels a2, b2). All of these simulations agree well with their corresponding
analytical solution, and the scatter produced for simulations with different time-step sizes are allcomparable.
However, while the agreement between simulations using different time-step sizes is good foralmost all of the simulations I tested, one set of simulations shows a marked change in the scatter
depending on the size of the time step used. Figure 3.22 plots simulations run for a range oftime-step sizes for the case of a = b = 2 and v0 = −3ν0 /2 compared to the analytical solutions at
two times. Looking at the summed-mass distributions (-b1 and -b2), it is clear that all of thesesimulations do distribute the particle ensemble appropriately for this diffusion test case. However,
the mass-fraction-per-bin profiles show significant, resonant-like scatter in the ∆t = 10 years andt = 5x103 yr

a=b=2

vo = −1.5 visco

Fraction per bin

0.012
0.010
0.008
0.006

dt=100 yr
dt=10 yr
dt=1 yr
dt=10−1 yr
dt=10−2 yr
dt=10−3 yr

(a1)

0.008
0.006
0.004

0.002

0.002
1.0

10.0

100.0 1000.0

1.0

Summed fraction

(a2)

0.010

0.004

0.000
0.1

0.000
0.1

1.0

10.0

100.0 1000.0

1.0
(b1)

(b2)

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0
0.1

t = 104 yr

0.012

1.0

10.0 100.0 1000.0
R (AU)

0.0
0.1

1.0

10.0 100.0 1000.0
R (AU)

Figure 3.22: Mass-distribution profiles (a1 and a2) and summed-profiles (b1 and b2) at t = 5 × 10 3
years (a1 and b1) and t = 104 years (a2 and b2) for the steady-disk case where a = b = 2, andv0 = −1.5ν0 . Simulations were run for a range of global time-step sizes (colored lines) and arecompared to the analytical solution (black/grey lines). In (b1) and (b2), the dark lines represent
summing from R = 0, and the pale lines summing from R = R0 . The colored lines of the simulations
are largely obscured by the analytical solutions (black/grey) plotted over top.
77

Figure 3.23: Schematics for producing resonant-like scatter of a diffused particle ensemble for (a)uniform diffusivity and (b) non-uniform diffusivity (increasing to the right). Vertical dotted lines
represent the type of grid-spacing appropriate for observing resonant-like scatter in these systems.
∆t = 1 year simulations; the scatter also appears a bit higher than usual in the ∆t = 10 −1
year simulation. Furthermore, the resonant-like-scatter region in the ∆t = 10 years simulation isrestricted to outward of about 15 AU, and this region moves outward with time. As shown above
in Figure 3.20 panel-a, inward of 15 AU in these simulations the Kepler time (1/Ω K ) is less than
10 years and decreases toward smaller R; the time-step sizes of individual particles at small AU
are restricted to smaller than the global 10-year time step. Therefore, the resonant-like-scatter
behavior is only observable when the same local time-step size is used for the entire region andwhen the source of all the particles is a single point in R. Nonresonant-like scatter at small AU
allows the particle distribution to spread out randomly, following the local diffusivity and removing
the single-source-point condition first at small AU and then moving outward.
In the simplest diffusion scenario, it is easy to produce a distribution displaying resonant-likescatter. My model for diffusion does not simulate the spread in velocities/step-lengths that real
diffusion produces, but rather uses a fixed diffusion step-length for a given time-step at a given
grid-point (Equation (3.31)). If the diffusivity is spatially constant, there is zero background
velocity, and diffusion occurs with the same time-step across a whole region, then particles from asingle source will only occupy discreet locations within the simulations space, as shown in panel-aof Figure 3.23. While this will produce resonant-like scatter of the distribution, it will only beobserved if the output grid onto which the grains are deposited is also appropriately spaced. This78

case would require linear spacing in R. Also, the displacement of each particle has to be at least
as long as the width of a grid-cell to be seen. Therefore, the effect should disappear for very small
time-stepping/displacements.
The simulations of Figure 3.22 are not of simplest-case diffusion. There is a non-uniformbackground velocity, non-uniform diffusivity, and logarithmically spaced grid cells. However, thisa = b = 2 case is special for two reasons: 1) The natural space for diffusion in this regime(b = 2 −→ ln (R)-space) is the same as the space wherein my disk-model grid cells are equal
width. Therefore we should be able to observe resonant-like scatter if it occurs. 2) Step-lengths√
for diffusion and advection both scale proportional with R: ∆Rturb ∝ ∆tν ∝ R2/2 = R, and∆Radvect ∝ ∆tv ∝ R2−1 = R. Therefore, if resonant-like scatter occurs, it should do so over a large
portion of the simulation space.
Because of the non-uniform diffusivity, particles in Figure 3.22 cannot simply take one step
forward and one step back to return to their starting location; they should end up interior totheir starting location, even without background advection. They might, however, take two steps
forward and one back, as depicted in the panel-b schematic of Figure 3.23, depending on the restof the parameters of the system and the time-stepping. For the Figure 3.22 simulation, a particle
that moves outward twice and inward once should be at a position:·

Rfin = Rstart 1 +√2 (∆tν0 )

1/2

¸
13
27
3
− ∆tν0 − ... −(∆tν0 ) ,2
8

where Rstart is the particle’s starting location. Therefore, for ∆tν0 ¿ 1, the system should approach√a resonant-like scatter distribution. Also note that the offset of R × 2∆tν0 is of the same order
as for the scaling of a diffusive step-length, so characterizing a distribution is still rather messy.
The grid spacing for my particle-transport simulations is ∆Rgrid ∼ 0.02R, and ν0 ≈ 2.2 × 10−12 s−1
for the Figure 3.22 simulations (so ∆tν0 < 1 for ∆t . 1.4 × 104 years). For ∆t = 0.1, 1, 10, and100 years, the offsets for resonant-like and diffusive stepping are ∼ 0.004R, ∼ 0.012R, ∼ 0.035R,
and ∼ 0.1R respectively. Therefore, we only see resonant-like scatter when the diffusive stepping
and resonant-like offset are on the order of the width of a grid cell. (Recall from Figure 3.22 that
resonant-like scatter is observed for ∆t = 1 and 10 years.
While this resonant-like-scatter behavior is curious and somewhat vexing, it is also a special

79Static t=0 Disk,

1.0

(a1)

Summed Fraction

Fraction per bin

0.020
dt=100 yr
dt=10 yr
dt=1 yr
dt=10−1 yr
dt=10−2 yr
dt=10−3 yr

0.015
0.010
0.005
0.000

1

10

100

0.8
0.6
0.4
0.2
1

10

100

1000

t=10 5 yr

1.0

0.020
(b1)

Summed Fraction

Fraction per bin

(a2)

0.0

1000
Steady Disk,

0.015
0.010
0.005
0.000

t=10 5 yr

1

10

100
R (AU)

1000

(b2)

0.8
0.6
0.4
0.2
0.0

1

10

100
R (AU)

1000

Figure 3.24: Mass-distribution profiles (a1, b1) and summed profiles (a2, b2) at t = 10 5 years forparticles initiated at 40 AU in the two static disk models employed in Chapter 4 for a variety of
global time-step sizes. These disk models do not have analytical solutions with which to compare
the results, though the steady-disk has a form similar to the a = b = 1, v0 = 0 disk case.
case and not a problem for the science simulations of this thesis. Figure 3.24 presents the resultsof time-step testing for particles in the two static-disk models used in Chapter 4; all timestepping
produces essentially the same particle distributions (within the limits of the natural scatter of thediscreet system).

3.3.3

Comparing Two Theories for Dust-Gas Coupling and Radial Particle Diffusion

In the simulations presented in this thesis, I use the most recent equations derived for the radial
particle diffusivity respective to the radial gas diffusivity (Youdin & Lithwick, 2007). However,in the literature there is still contention over whether the older relation (Cuzzi, Dobrovolskis, &Champney, 1993) may not be the correct one. Therefore, in this section, I compare simulations
using each of these diffusivity relations and show that for the results of Chapters 4, 5 & 6 the80

specific relation for Dp is unimportant; headwind drag tends to exclude grains from regions where
the two give noticeably different results.
In the papers that derive equations for Dg /Dp , the ratio is referred to as the Schmidt number,
Sc. However, Sc ≡ ν/Dg is the traditional fluid-dynamics definition, and is also an importantparameter in my simulations. Therefore, I retain Sc ≡ ν/Dg and simply refer to Dp and Dg
explicitly. The most important parameters for relating particle and gas diffusivities are: τ s = Ωtstop ,which describes the relative importance of gas drag versus orbital motions on the particle motions;τe = Ωteddy , which describes the frequency of turbulent kicks (one per eddy time) relative to theorbital timescale; and the Stokes number, St = tstop /teddy = τs /τe , which describes the frequencyof turbulent forcing relative to the gas-drag stopping time.
As given in Equation (3.30) above, the Youdin & Lithwick (2007) relation for radial diffusivityof particles in the case of isotropic turbulence is

Dp,YL2007 = Dg¡¢
1 + 4τs2(1 + τs2 )2.

(3.33)

It is noteworthy that the result is independent of τe . Youdin & Lithwick (2007) point out that thismay simply be consistent with long–stopping-time particles also having long mean-free-paths. The
authors also describe the diffusivity for anisotropic turbulence and note that a larger proportion of
azimuthal forcing leads to relatively more radial diffusion of particles. For all simulations, I haveassumed isotropic turbulence.
The older relation for particle diffusivity is given by Cuzzi, Dobrovolskis, & Champney (1993).

Dp,CDC1993 = Dg

1.
(1 + St)(3.34)

Not only does this relation depend on the eddy time, it also falls off more gradually as D p ∝τs−1 , compared to the Dp ∝ τs−2 of Equation (3.33). The full equation in Cuzzi, Dobrovolskis, &Champney (1993) also finds a small decrease in diffusivity in the case of a non-zero mean verticalvelocity for the particles. In my simulations, I assume that the mean vertical velocity is zero.
The two relations for Dp are plotted in Figure 3.25. Note that Youdin & Lithwick (2007) find aslight up-tick in diffusivity for marginal coupling (τs ∼ 1). This corresponds to a state of resonant81

0

Dp / D g

10−1
10

10−3
10−4
10−5
10

−2000

−2

−4000
Dp relation:
YL2007
CDC1993: taue=10
CDC1993: taue=1
CDC1993: taue=0.1

−6000
−8000

−6

10−4 10−3 10−2 10−1 100 101 102 103
taus

drift velocity (cm s −1)

100

Figure 3.25: Comparison of the Youdin & Lithwick (2007) (red line) and the Cuzzi, Dobrovolskis, &Champney (1993) (blue lines) relations for the radial particle diffusivity in a gas disk as a functionof the relative stopping time, τs . For isotropic turbulence, Dp,YL2007 is independent of the eddy
time, whereas Dp,CDC1993 ∝ τe . The grey curve plots an example of the radial-drift velocity of aparticle (scale on the right-hand-side) as a function of τs = πsd ρd /2Σg .
forcing with the particle orbit. Figure 3.25 includes an example of the local mean-radial-drift
velocity as a function of τs . Small τs occurs where the disk is denser, so toward smaller disk radii,
and large negative values of the drift velocity correspond to rapid inspiral of the grain toward small
R. Therefore, it is clear from this figure how grains may often be kept interior to regions of thedisk where the two relations for Dp diverge.
Figures 3.26 & 3.27 demonstrate the negligible effect the difference between Equations (3.33)& (3.34) have on the results of my particle-transport simulations. Often, the eddy time is assumed
to be equivalent to the orbital time, τe = 1, and I have used this assumption for these simulations.
Figure 3.26 plots Md,CDC1993 /Md,YL2007 , the ratio of the CDC1993-driven dust-mass distribution
to the YL2007-driven distribution for the fiducial model of Chapter 6. The map is for R and tof the evolving disk, and the green regions correspond to Md,YL2007 /Md,CDC1993 ∼ 1. The grey
regions correspond to (R, t) with insufficient resolution to report results. Within the scatter of thesimulations, the two diffusivity relations produce identical dust distributions. Figure 3.27 plots
the average positions and fractions beyond 25 AU for the fiducial model of Chapters 4 & 5 for82

6×106>2.0
2.0Time (years)

5×106
1.5

4×106
1.0

3×10

6

2×10

6

1×10

6

0.5
0.0

0
−1

1 g cm−2

notenoughres.

012log10R (AU)3

4

Figure 3.26: Md,CDC1993 /Md,YL2007 : A map in R and t of the ratio of the CDC1993-driven dust
distribution to the the YL2007-driven distribution. Dust distributions simulated in the fiducial
disk model of Chapter 6. Time-resolution of 25 points averaged per ∆t = 5 × 10 4 years. The blacklines trace the evolving-disk gas–surface-density contours in orders of magnitude.
simulations using both diffusivity relations. Even for the largest particles, which are least-coupled
to the gas motions, the two sets of results are identical within the noise. Therefore, while there issome disagreement over the correct relation to use for the particle diffusivity, for τ e = 1, the choice
has no impact on the results presented in this thesis.

3.4

A Cartoon Model for Grain-Growth Constraints

Judging by observations, protoplanetary disks are sites for rapid growth of dust grains and also
environments that harbor a significant small-grain population over a large fraction of their lifetimes(Bouwman et al., 2003; Kwon et al., 2009; Sargent et al., 2009). Grains observed in the ISM are submicron in size (Mathis, Rumpl, & Nordsieck, 1977), while every disk for which we have observations
shows signatures of grains that are microns, tens of microns, or even millimeters in size (Testi et al.,2003). In the global-dust-distribution simulations presented in Chapter 6, I use a pair of cartoon83Accretion Flow − R avg

100

1000

dust size:
0.2 m m
20 m m2 mm

R (AU)R (AU)

1000

Midplane Flow − R avg

10

4

5

10
10
Time (years)

6

10

7

Accretion Flow − R > 25AU

1.00000.10000.01000.0010104

105106Time (years)

107

104

Fraction beyond 25 AU

10Fraction beyond 25 AU

10
1

1

0.0001

100

105
106
Time (years)

107

Midplane Flow − R > 25AU

1.00000.10000.01000.00100.0001

104

105
106
Time (years)

107

Figure 3.27: Simulations run for several particle sizes in the fiducial model of Chapters 4 & 5 forboth the accretion-flow and midplane-flow gas-velocity cases. Red curves plot results using the
YL2007-relation for Dp used throughout this thesis. Blue curves plot results using Dp,CDC1993 withτe = 1. Particles are initiated between 0.5 and 10 AU. Top panels show the average positions ofparticles (solid curves) and their RMS spread (dashed curves). Bottom panels show the fraction of
the particles that are beyond 25 AU.

84

Figure 3.28: Depictions of grain growth in two cartoon models. The modeled, growing grain ishighlighted in red, while the background grains from which it grows are shown in blue with black
outlines. Arrows depict the relative magnitude and direction (random in the Brownian model)of grain motions. Growth in the raindrop model is dependent on grain height, z, above the disk
midplane, while the grain growing in the Brownian model is assumed to exist always at z = 0.
models for the early stages of grain growth to place constraints on the timing for which larger
particles (micron–centimeter sizes) may appear at various disk radii. The progression of grain
growth for these two models is depicted in Figure 3.28, and in this section I outline how thesemodels work, the physical mechanisms they do and do not describe, and how their depictions ofgrain-growth vary across different regions of the disk.
Aside from the supply of solid material from which to grow, the two most important parameters influencing grain-growth are the cross-sections for grain collisions, σ c (m1 , m2 ), and the relative velocities of potentially-colliding grains, ∆v (m1 , m2 ), (more generally, the velocity dispersion)(Dullemond & Dominik, 2005).
dm∝ σc ∆v .
dt(3.35)For a solid-sphere representation of the dust grains, the collisional cross-section is given by Dulle-85mond & Dominik (2005)
σc = π (sd1 + sd2 )2 ,(3.36)

where sd1 and sd2 are the radii of each of two colliding grains. Some sophisticated grain-growthmodels consider grain porosity and collisional compaction (e.g., Ormel, Spaans, & Tielens (2007)),
but the cartoon models I use assume constant-density spheres.
Different regimes of grain-growth are largely described by the method that produces the colliding
grains’ relative velocity. My cartoon models for providing grain-growth/size constraints consider
two methods separately. These methods are Brownian motion, where small particles have a large
velocity dispersion due to their kinetic energy alone; and differential settling, where larger grains
settle toward the disk midplane faster than small grains and so sweep up small grains in their path.
For the cartoon models, I do not consider differential radial drift, turbulent coagulation, collisional
fragmentation, or bouncing (collisions with zero-change in particle size). Many papers discuss theseeffects in detail (e.g., Zsom & Dullemond (2008); Okuzumi (2009); Zsom et al. (2010)). The main
points to consider concerning their omission are:
* While the velocity dispersion due to Brownian motion decreases as grains grow, the dispersion due to turbulent motions increases for larger grain sizes (less well-coupled to the gas
motions that are forcing them), allowing continued, efficient collision of equal-mass grains
and rapid growth at larger sizes.
* Both fragmentation and bouncing lead to a maximum grain size above which particles do not
grow. This tends to occur where grains are marginally coupled to the gas motions (τ s . 1).
* No single grain-growth process occurs in isolation. Therefore, models that consider only one
type of coagulation will tend to underestimate growth rates, no matter the regime.
These are the limitations of using a cartoon-model for grain-growth. However, my needs for thesemodels are only for setting relative thresholds for when larger grain sizes might appear in a disk,
and so an efficient, simple model set is generally sufficient. They give me upper limits on the
rapidity of grain-growth in the absence of turbulence.
The raindrop-growth model (so called because it mirrors the growth of raindrops falling throughthe atmosphere) has ∆v due to differential settling and is described in Dullemond & Dominik (2005),86where its simplified nature is used to illustrate grain-growth processes. In true growth by differential
settling, ∆v = |vsettle (m1 ) − vsettle (m2 )| and so is strongly dependent on the local distribution (andvertical distribution) of grain sizes. However, the maximum rate of growth for a single dust grain
via settling is easily defined by assuming that all other grains do not experience growth but remain
small and perfectly suspended in (and mixed with) the disk gas (σc = πs2d1 and ∆v = |vsettle (m1 )|).
In this limit, we have Equations (10) and (11) in Dullemond & Dominik (2005):
dz
dt
dm1
dt

= vsettle = −3Ω2K zm1
4cs ρg (z) σc (m1 )= Zρg (z) |vsettle | σc (m1 )(3.37)(3.38)

where z is the height abovesthe midplane, ρg is the local gas density (vertically isothermal profile,kB T
is the local sound speed, and Z is the assumed dust-to-gas mass
µmH
ratio, set to Z = 0.01 for all runs of my cartoon grain-growth model. v settle is defined for the case of
as described in §2.2), cs =Epstein drag. These equations are easily numerically integrated using the same integrator used forthe force-balance–integrated radial dust-grain trajectories of §3.2.1. Figure 3.29 matches Figure 1
in Dullemond & Dominik (2005) and plots the change in z, sd , and m for this raindrop–growthmodel using the parameters assumed in that paper. The initial grain size, sd (t = 0), is varied, andwhile initially larger grains grow somewhat faster, the maximum size achieved by the time the grainreaches the midplane is the same for each t = 0 size.
For the Brownian-growth model, I use the same assumption that all background grains remain
small, allowing only the particle-of-interest to grow. The collisional cross-section is defined as inEquation (3.36) (with sd2 = 0.1µm always) and the collisional velocity is set by Brownian motion:∆vB (m1 , m2 ) =s

8kB T (m1 + m2 ).
πm1 m2(3.39)For this model, I keep the growing grain fixed at the midplane, where densities are highest and so
growth is fastest, leading todm1= Zρg,mid ∆vB σc .
dt

(3.40)

Growth in the cartoon-Brownian-motion model is very rapid at small particle sizes when the velocity

87

4
Initial grain size:
0.2 microns
0.02 microns
0.002 microns

z/Hg

3
2
1
0

0

200

400

600

800

0

200

400

600

800

10−14 0200400Time (years)

600

800

sd (microns)

1000.0
100.0
10.0
1.0
0.1

10

−2

md (g)

10−4
10−6
10−8
10−10
10−12

Figure 3.29: Grain growth in the simplified raindrop growth model depicted in Figure 3 of Dullemond & Dominik (2005) and using the simulation parameters of that paper: z (t = 0) = 4H g , R = 1
AU, Σg = 100 g cm−2 , T = 204 K, M? = 0.5M¯ , ρd = 3.6 g cm−3 , µ = 2.3, and dust/gas-mass =0.01. Raindrop-growth results are plotted for a range of initial particle sizes in blue. Growth with
the Brownian-growth model (sd (t = 0) = 0.2µm) for these models parameters is plotted in red forcomparison.
dispersion is high and roughly equal-mass grains are colliding. However, even as the cross-sectionof the grain-of-interest grows, the mass flux continues to come from only very small particles.
Therefore, Brownian-growth in this model is initially faster than raindrop growth. Raindrop growthtends to reach the larger sizes fastest, but ceases once the model-particle reaches the midplane, so
if the cartoon-Brownian growth is allowed to persist for long periods of time, it will eventuallysurpass the raindrop model in maximum particle size.
In Figure 3.30, I plot raindrop growth and Brownian-motion growth using my cartoon models
for a series of different parameters. For all, the initial particle size is 0.2 µm (s d,0 = 0.1µm). My
88

varying the disk

102
100

disk model:
DD2005fiducial

sd (cm)sd (cm)

101
10−1
10−2
10−3
10−4

10

sd (cm)

100
10−1
10−2

10000

varying z0

102
101

100
1000
Time (years)

initial z (Hg):
1
2
3
4
5

102
101
2 mm

sd (cm)

10−5

10−3
2 m m

10−4
10−5

10

100
1000
Time (years)

102
101
100
10−1
10−2
10−3
10−4
10−5

100
10−1

varying rhod
rhod (g cm−3):
3.6
2
1

10

100
1000
Time (years)

10000

varying R
R (AU):
1.2
15
185

10−2
10−3
10−4
10−5

10000

101 102 103 104 105 106 107
Time (years)

Figure 3.30: Grain growth using the two cartoon-grain-growth models. Blue: the raindrop growthmodel. Red: the Brownian-motion growth model. In the upper-left panel, I retain ρ d = 3.6 g cm−3
but compare growth in the Dullemond & Dominik (2005) setup to that in the evolving fiducial disk
model of Chapter 6 at R = 1 AU. In that model (used in the rest of the panels), at 1 AU and t = 0,
Σg = 1500 g cm−2 , T = 500 K, µ = 2.34, and M? = 1M¯ . In the lower panels, ρd = 1 g cm−2 , andin the lower-left panel I plot only variations on raindrop growth as my cartoon Brownian-growthmodel assumes the particle remains at the midplane.
fiducial-model disks are denser and hotter than the disk model of Dullemond & Dominik (2005), andso grain-growth in them is faster. Like Dullemond & Dominik (2005), I find that fluffier particles(ρd smaller) tend to grow faster and larger due to larger collisional cross-sections. In the raindropgrowth model, grains grow the fastest and the largest if they are initiated at greater heights above
the disk midplane, but this effect saturates at around z (t = 0) = 4Hg . And finally, because theinner regions of a disk are hotter and denser, they grow particles the fastest. In my Chapter 6
simulations, I combine runs of a number of different grain sizes. To do this, I define a series ofradial zones and run these two cartoon-grain-growth models in each zone as the disk evolves. I theninitiate each particle size of interest in each zone when either of the growth-models reaches thatgrain size or larger. Using this method, mm-sized grains are initiated no further out than 10 AU,

89though they are observed in disks around other stars at hundreds of AU Testi et al. (2003).

3.5

Summary

In this chapter, I have outlined both the numerical setup for the particle-transport model that I use
in my simulations, and the physical basis for its construction. This model operates by calculating
the trajectories of an ensemble of simulation particles influenced by gas-drag advection and drift
and by radial turbulent diffusion within the disk. My methods for calculating both radial-drift
velocities and random-walk diffusion of the particle ensemble have been tested against direct forcebalance integrations and analytical predictions, respectively, and have shown good fidelity to theirtest cases.
Considering radial transport of grains within a 1D disk model allows me to explore the interaction between disk evolution and the small-dust distribution without large computational expense.
It does, however, require that I use single values (as a function of R) for the radial advection andgas-drag velocities, for which I have chosen the midplane values on the assumption that the bulk ofboth gas and solid material reside in the near-midplane zone. In reality, both the gas-velocity structure (likely) and the particle-drag velocities (definitely) vary with height above the disk midplane,
possibly producing an outflow of some small fraction of larger grains near the disk surface layers.
Taking a 1D (radial) modeling approach is considerably less computationally expensive than a 2D
(radial,vertical) approach, but therefore probes only a subset of effects in terms of bulk gas-dragadvection of particles.
Commonly, mixing simulations are performed in the fluid approximation for the dust distribution, so that the diffusion of individual grain populations is each modeled by a separate fluiddiffusion equation — much as the evolving disk surface-density is modeled as a viscously spreading
fluid. This method makes it simpler to add source/sink terms to the dust distribution, but loses
all information about the statistical history of individual grain motions and numerically must be
carefully handled to avoid problems with mass conservation of the dust solids. By modeling thedust as an ensemble of particles, I am able to examine the spread of individual grain motions and to
explore and compare mixing between populations originating in different regions of the disk within
a single simulation calculation. This method also provides an additional check and perspective on
results obtained using fluid-mixing calculations.
90

Chapter 4

Transport and Outward Mixing ofParticles in Evolving Protoplanetary
Disks: Implications for Results from
Stardust
The majority of this chapter appears in Hughes & Armitage (2010).

Samples returned from comet 81P/Wild 2 by the Stardust mission confirm that substantial quantities of crystalline silicates were incorporated into the comet at the time of its formation. I investigate
the constraints that this observation places upon protoplanetary disk physics, under the assumption that outward transport of particles processed at high temperatures occurs via a combinationof advection and turbulent diffusion in an evolving disk. I also look for possible constraints on
the formation locations of such particles. My results are based upon one-dimensional disk models
that evolve with time under the action of viscosity and photoevaporative mass loss, and track solidtransport using an ensemble of individual particle trajectories. I find that two broad classes of
disk model are consistent with the Stardust findings. One class of models features a high particle
diffusivity (a Schmidt number Sc < 1), which suffices to diffuse particles up to 20 µm in size outward against the mean gas flow. For Sc ≥ 1 such models are unlikely to be viable, and significant
outward transport appears to require that the particles of interest settle into a midplane layer thatexperiences an outward gas flow. In either class of models, the mass of inner disk material thatreaches the outer disk is a strong function of the initial compactness of the disk. Hence, models ofgrain transport within steady-state disks underestimate the efficiency of outward transport. Nei-

91

ther model results in sustained outward transport of very large particles exceeding a millimeterin size. I show that in most circumstances, the transport efficiency falls off rapidly with time.
Hence, high-temperature material must be rapidly incorporated into icy bodies to avoid fallback
to small radii. I suggest that significant radial transport may only occur during the initial phaseof rapid disk evolution. It may also vary substantially between disks depending upon their initial
mass distributions. I discuss how my model may inform recent Spitzer observations of crystalline
silicates in T Tauri star-disk systems.

4.1

Introduction

Comets are believed to be some of the most primitive bodies in the Solar System. As such, theypreserve information about Solar Nebula conditions during the early stages of planet formation.
Samples collected from the comet 81P/Wild 2 during the course of the Stardust mission show thatthe nonvolatile material of the comet is rich in relatively large grains (∼few–20 µm) of crystalline
silicates (Brownlee et al., 2006; Zolensky et al., 2006; Westphal et al., 2009). Crystalline silicatesare not detected in the diffuse interstellar medium (ISM) and, although there is debate as tothe formation mechanism of these grains in protoplanetary disks, they are not thought to form
as far out in the disk as comet 81P/Wild 2 is believed to have originated. The Stardust results
thus provide direct evidence for the outward transport of particles through protoplanetary disks.
Spectroscopic observations suggest that similar processes may be widespread in T Tauri disks.
Spitzer studies have revealed an apparent link between disk crystallinity, and grain growth andsettling, but otherwise these disks exhibit a wide range of crystalline-to-amorphous silicate mass
ratios and spatial distributions. This is suggestive of a great diversity in protoplanetary dust
processing (van Boekel et al., 2005; Watson et al., 2009; Olofsson et al., 2009).
Because no process has been proposed to form crystalline silicates out as far as the cometforming regions, a mechanism is needed that can explain the presence of particles processed throughhigh temperatures in cold regions of the Solar Nebula. One obvious possibility is radial mixing
induced by turbulence. Turbulence is probably necessary for gas within the disk to accrete (Shakura& Sunyaev, 1973), and that same turbulence will result in diffusion of gaseous tracers and small
particles coupled to the gas by aerodynamic forces (Morfill & Völk, 1984; Clarke & Pringle, 1988).
It is unclear, however, whether turbulent transport suffices to explain the observations, or whether
92other physical processes are also needed. Examples of such additional mechanisms include theballistic launching of particles in a wind from near the inner edge of the disk (Shu et al., 2001),
photophoretic gas-pressure forces acting on grains in an optically thin disk (Krauss & Wurm, 2005;
Mousis et al., 2007), and radiation pressure on larger grains near the surface of the disk (Vinković,2009).
The radial transport of both gaseous and particle species within a turbulent disk has beenstudied by several authors (Gail, 2001; Bockelée-Morvan et al., 2002; Keller & Gail, 2004; Boss,
2004; Dullemond, Apai, & Walch, 2006; Ciesla, 2007; Alexander & Armitage, 2007; Garaud, 2007;
Boss, 2008; Dullemond et al., 2008; Ciesla, 2009; Turner, Carballido, & Sano, 2010). Here, I use
a particle-based approach to model the advection and turbulent transport of non-interacting dust
grains within evolving protoplanetary disks. My goals are to identify the conditions under whichsignificant outward transport of particles can occur, bearing in mind the range of uncertainty in disk
physics and evolution. I quantify the extent of radial transport as a function of the size of the grains,the initial compactness of the disk, the relative diffusivity of disk-gas tracers, and on the verticalprofile of the gas flow and particle distribution. Initially compact configurations that expand rapidly
appear to have the greatest promise to explain the Stardust results, though unambiguous predictions
require accurate knowledge of the internal disk-flow structure and of crystalline-silicate formation
mechanisms. I find that the importance of the advection of solids within the gas flow means thatthe outward transport efficiency drops significantly for larger particles (∼few millimeters), and atlater times, thereby limiting the extent of mixing uniformity achievable within the disk.
In §4.2 I provide a summary of the relevant observations of silicates in both primitive Solar
System bodies and in other protoplanetary disks. In §4.3 I describe the modeling setup used forthese outward-mixing simulations. In §4.4, I step through a set of examples designed to illustrate
the basic effects of various model parameters. In §4.5, I present and analyze results for a baseline
case of particle transport in two scenarios of radial gas flow, then consider the effects of varying thesize of the particles, the diffusivity, and the initial compactness of the disk. I present my conclusionsin §4.6.

93

4.2

Observational Constraints

Observations of dust in the diffuse ISM show that the vast majority of silicates there are amorphous,with sub-micron grain sizes and a generally balanced Mg-Fe composition (Molster & Kemper, 2005;
Wooden et al., 2007). Therefore, the crystalline silicates that we observe in the Solar System andin disks around other stars are expected to have formed after the onset of the star-disk–system formation. Several mechanisms have been proposed: (1) The evaporation and condensation of silicate
vapor, requiring gas heated to temperatures of T ∼ 1250–1450 K (probably at less than an AU
from the parent star) (Lodders, 2003; Gail, 2004; Wooden, Harker, & Brearley, 2005; Bell et al.,1997); (2) Annealing of amorphous silicate grains, requiring T & 1000 K, (possibly occurring as far
out as ∼2–3 AU in very hot, early disks) (Nuth & Johnson, 2006; Wooden et al., 2007; Westphal etal., 2009); and (3) Shock-heating and annealing in disk spiral arms. The first two of these are equilibrium processes, whereas the third relates to transient events that rely on the disk being massive
enough to produce spiral arms. It may, however, result in the production of crystalline silicates out
as far as 10 AU (Harker & Desch, 2002; Scott & Krot, 2005; Wooden et al., 2007) 1 . These mechanisms have different chemical signatures. Grain formation in long-lived, high-temperature regionsof the disk is most likely to produce Mg-rich silicates due to the low oxygen fugacity expected toprevail in those regions. Fe-bearing and Fe-rich crystalline silicates likely require a water-rich regionof the disk in which to form, produced perhaps by migration and sublimation of icy bodies interior
to the snow line, or in shocks in the outer disk (Wooden, 2008). They may also require transientheating mechanisms to form without evaporating, unlike Mg-silicates, which can crystallize below
short-time-scale–evaporation temperatures (Nuth & Johnson, 2006). Pyroxene is thermodynamically favored over olivine when these minerals are formed by condensation, and, while annealing
will tend to produce olivine from forsterite and pyroxene from enstatite, forsterite will convert
to enstatite in long-lived (∼ 106 years to completion), high-temperature conditions (Gail, 2004;
Wooden, Harker, & Brearley, 2005).
Observationally, it has been known for some time that Oort Cloud comets contain crystallinesilicate material (Hanner et al., 1994; Harker et al., 2002; Honda et al., 2004; Wooden, Woodward,& Harker, 2004). These comets are believed to have formed primarily at distances of ∼ 5 − 10 AU
Note that although the strength of spiral shocks is expected to be greatest at many tens of AU(Clarke & Lodato, 2009), the gas in these regions is too tenuous and cold to anneal silicate grains.
194

from the Sun, in a region of the disk that may have overlapped that where in situ crystalline silicate
formation was possible. The high amounts of crystalline silicates recovered by Stardust, nonetheless,came as a surprise, since 81P/Wild 2 is a short period comet, that most probably formed in theouter disk, around the current orbits of Uranus and Neptune (Wooden, Harker, & Brearley, 2005).
This is beyond any plausible source region for crystalline silicates. Subsequently, observations of
the comet 9P/Tempel 1 have confirmed crystalline silicates to be an important component of that
Jupiter-family comet as well (Lisse et al., 2006). The compositional evidence from the Stardust
samples points to diverse formation environments for the high temperature materials (Wooden,2008). A predominance of Mg-rich silicate grains, together with the recovery of three calciumaluminum (CAI-type) minerals that almost certainly formed by evaporation and condensation at
T > 1400–2000 K, suggest a substantial contribution from the innermost, hottest, disk regions.
However, some Fe-bearing and Fe-rich crystalline silicates were also recovered. There are also hintsof igneous and aqueous alteration among some grains (Joswiak et al., 2010; Stodolna, Jacob &Leroux, 2010), which, if confirmed, would place additional constraints on the timing of the outward
transport and incorporation of these materials into the 81P/Wild 2 cometesimals.
Direct comparison between the Stardust Solar System results and observations of other disks
is difficult. Astronomical measurements are primarily sensitive to smaller orbital radii, and thus
provide more important constraints on the degree and nature of processing than on radial transport.
The observed mass ratio of crystalline-to-amorphous silicates varies greatly, both between disks (vanBoekel et al., 2005; Watson et al., 2009; Olofsson et al., 2009) and likely radially within a single
disk (van Boekel et al., 2004; Olofsson et al., 2009). A study by Watson et al. (2009) measured
crystalline mass fractions in the inner disks (.10 AU) of 84 classical T Tauri stars in the TaurusAuriga star-forming region. The measured mass fractions ranged from less than 0.5% to more than
80% despite the fact that the systems were all of similar ages, 1–2 Myr, and all were observed
within a single star-forming cluster. A significant correlation is observed between the crystalline
mass fraction and the extent of dust settling toward the disk midplane. This may be related to thelink other studies have found between disk crystallinity and grain growth (van Boekel et al., 2005;
Olofsson et al., 2009), with characteristic crystalline grain sizes of a few microns. No correlations
were found relating to the mass or luminosity of the star, the mass of the disk, or the mass ratioof the star-disk system, all properties expected to affect heating and thermal processing within the95

disks. These results agree with a previous study of Herbig Ae-star disks by van Boekel et al. (2005)
who found a correlation between disk crystallinity and stellar mass/luminosity that disappearedfor M? . 2.5M¯ .
In terms of composition, silicates within other protoplanetary disks show the same predominanceof Mg-rich grains as in the Stardust samples (van Boekel et al., 2004, 2005; Molster & Kemper,
2005; Watson et al., 2009; Olofsson et al., 2009). There are also marginally significant correlations
between crystallinity and stellar accretion rate. Watson et al. (2009) found that for pyroxene, thetrend in crystalline mass fraction was inverse to the mass-accretion rate onto the star, whereas forolivine, the trend was proportional to the accretion rate. A study by Olofsson et al. (2009) foundthat disks of higher crystallinity tend to be dominated by enstatite (pyroxene-type) grains, andof lower crystallinity by forsterite (olivine-type) grains. Olofsson et al. (2009) also suggest very
heterogeneous mixing of silicate particles in disks, reporting a higher rate of crystalline-featuredetection for the cold (.10 AU) than the warm (.1 AU) spectral features. I will discuss later
how some of these trends may be interpreted within the context of a turbulent transport model for
grains within the disk.

4.3

Methods

The simplest model for studying the radial redistribution of particles within a disk assumes that thedisk surface density is static and that the particles are small enough as to be perfectly coupled to thegas. Even in this limit, a range of other physical effects can be important and have been explored,e.g., mixing by (marginal) gravitational instability (Boss, 2004, 2008), the thermal evolution andproduction of annealed silicate grains (Gail, 2001; Bockelée-Morvan et al., 2002), and vertical mixing
and settling in an MHD model that includes dead-zone effects (Turner, Carballido, & Sano, 2010).
Many of these studies agree that turbulent mixing can, in principle, be strong enough to explain
the presence of crystalline silicates in comets.
My goal in this work is to systematically incorporate a range of additional physical processes
into what is still a relatively simple one-dimensional model for the evolution of particles within agas disk. I focus on three effects:(1) Imperfect coupling of particles to the gas. The largest particles captured by Stardust (about96

20 µm) may – depending upon the gas density – be only marginally well-coupled. Indeed,prior work that includes grain size and settling effects has found that large grains may settle
out to the midplane and experience outward advection in a 2D stratified disk model (Ciesla,
2007, 2009). Here I consider imperfect coupling to examine the feasibility of transporting large
grain sizes radially outward into the more tenuous outer disk and comet-forming regions.
(2) Disk evolution. The formation of material processed at high temperature almost certainlycommenced early in the disk lifetime, when the disk would have been more massive, hotter,and more compact than the typical T Tauri disk. As I will show later, the evolution ofsuch disks can have a substantial impact on the radial transport of particles. This has been
demonstrated explicitly by Dullemond, Apai, & Walch (2006) and Dullemond et al. (2008),
who found that disks that form in initially more compact configurations produce greater
outward mixing of hot material.
(3) Uncertainties in the radial flow of gas at the midplane. Although the vertically integrated
flow of gas in an active disk is assuredly inward at small orbital radii, the actual magnitude(and even direction) of flow at the midplane depends upon unknown aspects of the angularmomentum transport within the disk. Indeed a flared viscous disk model with little tono vertical mixing yields an outward midplane flow (Urpin, 1984; Takeuchi & Lin, 2002;
Tscharnuter et al., 2009), which could be very important for the transport of particles large
enough to have partially settled (Keller & Gail, 2004; Ciesla, 2007, 2009).

4.3.1

Model Disk Setup

For the work presented in this chapter, I implement these effects using the simplest-case–1D modelfor the gas-disk evolution. In this model, described in §2.2, the disk-temperature distribution
is static, and everywhere T ∝ T0 R−1/2 (T = 278.9K at 1 AU). This temperature structure isappropriate for a passive, flared disk after most of the gas infall has occurred and the initial
rapid phase of accretion has slowed. Note that very-early disk temperatures are expected to bemuch higher (Kenyon & Hartmann, 1987; Bell et al., 1997; Tscharnuter et al., 2009). However,as described in §2.4, a more precise treatment of disk temperature requires models or assumptions
involving, for example, disk chemistry and disk-forming infall rates. As described in §2.5, the97

disk is chosen so as to be consistent with observational estimates of the lifetime and accretion-rate
evolution of protoplanetary disks around low-mass stars, where final-stage clearing is effected by a1042 photons s−1 EUV-flux-driven photoevaporation. I do not here study the evolution of particles
during photoevaporation (long after the presumed epoch when crystalline material must have been
incorporated into comets or their progenitor bodies); I include photoevaporation only because it isessential for providing reasonable model-disk lifetimes.
As discussed in Chapter 2, the disk-evolution model uses a non-steady t = 0 surface-density
profile given by:Ṁ0Σg (R, t = 0) =
3πν

Ã

1−

r

Rin
R

!

¶
µ
R
,
exp −
Rd

(4.1)

where where Σg is the surface density, t is time, R is radial distance from the central star, ν isthe local disk viscosity, Ṁ0 is the t = 0 accretion rate onto the central star, Rin is the inner-diskboundary (set equal to the inner-grid boundary, R1/2 = 0.099 AU), and Rd is a variable controllingthe compactness of the t = 0 profile. Having assumed T ∝ R1/2 , Equation (2.9) for the disk viscosity
leads to ν ∝ R. I adopt α = 10−2 . This is also important for establishing the turbulent diffusion
properties for the particle ensemble, as discussed in §3.3. In my fiducial-disk model, Rd = 20 AU,
and the initial disk mass of 0.03 M¯ dictates Ṁ0 = 1.8 × 10−7 M¯ yr−1 . The evolution of thismodel is shown in Figure 4.1. Dissipation via photoevaporation occurs after ∼5.5 Myr.
In §4.5.4 I vary Rd (retaining Md,0 = 0.03M¯ ) to consider the effects of varied disk-forming
conditions on the transport of particles. To facilitate a comparison with others’ models, I also
consider two models of dust motion within a static disk, including a steady-disk model where
Rd = ∞ (retaining Ṁ0 = 1.8 × 10−7 M¯ yr−1 ). As shown in Figure 4.1, this steady-disk model
results in a disk profile that is matched to my t = 0 fiducial-disk profile in the inner-disk regions,but maintains a shallower profile and much larger gas surface densities in the outer disk.
The mean radial flow of gas experienced by solid particles within the disk depends on aspects
of disk turbulence that cannot yet be predicted from first principles. The outward-mixing studiesof this chapter maintain this uncertainty explicitly by considering two bounding cases for the gas
flow, the predominantly inward “accretion-flow” case (Equation (2.55) ) and the predominantly
outward “midplane-flow” case (Equation (2.57) ). Recall from §2.6 that the accretion-velocity caseincludes a region of outward-flowing gas in the outer disk where the disk is expanding, and that

98

10

t=0
t=105 yr
t=106 yr

SteadyCompact

4

102

St.

100

Myr
Com.

10−2

t=0

0.1

1.0

10.0

Mass AU−1

10−2

St.

Com.

10

100.0 1000.0

−3

10−4
Myr

10−5
t=0
−6

10 0.11.0

10.0 100.0 1000.0
R (AU)−3
Midplane density (# cm −3) Midplane density (g cm )Surface density (g cm −2)
Mass function (M sun AU−1)

Sigmag106

rhog,mid

10−6
10−8
10−10

Com.

Myr

10−12
St.

10−14
10−160.1

1.0

10.0

100.0 1000.0

ng,mid1016

Com.

mu=2.34

1014
1012

Myr

1010

St.

108
0.1

1.0

10.0 100.0 1000.0
R (AU)

Figure 4.1: Plots of the local properties of the disk gas, including Σg , the gas-mass distributions,
ρg,mid , and the local gas-particle number density at the disk midplane. The purple curves (A1, A2,
A3) track the time evolution of the fiducial-disk model. The yellow curve (B) is the t = 0 profile
for the most compact (Rd = 5 AU) disk model. The thick grey curve (C) is the profile of the static
steady-disk model.
the boundary of this region moves outward as the disk evolves and spreads. The transition frominward to outward flowing in the midplane-flow case, meanwhile, is roughly fixed at ∼ 0.33 AU in
the fiducial disk model. This flow is therefore highly conducive to the transport of inner-disk grainsto the outer disk. Note, however, that the outward expansion of the disk dominates both velocity
cases at large disk radii.

4.3.2

Particle Transport Setup

All of the particles run in these outward-mixing simulations are small enough that transport iscalculated (following the methods outlined in Chapter 3) for the case of Epstein drag only. This
does not, however, mean that the gas and dust motion are everywhere well-coupled; headwind dragdominates in the outer, tenuous regions of the disk.

99

The force of drag scales directly with the local gas density, ρg , and in my model, I assume that√ρg = ρg,mid = Σg / 2πHg , the gas density at the disk midplane. (Hg is the local scale-height of thedisk gas.) In the accretion-flow case, inputting the midplane density into the drag-force equation
is an approximation that selects for the tightest possible coupling between the gas and the particle
ensemble at a given R; it leads to the most conservative estimate of outward mixing in this gas-flowcase. However, the midplane density is the most appropriate choice for use in the midplane-flow
case, which presupposes that the entire particle ensemble has settled to the disk midplane.
As discussed in §4.5.2 (and depicted in Figure 4.9), very small grains are well coupled to thegas flow, but the crystalline silicates larger than 10 µm associated with the Stardust collection arelarge enough that at large distances from the star (tens of AU) (and also at late times in the disk
evolution) the dust motion significantly departs from the gas flow.
Turbulent diffusion of the particle ensemble is calculated as outlined in §3.3. In this chapter’s
fiducial models, I assume that because the disk viscosity is derived from turbulence within the disk,
Dg = ν, where Dg is the gas diffusivity. Recall that the dust (Dp ) and gas diffusion coefficients areequal throughout most of the disk, except in the outermost regions where the gas surface density
is very low, and Dp drops to zero. In §4.5.3, I vary Sc = ν/Dg to evaluate the impact of thisassumption on the potential for the outward mixing of inner-disk grains.

4.4

Effects of Individual Transport Processes

To illustrate how each of the model components influences particle transport, I first consider a setof reduced models of successively greater complexity. I begin with a diffusion-only case for perfectly
coupled particles, which I run keeping the gas disk profile fixed at its fiducial t = 0 form. I thenadd in the effects of radial gas velocity, particle size, and disk evolution. Unless otherwise stated,these simulations use 1000 particles initiated at 1 AU within the fiducial-disk model.
Figure 4.2 (a) displays an image of the relative number of particles in each radial bin over time
for the diffusion-only simulation2 . Diffusion results in rapid (within ∼ 105 yr) radial spreadingover tens of AU, such that the particle distribution closely approximates that of the gas within afraction of the disk lifetime. While some particles become lost past the inner edge of the simulated
Note that the bin size increases exponentially with R, so this representation looks somewhat
different from a plot of the particle surface density or concentration.
2100

5×105

5

2×105
1×105
0
−1

Fraction per bin

particles per bin

Time (years)

4×105
3×10

Mass Distributions

30

(a)

25
20
15
10
5
0

0

1
2
log10 R (AU)

3

(b)

0.1

4

10 yr

gas mass
t=104 yr
t=106 yr

106yr

1.0

10.0
100.0 1000.0
R (AU)

Figure 4.2: Basic diffusion: (a) Map of the number of particles per grid space as a function of radial
position and time. I initiated 1000 well-coupled particles at 1 AU (denoted by the green line) ina static disk with zero radial-gas velocity. (b) The number of particles per radial grid space at
two times, compared to the static gas-mass distribution (t = 0 profile of the fiducial-disk model).
Vertical scale is arbitrary. Note that the gas-mass distribution appears to peak at a distance ofseveral tens of AU from the central star. This appearance is an effect of grid spaces that increase
exponentially in size with R. In fact, the gas-mass-per-AU function peaks slightly at around 2 AU,
falls off gently in the main disk, and exponentially from beyond a few tens of AU
disk, the particle distribution comes to closely mirror that of the gas, as shown in Figure 4.2 (b).
Next, I include the effects of gas velocity by using the two cases for radial gas velocity, v acc andvmerid . The distribution maps for these simulations are shown in Figure 4.3 (a1, a2), and here thegross effects of mostly inward versus mostly outward gas velocities are clear. In the accretion-flow
case, where most of the gas flow is inward, the majority of the particles are quickly lost onto thecentral star. Only ∼2% of the particles remain in the simulation after 10 4 years. However, theparticles that do remain in the disk reach the outer-disk region and continue to move outward,
because the steep gas distribution in the outer disk causes the accretion flow there to be outward.
In the midplane-flow case, less than 10% of the particles are lost. Particles are instead swept rapidly
outward with the gas flow and become trapped at the edge of the static disk-gas distribution, wherethe value of the disk surface density falls rapidly toward zero (at about 340 AU).
After turbulent diffusion and gas velocity, I add in the effects of particle size. The results ofsimulations using 20 µm particles are shown in panels-b1 and -b2 of Figure 4.3 . The effects ofparticle size are most clearly visible in the midplane-flow case where particles are still advected
outward as before, but pile up at smaller radial distances (around 90 AU) than in the simulations
with the well-coupled particles. What is happening here is that beyond ∼115 AU, the disk gas
101

Accretion Flow Cases
Time (years)

5×10

5×10
well−coupled
static disk
from 1 AU
(a1)

4×105
3×105

Time (years)

4×105
3×105
2×105

1×105

1×105
1

0

2

3

5

4×105

20 microns
(b1)

0

4×105

20 microns
(b2)

3×105

2×105

2×105

1×105

1×105
0

4×105

evolving disk
(c1)

3×10

5

2×10

5

1

2

3

well−coupled
static disk
from 1 AU
(a2)

0
−1
5
5×10

3×105

0
−1
5×105

Midplane Flow Cases

5

2×105

5×10

Time (years)

5 10 15 20 25 30
particles per bin

5

0
−1

0
−1
2.0×106

1.5×10

6

1

2

3

1

2

3

1

2

3

1
log10R(AU)

2

3

0

evolving disk(c2)

1.0×106
5.0×105

1×105
0
−1

Time (years)

0

5×10

5

4×10

5

1

0

2

0
−1

3

2.0×10from 10 AU
(d1)

0

6

1.5×106

from 10 AU
(d2)

3×105
1.0×106

2×105

5.0×105

1×105
0
−1

0

1
log10R(AU)

2

3

0
−1

0

Figure 4.3: Diffusion scenarios: Maps of the diffusion of particles as in Figure 5.2(a) where the green
line denotes starting location. The right-side panels all include gas velocity in the accretion-flow
case, and the left-side are for the midplane-flow case. The simulations in (b1) and (b2) add in theeffects of non-zero particle size. In (c1) and (c2) the disk surface density is also evolved with time,
and the (d1), (d2) simulations are just like those above, but with the particles initiated at 10 AU.

102

is too diffuse to support the outward advection of particles of this size. Instead, headwind dragdominates the particle motion, pushing particles inward. In the accretion-flow case, the effects ofa non-negligible particle size are similar. Though the dominant behavior of particle motion here
remains the loss of most of the particles inward onto the parent star, those 20 µm particles that
do reach the region of outward accretion flow are also confined to smaller radii than their masslesscounterparts.
Finally, I include the effects of disk evolution. As the disk evolves, the surface density drops.
Panels-c1 and -c2 of Figure 4.3 show that in the accretion-flow case, this drop in density results
in a sustained inward loss of the particles; in the midplane-flow case, the majority of particles arestill retained and advected outward. In fact, some particles reach greater distances than in thestatic-disk model, because, as the disk expands, the surface density no longer drops off as steeply
around 100 AU. However, after about t = 0.5 Myr, the decrease in disk surface density begins todominate over the effect of the disk expansion. The gas can no longer support 20 µm particles atsuch large distances, and the particle distribution begins to move inward.
So far, I have presented simulations of particles initiated quite close to the parent star (R = 1
AU). In panels-d1 and -d2 of Figure 4.3 I see that if the particles are initiated further out in thedisk, the particle motions are roughly the same, with the major exception that significantly moreparticles are retained in the accretion-flow case. For particles initiated at 10 AU in this case, about
60% remain in the disk after 105 years. In the midplane-flow case, particles initiated at 10 AU reachthe furthest regions of the disk slightly faster, but otherwise behave the same as when initiated at
1 AU. Therefore, while both gas-flow cases can turbulently diffuse particles over wide regions of thedisk, the accretion-flow case selectively retains particles initiated at larger radii and rapidly loses
particles initiated at smaller radii.
To summarize, the basics of particle mixing in these transport simulations are that:
1. Turbulent diffusion rapidly spreads the particle distribution over many tens of AU.
2. Where advection tends toward the loss of particles inward onto the parent star, there will bea preferential rapid depletion in grains initiated close to the parent star.
3. Even in the case of outward-flowing disk gas (the midplane-flow case), small dust grains may
still spiral inward at radii (or epochs) where the disk is locally tenuous.
103Accretion Flow Case

Midplane Flow Case

100

100
t = 104 yr
t = 105 yr

80

Rnow (AU)

Rnow (AU)

80
60
40
20
0

60
40
20

0

2

4
6
8
Rstart (AU)

0

10

0

2

4
6
8
Rstart (AU)10

Figure 4.4: The position of 20 µm particles at two times in the fiducial-disk model plotted against
their starting locations. The black line marks R = 25 AU, and the yellow line denotes the particlestarting locations (from 0.5 to 10 AU).
Figure 4.4, which plots particle positions versus their starting positions at two times, demonstrates
how these effects will be reflected both in the availability of inner-disk particles in the comet-forming
regions and in the source-region composition of those particles.

4.5

Results

In this section, I study how several parameters of the system affect the efficiency of mixing innerdisk particles out to large distances. The parameters examined include the structure/directionof the radial gas flow, the sizes of the dust particles, the diffusivity relative to the disk viscosity,and the initial distribution and evolution of the disk gas mass. I vary these parameters relative
to a fiducial model that assumes 20 µm particles (all particle sizes assume a dust-particle internal
density of 3 g cm−3 ), a Schmidt number of Sc = 1 (Dg = ν), and an evolving fiducial-disk model.
The fiducial disk begins with a total gas mass of 0.03 M¯ and an exponential fall-off radius of
Rd = 20 AU, so that at t = 0, 61% of the disk mass is concentrated within 20 AU of the star. All
simulations presented here use α = 10−2 .
In each model, the particles (10,000 for the accretion-flow runs, 2,000 for the midplane-flow) are
initialized with a uniform radial distribution between 0.5 and 10 AU. Although this is not an exact
match to the initial gas distribution at these radii, it allows me to fully sample the likely source
regions for crystalline silicates within the disk. Figure 4.5 demonstrates how the evolution of the104Time (years)

1×106

Midplane Flow Case

Accretion Flow Case

3.0×106
2.5×106

8×105

2.0×106

6×105

1.5×106
4×10

5

1.0×106

2×105
0
−1

5.0×10501
log10R(AU)

2

0
−1

3

0

1
log10R(AU)2

3

0 5 10 15 20 25 30
particles per bin

0
50
100
150
particles per bin

Figure 4.5: Map of 20 µm particles per bin for the fiducial disk simulations. Particles initiatedbetween 0.5 and 10 AU. The black line indicates R = 25 AU. The accretion-flow simulations use fivetimes as many particles as the midplane-flow simulations, so the color-scales have been adjusted
accordingly.
particle distribution proceeds for this configuration of initial positions, in contrast to the R 0 -fixedruns of Figure 4.3. I am interested, primarily, in the question of what fraction of these particles
reach the “comet-forming region” (defined, for this paper, as the disk beyond 25 AU) at any time
during the evolution of the disk. However, in some physical models, crystalline silicate formationis confined to smaller radial extents. Therefore, I also consider a subset of my simulation particles
divided into three smaller source regions: the inner-quarter region (0.5–2.5 AU, 2106 particles in
the accretion-flow simulations, 421 particles in the midplane-flow simulations), the inner-half region(0.5–5 AU, 4737 and 947 particles), and the outer-half region (5–10 AU, 5263 and 1053 particles).
The particle-transport simulations presented below were run on a 2.4 GHz Intel Core2Duo (usedto run two simulations simultaneously when work-resource allocation permitted; my code does notuse parallel processing). Run-time for a simulation depends primarily on particle size, as this affects
the fraction and duration of simulation particles that persit within the disk throughout its lifetime.
Runs with larger, 0.2 mm particles required roughly half-a-day of processing, while simulations ofwell-coupled, 0.2 µm grains required up to about five days. Simulations within a more-compact disk
configuration run faster, roughly consistent with the shorter disk-lifetime of these systems. Also,
particles persist for a smaller fraction of the disk lifetime in the accretion-flow cases simulations,105

and so these simulations also tend to be slightly less computationally expensive.

4.5.1

Transport in Different Radial Gas-Flow Cases

Figure 4.6 shows the baseline results for particle mixing if the effective gas velocity seen by theparticles is that of the accretion-flow case. I plot the fraction of 20 µm particles beyond 25 AU thatoriginate from each of the source regions, together with the same quantity for perfectly coupled
particles (i.e. of negligible size). I also show the gas fraction beyond 25 AU. As expected from thereduced models, diffusion is fast enough to mix some particles upstream into the comet-forming
region. Indeed, the first particles pass 25 AU in less than 20,000 years. However, loss of particles
by accretion onto the star is also rapid, especially close to the inner edge of the disk. After 10 5
years only ∼30% of all 20 µm particles remain in the disk, and most of those originated in the outerhalf of my total source distribution (the peak fractions of particles in the comet-forming region are
2.6% from the inner-half zone, but 11% from the outer-half zone). The lifetime of particles thatfraction beyond 25 AU

do manage to attain large radii is also limited, first by the fact that the transition radius that

0.15
all particles
inner quarter particles
inner half particles
outer half particles
gas / 30.10
outer−
half
all

0.05

0.00

0

1×106
2×106
3×106
Time (years)

4×106

Figure 4.6: Fraction of simulation particles beyond 25 AU for different source regions compared to
the fraction of the total disk gas mass currently beyond 25 AU (divided by three). Pale-coloredlines denote particles well-coupled to the gas; dark colors denote 20 µm-sized particles. The innerquarter source region is 0.5–2.5 AU; the inner-half source region is 0.5–5 AU; the outer-half source
region is 5–10 AU. Simulation run in the fiducial disk model using the accretion-flow case.

106

normalized dust/gas beyond 25 AU

0.08
all
inner quarter
inner half
outer half

all

0.06

0.04

outer−
half

0.02

0.00

0

1×106
2×106
3×106
Time (years)

4×106

Figure 4.7: Normalized concentration of particles beyond 25 AU. [See Equation (4.2).] The palecolored lines denote well-coupled particles. The dark lines are for 20 µm particles. Fiducial disk
model. Accretion-flow case.
divides gas inflow from outflow itself moves out as the disk evolves, and second because declininggas densities in the outer disk eventually preclude retention of 20 µm particles. These effects meanthat by t = 106 years, less than 1% of 20 µm particles exist in the disk beyond 25 AU, even if they
started in the outer-half source region. By t = 2.1 Myr, all have been lost inward onto the parentstar.
To quantify the effect of these mixing patterns on the abundance of hot, inner-disk grains in
the comet-forming regions, I plot in Figure 4.7 the normalized concentration, C N , of source-region
particles beyond 25 AU as a function of time. I define the normalized concentration as the numberof particles in a region divided by the gas mass in that region, then normalized by the initial numberof source-region particles divided by the t = 0 gas mass of that source region:CN =(number particles)outer disk / (gas mass)outer disk
(number particles)source, t=0 / (gas mass)source, t=0(4.2)

With this definition, CN is like a scaled dust-to-gas ratio, where CN = 1 means that the ratio of107

particles-of-interest–to–gas in the region of interest (the outer disk) is the same as where those particles originated at the time that they formed. I note that while the denominator of Equation (4.2)
is almost constant across my source region (i.e. the initial dust to gas ratio is a fixed value towithin 10%), the normalized concentration beyond 25 AU is an average over a broad region. The
local values can vary widely, depending upon the parameters of the disk. For the accretion-flowcase simulations, the highest local values of the normalized concentration typically lie just outsideof 25 AU, with a lower local CN in the regions hundreds of AU from the parent star.
From Figure 4.7, note that the peak values of the normalized concentration – and thus the
maximum extent to which the outer disk can be contaminated by particles from the hot inner
regions – is relatively modest. For 20 µm particles, the peak in CN from the all source region
is approximately 6%. This is actually higher than the value obtained from the outer-half source
region, reflecting the fact that although a greater fraction of particles mix outward from the outerhalf region, there is less total mass there to start with. Observe also that particle size strongly
influences the lifetime of particles in the outer disk. For well-coupled particles, C N declines only
modestly from its peak at a few hundred thousand years out to several Myr. This reflects the factthat well-coupled particles that survive the initial evolution have thoroughly mixed with most of
the entire disk gas mass, so that their subsequent loss is largely in proportion with the disk-gas
accretion rate. By contrast, 20 µm particles are lost much faster, and only significantly contaminate
the outer disk between ∼ 5 × 104 and 8 × 105 years.
Dramatically higher efficiencies of particle retention are obtained if, instead of experiencing
the mean gas flow, vacc , the particles settle and experience an outward flow at the disk midplane(recall that this requires little or no vertical mixing of angular momentum, so that the steep radial
density gradient at the midplane results in outflow). Results for this midplane-flow case are plottedin Figure 4.8. In this limit, few 20 µm particles are initially lost inward onto the parent star.
98% remain in the disk after 105 years, and around 90% of particles from each source region aretransported to beyond 25 AU, with only a slight selection against particles from the innermost
regions of the disk. This means that the normalized concentration curves for the different source
regions are all nearly the same, only scaled relative to a given source region’s innate ability tocontaminate the outer disk - the relative fraction of t = 0 disk mass in that source region.
Because 20 µm and well-coupled particles are swept to large distances and remain there for108

normalized dust/gas beyond 25 AU

4
all: 20 microns
all: well coupled3
dissipationof disk2

drainingof disk
particle
infall1
outwardadvection00

1×106 2×106 3×106 4×106 5×106 6×106
Time (years)

Figure 4.8: Midplane-flow case: Normalized concentration of simulation particles beyond 25 AU
for all particles initiated between 0.5 and 10 AU. Fiducial disk model.
a long time, there is a period when the change of normalized concentration beyond 25 AU is afunction solely of the loss of disk gas mass via accretion. Without the concurrent loss of accreting
particles, the particle-to-gas ratio reaches values even higher than in the original 0.5–10 AU source
regions. This is particularly so for the well-coupled particles, which never leave the outer disk once
they have entered it. The normalized concentration of 20 µm particles drops to half-maximum ataround t ∼ 2.6 Myr, about three times the contamination time-window given in the accretion-flow
simulations.
Note, however, that my model cannot account for the radial mixing that would occur in thepresence of vertical mixing and a vertically stratified gas flow. If the gas flow is rapidly inward far
from the midplane and outward near the midplane, then vertical mixing will lead to radial mixing,
even in the absence of radial diffusion. Ciesla (2007, 2009) discusses in detail how a given degreeof vertical mixing can lead to enhanced mixing of grain populations or to the partial segregation of
inward-flowing and outward flowing populations. Therefore, the normalized concentration for the
midplane-flow case merely represents the extreme upper limit on the extent and duration of the109contamination of the outer disk by inner-disk material.
The results imply that the efficacy of the fiducial disk model (Rd = 20 AU, Sc = 1) for outward
transport of 20 µm grains depends upon both the location (and extent) of the the source region
for crystalline silicates, and upon the radial gas flow experienced by the particles. In the accretion
flow case, the peak of contamination of the outer disk (beyond 25 AU) would result in a crystallineto-amorphous ratio of ∼6%. Attaining this limit requires that at t = 0 all silicate grains in theentire inner disk out to 10 AU were crystalline (and likewise assumes that all beyond that were
amorphous). If, instead, the source of crystalline silicates is confined to inward of 2.5 AU then thepeak CN beyond 25 AU is only about 0.2%. Moreover, I find that even these modest degrees ofcontamination are relatively short-lived. 20 µm-sized grains (matched to the upper-size end of theStardust grains) only have a contamination lifetime in the outer disk of . 1 Myr. Unless the timingof the epoch when grains became assembled into cometary material happened to coincide with thepeak of the contamination, the crystalline fraction would be diluted, either by grains grown locallyin the cold comet-forming region or by those raining inward from the even-colder outer regions of
the disk. Finally, my results for the accretion flow case might be yet further reduced by the factthat some crystalline silicate formation mechanisms may not yield 100% crystalline fractions out
to large radii. For example, if the primary source of crystalline silicates is shocks out to ∼10 AU,
but a sizable fraction of silicates in that region remain amorphous, then that cuts into the potential
maximum in the crystalline-to-amorphous ratio produced in the outer disk.
In the midplane-flow case, on the other hand, substantial outward transport can occur almost
irrespective of the location of the source region. If there is an outward flow of gas at the disk
midplane, then grains that are sufficiently well-settled will readily reach large distances (Ciesla,2009). An outward-midplane flow also allows material that has been transported outward to persist
there for a longer period of time. My midplane-flow simulations produce an upper-limit maximum
CN in the outer disk of 30% for the inner-quarter source region. Even if vertical mixing means that
only a fraction of the maximum can be achieved, it is still plausible to believe that the CAI-likegrains captured during the Stardust mission may have reached the outer disk via mixing within thedisk gas alone. This scenario may also be consistent with the observations of Watson et al. (2009),
who find higher crystalline-to-amorphous silicate mass ratios in disks where the grains appear more
settled toward the disk midplane, and of Olofsson et al. (2009), who find that possibly a large
110fraction of crystalline silicates exist in the outer regions (.10 AU versus .1 AU) of some disks.

4.5.2

Transport of Differently Sized Particles

In §4.4, I demonstrated that non-negligible particle size bars particles from the outermost regionsof the disk, because at large distances, the gas is so tenuous that the particles decouple from thegas motions, and inward-pointing head-wind drag dominates. Figure 4.9 plots the dust-mean-radial
velocities for several different particle sizes in the fiducial-disk model at t = 0 for the accretion-flowcase. (In the midplane-flow case, the curves are similar, only shifted toward outward velocities,
as in Figure 3.9 and following the pattern of Figure 2.14.) Though turbulent diffusion can sendparticles upstream of their average radial-velocity flow, grains are effectively barred from the regionsof the disk beyond where their average-radial velocity falls to large, inward-pointing values. It isclear from Figure 4.9 that the range within the disk that dust grains may occupy depends strongly
on the particle size, and that the mixing achievable out to the comet-forming regions will become
sharply limited as one approaches millimeter particle sizes. Also, where this cutoff occurs varies,
not only in time as the disk thins, but also from disk to disk depending on the total mass and mass

Radial Velocity (cm s −1)

distribution of each system.

400
300
200

Accretion Flow Casegas
2 microns
20 microns
0.2 mm2 mm

2 microns

100
0
2 mm

−100
−200
0.1

1.0

10.0
R (AU)

100.0

1000.0Figure 4.9: The mean-radial velocity of particles of different sizes in the fiducial disk model at
t = 0, accretion-flow case. Sizes listed assume an internal-particle density of 3 g cm −3 .

111normalized dust/gas beyond 25 AU

102
10

all
inner quarter
inner half
outer half

1

100
10−1

all

10−2
10−3

inner−
quarter

10−4
10−5 −2

10

10−1 100 101 102 103 104
grain size (microns)

105

Figure 4.10: Maximum values of normalized concentrations beyond 25 AU as a function of grain
size. Solid lines denote simulations run in the accretion-flow
case; dashed lines with black markers√
denote the midplane-flow case. Error-bars mark the N noise of the peak values. Fiducial disk
model. Grey line marks the baseline case of 20 µm-sized particles. In the midplane-flow case for thesmallest particles, peak values shown are for t = 5.2 Myr, just prior to photoevaporative dissipationof the disk.
I parameterize the outward mixing of grains as a function of size by plotting in Figure 4.10
the peak values of the normalized concentration of grains beyond 25 AU as a function of grain
size. The error bars show statistical errors based on the number of particles beyond 25 AU in eachrun. I plot the maximum values of CN from each of the different source regions, as well as theupper-limit values provided by the midplane-flow simulations (dashed lines with black markers).
While the magnitude of outward mixing varies by orders of magnitude across these populations,the trends with grain size are the same across all sets: as seen previously, the highest normalized
concentrations of inner-disk particles in the outer disk occur for the smallest grain sizes, up to a
few tens of microns, and the peak-CN values drop off sharply at around millimeter grain sizes. The
results of Ciesla (2007, 2009) for a vertically stratified disk flow suggest that the loss of larger grain
sizes may be partially mitigated by settling toward the midplane, thereby increasing the relative
outward transport and retention of these particles. Still, the concurrent drop in my midplane-flow
upper-limit concentrations suggests that this can only hold back the loss of large grains from the112

Time (years)

107
allinner quarter
inner half
outer half

after
peak

106

105

beforepeak

104 −210−1 100 101 102 103 104
10
grain size (microns)

105

Figure 4.11: Time-windows for the peak values of normalized concentrations beyond 25 AU as
a function of grain size and for different source regions. The lower set of solid curves mark thelatest pre-peak time of half-maximum concentration in the accretion-flow case. The upper curves
mark the earliest post-peak time of half maximum concentration for the accretion-flow case (solidlines) and the midplane-flow case (dashed lines with black markers). Fiducial disk model. In the
midplane-flow case, the smallest particles remain in the outer disk longest and never drop below
half-maximum concentration beyond 25 AU.
outer disk to a point. For 2 mm-sized particles, for example, even the midplane-flow runs yield apeak normalized concentration beyond 25 AU for all simulation particles of only ∼3.4%.
The other important variable to consider is the timing of these mixing events. In Figure 4.11,
I depict, as a function of particle size, the time-frame over which the normalized concentrationof inner-disk particles in the comet-forming region is at a value of half-maximum or greater. For
each source region (for the accretion-flow case simulations) I plot the latest time of half-maximum
concentration prior to the peak and the earliest time of half-maximum after the peak. (Due tolimited particle statistics, some of these quantities are quite noisy.) I also include the boundary of post-peak half-maximum concentration in the midplane-flow simulations to illustrate when
headwind-drag inspiral will overwhelm other physical effects even in the most extreme gas-flow
scenario.
Figure 4.11 demonstrates that the smallest inner-disk particles remain mixed into the outer disk
for millions of years, and that even inner-disk grains as large as 20 µm may be noticeably present at
113

normalized dust/gas beyond 25 AU

102
10

1

t=106 years

all
inner quarter
inner half
outer half

100
10−1

all

10−2
10−3

inner−
quarter

10−4
10−5 −2

10

10−1 100 101 102 103 104
grain size (microns)

105

Figure 4.12: Normalized concentration beyond 25 AU at t = 1 Myr as a function of grain size. The
largest particles are all depleted from beyond 25 AU by 1 Myr. Dashed lines with black markers
denote the midplane-flow simulations. Fiducial disk model.
large distances for ∼1–2 Myr. However, while the mixing of all grains outward is relatively rapid inthese simulations, not only do the larger particles experience relatively less outward mixing, but thatmixing is similarly short-lived. Even in the midplane-flow simulations, post-peak half-maximum
concentration is reached by t ∼ 6.2 × 105 years for 0.2 mm particles and as soon as t ∼ 1.1 × 105
years for 2 mm particles. The fact that larger inner-disk particles are short-lived in the outer disk isdemonstrated more bluntly in Figure 4.12, where I plot the normalized concentration as a functionof grain size at t = 1 Myr. While the mixing effects are relatively uniform for small particles atthis time, no particles 0.2 mm and larger remain in the outer disk in the accretion-flow simulation,and the upper-limits of the midplane-flow simulations suggest that even in that extreme thoselarge-grain populations are in rapid decline, if not already disappeared.
These results imply that it is the upper envelope of particle sizes recovered by Stardust thatprovides the most stringent constraints on disk evolution and particle transport. In my simulations,particles at least as large as 2 µm appear to behave as though well-mixed with the disk gas. Therefore, the presence of high-temperature particles in this size range within a Jupiter-family comet114

can be largely explained by outward mixing of disk material aided by some disk expansion from an
initially compact state. However, the particles recovered by Stardust include crystalline silicates aslarge as 20 µm. While the outward transport of 20 µm particles to the comet-forming region does
occur in my models, these particles do not remain well coupled to the disk gas indefinitely. Thissuggests that either (1) mixing of particles and planetesimal formation in the Solar Nebula occurredin a disk massive enough for 20 µm particles to remain well coupled to the gas motions, or that(2) the formation of comets or cometesimals occurred within the first million years or so of disk
evolution. This second constraint may be in agreement with Stardust results. Early studies foundno aqueous minerals in the Stardust materials, suggesting that the outward transport and incorporation of these materials into the 81P/Wild 2 cometesimals occurred very early, before the primary
accretion and fragmentation of the chondritic-asteroid parent bodies (Wooden, 2008). However, thepossible identification of igneous materials in the Stardust samples may contradict this scenario or
else put interesting time-line constraints on the formation time-scales of planetesimals and cometesimals across the range of Solar System radii (Stodolna, Jacob & Leroux, 2010; Joswiak et al., 2010).
Another constraint linked closely with time is the large-end grain size for which my model predicts
measurable outward mixing. Not only do the peak (and midplane-flow–upper-limit peak) values of
CN beyond 25 AU drop off markedly for grains a few millimeters in size, but the time-frame forthat outward transport is also restricted (to barely more than 105 years for 2 mm-sized grains), sothat comet-formation would have to be quite rapid to capture such large inner-disk grains if they
did appear in the comet-forming regions.
The transport of different-sized particles is also of interest for models that include the effects
of grain growth. Some observational studies of disks suggest a link between disk crystallinity andgrain growth (van Boekel et al., 2005; Olofsson et al., 2009), with characteristic crystalline grain
sizes of a few microns (distinctly larger than typical ISM grains). At the very least, grain growthand crystallization likely occur on similar time scales. The results presented here suggest thatas dust grains grow, they should become less likely to enter and more likely to leave the outerregions of the disk. Large grains become mostly confined to small disk radii where the disk gas isstill dense enough to support them. We know that growth at centimeter and meter scales poses a
constraint to disk models and planetesimal formation, because particles of this size are expected tofall inward toward the parent star very rapidly, depriving the disk of these solids. However, growth115

to millimeter scales may also place constraints on cometesimal formation if the outer regions of thedisk cannot support particles of even these sizes. At the very least, it could restrict the compositionof bodies formed in the outer disk, if most of the particles available are those falling inward fromlarge AU.

4.5.3

Transport Varying the Diffusivity

Next, I explore how the outward mixing of inner-disk grains depends on the Schmidt number, whichis the ratio of the disk viscosity to the gaseous diffusivity (so that a lower Schmidt number means
more relative diffusivity.) In Pavlyuchenkov & Dullemond (2007) the authors argue for an Sc lowerlimit of 1/3. Therefore, I have run simulations of particle transport varying the Schmidt number bya factor of three, and in Figure 4.13 I plot the values of maximum normalized concentration beyond
25 AU for four particle sizes and two source regions as a function of Sc. The primary effect shown in
Figure 4.13 is that a higher diffusivity can lead to a substantially higher degree of outward transport
in the general accretion-flow case of my particle-transport simulations. Here there are variations

normalized dust/gas beyond 25 AUof an order-of-magnitude or more in the peak outer-disk normalized concentrations between the

102
10

all
inner quarter

1

all

100
−1

all

10−2

inner−
quarter10

2 microns
20 microns
0.2mm
2mm

inner−
quarter

10−3
10−4
10−5

0.1

1.0
Sc

10.0

Figure 4.13: Maximum values of normalized concentration beyond 25 AU as a function of Schmidt
number for a range of grain sizes and two source regions. Low diffusivity (large Sc) tends to restrict
particles from reaching the outer disk. Dashed lines with black markers denote the midplane-flow
simulations. Fiducial disk model.
116highest (Sc = 1/3) and lowest (Sc = 3) diffusivity simulations across the range of particle sizes
considered. This is less than the variation predicted in the model scenario of Pavlyuchenkov &Dullemond (2007) (σSc=1/3 /σSc=3 ∝ (R/Rsource )4 ), but is still quite a substantial effect consideringthat my normalized concentrations are reported for the whole of the outer disk and that my poolof contaminants is limited to those present in the source regions at t = 0 only.
More generally though, the Schmidt number controls the relative importance of advection versus
diffusion of particles. When the diffusivity is low, advection dominates, so that more grains reachthe outer disk if the bulk flow is outward, and fewer do so if their bulk flow is inward (including
the case of 2 mm particles in the outward-flowing–gas midplane-flow case whose radial advection isdominated by headwind drag). Conversely, high diffusivity pushes the system in the direction of aflow-independent state where diffusion, both within the disk gas distribution and inward onto theparent star, is the dominant term. The diffusivity can also have an important effect on outer-disk
contamination from the inner-most source regions. For 20 µm-sized particles from the inner-quartersource region, Sc = 1/3 simulations produce a peak normalized concentration in the comet-forming
region of 1.7%, up from 0.2% in the baseline Sc = 1 simulations. In the low-diffusivity simulations
inner-quarter source grains of all sizes are completely absent from the comet-forming region unlessdirectly advected there by outward-flowing gas in the midplane-flow simulations.
Finally, to some degree, the diffusivity also affects the timing of outward mixing. This is plottedfor 20 µm-sized particles in Figure 4.14. The higher diffusivity simulations allow higher normalized
concentrations in the outer disk to occur sooner and to last longer (∼ 4 × 10 4 –1.1 × 106 years
half-max to half-max), whereas for lower diffusivity the outer-disk normalized concentration peaks
later and more briefly (∼ 9 × 104 –5.5 × 105 years).
My results for the variation in transport efficiency with Schmidt number suggest that two
distinctly different disk models could be consistent with observations. One possibility is that the
diffusivity is high, as suggested by Pavlyuchenkov & Dullemond (2007). In this case, it seems
plausible that a significant fraction of grains, formed in high temperature regions close to the star,can end up in the comet-forming region, even if the disk is highly turbulent and no significant settling
occurs. The recovery of CAI-type grains by Stardust – whose condensation requires especiallyhigh temperatures – poses the strongest constraint on this scenario, and I have not demonstrated
explicitly that it is possible. However, given that the early disk was certainly hotter than the disk
117

all
inner quarter
inner half
outer half

107

Time (years)

20 microns

106

after
peak

105
before
peak

4
100.1

1.0
Sc

10.0

Figure 4.14: Time-windows for the peak values of normalized concentration of 20 µm particles
beyond 25 AU as a function of Schmidt number (Sc ≡ ν/Dg ). Lower solid curves mark the lastest
pre-peak time of half-maximum concentration in the accretion-flow simulations. Upper curves markthe earliest post-peak time of half-maximum concentration for the accretion-flow (solid lines) andmidplane-flow (dashed lines) simulations. Fiducial disk model.
which I have modeled, it is plausible that enough material from the innermost disk could reachlarge enough radii. Hot outflows near the disk midplane, which Tscharnuter et al. (2009) find very
early in the star/disk formation process, would assist outward transport from the innermost disk.
A qualitatively distinct disk model with low diffusivity is also conceivable. My results suggestthat if Sc is significantly larger than unity, there is negligible transport of crystalline silicates subject
to the accretion flow to the region where Jupiter family comets form. This is true even if silicates
can form at radii as large as 10 AU. A low diffusivity disk, however, could potentially be favorable
to the establishment of an outward midplane flow, which is able to move settled particles outward
efficiently. Ciesla (2009) showed that a transport scenario dominated by high-altitude–inward andlow-altitude–outward advection can lead to segregated grain populations with different processing
histories. This is possibly compatible with disk observations of Olofsson et al. (2009) that suggest
some disks are more crystalline at larger distances.

118

4.5.4

Transport in More/Less Compact Disks

Finally, I assess the role of the disk mass distribution and its evolution on outward transport. First,
I contrast the results of runs assuming a static disk with those of the evolving disk, and then moveon to exploring the effects of the initial compactness of the disk in the evolving-disk model. Of my
two static-disk models, the first uses the t = 0 profile of the fiducial-disk held static (the static0-disk
model). The second is the steady-disk limit of the t = 0 fiducial-disk profile, where the disk surface
density follows Equation (4.1), with Rd = ∞ (the steady-disk model).
Results from the two static-disk models are shown in Figure 4.15, which plots in the top panel
the total-fraction and in the bottom panel the normalized-concentration of 20 µm particles beyond
25 AU with time for two different source regions. The mass per AU of the gas disk does not drop
off in the outer disk for the steady-disk model but is instead continuous out to the edge of mysimulation space. Therefore, the normalized concentrations beyond 25 AU for this disk model arecalculated using the disk gas mass only out to 100 AU (for a total 0–100 AU disk mass of 0.162 M ¯ ;note that 0.03 M¯ of gas is contained within the first 21 AU in the steady-disk model). Figure 4.15
includes the results for the evolving fiducial-disk model for comparison. In a static disk, the upperlimit outward transport set by the midplane-flow case is that virtually all grains are advected out
to the outer disk and remain there.
The most obvious result from Figure 4.15 is that the two static-disk models, together with thefiducial evolving-disk model, form a hierarchy in outward mixing that is based on disk structure.
In the static0-disk model the region of outward-flowing accretion flow remains fixed relatively close
to the central star, and those outward velocities are fairly rapid. Therefore, more than twice thepeak fraction of fiducial-disk particles move to the outer disk (and stay there, as per the staticdisk-model tendencies seen in §4.4). The surface-density structure in the steady-disk model meansthat in the accretion-flow case the gas radial-velocity is everywhere inward, and hence the numberof 20 µm particles beyond 25 AU peaks at less than 40% of the value for the corresponding fiducial
evolving-disk simulations, and those particles eventually all fall back inward onto the parent star,— and more rapidly than in the evolving disk simulations. Roughly 23% remain in the disk after
105 years (when 30% remain in the fiducial-disk accretion-flow simulations).
However, in terms of the normalized concentration of inner-disk particles in the outer disk, the119Fraction of Particles Beyond 25 AU

fraction beyond 25 AU

all
inner quarter

evolving disk
static t=0 disk
steady disk

20 microns
midplane flow:

1.0000inner−
quarter

0.1000accretion flow:

all

all

0.0100inner−quarter

0.0010normalized dust/gas beyond 25 AU

0.0001 4
10

105

106

107

Normalized Concentration Beyond 25 AU
100
20 microns
accretion flow casesteady disk to 100 AU

10−1
10−2
10−3
10−4

all

inner−quarter

10−5
10−6 410105106Time (years)

107

Figure 4.15: For the static disk models, (top panel) the fraction of 20 µm particles beyond 25
AU and (bottom panel) the normalized concentration of 20 µm particles beyond 25 AU for twosource regions. Dark, heavy curves denote the steady disk model and assume an outer-disk gas
mass truncated beyond 100 AU for the normalized-concentration calculation. The curves for thefiducial, evolving disk model are included for comparison. Solid curves denote the results for theaccretion-flow gas-velocity case; dashed curves denote those for the midplane-flow case.

120

disk-gas mass distribution also really sets the steady-disk simulations apart from the rest. With somuch more mass in the outer disk than the other two models, the steady-disk model is that muchless intrinsically capable of contaminating the outer disk with inner-disk grains than are the diskswith a more compact disk mass distribution. Finally, for the upper limits placed by the midplaneflow simulations both static-disk models send more than 90% of the simulation particles from all
source regions to beyond 25 AU; the outward advection is only more rapid in the static0-disk
model. Therefore, the upper bounds on the mixing in these models are set only by their relative
source-region and comet-forming region disk-gas mass ratios.
Although some trends are the same for static as compared to evolving disk models, the results
imply that the evolution of the disk – particularly at early times – has a substantial impact on particle transport. Steady-disk models tend to underestimate the degree of outward particle transportpossible in protoplanetary disks in the early stages of disk evolution, when the most processingof high-temperature materials occurs. The outward transport of particles is substantially more
efficient at early times, consistent with the results of Bockelée-Morvan et al. (2002) and Ciesla(2009), even here assuming a fixed diffusivity. How strong the effects of evolution are will depend,of course, on the details of the initial conditions for the disk. To examine this dependence, I vary
the initial compactness of my evolving-disk model (parameterized by Rd in Equation (4.1) ). I
consider Rd = 5, 10, and 40 AU, in contrast to Rd = 20 AU in the fiducial-disk model. I retain 0.03
M¯ as the starting mass of the disk, so that the lifetimes of the model disks vary between 3.6–6.9
Myr, and the t = 0 mass accretion rates between 8.3 × 10−7 –8.6 × 10−8 M¯ yr−1 . I do not consider
possible variations in heating for disks forming in more or less compact configurations, retaining
the static temperature profile described in §4.3.13 .
Reducing the compactness of the model disk has two consequences: the disk initially expands
more rapidly, and it does so from an initial state where a greater fraction of the mass is at small
radii (and thus hot and potentially able to form crystalline material). The combined magnitude of
I also continue to employ the same uniform radial distribution to initiate my simulation particles.
One should note, however, that the gas-mass–per-AU is no longer roughly uniform across the source
regions in the Rd = 10 and 5 AU disks, as shown in Figure 4.16. For Rd = 10 AU, the ratio of
the average-gas-mass–per-AU for the inner-quarter source region to the outer-half source region is∼ 1.5, and for Rd = 5 AU it is ∼ 2.6. This affects the accuracy of the normalized concentration
values that I report for these disks (especially for the widest source regions), but it does not alter
the trends with Rd repeated across source-region populations.
3121

Mass Distribution (M sun AU−1)

Rd=5 AU
Rd=10 AU
Rd=20 AU
Rd=40 AU

0.005
0.004
inner−quarter
inner−half

0.003

all
outer−half

0.002
0.001
0.000

0

2

4

6
R (AU)

8

10

20 microns

fraction beyond 25 AU

Peak Fraction
1.000

(a)

all

0.100
inner−
quarter

0.010

0.001

10
Rd (AU)100

norm. dust/gas beyond 25 AU

Figure 4.16: Gas mass distributions at t = 0 for different intial disk compactnesses (R d ). The
different source regions considered are marked by colored boxes: green - all; red - inner quarter;orange - inner half; blue - outer half. Note that the fiducial disk model (Rd = 20 AU) has a mostly
uniform distribution of mass across the different source regions, but that the more compact disk
models contain significantly more mass per AU in the inner regions than out around 10 AU.

102

Peak Concentration
(b)

101
100
10−1
10−2
10−3
10−4
10−5

all
inner−
quarterallinner quarter
inner half
outer half10
Rd (AU)

100

Figure 4.17: (a) Maximum values of the fraction of 20 µm particles beyond 25 AU and (b) maximum
values for the normalized concentration of 20 µm particles beyond 25 AU as a function of initial
disk compactness. Dashed curves with black markers denote the midplane-flow simulations. The
grey line marks the baseline case of Rd = 20 AU. Note that the most compact disks both send alarger fraction of simulation particles beyond 25 AU and have a larger fraction of disk material in
the source regions at t = 0.

122

all
inner quarter
inner half
outer half

107

Time (years)

20 microns

106

after
peak

105
before
peak

104

10
Rd (AU)

100

Figure 4.18: Time windows for the peak values of the normalized concentration of 20 µm particles
beyond 25 AU as a function of initial disk compactness. Lower solid curves denote the latest
pre-peak time of half-maximum concentration in the accretion-flow case. Upper curves denotethe earliest post-peak time of half-maximum concentration for the accretion-flow (solid lines) andmidplane-flow (dashed lines with black markers) simulations.
these effects is shown in Figure 4.17, where I plot first (a) the simple peak-fraction and second (b)the peak normalized-concentration of 20 µm particles beyond 25 AU as a function of R d . Variationsof approximately one order of magnitude in Rd result in nearly two orders of magnitude differencein the peak concentrations for each source region. For the smallest disk – with R d = 5 AU –the normalized concentration of all simulation particles peaks at ∼68%, more than an order ofmagnitude in excess of the value for the fiducial simulations. A substantial part of this increase isdue to the larger reservoir of mass that a compact disk has at small radii, which can potentiallycontaminate the (proportionally less massive) outer disk. That this mass-distribution effect isimportant is made clear by the fact that even the upper-limit concentrations from the midplane-flow
simulations show a strong (order-of-magnitude) trend with Rd . However, as shown in Figure 4.18,
there is little variation in the timing of outer-disk contamination with Rd . The timing may beslightly faster for smaller Rd . However, this is largely an effect of the disk lifetime, which isnoticeably shorter for an initially more compact configuration.
The fact that the outer disks of the initially more-compact-disk models are less massive (and123

fraction beyond 25 AU

Peak Fraction
1.0000(a)0.10000.01000.00100.0001

all

10
Rd (AU)100

norm. dust/gas beyond 25 AU

2mm

Peak Concentration102
allinner quarter
inner half
outer half

101
100

(b)

10−1
10−2
10

all

−3

10−4
10−5

10
Rd (AU)

100

Figure 4.19: Maximum values for (a) the total-fraction and (b) the normalized-concentration of 2
mm particles beyond 25 AU as a function of initial disk compactness. Dashed curves with black
markers denote the midplane-flow simulations. In the accretion-flow case, almost no 2mm particles
from the inner-quarter source region reach beyond 25 AU.
more tenuous) does place some constraints on the outward mixing of large grains in my particletransport simulations. In Figure 4.19 (b), I plot the peak normalized concentration of 2 mmparticles beyond 25 AU as a function of Rd . All of the disks, of course, have less outward transportof these larger grains, but in my simulations, the peak in the fraction of particles sent to beyond 25
AU occurs for Rd = 10 AU (as shown in panel-a), which sends up to 0.25% of all 2 mm-simulation
particles beyond 25 AU (4% in the midplane-flow case) compared to less than 0.1% for R d = 5
AU (less than 2% in the midplane-flow case). Because of the larger source-region mass for R d = 5
AU, this drop in fractional outward transport constitutes a leveling off in the maximum-achievablenormalized concentrations of inner-disk particles in the outer disk.
Comparing these results to observations of crystalline silicates in other disks allows us to construct a possible scenario for the formation and transport of high-temperature minerals in thosedisks. In the Watson et al. (2009) survey, the disks observed were around T Tauri stars ∼1–2 Myr
old and showed a slight correlation between the crystalline-silicate mass fraction and the measured
accretion rate onto the star. As shown in Figure 4.20 for my model disks, t = 1 Myr corresponds toa time when those disks that were initially most compact (with high t = 0 accretion rates) now havelower accretion rates than the initially more-extended disks. Therefore, small grains that formed
at small AU near t = 0 should be more broadly distributed in disks that now have lower accretion124Accretion Rate (M sun yr−1)

10−6
Rd = 5 AU
Rd = 10 AU
Rd = 20 AU
Rd = 40 AU

10−7
10−8
10−9
10−10 410105106Time (years)

107

Figure 4.20: Mass accretion rates for disk models initiated with the same total disk mass, but withdifferent initial exponential cutoff radii (Rd ). Note the reversal in ordering from early to late times.
All disks were initiated at t = 0 with a total mass of 0.03 M¯ .
rates. However, lower accretion rates also mean disks that are more tenuous. The initially mostcompact disks lose mass the fastest, causing their outward transport efficiency to drop rapidly withtime. Therefore, grains that formed at small AU after t ∼ 2 × 10 5 yr may be better distributed in
the initially more-extended disks, which now have higher accretion rates.
These properties could potentially explain the opposing trends between crystallinity and observed accretion rate reported by Watson et al. (2009). Pyroxene is favored over olivine when theminerals are formed by condensation (Gail, 2004; Wooden, Harker, & Brearley, 2005). Therefore, apossible scenario is that pyroxene formed primarily at very early times when the disks were hottest
and became most broadly distributed throughout the initially most-compact disks. Most of theolivine, on the other hand, may have formed over a longer period of time. It might, therefore,
have become more broadly distributed within disks that retained significant surface densities forlonger times (the initially more-extended disks). This hypothesis does not, however, explain why
the Watson et al. (2009) survey found no correlation between crystalline mass fraction and disk
mass.

125

4.6

Conclusions

Observations of comets (Hanner et al., 1994; Wooden, Woodward, & Harker, 2004) and sample
return from the Stardust mission (Brownlee et al., 2006) suggest that outward transport of hightemperature solids to the comet-forming regions is a necessary component of any successful modelof the early Solar Nebula. Some proposed mechanisms for such outward transport include thelaunching of grains from near the star out to large distances in a disk jet or wind (Shu et al.,2001), the outward flow of mid-sized particles along the surface of the disk due to photophoretic
heating and gas-pressure forces (Mousis et al., 2007), and the outward diffusion of the hot grain
population within the disk gas due to turbulent mixing. I have focused on this last mechanism, as
turbulence is believed to be intrinsic to protoplanetary disks of all types, and therefore tubulent
mixing of grain populations is likely important within all disks, reguardless of the degree of outerdisk contamination finally achieved. A turbulent mixing hypothesis is also free of concerns regarding
grain survivability within potential wind launch flows initiated extremely closely to the central
star, and of potential special-condition local optical-depth requirements for grains to reach large
distances due to photophoretic forces. This is not to say, however, that these other mechanisms
play no roles protoplanetary disk compositions; they are simply neglected here in favor of more
univeral disk-turbulence and gas-drag effects.
I have constructed a 1D model of viscous disk evolution and particle transport that I use to
study the relationships between the evolving disk mass distribution and the local conditions thatare supportive of the aerodynamic outward mixing of grains, specifically including self-consistenttreatment of the dust grains’ surface-area-to-mass ratios in my aerodynamic transport terms. I
have examined patterns of outward mixing considering variations in the radial gas-flow structure,
the sizes of dust grains, the model’s Schmidt number, and the initial compactness of the model
disk’s gas mass distribution. The differences in outward transport and outer-disk contaminationaccording to these parameters are summarized in Figures 4.21–4.24 for both the 1D-gas-accretionvelocity and outward-midplane-flow scenarios.
I find that a range of disk models are able to account for the presence of high-temperature–
inner-disk particles in the comet-forming region. As in Dullemond, Apai, & Walch (2006), the mostfavorable models involve rapid early expansion of initially quite compact disks. These conditions can126Fraction Beyond 25 AU
Fraction Beyond 25 AU

Varying Source Region
0.12

0.20

Varying Grain Size
0.08

all
inner quarter
inner half
outer half

outer−
half

0.10
0.08

2 mm
0.2mm
20 microns
2 microns
0.2 microns
well coupled

0.06

0.06

0.04

all

0.04
0.02

0.02
0.00

0.2mm
20 microns

0

1×10

6

2×10

6

3×10

6

0.00

Varying Diffusivity
1/3

0.15

2×106

3×106

Varying Disk Compactness

Sc=3
Sc=1
Sc=1/3

0.15

5AU

0.10

0.10

Rd=40 AU
Rd=20 AU
Rd=10 AU
Rd=5 AU

0.05

0.05
0.00

1×106

0

2 microns

0

1×106
2×106
Time (years)

3×106

0.00

0

1×106
2×106
Time (years)

3×106

Figure 4.21: Accretion-flow case; Total fraction beyond 25 AU, each panel varying a different
set of parameters. Unless otherwise marked, the baseline is: 20 µm particles from the “all” source
region (0.5–10 AU) simulated in the fiducial (Rd = 20 AU) evolving disk model with diffusivity
equal to the disk viscosity (Sc = 1).
result in inner-disk material flowing to the outer disk on a short time scale. In agreement with Ciesla(2009), I find that outward transport is more efficient at early times due largely to stronger gas
radial outflows. Even more favorable are cases involving outward-flowing gas at the disk midplane(Keller & Gail, 2004; Ciesla, 2007, 2009), which are capable of delivering substantial fractions ofgrains inward of 1 AU to the outer regions of the disk. It is not known whether such outward flows
exist in real disks (or, whether the particles are sufficiently settled to experience them consistently).
Whether such midplane flows are required to explain the Stardust results depends upon the Schmidt
number, which at its lower limit (highest relative diffusivity) could allow for a relatively high degreeof general outward mixing, even from relatively near the parent star. While the mean accretion flow
is less efficient at transporting particles outward, significant outward contamination by crystalline
material would be possible if the disk were initially both compact and able to form silicates out to127

Varying Grain Size

Fraction Beyond 25 AU
Fraction Beyond 25 AU

Varying Source Region
1.0

1.0

1.0
all
inner quarter
inner half
outer half

0.8
0.6

0.8

0.4

0.2

0.2
0

1×106 2×106 3×106 4×106 5×106

2 mm
0.2 mm20 microns
2 microns
0.2 microns
well coupled

0.6

0.4

0.0

20 microns

0.0

0.2mm

0

Varying Diffusivity

1×106 2×106 3×106 4×106 5×106
Varying Disk Compactness

1.0
Sc=3
Sc=1
Sc=1/30.8

Rd=40 AU
Rd=20 AU
Rd=10 AU
Rd=5 AU

0.8

0.6

0.6

0.4

0.4
1/3

0.2
0.0

5AU

0.2
0

1×106 2×106 3×106 4×106 5×106
Time (years)

0.0

0

40AU

1×106 2×106 3×106 4×106 5×106
Time (years)

Figure 4.22: Midplane-flow case; Total fraction beyond 25 AU. Unless otherwise marked,
the baseline is: 20 µm particles from the “all” source region (0.5–10 AU) simulated in the fiducial(Rd = 20 AU) evolving disk model with diffusivity equal to the disk viscosity (Sc = 1).
a distance of several AU.
My results suggest that it is the largest particles recovered by Stardust that place the tightest
constraints on the evolution of the early Solar Nebula. Particles as large 2 µm are well coupled tothe disk gas, and plausible levels of turbulent diffusivity allow them to reach the outer disk fromsmall radii in a wide variety of models. My simulations show that 20 µm particles can also reachthe outer disk — consistent with the Stardust results — as long as they were incorporated intolarger bodies within the first 1–2 Myr of the disk evolution. For even larger particles, those of a few
mm or larger, none of my disk models admit significant outward transport to the comet-forming
region, regardless of the proposed source region. Even grains of a few hundreds of microns in size
should be rare, since their residence time in the outer disk is very limited. Discovery of such grains
would, within the context of the model developed here, require a more massive disk during the time128

norm. dust/gas beyond 25 AUnorm. dust/gas beyond 25 AU

Varying Grain Size

Varying Source Region
0.08

0.10
all

0.06

all
inner quarter
inner half
outer half

2mm
0.2mm
20 microns
2 microns
0.2 micronswell−coupled

0.08
0.06

0.04
outer−
half

0.02
0.00

0.04
0.02

inner−half0

5.0×105 1.0×106 1.5×106 2.0×106

0.00

0

20 microns

5.0×105 1.0×106 1.5×106 2.0×106

Varying Disk Compactness

Varying Diffusivity

0.20

0.8
1/3

0.15

5AU

Sc=3
Sc=1
Sc=1/3

0.6

0.10

0.4

0.05

0.2

0.00

2 microns
0.2mm

Rd=40 AU
Rd=20 AU
Rd=10 AU
Rd=5 AU10AU0

5.0×105 1.0×106 1.5×106 2.0×106
Time (years)

0.0

0

5.0×105 1.0×106 1.5×106 2.0×106
Time (years)

Figure 4.23: Accretion-flow case; Normalized concentrations beyond 25 AU. Unless otherwise marked, the baseline is: 20 µm particles from the “all” source region (0.5–10 AU) simulated in
the fiducial (Rd = 20 AU) evolving disk model with diffusivity equal to the disk viscosity (Sc = 1).
of their formation, transport, and incorporation into cometesimals.
The existence of substantial uncertainties in the disk physics currently precludes an observational determination of the characteristic radii at which crystalline silicate formation occurs. The
strongest constraints come from observations of CAI-like grains, which probably formed at fractionsof an AU from the Sun. Only a subset of my accretion-flow simulations are likely compatible withthis finding. A broader range of models in which the midplane gas flow is outward are viable, sincethese models retain particles in the disk for a longer period of time and facilitate the co-existenceof grain populations of markedly different processing histories. Generically I expect that, provided
sufficient outward transport is possible, turbulent diffusion will cause particles to become radiallywell-mixed over tens of AU, so that incorporation of CAI-like grains and Fe-rich crystalline silicates(possibly formed in water-rich shocks out to 10 AU) into the same body is plausible.

129norm. dust/gas beyond 25 AUnorm. dust/gas beyond 25 AU

Varying Grain Size

Varying Source Region
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0

3.0
all
inner quarter
inner half
outer half

all

2.5
2.0
1.5
20 microns

1.0
0.5

0.2mm

inner−quarter0

1×106 2×106 3×106 4×106 5×106

0.0

2mm
0.2mm
20 microns
2 microns
0.2 micronswell−coupled

0

Varying Diffusivity

1×106 2×106 3×106 4×106 5×106

Varying Disk Compactness

2.0

6
3

1.5

Sc=3
Sc=1
Sc=1/3

5

5AU

4

1.0

Rd=40 AU
Rd=20 AU
Rd=10 AU
Rd=5 AU

3
1/3

10AU

2
0.5
1
0.0

0

1×106 2×106 3×106 4×106 5×106
Time (years)

00

1×106 2×106 3×106 4×106 5×106
Time (years)

Figure 4.24: Midplane-flow case; Normalized concentrations beyond 25 AU. Unless otherwise marked, the baseline is: 20 µm particles from the “all” source region (0.5–10 AU) simulated in
the fiducial (Rd = 20 AU) evolving disk model with diffusivity equal to the disk viscosity (Sc = 1).
For astronomical observations, my results suggest that the mass and size of the disk at an
early epoch are critical parameters in determining the viability and extent of outward particle
transport. These disk properties are inherited from the mass and angular-momentum structure of
the cloud that collapses to form the star, and would vary from system to system. I expect thisto result in variations in the crystalline fraction at large radii in disks around different stars. I
have also noted that the time scale during which conditions allow substantial outward transport
is only a small fraction of the disk lifetime. A limited time for outward mixing is both interestingand puzzling when paired with the emerging evidence for aqueous and/or igneous grains among theStardust samples (Wooden, 2008; Joswiak et al., 2010; Stodolna, Jacob & Leroux, 2010). However, it
may allow the Stardust results to be reconciled with other observations that require compositional
gradients within the disk to be maintained. A possible scenario is that an early stage of rapid

130expansion drove particles into the outer disk where they were rapidly locked up into larger icy
bodies. Once the disk had expanded, the gas flow became more uniform, limiting the outward
transport of solids.

131

Chapter 5

Particle Mixing and Outward
Transport: Extended Parameter
Study
In Chapter 4, I explored the outward mixing of dust grains due to aerodynamic forces within aprotoplanetary gas disk in the context of the Stardust results, specifically underlining the advantages
and constraints imposed by protoplanetary disk evolution. I explored outward mixing as a functionof some of the most obviously relevant parameters: the relative diffusivity of the particle ensemble,
the particular sizes of transported grains, the relatively unknown gas-velocity structure of the disk,
and the initial compactness (and subsequent expansion) of the evolving disk. All of these parameters
are important to the outcome of outward mixing via gas-drag advection and turbulent diffusion. In
this chapter, I use the same models and simulation setup to expand my exploration of parameter
space in relation to outward mixing. Here, I consider those parameters not as immediately obvious
to the basic question posed in Chapter 4, but equally fundamental to the efforts of modelingparticle-transport within a gas disk. I find that constraints imposed by these new parameters
on aerodynamic outward mixing of hot grains remain in keeping with the observations of disk
crystallinity in disks around other stars. I also find that the results outlined in Chapter 4 are
not unique to the disk models used in those simulations, and that my conclusions remain robust
within the context of a one-dimesional, evolving-disk-model scenario: Disk mass distribution andcontaminant production at the earliest times are the primary factors determining disk composition,and the process of disk evolution is tightly coupled to the outcome of particle transport within aprotoplanetary disk.

132

5.1

Introduction

In this chapter, I continue to explore the outward mixing of hot grains in relation to the Stardust
results and observations of disk crystallinity discussed in Chapter 4. Therefore, the simulationsin this chapter are performed using the same disk-model and particle-transport setup as described
for the Chapter 4 simulations. I continue to use a static, power-law disk-temperature profile andnormalize my results to the same fiducial-disk model where MD,0 = 0.03M¯ , Rd = 20 AU, and
α = 10−2 . See §4.3 for details. In some instances, I also use the initially-most-compact disk setup asa baseline (Rd = 5 AU) but always in parallel with the fiducial model, and I include in this chapter
results of varying MD,0 and α explicitly. I use the same model-disk grid setup as in Chapter 4,
except in §5.2.2, where I test the effect of moving Rin outward by removing grid cells from the inneredge of the grid (so that the grid as a whole is static for these tests).
As in Chapter 4, the primary measure I use to assess the degree of outward mixing in thesesimulations is the normalized concentration of source particles in the outer disk (beyond 25 AU),
CN , as a function of time, or as the peak value of that curve. Please refer back to §4.5.1 andEquation (4.2) for the definition and discussion of this quantity. Again, the source regions that
I consider for outwardly-mixed grains are the inner-quarter source region, extending from 0.5–
2.5 AU, the inner-half source region, 0.5–5 AU, the outer-half source region, 5–10 AU, and theall source region, 0.5–10 AU. Simulation particles are initiated evenly spaced between 0.5 and 10
AU, and I continue to assume a rocky internal density for the particles of ρ d = 3 g cm−3 . Onceagain, I implement 10,000 particles within the mostly inward-flowing accretion-flow case, and 2,000
particles within the outward-flowing midplane-flow case. See §2.6 for details of the disk-gas radial
velocities represented by these two cases. In general, the accretion-flow case should be thought ofas providing the primary results of the simulations, tracing grain transport within the bulk flow ofthe disk, whereas the midplane-flow case provides the extreme upper limit on both the magnitude
and temporal extent of grain transport to the outer disk.
Because I would like to test the results of outward transport not only on disk characteristics but
also on my model setup, I begin in §5.2 by varying a couple (mostly) numerical parameters, namely
the seed number used to initialize random-walk diffusion of the particle ensemble, and R in = Rmin ,the inner boundary of both the model-disk grid and disk surface-density profile. In §5.3, I explore133

outward transport as it depends on the time (according to the clock of my disk-model evolution) at
which grains are initiated. In §5.4 and §5.5 I vary MD,0 and α, respectively, and in §5.6 I explore
outward transport as a function of the disk-temperature–model parameters. I provide conclusionsin §5.7, tying the results of this chapter to those of Chapter 4, the compositions of extrasolar disks,and solar-system comets.

5.2

Some Numerical Considerations

While the simulations of Chapter 4 tested the results of outward transport on a number of different
effects, they neglected to examine the effects of the simulation set-up itself. In this section, I test
the results for outward transport under the influence of two effects: the random error introduced
by using a relatively small particle sample, and the location chosen for the inner-disk edge, which
is set to Rin = 0.1 AU in the fiducial runs but which has a relatively minor effect on the evolutionof the global gas disk.

5.2.1

Randomized Diffusion

To look for effects of statistical error on the output of my particle-transport simulations, I have run
simulations in the fiducial-disk model varying the seed number inputed into the random-number
generator that controls the random-walks for diffusion of the particle ensemble. In the fiducial
simulations, I use an input seed of -34. For this section, I consider also runs with seeds of -33 and-35.
Figure 5.1 plots the normalized concentration beyond 25 AU as a function of time for 20 µmsized grains from the all and inner-quarter source regions. From these curves, one can see that eachof the separate runs produce qualitatively the same results, but that quantitatively the curves may
differ by several percent at least at any given time-step. Unsurprisingly, this type of divergence isstrongest for grains from the inner-quarter region in the accretion-flow case, for which the smallest
fraction as well as the smallest total number of grains reach the outer disk (despite having fivetimes more simulation particles initiated than in the midplane-flow case).
However, the primary quantitative result that I have used to characterize the outward-transport
results of my simulations is the peak value of the individual CN curves. As shown in the plots of
Figure 5.2, this peak value is fairly well-characterized using the fiducial setup of my transport134normalized dust/gas beyond 25 AU

normalized dust/gas beyond 25 AU

Accretion Flow Case

10.0000s# = −35
s# = −34 (fiducial)s# = −33

1.00000.1000all

0.0100inner quarter

0.00100.00010

5.0×105 1.0×106 1.5×106 2.0×106
Time (years)

Midplane Flow Case

10.0000
1.0000all

0.1000inner quarter

0.01000.00100.00010

1×106

2×106 3×106
Time (years)

4×106

102

20 microns
a)

101
100
10−1
10−2
10−3
10−4
10−5−36allinner quarter
inner half
outer half−35

−34
seed #−33−32

normalized dust/gas beyond 25 AU

normalized dust/gas beyond 25 AU

Figure 5.1: Normalized concentration beyond 25 AU for 20 µm grains in the fiductial disk modelfrom the all and inner-quarter source regions. Three separater run-sets varying the seed number
used to initialize random-walk diffusion of the particle ensembles.

102
101

b)

seed # = −35
seed # = −34
seed # = −33

100
10−1
10−2
10−3
10−4
10−5 −2
10−1
10

100 101 102 103 104
grain size (microns)

105

Figure 5.2: Panel-a) Peak CN values for 20 µm particles in the fiducial disk model as a function of
randomization seed number. Panel-b) Peak CN values as a function of grain size for the three setsof initializing seed number. Thick lines plot values for the all source region, thin lines for the innerquarter region. Both panels) Accretion-flow case: solid lines and colored markers. Midplane-flow
case: dashed lines and black markers.
simulations. In panel-a, I plot the peak values for each of the source regions as a function of thesimulation seed number and find that the spread in values is within the error-bars estimated using
Poisson statistics. In panel-b, I plot the CN -max curves as a function of grain size for each of theseed-number run sets, with results that suggest the peak CN value I extract from a simulation is
fully adequate for describing trends in outward grain-transport across simulation parameters.
135

Time (years)

allinner quarter
inner half
outer half

106

105

−36

after
peak

before
peak

−35

−34
seed #−33−32

Figure 5.3: Times marking half-maximum CN values for 20 µm grains before and after the peakfor the accretion-flow case (solid lines, colored markers) and after the peak for the midplane-flowcase (dashed lines, black markers), as a function of randomizing seed number.

Another measure I have used for examining outward transport in my simulations is the timewindow between half-maximum values of the CN curve. As the scatter in previous time-windowplots has suggested, this measure is less well characterized in my simulations, and this is verified
in Figure 5.3. While the source regions that send large fraction of grains to beyond 25 AU, suchas those in the midplane-flow case, have well-characterized boundaries for the peak in outer-disk
concentrations, small-number statistics in some other cases leads to extremely poorly characterized boundaries. The boundary for post-peak half-maximum for the inner-quarter region in theaccretion-flow case ranges from < 3 × 105 to nearly 106 years across the 20 µm-grain runs using
three different seed numbers. Therefore, trends deduced from time-window plots should mostly
consider the largest source-region sets in the accretion-flow case, and the upper-bounds given by
the midplane-flow case.

5.2.2

Varying the Location of the Inner Disk Edge

The inner edge of the grid and model disk in my simulations is set, somewhat arbitrarily, to
Rin = 0.1 AU. As the mass flux from large AU has primary control over the disk evolution, varying
this inner edge by a few tenths of an AU changes the computed disk lifetime by a percent or less.
However, as I have set the inner edge of my total-particle-ensemble distribution to 0.5 AU, the

136

1000
100
10
1
0.1

1.0

10.0
R (AU)

100.0

Radial Gas Velocity (cm s −1)

Surface Density (g cm −2)

10000

50
0

Rin = 0.1 AU
Rin = 0.2 AU
Rin = 0.3 AU

−50
−100
−150
0.1

1.0

10.0
R (AU)

100.0

Figure 5.4: Surface-density and accretion-flow velocity curves for the disk gas in the fiducial disk
model varying Rin , the inner edge of the disk and simulation grid-space. Dark curves correspondto t = 0, light curves to t=0.5 Myr.
relative position of Rin may have a noticeable effect on the fraction of particles that reach large
distances, particularly for the inner-quarter source region. In this section, I test outward mixing
in both my fiducial-disk model and in the initially compact, Rd = 5 AU-disk model for cases whenRin = 0.2 and 0.3 AU. (Note that shifting Rin outward is less expensive than inward because
the disk-evolution time-step is controlled by the size and viscosity of the inner-most grid cell.)Figure 5.4 provides plots of the disk surface-density and accretion-flow velocities for R in = 0.1, 0.2
and 0.3 AU. Shifting Rin outward also increases the rate of inward-flow of the gas in the inner disk.
It seems plausible, then, that for larger Rin , the combination of a nearer sink boundary and fasterinward advection should lead to faster inward loss of particles, to some degree, and that therefore
larger Rin should correspond to somewhat less mixing to the outer disk. It is surprising, then, that
this is not precisely the case.
In Figure 5.5, I plot peak CN values as a function of Rin for the fiducial and Rd = 5 AU disks
for 20 µm and 2 mm sized grains. Each simulation run in the accretion-flow case for each source
region (except the inner-quarter region for 20 µm grains when Rd = 5 AU) shows a peak in outward
transport for Rin = 0.2 AU. Note that I plot here the CN values for consistency with the results
reported in other sections, but that this peak occurs equally if I plot the simple fraction of particles
transported to beyond 25 AU; this is not an effect of disk-mass redistribution. Note also, that thispeak-at-Rin =0.2-AU effect is dominant for generally inward-flowing populations. It is more marked

137

normalized dust/gas beyond 25 AU normalized dust/gas beyond 25 AU

Rd=20 AU;102

20 microns

102

101

101

100

100

10−1

10−1

10−2

10−2

10−3

10−3

10−4

10−4

10−50.0102

0.1

Rd=5 AU;

0.2

0.3

0.4

20 microns

102
101

100

100

10−1

10−1

10−2

10−2

10−3

10−3

10−4

10−4
0.1

0.2
Rin (AU)

0.3

0.4

2 mm
all
inner quarter
inner half
outer half

10−50.0101

10−50.0

Rd=20 AU;

0.1

Rd=5 AU;

10−50.0

0.1

0.2

0.3

0.4

0.2
Rin (AU)

0.3

0.4

2 mm

Figure 5.5: Maximum concentration beyond 25 AU as a function of Rin for the fiducial and Rd = 5
AU disk models and for 20 µm and 2 mm sized grains. Accretion-flow case: solid lines and coloredmarkers; Midplane-flow case: dashed lines and black markers. Grey line marks the fiducial setup
where Rin = 0.1 AU.
for the 2 mm grains and is evident even in the midplane-flow case for these large grains. It is only
for the dominantly outward-flowing populations (20 µm grains in the midplane-flow case) where
intuition holds and increasing Rin provides a steady, if minor, decrease in peak outward transport.
One possible explanation for these results relates to the headwind drag experienced by particles
in the inner disk. Headwind drag is strongest for the larger particle sizes (dominate over radialadvection drag) and could be lessened somewhat by shifting Rin outward (because this lowers Σgin the inner disk and also shifts azimuthal gas velocities to slightly closer to Keplerian at the inneredge of the particle-source region). Potentially, then, larger grains may be able to last slightly
longer in the inner disk and so remain available for turbulent diffusion to the outer disk. Perhaps,138

Accretion Flow Case

0.1000Rin (AU):
0.1
0.2
0.3

Midplane Flow Casea)1.0000CN (beyond 25 AU)

CN (beyond 25 AU)

1.0000all

0.0100inner
quarter
0.00100.0001

104

105
Time (years)

Time (years)

105

4
100.0

after
peak

all

0.1000inner
quarter

0.0100Rin (AU):
0.1
0.2
0.3

0.00100.0001

106

104

105
Time (years)

106

20 microns

c)
106

b)

source regions:allinner quarter
inner half
outer half

before
peak

0.1

0.2
Rin (AU)

0.3

0.4

Figure 5.6: Panels-a and -b) CN as a function of time for 20 µm grains in the fiducial disk model,
varying Rin , for the inner-quarter and all source regions - accretion-flow and midplane-flow cases,
respectively. Panel-c) Times of CN half-maximum before and after the peak for the accretion-flowcase (solid lines, colored markers) and after the peak for the midplane-flow case (dashed lines, black
markers).
then, the nearness of the innward-loss boundary in the Rin = 0.3 AU case overcomes whateveradvantage is presented by lessing this headwind drag.
This explanation is not entirely sufficient, however, to explane the results shown in Figure 5.6
that, even for the outward-flowing grains in the midplane-flow case, transport to beyond 25 AU
is also markedly faster for Rin = 0.2 AU than for the fiducial setup. Because the probability ofoutward versus inward diffusion is dependent on the gas mass available in both directions, it ispossible that preferential outward diffusion is playing a role in the speed with which grains reachthe outer disk. It is still surprising that such effects could be so strong for such a small redistribution
to the disk mass, but for Rin = 0.3 AU at least, faster inward advection does appear to dominate,allowing fewer particles to reach the outer disk, as expected.
As a summary, I plot in Figure 5.7 the peak CN curves as a function of grain-size for the139Accretion Flow Case
Rin = 0.1 AU
Rin = 0.2 AU
Rin = 0.3 AU

101
100
10−1
10−2
10−3
10−4
10−5 −2
10

10−1

100 101 102 103 104
grain size (microns)

105

normalized dust/gas beyond 25 AU

normalized dust/gas beyond 25 AU

102

102

Midplane Flow Case

101
100
10−1
10−2
10−3
10−4
10−5 −2
10

10−1

100 101 102 103 104
grain size (microns)

105

Figure 5.7: Maximum concentration beyond 25 AU as a function of grain size for the fiducial-disk
model, varying Rin , for the all and inner-quarter source regions.
different Rin cases of the fiducial-disk model. In general, Rin has a fairly minor effect on the trends
for outward mixing simulated. However, Rin = 0.2 AU seems to provide a peak in outward mixing
for large grain sizes, therefore possibly shifting the cut-off grain size for outward transport to aslightly larger value. In general, this seems to be a ”fine tuning” sort of effect, that is mostly
mysterious but should not affect the larger-scale trends reported for these simulations.

5.3

Varying the Time of the Particle Initialization

In this section, I examine outward mixing of grains initiated at different times during the evolutionof the fiducial disk model. Under the simulation setup used in Chapter 4, I essentially assumed that
all crystalline silicates and high-temperature minerals were already formed in the disk at t = 0, withno further production at later times. While this assumption makes testing over a wide parameter
set more tractable, it is flawed on a number of fronts, neglecting many of the nuances that makedisentangling the compositions of bodies in our solar system such an interesting puzzle.
* First, of course, is that the creation of high-temperature minerals may have occurred overa relatively extended period, though still likely near the beginning of the disk evolution.
Formation via condensation requires temperatures over 1000 K and so likely takes place only
within the high-temperatures generated while the disk is initially forming. However, sometheories predict the formation of crystalline silicates via annealing within disk shocks (Harker

14020 microns

101
100
10−1
10−2
10−3
10−4
10−5

105
t0 (years)

106

normalized dust/gas beyond 25 AU

normalized dust/gas beyond 25 AU

102

102
101
100

2 mm
all
inner quarter
inner half
outer half

10−1
10−2
10−3
10−4
10−5

105
t0 (years)

106

Figure 5.8: Peak outer-disk concentrations as a function of t0 for 20 µm and 2 mm sized grains in
the fiducial disk model. Solid colored lines and markers plot CN -max for the accretion-flow casesimulations, and dashed lines with black markers show the upper-limits from the midplane-flow
simulations.
& Desch, 2002), and this could conceivably occur over a longer period.
* Next, when minerals formed and when they experienced transport to the outer disk may
represent two different epochs. Some of the samples gathered from the comet 81P/Wild 2
show possible signs of minor aqueous processing (Stodolna, Jacob & Leroux, 2010), which
would imply residence on an asteroid-like body, one both large enough and warm enough tocontain liquid water. Depending on where and how such bodies may have formed and become
disrupted, this could potentially represent a significant delay between the formation of thedisk and the transport of some minerals out to the comet-forming regions.
* Finally, disk formation itself was not an instantaneous event, and by testing outward transport
starting from t > 0, one may get a sense of how a disk’s (t < 0) formation affects it’s latercomposition. This is particularly important considering that the high temperatures providing
some of the best conditions for forming crystalline silicates occur before the t = 0 of my model
disks.
Figure 5.8 shows the primary results for outward transport of grains initiated at different times,
t0 > 0, during the model-disk lifetime for grains that are 20 µm and 2 mm in size. It plots the peakin the normalized concentration beyond 25 AU (defined in §4.5.1, Equation (4.2)) as a function
141

b)

a)
Accretion−flow
case
0.06

0.04

t=0
t=5x104 yr
t=105 yrt=2x105 yr
t=5x105 yr
t=106 yr

Time (years)

normalized dust/gas beyond 25 AU

20 microns

0.08

0.02

0.00

0

5.0×105 1.0×106 1.5×106 2.0×106
Time (years)

106

105

after
peak

beforepeak

allinner quarter
inner half
outer half105
t0 (years)

106

Figure 5.9: Panel-a: Normalized concentration beyond 25 AU as a function of time for particles in
the all source region in the accretion-flow case of the fiducial disk model with different t 0 s. Panel-b:Time windows for peak outer-disk concentrations as a function of t0 - times of CN -half-maximum
both before and after the peak for the accretion-flow case (solid colored lines and markers) andhalf-maximum after the peak for the midplane-flow case (dashed lines and black markers). All
curves reference the results for 20 µm sized particles.
of t0 for the four source regions considered and for both gas-flow cases. From this, one can see
a steady fall-off in outer-disk contamination achievable for grains formed/initiated at later times.
There is more than an order-of-magnitude difference between peak contamination of 20 µm grains
formed between t = 5 × 104 and 106 years, and no transport of 2 mm grains to the outer disk after2 × 105 years.
Furthermore, the window of time during which peak outer-disk contamination occurs shrinks asa function of t0 . In Figure 5.9 panel-a, I plot the CN curves as a function of time for 20 µm grainsin the general accretion-flow case. In panel-b, I plot the times of CN half-maximum before andafter the peak as a function of t0 , bounded by a grey line indicating the times of particle initiation.
This time-window, for the accretion-flow case, narrows considerably for larger t 0 , though the wholeis shifted to slightly later times for t0 = 5 × 105 and 106 years. After-peak half-maximum for the
maximum-transport, midplane-flow case, however, presents a fairly fixed barrier in time (t ∼ 2.6
Myr for this disk model). Mid-sized dust grains do not remain in the outer disk once it has evolved(viscously accreted and thinned) past a certain state.
Finally, in Figure 5.10, I plot the CN peak values as a function of grain size for three different
t0 -simulation sets. The limited state of outward transport at late times in the fiducial disk model

142

normalized dust/gas beyond 25 AU

102
101

all
inner quarter

t=0
t=105 yrt=5x105 yr

100
10−1
10−2
10−3
10−4
10−5 −2
10−1 100 101 102 103 104
10
grain size (microns)

105

Figure 5.10: Peak CN values as a function of grain size for the inner-quarter and the all source
regions and for three different t0 initialization times.

can best be summed up here, where one sees that by t0 = 0.5 Myr the outer-disk contaminationachievable by all grains initiated between 0.5–10 AU is about the same or less than the contamination from the inner-quarter region alone for t0 = 0. Therefore, outward mixing at later times isstill possible, but must be understood to be significantly less efficient.
These results agree with Ciesla (2010), who found that at some late time more CAI-type grains
within a disk are those formed at earlier rather than later times. This is due to the greater
prevalence of CAI formation at early times, but also to the stronger outflows produced in a hotter,
expanding, young disk. In this way, comparing outward transport as a function of t 0 is similar tothe study of §4.5.4 considering transport as a function of Rd , the compactness of the disk at t = 0;
both greater source mass and outflow are in favor of greater outward mixing. The question, then,is how this might extend back in time to include the initial cloud collapse and formation of thestar-disk system. Dullemond, Apai, & Walch (2006) found in their simulations of disk formationthat heating and outflows were so extensive, they questioned primarily why not all disks aren’t
mostly crystalline, as opposed to the huge range (0–80%) of disk crystallinity observed (Watson
et al., 2009). Ciesla (2010) found the greatest outward mixing for the peak in disk mass, rather
than the very earliest times, so the true answer, as indicated by observations, lies in some happy143

medium. However, it is clear that disk formation and the earliest times likely play an importantrole in later-disk composition.

5.4

Varying the Initial Disk Mass

In this section, I consider the range in outward mixing achievable if I vary the initial mass of mymodel disk. For my fiducial model, I have used a disk with mass at t = 0 of MD,0 = 0.03M¯ .
This is a reasonably average choice in terms of both the minimum-mass Solar Nebula (lower boundof ∼ 0.01M¯ ) and in terms of the masses of disks observed around other stars. Furthermore,
my fiducial model has a disk lifetime, within the rest of the disk parameters I have chosen, thatmatches well the frequency of disks observed around stars of different ages. I have chosen sucha disk as my fiducial model both in order to produce results widely applicable to observations ofdisks around other stars and because the prevalence of stars with debris disks and of stars foundto host extra-solar planetary systems (Currie, Plavchan, & Kenyon, 2008; Cumming et al., 2008;
Sierchio et al., 2010; Wittenmyer et al., 2011) suggests a low probability that our own solar systemformed under special-case circumstances.
However, the range and variety in disk masses observed is high, even for neighboring stars ofsimilar ages (e.g., Watson et al. (2009); Dahm (2010)), and the gas mass of the early Solar Nebula
cannot truly be known at this time. Furthermore, the study by Watson et al. (2009) found nocorrelation in a census of 84 classical T Tauri stars between the observed disk crystallinity andeither the mass of the disk or the disk-star mass ratio. In order to offer aerodynamic mixing
as a viable option to explain the crystalline compositions of disks, this mechanism must meetthis non-correlation criterion, else invoke some counter-balancing mechanism, such as unknowns
in the production rate of crystalline minerals. To study outward transport as a function of M D,0 ,I have run simulations in the fiducial-disk setup and in the compact-Rd = 5AU setup for disk
masses of MD,0 = 0.01 and 0.09M¯ . For the fiducial setup, MD,0 = 0.09M¯ produces a diskthat is somewhat longer-lived than 10 Myr under the constant-viscosity processes modeled in my
disk-evolution model. However, for the purposes of characterizing dependences observable for mysimulation setup, it offers a decent range in the MD,0 parameter space.
In Figure 5.11 panel-a, I plot the peak values of normalized contaminant concentration in theouter disk as a function of MD,0 for 20 µm grains in the fiducial setup, where Rd = 20 AU and144

20 microns

107

a)

20 microns
b)

101
100
10

Time (years)

normalized dust/gas beyond 25 AU

102

−1

10−2
10−3
10−4
10−5

all
inner quarter
inner half
outer half
0.01

106

105

after
peak

before
peak

104

0.10
MD,0 (Msun)

0.01

0.10
MD,0 (Msun)

Figure 5.11: Panel-a) Peak CN values as a function of MD,0 for 20 µm particles and the four source
regions. Panel-b Times of half-maximum CN before and after the peak for the accretion-flow case(solid lines and colored markers) and after the peak for the midplane-flow case (dashed lines andblack markers).

α = 10−2 . From this plot, it is clear that outward mixing is largely independent of MD,0 . A
slight slope toward higher peak CN for higher MD,0 is noticeable in the results of the midplaneflow simulations, but this represents less than a factor of two difference between transport in the
0.01M¯ and 0.09M¯ disks. The amount of time over which the disks experience high outer-disk
contamination, however, is strongly dependent on the initial disk mass, as shown in Figure 5.11
panel-b. Initially more massive disks are longer lived, with similarly longer-lived conditions in whichouter-disk surface densities are sufficient to sustain a population of mid-sized 20 µm particles.
Furthermore, when it comes to the large particle sizes, variation in outer-disk surface densities
means there is a strong dependence of MD,0 on outward mixing. This is shown in Figure 5.12
panel-a wherein I plot peak CN values as a function of MD,0 for 2 mm sized grains. For an initially0.01 M¯ disk, these grains do not reach the outer disk at all, even under the influence of outward
advection in the midplane-flow case. Therefore, the influence of MD,0 on outward mixing is summed
up in panel-b. Here, I plot maximum CN for the all source region as a function of grain size forthe three MD,0 disk models in the fiducial setup. In general, small grains experience equal levelsof mixing independent of the initial disk mass. However, the max-CN curves diverge toward larger
grain sizes and the upper-limit in grain size that can be transported to beyond 25 AU is notably
smaller for the less massive disk than for the more massive one.
145

101
100

2 mm
a)

all
inner quarter
inner half
outer half

10−1
10−2
10−3
10

−4

10−5

0.01

0.10
MD,0 (Msun)

normalized dust/gas beyond 25 AU

normalized dust/gas beyond 25 AU

102

102
b)
101
100
10−1
10−2
10−3
10−4
10−5 −210

MD,0=1% Msun
MD,0=3% Msun
MD,0=9% Msun
10−1

100 101 102 103 104
grain size (microns)

105

Figure 5.12: Peak CN values as a function of MD,0 for 2 mm particles (panel-a) and as a functionof grain size in the all source region for different disk masses (panel-b). Accretion-flow case: solidlines and colored markers. Midplane-flow case: dashed lines and black markers.

The figures I have shown here are only for the fiducial disk setup with Rd = 20 AU, but theresults of this experiment using a more compact disk configuration with Rd = 5 AU are essentially
identical. These results suggest that a lower-mass disk with MD,0 = 0.01M¯ should be equallyconsistent with the transport of 20µm sized grains to the comet-forming regions as the fiducial-disk
model and so conform to the Stardust results equally well. This is reassuring given our uncertainties
in the mass of the Solar Nebula, though this model cannot then offer new constraints to reduce
those uncertainties. Holding the other disk-model parameters constant, a lower-mass disk offers asmaller window in time in which to form a comet enhanced in hot minerals via turbulent mixing.
However, the disk lifetime as a whole is shorter and so a lower-mass disk offers a smaller windowof time for forming most solar-system architecture.
Qualitatively, these simulation results are also a good match to the observed non-correlation
between disk crystallinity and mass for systems around other stars, though there are a number ofcaveats. Using MD,0 as a mixing measure presumes that the correlations that exist for that initialcondition will hold for disk masses measured at a later given time, despite the fact that the ratesof evolution and disk-mass loss between different-MD,0 disks may vary. In the static-temperature
viscous-disk model, at least, the rate of relative mass loss is set by the initial distribution of mass
(Rd ), rather than its sum (MD,0 ). Also, because of the Stardust results with their clear constraints

146

on grain size, I have focused on transport of hot grains to the outer disk, defined as beyond 25 AU.
However, observations of disk-composition for extra-solar disks tend to probe smaller distances,
10 AU or less. While this likely shifts the maximum grain size for which my results predict nocorrelation to slightly larger sizes, it is still not known what grain-sizes transport and mix the bulkof material in disks around other stars. My simulations suggest that turbulent mixing is consistent
with the compositions of extra-solar disks if most high-temperature material is held and transported
within the small-grain population.

5.5

Varying the α-scaling of the Disk Viscosity

The most poorly constrained parameter of the disk-evolution model is the α-parameter of the disk
viscosity, whose biggest effect is on the rate of mass transfer through the accreting disk. Assuming
the Shakura & Sunyaev (1973) model to describe the viscosity, many attempts have been made totest and to measure α through observations and modeling of disk outburst events (e.g., Pringle,
Verbunt & Wade (1986)) or through censuses of disk masses, ages, and accretion rates (Hartmann
et al., 1998; Kitamura et al., 2002). In the simplest disk-evolution models, including the one usedin this thesis, α is held spatially and temporally constant. However, some studies pairing diskevolution–similarity-solution models and disk observations suggest α should decrease toward larger
radii (Isella, Carpenter, & Sargent, 2009), while models involving episodic accretion events to
explain rapid assmebly of stellar masses necessarily require short periods of time when the effective
disk-evolution α is quite high relative to the mean (e.g., Dunham et al. (2010)). Models that attempt
to replicate disk viscosity through phenomena such as the magneto-rotational instability (MRI)(Balbus & Hawley, 1991) sometimes predict α s that seem inconsistent with observations (King,
Pringle, & Livio, 2007), but also predict phenomena such as time-variable turbulence (Reyes-Ruiz,2007), and effective α s that varying with height above the disk midplane (e.g., Flaig, Kley, &Kissmann (2010); Kretke & Lin (2010)). The dead-zone model of MRI turbulence predicts a region
near the midplane in the inner disk where the effective α drops toward zero (e.g., Hasegawa &Pudritz (2010)).
For the fiducial-disk model and the majority of simulations presented in this thesis, I have usedan α-viscosity (always) with α = 10−2 (almost always). This value produces a relatively high level of
turbulence which may or may not be compatible with grain settling observed in some disks (Miyake
147

20 microns

101
100
10−1
10−2
10−3
10−4
10−5

0.001

0.010
alpha

0.100

normalized dust/gas beyond 25 AU

normalized dust/gas beyond 25 AU

102

102
101
100

2 mm
all
inner quarter
inner half
outer half

10−1
10−2
10−3
10−4
10−5

0.001

0.010
alpha

0.100

Figure 5.13: Maximum normalized concentration beyond 25 AU as a function of α for two grain
sizes. Accretion-flow case: solid lines and color markers. Midplane-flow case: dashed lines andblack markers.

& Nakagawa, 1995; Dullemond & Dominik, 2004) and that modeled for some particle-interactiondynamics simulations (e.g.Cuzzi, Hogan, & Shariff (2008)). However, α = 10 −2 is compatible with
disk-census observations and produces a reasonable 5.5 Myr lifetime for my fiducial-disk model.
On the other hand, my particle-transport model assumes that the diffusivity of the ensemble isdirectly proportional to the disk viscosity (meaning Dp ∝ α) and in §4.5.3, I showed that varying
the magnitude of the diffusivity by less than half an order-of-magnitude produced strong variationin the outward mixing of the particle ensemble. Therefore, in this section, I test the dependance ofoutward mixing on α by running particle-transport simulations in disks with α = 10 −3 , 5 × 10−3 ,and 2 × 10−2 (and otherwise retaining MD,0 = 0.03M¯ and Rd = 20 AU, as usual).
In Figure 5.13, I plot the peak outer-disk concentration for these simulations for 20 µm and
2 mm-sized grains as a function of α. For the 20 µm grains, the dependance on alpha is small.
There is a slight increase in peak outward mixing for higher α, but it is less than a factor of two
across more than an order-of-magnitude in α for the accretion-flow case and about a factor of
two across one order-of-magnitude in the midplane-flow case. For 2 mm-sized particles, however,outward mixing depends strongly on α. In low-α disks, any particles from the inner source regions
are less likely to reach the outer disk in the accretion-flow case, and no 2 mm particles in either
gas-flow case reach the outer disk for α = 10−3 . Therefore, as seen with the initial disk mass,

148

normalized dust/gas beyond 25 AU

102
101
100
10−1
10−2
10

−3

10−4

alpha = 10−3alpha = 5x10−3alpha = 10−2alpha = 2x10−2

10−5 −2
10−1 100 101 102 103 104
10
grain size (microns)

105

Figure 5.14: Maximum normalized concentration beyond 25 AU for the all source region as afunction of grain size. Simulations run in the fiducial disk setup with MD,0 = 0.03M¯ and Rd = 20
AU, but varying α. Accretion-flow case: solid lines and color markers. Midplane-flow case: dashedlines and black markers.

varying α primarily changes the upper-limit in grain-size that can be transported to beyond 25
AU, as summarized in Figure 5.14.
α is directly proportional to the diffusivety of the particle ensemble and also to the outwardflow velocity of the gas in the midplane-flow case. That the dependence on α of general outward
mixing of well-coupled grains is so low then, suggests that these effects are largely counteracted
by the more-rapid disk evolution that also takes place in higher-α disk models. Particles wellmixed with the gas will be rapidly accreted onto the central star at the rate of that disk mass. As
with lower-mass disks, higher-α disks have shorter lifetimes and so the time-window for outer-disk
contamination is also shorter. Unlike in low-mass disks, however, the high diffusivity that comeswith high α means that outward mixing is also faster, so that the entire transport scenario shifts
to earlier times for higher-α disks. This is shown in Figure 5.15, where I plot, for 20 µm sizedparticles, both the outer-disk contamination as a function of time, and the time-window for peak
contamination as a function of α.
The primary conclusions, then, about outward mixing as a function of α are that the α-

149

107

a) Accretion−flow
case
0.06

−3

b)

alpha=10
alpha=5x10−3alpha=10−2
alpha=2x10−2

Time (years)

normalized dust/gas beyond 25 AU

20 microns
0.08

0.04

106
after
peak

105
all
inner quarter
inner half
outer half

0.02

0.00

105

106
Time (years)

107

104

0.001

0.010
alpha

beforepeak

0.100

Figure 5.15: Panel-a) Normalized concentration beyond 25 AU for 20 µm particles from the all
and inner-quarter source regions. Panel-b) Times of CN half-maximum as a function of α before
and after the peak in the accretion-flow case (solid lines and colored markers) and after the peakin the midplane-flow case (dashed lines and black markers).

dependence is small, but not non-existent. For the moment the α model for disk viscosity anddiffusivity is likely sufficient for the purposes of particle-mixing simulations — as long as diffusivity and disk evolution are considered together. However, more complete models of disk viscosity
will likely yield at least one or two interesting results from future work, perhaps particularly as
researchers come to better understand the earliest, potentially outburst-driven stages of disk formation and evolution.

5.6

The Disk Temperature Distribution

For all of my outward-mixing simulations, including those discussed in Chapter 4, I use a static,
power-law model for the disk temperature distribution that roughly corresponds to the temperature
profile of a passive, flared disk around a 5 L¯ star (see §2.4). This temperature model has only two
parameters, the temperature of the disk at 1 AU, TAU , and the slope of the disk-temperature asa function of R, qT . In this section, I explore the dependence of the outward mixing of inner-diskgrains on these two parameters.

150

5.6.1

Varying the Baseline Temperature

The primary reasons to consider varying the baseline disk temperature, TAU , are that: 1) disks
around stars of various luminosity will be heated by stellar irradiation to various base temperatures;and 2) accretional heating in the disk at the earliest times when accretion rates are high tends toraise a disk’s temperature, particularly in the inner disk (see §2.4). Varying T AU is similar tovarying the α parameter of the disk viscosity. Since, from §2.2,
ν = αΩK,mid Hg2 ,where ΩK,mid is the Keplerian angular velocity at the disk midplane and Hg is the disk-gas scaleheight:
Hg =

cs
1
=
ΩK,midΩK,mid

s

kB T
,
µmH

where cs is the sound speed in the gas and µmH is the average mass of a gas particle. Therefore,ν ∝ αTAU ,(5.1)

and it is the αTAU combined quantity (in the static, power-law–temperature disk model) that
determines the rate of viscous evolution of the disk surface density.
Using Equation (5.1) and the results of §5.5, one might then predict that outward mixing
depends on TAU in the same way it depends on α: that generally mixing of well-coupled grains ismostly independent of TAU , but that a colder, more slowly evolving disk will mix fewer large grainsto the outer disk. However, Figure 5.16, shows that this is not the case, that instead the outer-disk
contamination achievable within a particle-transport simulation is essentially independent of T AU ,no matter the particle size. TAU affects disk evolution and particle diffusivity in the same way as
α. However, disk temperature also plays an important role in the drag a particle feels against thegas flow, particularly in the Epstein-drag regime where (from §3.2.1)
FD = −mdCRρg vtherm ∆v ,3

where md and CR are the particle mass and surface-area-to-mass ratio, ρg and vtherm are the local15120 microns

101
100
10−1
10−2
10−3
10−4

all
inner quarter
inner half
outer half

10−5100

1000
TAU (K)

normalized dust/gas beyond 25 AU

normalized dust/gas beyond 25 AU

102

102

2 mm

101
100
10−1
10−2
10−3
10−4
10−5100

1000
TAU (K)

Figure 5.16: The maximum nomralized concentration beyond 25 AU as a function of the baseline
disk temperature, TAU for two particle sizes. In the fiducial disk model, TAU = 278.9 K. Accretionflow case: solid lines and color markers. Midplane-flow case: dashed lines and black markers.

gas density and thermal velocity, and ∆v is the difference between the particle and gas velocities.
Next,
vtherm = cs

r

8 √
∝ T,
π

and, in a vertically isothermal disk like that used in my model,

ρg ∝

Σg
1
∝√ .
Hg
T

Therefore, varying the disk temperature has no effect on the gas drag a particle feels in the Epsteindrag regime, so this aspect of the particle transport also remains balanced.
The balance, when varying TAU , between disk evolution, and particle advection and diffusion isshown in Figure 5.17, where I plot both the outer-disk contamination as a function of time varying
TAU and the time-windows for peak contamination as a function of TAU . Because disk evolution
is faster in a hotter disk, the function for outward mixing is also shifted to earlier times, but theshape of that function remains essentially unchanged. The timing of outward transport still varies,
but the net result is the same. This is a good illustration of why it is important to consider temperature/viscosity/diffusion and disk-evolution simultaneously. Higher disk temperatures should
produce faster outward mixing, but not necessarily more.

152

Accretion−flow case

107

a)

b)

0.06
0.05

Time (years)

normalized dust/gas beyond 25 AU

20 microns
0.07

0.04
0.03
0.02

105

TAU=139.45 K
TAU=278.9 K
TAU=557.8 K

0.01
0.00

106

105

106
Time (years)

allinner quarter
inner half
outer half
107

4
10100

1000
TAU (K)

Figure 5.17: Panel-a) Normalized concentration beyond 25 AU for 20 µm particles from the all
and inner-quarter source regions. Panel-b) Times of CN half-maximum as a function of TAU before
and after the peak in the accretion-flow case (solid lines and colored markers) and after the peakin the midplane-flow case (dashed lines and black markers).

5.6.2

Varying the Radial-slope of the Disk Temperature

The fiducial, static-temperature model uses qT = −0.5, which is appropriate for a simple passivemodel of disk temperature where the disk is heated only by intercepted radiation from the central
star and where the disk flares, increasing scale-height with distance from the star so that the disk
surface sees the starlight at the same interception angle for all disk radii (Armitage (2010), pp 45–
47). (A flat disk structure where the incidence of starlight becomes ever shallower with increasing
R has qT = −0.75, Armitage (2010), pp43–45.) Even the energy-balanced temperature model
discussed in §2.4.1 that includes accretional heating also includes a temperature-slope of q T = −0.5
in the cold outer disk where stellar irradiation still dominates the temperature profile. However, as
discussed in §2.4.1, how a disk is heated can be fairly complex and there are a number of reasons
to consider variations in qT . These include:* Dominance of different heating mechanisms: As shown in the evolving-temperature modelsof §2.4.1, the dominance of accretional heating in the early, inner disk while the outer disk
remains cold, can lead to relatively complex temperature structure. In these models, suchas the one shown in Figure 2.6, both the innermost disk and the outermost disk heated by
the ambient temperature of the parent molecular cloud have shallow temperature gradients,
153

while the transition region between accretion and radiant heating has a sharp fall-off in disk
temperature and a steep qT .
* Variations in types of disk opacity: As suggested by some of the complex temperature structure in the evolving temperature model (Figure 2.6), regional variations in disk opacity, largely
dominated by dust opacities, can strongly affect the fall-off rate of disk temperature. For themodels of Keller & Gail (2004) (assuming the flat disk-structure model normalized aroundqT = −0.75), the authors report qT ≈ −1 where ice-coated grains dominate the opacity, andqT ≈ −0.62 where bare grains do so.
* Dust settling to the midplane: Many disk observations suggest that for some disks the dustgrain population has settled toward the midplane (e.g.,Miyake & Nakagawa (1995); Furlan
et al. (2006); Tsukagoshi et al. (2011)). This is often attributed to grain growth, but could
also occur if disk turbulence dropped to relatively low levels. In either case, the lack of small
grains in a disk’s atmosphere means that the disk structure flattens (Dullemond & Dominik,

104

Disk surface density
a)
t=0

102
100
10−2
10−40.1

t=1 Myr
qT=−1
qT=−0.75
qT=−0.62
qT=−0.5 − fiducial
qT=−0.4

1.0

10.0 100.0 1000.0
R (AU)

accretion velocity (cm s −1)

surface density (g cm −2)2004) and less starlight is intercepted, providing a disk with a steep qT .

Disk accretion velocity

400
b)

200

t=0

0
t=1 Myr

−200
−400
0.1

1.0

10.0 100.0 1000.0
R (AU)

Figure 5.18: Panel-a) Surface-density profile and Panel-b) Gas radial-flow accretion velocity
for disk models varying the radial slope of the disk temperature, qT . Curves plotted for t = 0(dark curves) and t = 1 Myr (light curves). Disk models using the fiducial model setup with
MD,0 = 0.03M¯ , Rd = 20 AU, α = 10−2 , and TAU = 278.9 K.

154

To test the dependence of outward mixing in my simulations on qT , I have run particle-transport
simulations in disks with qT ranging between -1 and -0.4. One important note is that varying qT
also varies the surface-density structure of the disk. In the α model for disk viscosity with a powerlaw disk structure, qν , the power-law constant of the viscosity, goes as qν = 3/2 − qT . And, in thesteady-state approximation of disk structure, Σg ∝ 1/ν. Therefore, a steeper temperature profile
leads to a shallower surface-density profile. This is illustrated in panel-a of Figure 5.18, where I
plot the disk–surface-density profiles for these simulations at t = 0 and t = 1 Myr. In these disk
models, I have left TAU fixed to the fiducial value of 278.9 K. Therefore, the disks with steeper
qT are also generally colder and evolve more slowly than the fiducial model. This leads to bothslower outward flow at the outer disk edge, and slower main-inward accretion, as shown in panel-bof Figure 5.18.
In some ways, varying qT can be compared to varying the initial disk compactness, Rd , (discussedin §4.5.4). Both initially compact disks and disks with shallow qT (steep qν and Σg ) have more mass
distributed at small disk radii and so have faster outer-disk outflows and evolve more rapidly. Thiscan lead to a greater degree of outer-disk contamination, as shown in Figure 5.19 panels-b1 and -b2.
However, while §4.5.4 showed that outward mixing is strongly dependent on R d , Figure 5.19 showsthat it is only moderately dependent on qT . This is because qT , also has a strong affect on theprobability for a particle diffusing outward versus inward and on the relative strength of headwind
drag on a particle orbit, both of which favor outward transport in steep, rather than shallow, q T
disks.
In the fiducial disk model with qT = −0.5, the roughly Σg ∝ 1/R profile in the main, quasisteady part of the disk produces an even distribution of mass per AU. This means that for a given
diffusivity (proportional to the disk viscosity) a particle in the ensemble is equally likely to diffuse
inward as outward. However, if Σg follows a shallower profile, as with steep qT , then in the main
disk there is locally more disk mass outward and a particle is more likely to be diffused outward.
A steep qT and cold outer disk do mean that diffusion in the outer disk is slower, but the mass
distribution still strongly influences where well-mixed particles are held within the disk.
Also, if qT is steep, leading to a shallow disk–surface-density profile, then the radial-pressure
gradient is relatively more shallow as well. (Even if the surface-density profile is unchanged, colder
temperatures still lead to denser gas at the disk midplane and a shallower radial pressure gradient).
155

2 mm

20 microns
fraction beyond 25 AU

1.00001.0000a1)

0.10000.10000.01000.0100all
inner quarter
inner half
outer half

0.0010normalized dust/gas beyond 25 AU

0.0001

102
101

−1.2

−1.0

−0.8

qT

0.0010

−0.6

−0.4

−0.2

20 microns

0.0001

102

b1)

101

100

100

10−1

10−1

10−2

10−2

10−3

10−3

10−4

10−4

10−5

−1.2

−1.0

−0.8

qT

a2)

−0.6

−0.4

−0.2

10−5

−1.2

−1.0

−0.8

−1.0

−0.8

qT

−0.6

−0.4

−0.2

−0.6

−0.4

−0.2

2 mm
b2)

−1.2

qT

Figure 5.19: Panels-a1 and -a2) Maximum fraction of source particles mixed outward to beyond
25 AU as a function of qT . Panels-b1 and -b2) Maximum normalized concentration of particles
beyond 25 AU as a function of qT . Accretion-flow case: solid lines and color markers. Midplane-flow
case: dashed lines and black markers. Note that a steeper disk-temperature profile (q T ) corresponds
to a shallower disk surface-density profile.

Steep radial-pressure gradients cause disk gas to orbit more slowly than Keplerian and lead toheadwind drag, so steep qT and shallower pressure gradients means a lessening of the headwinddrag effect. This is demonstrated most clearly in panel-a2 of Figure 5.19, plotting the peak fractionof 2 mm grains transported to the outer disk. In the accretion-flow case, a larger portion of theselarger grains reach the outer disk for steep qT because their range within the disk is less restricted
due to headwind drag. This does not lead to more outer-disk contamination of 2 mm grains in thesteep qT cases, however, because shallow qT , with more mass distributed into the hot, inner disk,
still has a larger source mass with which to contaminate the outer disk.

156

20 microns − Accretion−flow case
a)

qT=−1
qT=−0.75
qT=−0.62
qT=−0.5
qT=−0.4

0.08
0.06
0.04
0.02
0.00 410105106Time (years)

107

normalized dust/gas beyond 25 AU

normalized dust/gas beyond 25 AU

0.10

102
b)

101
100
10−1
10−2
10−3
10−4
10−5 −2
10

10−1

100 101 102 103 104
grain size (microns)

105

Figure 5.20: Panel-a) Normalized concentration beyond 25 AU of 20 µm particles from the all
and inner-quarter source regions in the accretion-flow case in disks varying q T . Panel-b) Maximum normalized concentration beyond 25 AU as a function of grain size for the all source region.
Accretion-flow case: solid lines, colored markers; Midplane-flow case: dashed lines, black markers.

The results of the outward-mixing dependence on qT are summarized in Figure 5.20. General
outward mixing, particularly of small grains, is stronger for shallower disk-temperature profiles
because a hotter disk has faster outflows and disk expansion and, importantly, because more massis initially distributed toward small AU, creating a larger source mass for outer-disk contamination. However, the outward mixing of larger, more poorly coupled grains, is less dependent on q T ,because a cold outer disk has a shallower radial pressure gradient providing weaker headwind-dragconfinement of the particle ensemble. Colder disks also evolve more slowly and may retain grainsat large AU for longer periods of time. This suggests that disks with grains settled toward the disk
midplane may retain a larger mass reservoir (of both gas and dust) in the outer disk, than hotter
flared disks. However, in terms of the Stardust results, a hotter disk with a shallower q T would be
more likely to process a larger mass of grains at high temperatures and to send moderately coupled
20 µm grains out to contaminate the comet-forming regions.

5.7

Conclusions

The primary motivation for examining the aerodynamic outward mixing of dust grains in protoplanetary disks came from the results of the Stardust mission that found grains processed at hightemperatures mixed within the material of a comet formed in the cold regions beyond the orbit157

of Neptune. Another important motivation has been the large scatter and poorly defined trends
in crystallinity (the relative fraction of observed silicates processed at high temperature) observedin disks around other stars. Therefore, the primary focus of these two chapters has been to systematically tie together aerodynamic transport processes with disk evolution in order to take into
account the effect of the variable dynamic nature of disk systems on mixing-transport outcomes.
In Chapter 4, I examined the dependence of outward mixing on four parameters most directlytied to the questions of particle-transport and disk evolution. Two of the biggest unknowns tying
protoplanetary disk models to particle-transport effects are (1) the velocity structures of evolving
disks, directly important for the advection of grains, and (2) the portioning between diffusivity
and viscosity within the disk. Next, (3) the actual physical sizes of dust grains not only help
to tie transport results to the Stardust findings, for which real grain sizes are provided, but also
link directly to disk aging; an older disk will no longer support larger grain sizes as it loses massand becomes more tenuous. Finally, (4) the initial compactness of the disk is a direct example
of variation between different disk systems that also has a strong influence on how disk evolution
proceeds.
All four of the parameters explored in Chapter 4, were found to have important impacts on the
degree of outward transport and outer-disk contamination achievable within a particle-transport
simulation. This, then, begs the question of what other parameters might be equally important,both in terms of the physical properties of disks and of the particular model used to calculate thedisk structure and evolution. Therefore, in this chapter, I have extended my search of parameter
space to include other disk-model parameters of concern. My primary conclusions are these:
1. The relative mass of source material is the primary factor controlling resultant
outer-disk contamination. In terms of the parameters tested, this source-mass importanceis demonstrated relating to: the initial compactness of the disk, Rd , the timing of particle
formation/initiation, t0 , the slope of the disk temperature (and therefore initial-mass) distribution, qT , and, of course, the various possible source regions for hot grains transported
within disks.
2. However, a number of disk-model and transport parameters are as yet poorly constrained
and several of these also have a strong impact on simulated outward mixing. Therefore, to158

rank the parameters I have tested that have the strongest impact on outer-disk
contamination: The initial compactness of the disk has the strongest impact on the amountof hot material that may be mixed into the outer disk. Of next greatest importance are theuncertainly constrained diffusivity of a dust-grain population and the velocity structure of
the disk gas. The most outward mixing occurs for an initially compact disk with regionsof outward-flowing gas and high diffusivity relative to the disk viscosity. Finally, from thischapter, early-time (t0 . a few ×105 years) production of hot minerals is also key to reachinghigh crystallinity fractions. Crystalline silicates formed at small AU at late times simply will
not reach the outer disk in sufficient quantities to match observations.
3. Finally, each parameter I have tested emphasizes the importance of including
disk evolution along-side particle-transport and mixing calculations. For example,
my simulations measure faster outward mixing at t = 0 for both a more compact disk (small
Rd ) and a hotter baseline disk temperature (TAU ). However, I measure no dependence in peakouter-disk contamination on TAU . A static-disk simulation would reveal only that increase
in initial mixing rate and could lead to the conclusion that, for example, disks orbiting moreluminous stars should experience a greater degree of outward mixing, when in fact more rapid
disk evolution for higher TAU entirely counter-balances the mixing-rate effect over the lifetimeof the system.
Next, to specifically relate the parameters tested in this chapter to the results from Stardustand ongoing attempts to define our Solar Nebula, I find that:
1. The discovery of tens-of-microns crystalline silicates in comet Wild 2 material does not help
to constrain α of our Solar Nebula (presuming the α-model of disk viscosity to be well representative of protoplanetary disks). The α parameter has a low impact on the outward mixingof grains, as long as a direct correlation between disk viscosity and dust-particle diffusivity is
maintained, as the simulations presented here assume.
2. Outward mixing of 20 µm grains is compatible with a range of disk masses, down to at least
MD,0 = 0.01M¯ . Outward mixing in my simulations is largely independent of initial disk
mass. This means that one need not invoke a special case wherein our Solar System formed

159

from an unusually massive disk. However, aerodynamic mixing in this case also does not
supply constraints as to what the mass of our Solar Nebula may have been.
3. In agreement with the findings of Ciesla (2010), high temperature minerals found withincomet material are statistically more likely to have formed at the earliest times of Solar
Nebula history. Not only does formation of the highest-temperature material (such as CAIs)
require a young, hot disk, but transport of inner-disk material outward is significantly more
efficient at early times.
These points are in addition to the Chapter 4 findings that: (1) outward transport of grains inthe size-range relevant to the Stardust crystalline silicates (up to tens of microns) is feasible via
aerodynamic mixing processes, but that (2) grains formed very near the sun, such as, probably,
CAIs, likely require some outward gas flow for aerodynamic effects to explain their presence in acomet like Wild 2.
For comparison to observations outside our Solar system, the results of Chapter 4 found apotentially favorable comparison between increased outward mixing in more compact disks andthe potential disk-crystallinity–accretion-rate correlation observed by Watson et al. (2009). Disk
parameters tested in this chapter also show favorable comparisons between mixing trends and the
Watson et al. (2009) survey results. Specifically, outward mixing appears largely independent ofboth the initial mass of the disk and the baseline disk temperature, while the Watson et al. (2009)
survey found no correlations between disk crystallinity and either observed disk mass or stellar
luminosity (which sets the baseline disk temperature in a passive-disk model). As discussed bythose authors, such non-correlations are puzzeling, as both disk mass and heating by the central
star are believed to be potentially important for the production of crystalline silicates. However,
the results of my simulations suggest that the resolution of these puzzles should not require taking
basic aerodynamics-mixing effects into account.
One note, however, is that comparing my simulation peak-mixing results to these survey results
possibly may not properly account for timing effects present within the observations. The surveyed
disks are observed at 1–2 Myr of age, which generally falls during the infall stage of 20 µm andlarger grain sizes in my simulations. However, more massive disks, for example, support slower
infall and higher outer-disk concentrations at a given time than simulated less massive disks. This160

could have consequences for the relative observed crystallinities between different disk-mass cases,
though the extent of the effect is unclear as crystalline and non-crystalline grains should experience
infall roughly equally. However, my simulations do predict that the early-time outward mixing of
the hot-grain population should proceed roughly identically for disks of different masses.
Whether or not aerodynamic mixing is the primary explanation for the existence of hot minerals
in the cold outer-regions of disk systems, my simulations show that it is an important aspect determining the composition and compositional gradients found within a disk. Furthermore, throughaerodynamic mixing, variation between the initial conditions found within young disks (e.g., mass
distribution, timing and location of mineral formation) should lead to a wide range in disk compositional properties observed at later times. This prediction agrees well with studies that find wide
scatter in observed disk compositions even within disk populations believed to have formed within
relatively quiescent environments, such as those of the Taurus-Auriga young cluster (Watson et al.,2009).
I have shown that the interactions between turbulence and evolving disk structure must haveimportant consequences within a protoplanetary system. However, the precise outcome of mixing
within a given, naturally evolving disk environment remains poorly resolved, largely due to presistent uncertainties concerning disk structure and mechanics. Nevertheless, there are a numberof possibly important effects, not explored here, that should and can be pursued in future studies.
Some specific avenues of research include:* Mixing within a 2D disk structure. Mixing and transport of grains within a 2D (radial,vertical) disk structure have been explored by a number of authors (e.g., Takeuchi & Lin(2002); Keller & Gail (2004); Ciesla (2009)). However, these studies have either focused on
the bulk transport of grains, or have neglected the specifics of grain-size effects. Of particularinterest for the outward mixing of large grains is the outward “tail-wind” drag on particles
high within a flared-disk atmosphere. Large grains are expected to mostly settle-out to thedisk midplane, as discussed by Ciesla (2009). However, moderate to high turbulence (such asassumed in the α = 10−2 disk models used in my simulations) should allow a non-negligible
fraction of grains at least tens-of-microns in size to experience outward drag at high z within
the disk (§3.1). Turbulent vertical diffusion of grains should then lead to some degree of161circulation-style radial diffusion (Pavlyuchenkov & Dullemond, 2007) of grains in excess of
the radial diffusion employed within my simulations. Whether such excess outward transport
is significant remains to be tested.
* Mixing during disk formation. Work by, e.g., Tscharnuter et al. (2009), Dullemond, Apai,& Walch (2006), and Ciesla (2010), as well as simulations presented in this chapter, all suggestthat the greatest outward transport should occur early in a disk’s lifetime, possibly as the stardisk system is forming. Therefore, mixing of imperfectly coupled grains during disk formation
merits further study. While disk-formation simulations tend to be computationally expensive,
the window of interest for such a study would be on the order of 105 years, rather than theseveral 106 year disk lifetimes explored in my current simulations. Furthermore, current
theories of star formation suggest that the birth of star-disk systems may be characterized
by episodic-accretion events of high mass processing set within a more quiescent accretion
environment. High-accretion events would provide periods of stronger gas inflow, but also
higher disk temperatures and enhanced turbulent diffusion. Both the formation of crystalline
silicates and the transport of grains within such a changeable early-disk environment are prime
candidates for future study that could shed new light on dust processing with protoplanetarysystems and our own Solar Nebula.
* Mixing within new and alternative disk models. While a large focus of my work has
been the influence of disk evolution and a changing disk environment on grain transport, I
have used a single, simple model (the α-disk model) to describe that evolution, and studyof particle mixing within alternative as well as more-sophisticated disk models is warranted.
For example, Boss (2004) considered the mixing of trace particles within a marginally selfgravitating disk, and disks involving the magnetorotational instability and possible dead-zones
of low turbulence at the disk midplane are often discussed in the literature as, for example,
structures that might preserve larger grain-sizes within the disk for longer periods of time
(Jacquet, Fromang, & Gounelle, 2011). Dead-zone disk models, however, have yet to be
fully explored in terms of their particle-transport properties. Also, it is likely that our own
Solar Nebula was born within a rich star-cluster and experienced periods of strong external
photoevaporation (Throop & Bally, 2005). As low surface-density in the outer disk can drive162

outward gas flows, such episodes may have been important for the transport of Solar Systemminerals and should be explored in terms of their particle-mixing characteristics.
Limited understanding of disk structure is not the only obstacle to achieving a full understandingof the observed distribution of high-temperature grains within our own and other circumstellar
systems. Formation mechanisms for high-temperature materials must also be more clearly outlined.
Given the variety of material recovered by the Stardust sample-return mission, these processes
likely relate not only to the local temperature evolution within a disk, but also to grain-growth andpossibly even early planetesimal-formation processes. Furthermore, as will be discussed in the next
chapter, our understanding of the transport of marginally-coupled grains remains incomplete. That
mid-sized particles experience headwind-drag infall as a disk ages and thins is potentially in directconflict with mm-sized grains observed at large distances within many extrasolar protoplanetarydisks (Testi et al., 2003). Until this conflict is resolved, mixing of large grains within a disk willremain an open area of study.

163

Chapter 6

The Necessity of Local Structure andSmall-scale Concentration forPlanetesimal Formation
Several recent theories of planetesimal formation (e.g., Youdin & Shu (2002); Johansen et al. (2007))
consider the formation of large bodies directly from the collapse of ensembles of small (millimeter–
decimeter) dust grains and rocks. Some of these theories either require or are greatly aided byenhancements above solar composition in the local dust-to-gas ratio of the disk, and early calculations suggest that radial drift of small grains due to headwind drag against the disk gas might
easily supply such an enhancement (Youdin & Shu, 2002). Here, I test the radial-drift hypothesis byrunning simulations of radial particle transport within a viscously evolving gas disk, including bothadvection with the gas accretion flow and turbulent diffusion of the particle ensemble. I find for arange of disk conditions (varying the disk mass, initial disk size, and α-scaled viscosity) that the
dust-to-gas ratio throughout most of the disk remains roughly uniform for the first 40–60% of thedisk lifetime, excepting a tapering-off depletion of solids in the outer disk. Enhancements, where
they do occur, are generally insufficient to drive collapse of solids into large bodies, even considering recent evidence for possibly lower values of the critical Richardson number (Lee et al., 2010).
Therefore, enhancements necessary to drive planetesimal formation must come from local, nonaxisymmetric disk structure and solids concentration. Furthermore, previous studies have notedthat the rapid aerodynamic inspiral of large grains (∼ 1mm) fits poorly with observations of large
grains at large AU in disks around other stars (e.g., Brauer et al. (2007)). My simulations corroborate this result, and also indicate that combined advection and headwind drag lead to a rapid loss164

of large grains from the inner disk as well. In my simulations, all mm-sized grains are lost onto thecentral star within half a million years, even though current understanding of chondrule-bearing
meteorite parent bodies is that these bodies formed over a 1–3 Myr period, incorporating both
calcium-aluminum-rich inclusions (CAIs) and chondrules with at least 1 Myr difference between
their ages of formation (see, e.g., Cuzzi, Davis, & Dobrovolskis (2003)). I discuss the implicationsof these results for efforts of protoplanetary-disk modeling and the necessity of considering complex
local structure and local concentration.

6.1

Introduction

Understanding the growth evolution of the small dust population in protoplanetary disks has long
been troublesome, particularly for studies seeking to understand the formation of planetesimals,
the building-blocks of planets, tens to hundreds of kilometers in size. Early research identified two
primary problems: First, the headwind drag on particles of roughly meter size should cause thesebodies to spiral in to the parent star on time-scales of a thousand years, and therefore growth acrossthis size regime must be rapid to avoid large-scale depletion of solids from the early Solar Nebula.
Second, the settling of solids toward the disk midplane where they might rapidly form large bodies
via gravitational collapse will be stymied by the Kelvin-Helmholtz instability. This instability iscreated by the shear between a layer of solids (orbiting at Kepler velocities) and the disk gas, which
is pressure supported and orbits at sub-Kepler velocities (Weidenschilling, 1995; Youdin & Shu,
2002).
Currently, many models of collisional grain growth find that often, around centimeter–metersizes, colliding particles may not stick, or may collisionally fragment, leading to a bottleneck rather
than rapid growth across this size range (e.g.,Brauer, Henning, & Dullemond (2008); Zsom et al.
(2010)). However, recent refinements in the understanding of aerodynamics in disks have led toseveral theories of planetesimal formation via the concentration and mass collapse of small particles(millimeters–decimeters in size) into large bodies. These theories include turbulent concentration(Cuzzi, Hogan, & Shariff, 2008; Johansen et al., 2007), shearing-instability binding of local clumps
of solids (Johansen, & Youdin, 2007; Johansen, Youdin, & Mac Low, 2009; Bai & Stone, 2010a),and the precipitation of bound clumps at the disk midplane under conditions of solid over-density(Youdin & Shu, 2002). Planetesimal formation by these sorts of mechanisms is supported by165

recent dynamical studies that suggest the asteroid-belt population formed at 100–1000 km sizes(Morbidelli et al., 2009b), rather than building up incrementally, as in the traditional coagulativeaccretion picture of planetesimal formation.
While not strictly a requirement for the Cuzzi, Hogan, & Shariff (2008) model of planetesimalformation, most of the formation-via-collapse models require local enhancement of the heightaveraged dust-to-gas ratio (the local disk metallicity) above solar values in order for the clumpingof solids to overcome the dispersive forces of the gas. These enhancement requirements vary between
roughly 2–10 times standard metallicity (Youdin & Shu, 2002; Johansen, Youdin, & Mac Low, 2009;
Bai & Stone, 2010a), though either a local buildup of solids or a local depletion of gas (such asby photoevaporation, as suggested for planetesimal formation by Throop & Bally (2005)) are often
cited as equally effective. Furthermore, early calculations by Youdin & Shu (2002) suggest that
headwind-drag infall of outer-disk solids could lead to large-scale radial-concentration of solids well
above the threshold necessary to produce precipitation of bound solids at the midplane, thereby
using the headwind-drag effect to advantage. However, this early calculation was performed withina static, steady-disk model, neglecting turbulent diffusion of small solids, which might counteract
the inward sedimentation of the dust population, and neglecting any radial dependence on theinitial grain-size distribution.
In this chapter, I test the hypothesis of large-scale dust concentration via headwind-drag infall
by performing simulations of global particle transport within a viscously evolving protoplanetary
disk model. My model for particle transport includes turbulent diffusion of the particle ensemble,
as well as precisely-calculated radial velocities that account for both headwind drag and radialadvection drag within the flow of inwardly accreting disk gas. While each model run considers onlya single grain size, I combine runs for a range of particle sizes into a composite map of the global
evolving distribution of dust solids. For each grain size, I use a simplified model of grain growth toconstrain the radial extent and the timing that grown solids will appear within the model disk.
In §6.2, I outline the specific models used in these simulations, including the disk-evolution andparticle-transport models, as well as my method for combining simulations of individual grain sizes
into a composite whole. In §6.3, I show the dust distributions resulting from my simulations andin §6.4 compare those distributions to the dust-to-gas enhancements thought necessary to lead toprecipitation of large bodies. I find that dust particles in the larger size bins are lost inward onto
166the parent star increasingly rapidly, and I discuss the implications of this effect and my results on
the whole in §6.5, providing conclusions in §6.6.

6.2

Methods

Several previous studies have focused on a range of mechanisms for accomplishing large-scale enhancements of the dust-to-gas ratio that could potentially drive planetesimal formation. These
include: headwind drag in a static disk (Youdin & Shu, 2002), infall of settled solids under the
influence of collective effects (Weidenschilling, 2003; Bai & Stone, 2010a) (most important once
dust-to-gas ratios are already high), and the photoevaporative loss of disk gas at large AU (Throop& Bally, 2005) (most important at the outer disk edge or near the end of the disk lifetime), witha range in potential outcomes reported. Because the focus of this chapter is to test radial dust
concentration via aerodynamic forces within an evolving disk, I have focused my modeling effortson three primary effects:
1. Self-consistent treatment of gas drag on particles of a specified size within the evolving protoplanetary disk structure. For this, I use the same particle-transport code described in
Chapter 3 and employed in the outward-mixing studies of Chapters 4 and 5. I also use
the updated model for the disk structure and viscous evolution that includes an evolving
disk-temperature profile.
2. Diffusion of the particle ensemble. Turbulent diffusion can be important for moving grains
both radially inward and outward within a protoplanetary disk structure. In the outwardmixing simulations of the previous two chapters, the degree of particle diffusivity was foundto have an important impact on the large-scale mixing of grains. Likewise, diffusivity should
not be neglected in calculations of the global dust-grain distribution, and I have included
diffusion of the particle ensemble in the simulations of this chapter, using the random-walkmethod set up in my particle-transport model.
3. Placing constraints on grain-growth and allowing for a radial dependence to the grain-size
distribution. While grain growth in disks is believed to be quite rapid, both theoreticalmodels and observations (e.g., Birnstiel, Dullemond, & Brauer (2009); Kwon et al. (2009))
suggest that it is more rapid in the denser regions of the disk close to the central star. The
167

full treatment of coupled radial transport and grain growth is beyond the scope of this work.
However, I consider that coagulation should produce the largest grains only at small disk
radii, and that their appearance at larger radii must not occur until later than t = 0 times.
I use the cartoon grain-growth model described in §3.4 to place constraints on the initiation
timing and radial extent of the various grain sizes followed in these transport simulations.
Below I describe the specifics of the disk-evolution model used in these large-scale dust-distribution
simulations, the setup of the particle-transport model and the initial distributions of simulation
dust-grains of various sizes, and the method used to combine simulations of separate grain sizes
into a composite map of the evolving dust-mass distribution.

6.2.1

Model Disk Setup

For the work presented in this chapter, I continue to use the 1D, viscously evolving model for the
disk surface-density profile discussed in §2.3, but lift the static-temperature constraint of Chapters 4& 5. As discussed in §2.4, I can calculate an energy-balanced temperature for the disk midplanethat accounts for heating from both external illumination and energy dissipation by accretion.
However, running the model to calculate this energy-balanced temperature is computationallyexpensive. Therefore, for the particle simulations run for this chapter, I run disk models using thetime-evolving, power-law fit to the temperature distribution discussed in §2.4.2. For this fit
T (R, t) = TAU (t)

µ

R
1AU

¶qT (t),(6.1)(truncated to greater-than or equal-to the local-cloud background temperature, T cloud ) where
TAU (t) and qT (t) both follow the functional form
" µ ¶ #
t bx
x (t) = (x0 − x∞ ) exp −+ x∞ .
tx(6.2)

The x0 , x∞ , tx , and bx constants for both TAU (t) and qT (t) are chosen to match the temperature
evolution produced for disks run using an energy-balanced temperature. For the energy-balanced
models, I assume that Tcloud = 10K, L? = 5.0175L¯ (which, in the absence of accretional heating,matches the static-temperature value at 1AU used in the previous chapters), and that the starlight
strikes the disk at an angle of φ = 0.05 at all disk radii (see §2.4.1). Table 6.1 provides the168

MD,0 (M¯ )Rd (AU)αTAU,0 (K)qT,0TAU,∞ (K)qT,∞tTAU (yr)tqT (yr)

bTAU
b qT

0.01

20

10−2

0.03

5

10−2

0.03

20

2 × 10−2

0.03

20

10−2

0.03

20

5 × 10−3

0.03

40

10−2

0.09

20

10−2

458.184-0.59066
500
-0.61
500
-0.61
500
-0.61
500
-0.61
472.431-0.596439
600
-0.65

279.94-0.497683
279.94-0.497683
279.94-0.497683
279.94-0.497683
279.94-0.497683
279.94-0.497683
279.94-0.497683

1.9471788e52.2240062e5
2.3981058e5
2.7206114e5
2.2784514e5
2.6107508e53.494091e5
3.9992166e55.066708e55.8107436e55.0936365e55.8878103e5
7.2640871e58.1425523e5

1.02847
1.08899
0.966192
1.006910.917239
0.958942
0.924903
0.965539
0.906569
0.942996
1.0102
1.074811.00336
1.02683

Table 6.1: Disk parameters and power-law temperature fitting constants (Equations (6.1) & (6.2) )for the disk models used in this chapter. The t = 0 temperature is lower if there is initially muchless disk mass at small AU. The time-scaling constants, tTAU and tqT give a sense of the relative
lifetimes of the various disk models, though the lifetimes are more than an order-of-magnitude
longer than either of the scaling-constants. The parameters of the fiducial model are in bold.
values of the fitting constants used for each of the disk models run in the simulations of thischapter. While the disk surface-density evolution is very similar using either the energy-balancedor the parameterized-power-law temperature models, changes in disk temperature also correspondto changes in diffusivity for the particle ensemble. Therefore, in §6.3, I include for comparison
a simulation of particle distributions run using the fully energy-balanced temperature disk model
within the fiducial setup for MD,0 , Rd , and α.
As previously, I use t = 0 disk profiles that follow the form
Ṁ0
Σg (R, t = 0) =3πν (R)

Ã

1−

r

Rin
R

!

µ

R
exp −
Rd

¶

,

(6.3)

where, for this initial profile, I use an assumed function for the t = 0 viscosity: ν (R) = ν 0 αR.

169

With this ν (R) profile, I can constrain the initial mass of the disk as
2Ṁ0 Rd
MD,0 =
3ν0 α

Ã

1−

r

πRin
Rd

!

,

(6.4)

and define the magnitude of the t = 0 surface-density profile appropriate to my chosen initial disk
mass. In these simulations of particle (re)distribution, I consider results for a range in the diskmodel parameters. From the fiducial state (fiducial-model parameters shown in bold), I vary the
initial disk mass: MD,0 = 0.09, 0.03, and 0.01M¯ , the initial disk compactness: Rd = 5, 20, and40AU, and the alpha-scaling of the disk viscosity: α = 5 × 10−3 , 10−2 , and 2 × 10−2 .
Unlike in Chapters 4 & 5, I consider only the accretion-flow case for the radial velocity ofthe disk gas (§2.6) with gas-drag parameters set to the midplane values (as in Chapters 4 & 5).
While the flow structure of both gas and dust is most certainly complex with height above thedisk midplane (as discussed in §3.1), these choices represent the simplest with which to examine
global dust distributions within a 1D disk model. Furthmore, as I am testing the hypothesis that
headwind-drag infall of particles may trigger planetesimal formation, it is appropriate to consider
gas drag at the midplane, where headwind-drag motions are most consistently inward and where
said planetesimal formation takes place.

6.2.2

Particle Transport Setup

For the simulations of this chapter, I consider grain sizes of 0.2 µm – 2 cm (with ρ d = 1 g cm−3 ),meant to represent the general range of grain sizes produced by coagulative grain-growth processes.
Above centimeter sizes, bouncing and collisional fragmentation represent possible barriers to further
growth (Brauer, Dullemond, & Henning, 2008; Zsom et al., 2010), and so I assume 2 cm as theupper-limit grain size for tracking a global dust distribution. Here, I continue to calculate grain
trajectories based on Epstein-drag forces only, as described in Chapter 3, though there is a brief
period near t = 0 where 2 cm grains for all disk models (and 2 mm grains for the initially mostcompact disk model, Rd = 5 AU) are technically in the Stokes-drag regime (§3.2.1) for peak gas
densities (at . 1 AU in the fiducial disk model) at the disk midplane. As I am here interested in
global transport over much larger than AU scales, however, the use of Epstein drag throughout istaken as sufficiently accurate.

170

There are two primary differences between the simulations of this chapter and those of Chapters 4 & 5. The first is that here I consider motions of globally distributed dust grains, rather
than only those initiated close to the parent star. The second is that, rather than examine the
distributions of individual grain of a given size separately, here I build a simulation set from acompilation of simulations run for a a range of grain sizes: 0.2 µm–2 cm, spaced per decade in
grain size. I describe the method for building these composites of the dust-grain distribution in§6.2.3. Here, I will elaborate on the initial distribution of dust grains in these simulations as it
depends on particle size.
Both observations and models (e.g. Kwon et al. (2009); Dullemond & Dominik (2005)) support
the idea that grain growth is more rapid in the inner, denser regions of a disk, than the outer,colder, more tenuous regions. Subsequently, one would expect a time lag in the formation of grainsof a certain size as one moves out in the disk. In these dust-distribution simulations, I represent
this lag to first order by dividing the model disk up into radial zones and initiating grains in eachzone according to the cartoon model of grain-growth described in §3.4 (which predicts when grainsof a given size should appear at a given location in the disk). I run this grain-growth model at
a representative radial location for each zone for each simulated particle size from 2 µm to 2 cm;
all 0.2 µm grains are initiated at t = 0, randomly distributed throughout the entire disk gas-mass
distribution, under the assumption that 0.2 µm is the base size of the grains. I run a total of 20,000
particles for each grain-size transport simulation.
For the zone-initiated grain sizes, I consider 10 zones in the disk, with representative points
spaced logarithmically from 0.5–1000 AU. The zone boundaries are places logarithmically between
the representative points, though the inner-most zone boundary is extended to the full innerboundary of the disk grid. The number of grains initiated in each zone, is taken as 20,000 divided
by the number of zones used for that grain size. No growth to 2 µm or above is found to occur pastzones 7 or 8 (depending on the disk model used), extending out to 121 or 282 AU, respectively. 2
cm-sized grains are initiated only out as far as zones 3 or 4, extending to 4.1 or 9.6 AU. The zone
ranges used for each disk model and particle size (dictated by the grain-growth model of §3.4) are
given in Table 6.2. Naturally, the smaller grain sizes are initiated the most quickly, with 2 µm-sizedgrains initiated out to 22 AU within the first 2000 years for the fiducial-disk model, and the full
121 AU within 4000 years. Alternately, in the fiducial-disk model, 2 cm grains aren’t initiated into171

MD,0 (M¯ )Rd (AU)

α

2 µm

20 µm

0.2 mm

2 mm

2 cm

0.01
0.03
0.03
0.03
0.03
0.03
0.09

20
5
20
20
20
40
20

10−2
10−2
2 × 10−2
10−2
5 × 10−3
10−2
10−2

1–7
1–7
1–8
1–7
1–8
1–8
1–8

1–6
1–6
1–7
1–7
1–7
1–7
1–7

1–5
1–5
1–6
1–6
1–6
1–6
1–6

1–4
1–4
1–4
1–4
1–5
1–4
1–5

1–3
1–3
1–3
1–3
1–4
1–3
1–4

Table 6.2: Table of radial zones (right-hand columns) into which grains of different sizes were
initiated for the different disk models used (parameters in left-hand columns). Ten zones arelogarithmically spaced with representative points from 0.5–1000 AU. Parameters for fiducial model
are presented in bold text.
zone 3 (out to 4.1 AU) before t =42,000 years.

6.2.3

Forming Composites of Multiple Single-grain-size Simulations

Here, I outline the method used to combine a set of particle-transport simulations for different
grain sizes into a composite map of the distribution of disk solids. Because one cannot simulate
the full number of particles expected to reside within a disk, this essentially comes down to a massallocation scheme, where each simulation particle represents a large mass of disk solids. Because
the primary focus of this chapter is on changes in the dust-gas composition of the disk, whenassigning mass to simulation particles, I use a gas-equivalent mass, MGE . Gas-equivalent mass canbe compared directly to the disk-gas mass, Mg , in a given region to define the local disk metallicity,with MGE = Mg for original (solar) metallicity and MGE > Mg in regions enhanced in solids
relative to the gas. For this work, I seek regions of the disk where the enhancement is above acertain threshold (usually about a factor of 2–10)1 . I use MGE to define the enhancement factor as

E=

MGE
.
Mg

(6.5)

For this work, I assume that at t = 0, the entire mass of disk solids is in 0.2 µm particles that
The required enhancements quoted in the results of planetesimal-formation papers often fall
between about 2–10. However, the specific requirements depend on both the planetesimal-formation
model and the specific local conditions within a disk. Required enhancement factors are discussed
in more detail in §6.4.
1

172

are evenly and randomly distributed throughout the entire disk mass (grains observed in the ISM
are sub-micron in size (Mathis, Rumpl, & Nordsieck, 1977)). Radial transport of each dust size
is simulated using an ensemble of 20,000 particles. Therefore, at t = 0, all of the 0.2 µm-sizedparticles are assigned mGE = MD,0 /20, 000, where mGE is the gas-equivalent mass assigned to anindividual simulation particle, and MD,0 is the total disk mass at t = 0, defined in §6.2.1.
Next, to add in the transport simulations for the larger particle sizes (binned logarithmically,
with bin boundaries placed logarithmically between each size simulated), I consider that graingrowth occurs separately within the disk zones defined in §6.2.2. Each time new particles are
initiated into a new size bin within a given zone, I first add up the total MGE of the simulation
particles currently located within that zone. I do this for particles up to and including the grain
size that has just locally been initiated. I do not include the mass from any larger particles that
may have been transported into that zone from one at smaller disk radii, as these large-particle
interlopers are considered outside the grain-growth processes occurring within the zone of interest.
Next, I calculate the fraction of the total MGE that should be portioned into each dust-size bin,based on a power-law distribution of particle sizesdnp∝ sqds ,
dsd

(6.6)

where np is the number density of grains of a given radius sd , and qs is the power-law index of the
distribution. For the fiducial simulations, I take qs = −3.5, corresponding to the size distributionof grains measured for the ISM (Mathis, Rumpl, & Nordsieck, 1977). However, flatter grain-size
distributions are often observed in disks around other stars and interpreted as a sign of grain-growthwithin the disk (e.g., Ricci et al. (2011)), while collisional size-distributions tend toward steeper
trends (Dohnanyi, 1969), often approximated as qs = −4 (e.g., Bai & Stone (2010a)). Therefore,in §6.4, I also present composite simulations compiled with qs = −4 and -2.5 for comparison withthe fiducial qs choice.
To continue, the volume mass density of particles may be found by (Garaud, 2007)

ρp =Z smax
smin

³
´
dnp
4+qs
s
m (sd ) dsd ∝ s4+q
−
s
,
max
min
dsd

173

(6.7)

4
where ρp is the local volume density of particles and m (sd ) = πs2d ρd is the mass of a dust particle
3of radius sd . Therefore, the fraction of dust mass in a given size bin can be calculated as
³

´
4+qs
4+qs
s
−
s
bmax
bmin
ρp,sd
´,
= ³
4+qs
4+qs
ρp
smax − smin

(6.8)

where smin and smax are the lower and upper bounds, respectively, on the entire size distribution,and sbmin and sbmax are the bounds on a given size bin. I define size bins logarithmically, so that,e.g., redistributing mass between 0.2µm and 2µm size bins (sd = 0.1 and 1µm) has bin boundaries
at sd = 0.0316, 0.316, and 3.16µm. Finally, note that qs = −4 is a special case for which
ρp,sd[ln (sbmax ) − ln (sbmin )]=,ρp[ln (smax ) − ln (smin )](6.9)

distributing an equal amount of mass into any set of logarithmically defined size-bins.
Once I know the fraction of gas-equivalent mass to be portioned into each size bin, I divide
it up evenly between the particles of that size currently in the initialized zone, and allot the newmGE to each particle. Note, that because of the wide range in the number of particles in a zone
between different size bins (ranges as large as 230–6666 particles between bins) and the fact thatsuccessive size bins tend to have roughly half-an-order-of-magnitude difference in mass allotted tothem, individual simulation particles may be allotted rather different values of mGE to carry within
the transport simulation. This effect is mitigated somewhat by the fact that the larger size bins
receive a larger share of the mass but also tend to have more particles to distribute it amongst
because their initialization range has a smaller radial extent, dividing 20,000 total particles between
fewer zones.
Finally, once MGE has been allotted among all the particles and particle sizes at each time step,
I can calculate a map in time and disk radius of the local enhancement factor using Equation (6.5)³´
22
Σg,i , in each radial gridby adding up the total MGE and gas mass, Mg,i = π Ri+1/2− Ri−1/2

cell. I also combine Ei information in both radial and time bins (Ri and δt) in order to reduce thenoise of this output. I consider the enhancement, E, averaged across (usually) 3 grid cells so that

E i0 =(MGE,i + MGE,i+1 + MGE,i+2 ),(Mg,i + Mg,i+1 + Mg,i+2 )174(6.10)

6×106
Enhancement of dust/gas

2.0Time (years)

5×106
4×106
0.1

3×106
1

2×106
10

1×10

6

100

1
2
log10 R (AU)0

2 microns

20 microns

0.2 mm

6×106

6×106

5×106

5×106

5×106

4×10

6

6

4×106

3×10

6

3×106

3×106

2×10

6

2×106

2×106

1×10

6

1×106

1×106
0
4
−1

4×10

1

2

0.5

4

3

6×106

0

1.0

0.0

0
−1

0
−1

1.5

3

4

0
−1

0

1

2

3

0

1

2

3

4

Figure 6.1: Example composite map of dust/gas enhancement factor (top panel) and maps of someof the contributing size bins (bottom panels). The black contours trace the evolving gas surface
density spaced by orders of magnitude between 10−4 and 103 g cm−2 . The grey regions mask out
bins for which the particle statistics were too low to report dust/gas enhancement values.
where i0 indicates the index value of the resolution-reduced grid, and I average over (usually) 250
output time-steps so that
∆t/δt

X
1
Ei,j .
Ei (∆t) =∆t/δt(6.11)

j=1

The transport simulations for this chapter output particle positions every δt = 200 years, so thecomposite-enhancement maps in the fiducial case have a time resolution of ∆t = 5 × 10 4 years.
An example composite map of the enhancement factor is shown in the top-panel of Figure 6.1,
overlain with disk-gas surface-density contours to provide context of the gas disk evolution. The

175

bottom-panels plot example maps of the individual grain-size components of the whole. Note thatthe scalloped artifacts visible in the composite map mark the boundaries of final particle infall for
successive dust-grain sizes.
Last, while the enhancement factor is probably the more fundamental output of these simulation
composites, it can also be translated into a value for the dust-solids surface-density, Σ p , using themore general definition of the enhancement factor

Ei,j =

µ

Σp
Σg

¶

i,j

/

µ

Σp
Σg

¶

,

(6.12)¯

where (Σp /Σg )¯ = Z¯ refers to the initial (solar) dust-gas composition of the disk, the baseline
metallicity. Z¯ used in this work is taken from Lodders (2003) and is broken into two components:

Z¯ = Z0 Zrel (T ) ,(6.13)

where Z0 = 0.0149 is the total fraction of condensable material thought to be present in a solarcomposition gas, and Zrel (T ) is the fraction of that material believed to actually be condensed
based on the local gas temperature. The values of Zrel used are

Zrel = 0.3289 for T > 182K ,Zrel = 0.7129 for 182K > T > 41K ,Zrel = 1 for T < 41K ,with transitions corresponding to the condensation of water ice (the snow line) and to the condensation of methane (and ammonia) ice. Therefore, from Equations (6.12) & (6.13),

Σp,ij = Eij Σg,ij Z¯ Zrel (Tij ) .

6.3(6.14)

Results: The Global Distribution of Solids

Note that the simulations presented below were run (like the outward-mixing simulations of Chapters 4 & 5) on a 2.4 GHz Intel Core2Duo. A full set of simulations including runs with 0.2 µm – 2176

Enhancement of dust/gas
0.0
0.5
1.0
1.5
2.0

Time (years)

6×106

Energy−Balanced Temp.

Fiducial
6×106

5×106

5×106

4×106

4×106

3×106
2×106
1×106
0
−1

3×106

0.1

2×106

1
10

1×106

100

0

1
2
log10R (AU)

3

4

0
−1

0

1
2
log10R (AU)3

4

Figure 6.2: Enhancment maps for dust transport simulations within the fiducial disk model where
MD,0 = 0.03M¯ , Rd = 20 AU, and α = 10−2 . The fiducial-model setup uses a fitted–power-law
temperature distribution during the grain-transport simulations. The results of simulations run
using the energy-balanced temperature distribution are included for comparison. Both maps werecompiled using three radial-grid-points for every radial-bin and averaging 250 time-points for every∆t = 5 × 104 years time-bin. Black on the map indicates regions of ×2 enhancement or greater.
Grey indicates regions where the particle-to-gas resolution was too low to report results. Black
contours range from 10−4 –103 g cm−3 , tracing the gas-disk surface-density evolution.
cm-sized grains required roughly 2–2.5 weeks of computation time in the fiducial-disk setup. Moremassive disks required longer run times due to not only a longer disk lifetime but also to more
extended retention of grains within the denser disk-gas medium. Runs using a larger value of α for√the disk viscosity have model disks with shorter lifetimes (roughly as a function of α) but also
take shorter disk-evolution time-steps (∝ α), and are therefore more computationally expensive on
the whole.
In Figure 6.2, I present the composite–enhancement-map results for the fiducial disk model aswell as the map for the same disk model run using an energy-balanced temperature during thetransport of the grains. The two are quite similar, providing reassurance that the fitted–powerlaw temperature distribution allows for the representation of global dust transport to sufficientaccuracy within an evolving disk. In these maps, green areas represent no change from the initial177

Size−binned components
a)

2 cm
2 mm
0.2 mm20 microns
2 microns
0.2 microns
total

1.5
1.0
0.5
0.0
0.1

1.0

10.0 100.0 1000.0
R (AU)

Composite Enhancement

Enhancement Portion

2.0

2.0

Composites

to 2cm (fiducial)to 2 mmto 0.2 mm
to 20 microns
to 2 microns
0.2 microns only

1.5
1.0
0.5
0.0
0.1

1.0

10.0 100.0 1000.0
R (AU)

Figure 6.3: Measured enhancement of dust/gas for the fiducial-disk model simulations at t =1.5 Myr. Panel-a) Portions of the enhancement factor contributed by each of the grain-sizebins. Panel-b) Composite enhancements varying the largest grain size assumed for particle-size
distribution. For no grain growth (0.2 µm grains only) the dust-to-gas ratio remains everywhere
near unity. Allowing larger grain sizes also allows inward drift of the dust population.
dust-to-gas ratio, red areas represent depletion of dust relative to the gas, and black areas represent
×2 enhancement of the dust or greater.
From these plots we see that the fiducial-disk simulations do not produce large radial concentrations of dust in the main and inner disks until near the end of the disk lifetime. In §6.4, I willcompare these measured enhancement values to predicted requirements for models of planetesimalformation in more detail. Recall, however, that a common minimum requirement quoted to aid
planetesimal formation is a doubling of the dust-to-gas ratio above solar metallicity and that thecore-accretion scenario for giant-planet formation requires the formation of large cores when a substantial fraction of the disk gas yet remains. At very early times, some small grains are pulled
outward with the outwardly expanding disk edge. Afterward, one sees that relative inward driftof the grain population does occur in these simulations, as marked by the depletion of solids inthe outer regions of the disk. However, this influx of mass to the inner disk is simply offset by
concurrent loss, particularly of the large particles, onto the parent star.
The importance of the large amount of dust mass contained in large particle sizes is shown in
Figure 6.3 where, for t = 1.5 Myr, I plot in panel-a the portion of the enhancement corresponding
to each grain-size bin, and (in panel-b) the compiled enhancement-factor for grain-size distributions

178

truncated at successively smaller particle sizes. From this second plot, one can see that graingrowth to roughly 20 µm sizes results in inward drift that depletes the outer disk of dust, and thatgrowth to larger sizes subsequently depletes the inner disk of dust mass.

Fiducial

Enhancement of dust/gas

6×106
5×106
4×106
3×106

0.1

2×106

1
10

1×106

100

0
−1

6×106

0

1

2

3

t=1.5x106 years

2.5

Fiducial
Sc = 1/2
Sc = 2
Energy−balanced

2.0
1.5
1.0
0.5
0.0
0.1

1.0

10.0 100.0 1000.0
R (AU)

4

Low Diffusivity (Sc=2)

High Diffusivity (Sc=1/2)

Time (years)

5×106
4×106
3×106
2×106
1×106
0
−1

0

1
2
log10 R (AU)

3

4

0
−1

0

1

2

3

4

1.5
0.0
0.5
1.0
2.0
Enhancement of dust/gas
Figure 6.4: Enhancement maps from the fiducial simulations, as well as from simulations with
Sc = 1/2 and 2. The simulations for different Schmidt number were run with output at lower
resolution than the fiducial setup. All composite maps here were compiled using six radial-gridpoints per radial-bin and averaging 25 time-points for every δt = 5 × 10 4 years time-bin. Top-rightpanels plots the enhancement values as a function of R at t = 1.5 × 106 years for these three cases,
as well as for the energy-balanced-model simulations shown in Figure 6.2. Maps following the same
conventions as in Figure 6.2.
179

Next, in Figure 6.4, I plot a comparison between the results for the fiducial simulations, andsimulations run varying the diffusivity of the particle ensemble by a factor of 2 in either direction.
Here one finds little difference between the global dust distributions produced for the different cases.
A lower diffusivity leads to greater segregation of the particle-size distribution and emphasizes
the discreet particle sizes used for these simulations. However, a real disk should have a mostly
continuous particle-size distribution, and therefore the results for the global distribution of solids
cannot be said to be qualitatively different from the other two cases. The higher-diffusivity case

5×106
4×10

alpha = 2x10−2

6

3×106

3×106

2×106

MD,0 = 0.01 Msun

2×106

0.1

0.1

1×106

10

1×106

1

100

100

0
−1

4×106

1

0

2×10
1×10

6

3

4

0
−1

10

1

0

2

3

4

6

fiducial 6×10
model
6

6×106

4×106

4×106

5×10

Rd = 5 AU
0.1

3×106
6

2

Rd = 40 AU

1

0.1

3×106

0.1

1

1

2×106

2×106
1×106

10

100

100

100

0
−1

0

1

2

3

4

0
−1

1×10

MD,0 = 0.09 Msun

0.0 0.5 1.0 1.5 2.0
Measured Enhancement
1.0Time (years)

1

10

10

0

1

2

3

4

0
−1

0

1

2

3

4

3

4

7

9×106

alpha = 5x10−3

8×106

8×106

7×106

7×106

6×106

6×106

0.8

5×106

0.6

4×106

0.4

3×106

0.2

2×106

0.1

5×106
4×106

1

0.1

0.0
−1 0

3×106
1

10

2×106

100

1 2 3
R (AU)

4

1×106

1×106
0

0

1

2

3

4

0
−1

10
100

0

1

2

Figure 6.5: Enhancement maps for simulations run in several different disk models, separately
varying Rd (horzontal), MD,0 (vertical), and α (diagonal). Fiducial-model map (MD,0 = 0.03M¯ ,Rd = 20 AU, α = 10−2 ) shown in the center panel. Maps follow the same conventions as in
Figure 6.2.
180

allows large grains to remain in the disk slightly longer than usual (all 20µm grains are lost after
4.1 Myr rather than by 3.8 Myr as in the fiducial simulations), but again, this is insufficient tosubstantially impact the global distribution of solids. That the diffusivity plays such a small rolein the global solids distribution suggests that, on this scale, the outward boundary defined byheadwind drag for each particle size is the stronger of the two effects.
Finally, in Figure 6.5, I plot the composite enhancement maps for simulations run varying the
disk-model parameters, MD,0 , Rd , and α. Qualitatively, the patterns of evolving global enhancement are roughly constant across the different disk-model setups. The infall of the smaller grain
sizes causes depletion in the dust relative to the gas at the outer edge of the disk, while in the main
disk enhancement values remain near unity for somewhere between 40 and 60% of the total disk
lifetime. The initially more compact disk shows somewhat greater outer-disk depletion because alarger fraction of grains are processed to larger sizes at early times, while the disk using α = 5×10 −3
shows some of the enhanced segregation of the binned particle sizes typical of the lower-diffusivity
simulations. But the basic shape of dust-to-gas values within the disk as controlled by aerodynamic

2.5

t=1.5x106 years

Enhancement

2.0
1.5

fiducial
MD,0=0.01 Msun
Rd=5 AUalpha = 2x10−2
Rd=40 AUalpha=5x10−3
MD,0=0.09 Msun

1.0
0.5
0.0
0.1

1.0

10.0
100.0
R (AU)

1000.0Figure 6.6: Radial enhancement profiles at t = 1.5 Myr for the different disk models shown in
Figure 6.5. The enhancement profiles all follow the same general trend. The greatest deviation fromthat trend here is seen for the most-quickly evolving systems. In particular, the MD,0 = 0.01M¯
system is aged more than 50% of it’s lifetime at t = 1.5 Myr and so has some concentration of thesmaller grain sizes as small AU, while the Rd = 5 AU system has a larger fraction of it’s dust massprocessed at small AU to larger grain sizes and so experiences greater dust depletion in the outer
disk.

181

a)

100
10−2
10−4
10

−6

10−80.1

1.0

10.0 100.01000.0
R (AU)

Mdust per AU (ME AU−1)

Sigmadust (g cm−2)

102

t=0
t=2.5x104yr
t=5x105yr
t=2.5x106yr
t=5x106yr
t=5.5x106yr

102

b)

101
100
10−1
10−2
10−3
10−4
10−50.1

1.0

10.0 100.0 1000.0
R (AU)

Figure 6.7: Dust surface-density (a) and mass-per-radius (b) distributions for the fiducial disk
simulations.
forces appears fairly universal, and in Figure 6.6, I plot the radial cross-section of E at t = 1.5 Myr
for each of these disk models for comparison. The primary result, of these simulations then, is thata smoothly-defined, azimuthally-symmetric, evolving disk model tends to lose dust-mass smoothly
inward onto the parent star at a rate that generally keeps pace with the loss of accreting disk-gas
mass, and that variety in basic disk properties/parameters does not tend to produce special-casedisks with respect to this process.
The results presented above all look at the distribution of the dust-to-gas ratio in evolving
protoplanetary disks, which is an important parameter for certain models of planetesimal formation.
However, important to any study of planetary-system formation is simply the distribution of thesolid material itself. In Figure 6.7 panel-a, I present surface-density profiles for the dust solids inthe fiducial-disk simulations at several times. In panel-b, I translate those profiles into distributionsof solid-mass per AU. Jumps in the distributions at around a few AU correspond to the snow-line,
beyond which more material should be condensed into solids for a given disk metallicity. At t = 0,
my fiducial disk has just over 100 Earth-masses of solid material spread throughout the entire disk,
with the greatest concentration just beyond the snow line at around 5–10 AU. As the disk evolves,the solids distribution spreads outward with the disk and decreases in total quantity, to 40 M Earthat t = 5 × 105 years, and 10 MEarth at t = 2.5 Myr. Presumably, planetesimal formation at someearlier time would allow for the retention of a larger fraction of this solid material. However, my

182

t = 5x105 yr

100
10−2
10−4
10−60.1

1.0

10.0 100.0 1000.0
R (AU)

dust surface density (g cm −2)

dust surface density (g cm −2)

102

102

t = 2x106 yr
MD,0=1% Msun
Rd=5 AU
fiducial
Rd=40 AU
MD,0=9% Msun

100
10−2
10−4
10−60.1

1.0

10.0 100.0 1000.0
R (AU)

Figure 6.8: Dust surface-density profiles at two times for variations on the disk-model physicalcharacteristics. The fiducial model has MD,0 = 0.03M¯ and Rd = 20 AU.
particle-transport simulations record only the loss of material as fine grains along with the accreting
disk gas itself. Of course, different disk models must, by their nature, exhibit different Σ p profiles,
and a few of these (corresponding to different disk models of Figure 6.5) are shown in Figure 6.8.

6.4

Comparison with Required Enhancement Factors

While the enhancement factors for the dust-to-gas ratio produced by my transport simulations aremostly fairly low, there are a number of different factors affecting the enhancement required under
different models of planetesimal formation so that, in Youdin & Shu (2002) for example, very low
to no enhancement may be required for certain disk types and conditions to produce planetesimals.
In this section, I compare my simulated enhancement factors to those required by theories ofplanetesimal formation, with emphasis on the Youdin & Shu (2002) collapse/precipitation model,
which has relatively straight-forward theoretical requirements. In §6.5, I include discussion of someof the specific requirements of the streaming-instability model for planetesimal formation. For the
moment, note that there are multiple parallels between the two and that, for example, both models
require increasingly large enhancement factors to accomplish planetesimal formation in regions of
the disk that have a steep pressure gradient.
In the Youdin & Shu (2002) model, planetesimal formation is accomplished by the gravitational
collapse of solids settled out to the disk midplane. It assumes that the disk is quiescent, with no183

global turbulence to stir the particles upward, but also that, as particles settle, shear between themostly Keplerian dust layer and the more slowly rotating gas will produce Kelvin-Helmholtz instabilities that will tend to mix particles upward. A dust disk in equilibrium with Kelvin-Helmholtz
effects should then settle out only to a degree, characterized by the critical Richardson number,
Ric , where the Richardson number in general characterizes the balance between buoyancy andshearing effects across the vertical profile with Ri ∝ ρg z/ (∂ρp /∂z) (Youdin & Shu, 2002), and thecritical value is typically assumed to be Ric ≈ 1/4 by inspection of conditions for instability (seePringle & King (2007) pp144–145). Youdin & Shu (2002) point out, however, that for midplane
dust densities in this regime above a certain mass threshold, the gravity of the dust sub-disk should
dominate and lead to the collapse of solids at the midplane. The criteria for this collapse, given in
their Equation (15), isΣp,c = 2p

Ric ηδφ Rρg s (ψ) ,(6.15)

where Σp,c is the critical surface density of the solids for collapse, ηδφ relates the radial pressure
gradient in the disk and the shear between gas and dust, and s (ψ) is a correction factor accounting
for the self-gravity of the gas. In the paper, these later two variables are defined asηδφ ≡ −(∂P/∂R),2ρg RΩ2(6.16)

and
"¡
¢#
√
p
1 + ψ + 1 + 2ψs (ψ) ≡ (1 + ψ) ln− 1 + 2ψ ;
ψ
ψ ≡

4πGρg
.
Ω2K

(6.17)

Assuming a gas density of ρg ≈ Σg /2Hg , where Hg = cs R/vK in a vertically isothermal disk model,
I can convert Equation (6.15) into a critical dust-to-gas ratio:
µ

Σp
Σg

¶

c

=

p

Ric ηδφ s (ψ)

vK
.
cs

(6.18)

Finally, to report this as a required enhancement factor, I must include the values for solar (initial)

184Youdin & Shu 2002 disks
100

H
A
B
Af
Af5

a)

Required Enhancement

Required Enhancement

100

10

1

1

10
R (AU)b)10
t=0
t=2x103 yr
t=2x104 yr
t=2x105 yr
t=2x106 yr
1

100

Fiducial disk model

1

10
R (AU)

100

Figure 6.9: Required Enhancements for midplane precipitation with Ric = 1/4 for the disks ofYoudin & Shu (2002) and my fiducial-disk model.
metallicity and the fraction of solids condensed out at different disk temperatures.

Eprecip ≥p(Σp /Σg )cvK ηδφ s (ψ)= Ric,(Σp /Σg )¯
cs Z0 Zrel (T )(6.19)

where my conventions for Z0 and Zrel (T ) are defined in §6.2.3.
This collapse criterion is separate from the simpler gravitational-collapse criterion using theToomre-Q parameter, Qp < 1. From Youdin & Shu (2002),Ω2 Hp,c;πGΣp,cpHp,c ≡
Ric ηδφ Rh (ψ) ;
"¡
¢#
√
p
1 + ψ + 1 + 2ψh (ψ) =
1 + 2ψ − ψ ln
.
ψ
Qp ≈

(6.20)The enhancement criterion for Q-collapse is then

EQp ≥

p
Ric

2 η h (ψ)
vK
δφ.
πGRΣg Z0 Zrel (T )(6.21)In Figure 6.9, I plot the required enhancements calculated using Equations (6.19) & (6.21)(Ric = 1/4) for the disk models considered in Youdin & Shu (2002) (panel-a) and for my fiducial

185

modelΣAU (g cm−2 )qΣ

TAU (K)

qT

H
A
B
Af
Af5

1700
1700
1700
1700
8500

−3/2
−3/2
−3/2
−1
−1

280
170
100
170
170

−1/2
−0.63
−3/4
−0.63
−0.63

fiducial

1176

-1

278.9

−1/2

Table 6.3: Parameters for disk models used in Youdin & Shu (2002) compared to rough steady-disk
equivalents for t = 0 of my fiducial model. In general, Σg (R) = ΣAU (R/1AU)qΣ and T (R) =TAU (R/1AU)qT . ”H” designates the minimum-Solar-Nebula model of Hayashi-1981 (see Armitage(2010) pp 4–5).
disk model at several times (panel-b). Panel-a does not provide a perfect replica of the Youdin &Shu (2002) results, in part because I’ve used my conventions for Z0 and Zrel (T ), taken from Chiang& Youdin (2010) and Lodders (2003). In Table 6.3, I list the main parameters defining the different
Youdin & Shu (2002) disk models, as well as the roughly-corresponding values (at t = 0) for my
fiducial disk model (Youdin & Shu (2002) use simple power-law disk models while my models haveboth a roll-over in Σg at the inner disk edge and an exponential tail-off of the outer disk).
Figure 6.9 demonstrates several interesting features of these enhancement requirements.
* First, note that my fiducial model calls for remarkably high Eprecip in the outer disk, particularly at early times. This is due to the strong dependence of Eprecip on the dust-gas shearing
term, ηδφ . When the radial pressure gradient is steep, as near the outwardly expanding diskedge, this shearing is strong, requiring large dust-to-gas ratios to be overcome. The similar
importance of ηδφ in streaming-instability models (e.g., Bai & Stone (2010b)) is discussed in§6.5.
* Second, while the Youdin & Shu (2002) paper de-emphasizes the roles of the disk gravity,
represented by s (ψ), the time-series for Eprecip of my fiducial model, particularly in the inner
disk, demonstrates the importance of the local disk mass for meeting dust/gas-precipitation
requirements. s (ψ) (and therefore Eprecip ) becomes large, when ψ is small, which may occur
for either a low local disk surface density, or for a locally hot disk.
* Finally, the Eprecip curves for both sets of disk models emphasize the importance of the186ice lines, particularly the snow line, as places of potential local minimum in metallicity requirements for planetesimal formation. While my particle-transport model does not treat
evaporation and vapor-transport effects, the simulations of Ciesla & Cuzzi (2006) do show
an additional peak in the dust population just outside the snow line due at least partially to
local concentration of diffused and recondensed vapor. This also corresponds to a peak in thedisk opacity and heating, however, so it is not clear what the degree of advantage the region
just outside the snowline lends to planetesimal formation via gravitational precipitation.
Next, while the calculations above assume Ric = 1/4, it is not entirely clear what the proper
value should be. Lower values of Ri mean that the dust is settled more densely toward the midplane.
Therefore, lower Ric would mean that greater settling was allowed to take place, without stirring by
Kelvin-Helmholz instabilities, thereby increasing the mass of the dust sub-disk and the importanceof its gravitational influence. Lee et al. (2010) ran 3D simulations of vertically-stratified disks withwell-coupled dust particles to test for physical variables of Ric . From these simulations, they findthe onset of Kelvin-Helmholtz instability at

Ric ≈

²ρ,0
,
36

(6.22)

where ²ρ,0 is the dust-to-gas volume-density ratio at the disk midplane. This means that for many
values of disk metallicity, the critical Richardson number could potentially be much lower than
1/4, and possibly that the dust/gas requirements for planetesimal formation could be lower aswell. ²ρ,0 can be written as ²ρ,0 ≈ (Σp /Σg ) / (Hg /Hp ) and, assuming Hp follows from Hp,c inEquation-set (6.20),
Ric ≈cs (Σp /Σg )cs EZ0 Zrel (T )√√=.
36vK ηδφ Rih (ψ)
36vK ηδφ Rih (ψ)(6.23)

This describes a critical Richardson number that is specific to the midplane density produced for agiven enhancement factor, E, with a vertical dust distribution specified by a given Ri. If, for fixed
E, the dust distribution settles further toward the disk midplane, Ri decreases and Ri c given in
Equation (6.23) increases. At some point, defined by Ric = Ri = Ri0c , this effect should balanceout. One may then define a critical Richardson number based on the Lee et al. (2010) simulations

187

Youdin & Shu 2002 disks
100

H
A
B
Af
Af5

a)

Required Enhancement

Required Enhancement

100

10

1

1

10
R (AU)

Fiducial disk model
b)

10
t=0
t=2x103 yr
t=2x104 yr
t=2x105 yr
t=2x106 yr
1

100

1

10
R (AU)

100

Figure 6.10: Required Enhancements for midplane precipitation using the results of Lee et al. (2010)and Equation (6.24) to calculate Ric for the disks of Youdin & Shu (2002) and my fiducial-disk
model.
as a function of the enhancement factor alone:

Ri0c =µ

cs EZ0 Zrel (T )36vK ηδφ h (ψ)

¶2/3

.

(6.24)

Using Ri0c in Equations (6.19) & (6.21), I define alternate enhancement criteria for precipitation/collapse of solids at the midplane:0
Eprecip
≥

and
0
≥
EQ
p

vK ηδφ s (ψ)3/2
p,
6cs Z0 Zrel (T ) h (ψ)µ¶√
cs vK ηδφ h (ψ)ΩK 3/2.
6Z0 Zrel (T )πGΣg

(6.25)(6.26)

Figure 6.10 plots the required enhancement factors using these new definitions for the same disk
models considered in Figure 6.9. The result is enhancement requirements that are generally somewhat lower than those assuming Ric = 1/4 (about a factor of 2 lower at the snow-line), particularly
at early times. However, in the very-outer-disk regions where radial pressure gradients are high,Equations (6.25) & (6.26) actually produce somewhat higher enhancement requirements. The enhancements required for precipitation in the expanding outer disk are so high in under both Ri c

188

Enhancement

100.0

t = 2.5x104 yr

10.0

10.0

1.0

1.0

0.1
0.1

Enhancement

100.0

1.0

10.0 100.0 1000.0

t = 2x106 yr

100.0

10.0

1.0

1.0

100.0

1.0

10.0 100.0 1000.0

t = 5x105 yr

0.1
0.1

10.0

0.1
0.1

Enhancement

100.0

1.0

10.0 100.0 1000.0

t = 5x106 yr

0.1
0.1

1.0

10.0 100.0 1000.0
R (AU)

t = 5.5x106 yr

Requiredqs = −2.5
qs = −3.5 − fiducial
qs = −4

10.0

1.0

0.1
0.1

1.0

10.0 100.0 1000.0
R (AU)

Figure 6.11: Measured, compiled enhancement values for the fiducial-disk model at several times
and using several particle-size distributions compared to enhancement factors required for precipitation/collapse as given by Equations (6.25) & (6.26).
assumptions, however, as to be virtually unobtainable. Therefore, for the rest of this section, I
compare the dust/gas enhancements produced by my particle-transport simulations to the slightly
more forgiving requirements from Equations (6.25) & (6.26).
The first such comparison is shown in Figure 6.11 for the fiducial-disk model at several times. In

189

MD,0 = 9% Msun
100.0

Enhancement

Enhancement

100.0

10.0
5

5x10 yr
107 yr
1.0

0.1
0.1

1.0

10.0 100.0 1000.0
R (AU)

Rd = 40 AU

10.0

5x105 yr
6x106 yr

1.0

0.1
0.1

1.0

10.0 100.0 1000.0
R (AU)

Figure 6.12: Eprecip (dark lines) and measured E (light lines) for two larger disks at early and latetimes. Left side: MD,0 = 0.09M¯ , Rd = 20 AU, α = 10−2 . Right side: MD,0 = 0.03M¯ , Rd = 40
AU, α = 10−2 .
this figure, I compare enhancement factors required for planetesimal precipitation to enhancement
results from my simulations compiled using three different assumed particle-size distributions. The
fiducial distribution, with qs = −3.5, uses the particle-size distribution measured for the ISM(Mathis, Rumpl, & Nordsieck, 1977), whereas qs = −4 is often used for a collisional size distribution(e.g., Bai & Stone (2010a)), and qs = −2.5 corresponds to shallow distributions, used as indicators ofgrain-growth, measured in some protoplanetary-disk systems (Ricci et al., 2011). From Figure 6.11,
it is clear that the size distribution used to compile my simulation results has little impact on the
conclusions of the fiducial runs. Radial drift of particles does not produce enhancement sufficient
to lead to planetesimal formation via precipitation for this model disk. The exception occurs at
the very end of the disk lifetime when photoevaporative clearing of the disk leads to an outwardsweeping pressure-maximum point at large AU where ηδφ goes to zero and Eprecip drops to smallvalues.
The enhancement maps of Figure 6.5 show that the dust-to-gas distributions for the more
extended or more massive disk models are qualitatively like that of the fiducial model. However,
the enhancement criteria for these disks should differ from the fiducial case and, perhaps, be morefavorable to collapse. In Figure 6.12, I plot measured versus required enhancements for the larger(MD,0 = 0.03M¯ , Rd = 40 AU, & MD,0 = 0.09M¯ , Rd = 20 AU) disk models at two times. Whilethe collapse criteria are slightly less stringent than in the fiducial case, this is generally insufficient
to allow the simulated enhancement factors to cross the threshold for precipitated planetesimal
190

10.0

1.0

0.1
0.1

100.0

Enhancement

100.0

Enhancement

Enhancement

100.0

MD,0=1% Msun; t=2.5x106 yr

1.0

10.0 100.0 1000.0
R (AU)

Rd = 5 AU; t=3.37x10 6 yr

10.0

1.0

0.1
0.1

1.0

10.0 100.0 1000.0
R (AU)

Rd = 40 AU; t=6.87x10 6 yr

Measured E
Eprecip

10.0

1.0

0.1
0.1

1.0

10.0 100.0 1000.0
R (AU)

Figure 6.13: Eprecip versus measured E in three non-fiducial disks close to the onset of gap opening
via EUV photoevaporation. Top-left: MD,0 = 0.01M¯ , Rd = 20 AU, α = 10−2 . Top-right:
MD,0 = 0.03M¯ , Rd = 5 AU, α = 10−2 . Bottom: MD,0 = 0.03M¯ , Rd = 40 AU, α = 10−2 .
formation. At early times, measured E is still close to unity, and near the end of the disk lifetime
both Eprecip and measured E show similar behavior to the fiducial case. Once again, it is justbefore and during disk clearing by photoevaporation that precipitation criteria are the closest to
being met, with equally strong potential for this to happen in the smaller-disk cases, as shown in
Figure 6.13. In fact, near the end of the disk lifetime, the smaller disks have a slight advantage inthat the more rapid loss of disk gas allows for infall of the smaller dust-grain population.
Of course, it can be difficult to rapidly pile-up large grains via radial drift when most of thosegrains originate at small AU. The observations of Kwon et al. (2009) of a collapsing cloud suggestthat grain-growth may be very rapid indeed, and so the restrictions I have placed on the extent
and timing of the large particle sizes may be too conservative. Furthermore, observations of protoplanetary disks tell us that large grains do exist at large AU (Testi et al., 2003). Therefore,

191

6×106

0.2 micron − 2 cm grains

6×106

0.2 − 20 micron grainsa)Time (years)

5×10

b)

6

5×10

6

4×106

4×106

3×106

3×106

2×106
1×10

1

10

6

0
−1

2×106

1

1×10

100

0

6

10
100

1
2
log10R (AU)

0.0

3

4

0
−1

0

1
2
log10R (AU)

3

4

1.5
0.5
1.0
2.0
Enhancement of dust/gas

Figure 6.14: Enhancement maps for the case when all particles are initiated at t = 0 randomly
distributed throughout the entire t = 0 disk-gas mass distribution. Panel-a) composite assuming
the entire 0.2 µm – 2 cm grain-size set. Panel-b) composite using grains only up to 20 µm in size.
Maps follow the same conventions as in Figure 6.2.
I next present a simulation set in which all grain sizes were initiated at t = 0 and entirely randomly (evenly) distributed throughout the mass of the t = 0 gas disk so that the initial grain-size
distribution is entirely independent of R.
In Figure 6.14, I plot composite enhancement maps from this t0 = 0 simulation for two cases:
one in which the particle-size distribution extends from 0.2 µm to 2 cm, and one in which only thesmaller 0.2–20 µm grains are included. Because qs = −3.5 distributes more mass into larger sizes
than small, in the first case, most of the dust-mass in the disk rapidly falls inward to small AU. It
is thus quickly lost onto the parent star, but not before creating a period if distinct enhancementin the dust-to-gas ratio in the inner-main disk. This enhancement is plotted in Figure 6.15 panel-a
relative to Eprecip . While the enhancement peaks around ×5 at t ∼ 4×104 yr, this is still insufficient
to cross the threshold of Eprecip in the fiducial-disk model. The infall of smaller particle sizes canalso be seen at larger disk radius in the t = 4 × 104 year panel, but the net enhancement in thedisk still declines swiftly after the t = 4 × 104 year peak, and by t = 5 × 105 years the metallicity
across the whole disk is at a fraction its initial value.
192

t0=0 full set
0.2 microns − 2 cm

t=4x104 yra)Enhancement

Enhancement

100.0

t=0 R−independent grain−size
distribution
100.0
t=5x105 yr

10.0

1.0

0.1
0.1

1.0

10.0
measured E
Eprecip

1.0

0.1
0.1

10.0 100.0 1000.0
R (AU)

b)1.0

10.0 100.0 1000.0
R (AU)

Figure 6.15: Eprecip versus measured E at two times for the case when all particles were initiated at
t = 0 equally distributed throughout the gas disk. The peak in measured E (occurring at t = 4×10 4
years) is shown using a composite made with only 25 time-outputs per time-bin with ∆t = 5 × 10 3
years.
t0=0 sub−set

0.2 − 20 microns only
100.0
t=5x106 yr

100.0

Enhancement

Enhancement

t=2x106 yr

10.0

1.0

0.1
0.1

1.0

10.0 100.0 1000.0
R (AU)

10.0

measured E
Eprecip

1.0

0.1
0.1

1.0

10.0 100.0 1000.0
R (AU)

Figure 6.16: Eprecip versus measured E at two times for the case when t0 = 0 for all grain sizes,
having a radially independent grain-size distribution at t = 0. Composite made using grains only
up to 20 µm in size.
For the smaller-maximum–grain-size t0 =0 case shown in the second panel of Figure 6.14, substantial dust mass remains in the disk for a longer period of time and is not concentrated toward
small disk radii until the disk is tenuous enough for 20 µm sized grains to experience substantial
infall. Again, this infall does cause a distinct increase in the inner-disk metallicity, but again, as
shown in Figure 6.16, it is insufficient to reach the requirements of Eprecip . At later times, moredust mass in small particle sizes remains in the disk, but peak enhancements toward the end of thedisk lifetime of Emax < 10 remain below Eprecip . Therefore, the potential of dust in these t0 = 0

193

a)

102
10

0

10−2
10

−4

10−6
10−80.1

10.0

dust / gas(t=0)

surface density (g cm −2)

Headwind drag only

104

1.0

b)

1.0

0.1
0.1

10.0 100.01000.0
R (AU)

t=2.5x104 yr
t=5x105 yr
t=2x106 yr
t=5x106 yr
t=5.5x106 yr

1.0

10.0 100.01000.0
R (AU)

Figure 6.17: Dust distributions for the case where particles fall in due to headwind drag only,
without radial gas-drag advection. Panel-a) Surface-density profiles of the dust (thick lines) andgas (pale, thin lines). Panel-b) Enhancement of dust relative to the t = 0 dust distribution.
Simulation includes only grain sizes 0.2µm–2 mm.
simulations to reach Eprecip concentrations are qualitatively the same as in the fiducial simulations
with large grain sizes confined to smaller disk radii at t0 .
Finally, in my particle-transport simulations, the majority of dust mass is lost onto the parent
star via simple advection within the accreting disk gas. In order to examine whether the patternof this accretion inflow has an important impact on the relative distribution of grains, particularlyon the rapid loss of the large grains, I present in Figure 6.17 a simulation in which particles were
initiated in the standard way based on their grain size, but were then allowed to undergo inward drift
based only on the difference between the dust and gas azimuthal velocities (just due to headwind
drag by itself). The gas disk is evolved as normal so the dust distribution diffuses outward as thedisk expands, as shown in panel-a. However, because the gas disk is rapidly losing mass while thedust disk is not, the standard measure of E, of course, reaches values of tens and even hundreds.
Therefore, in panel-b, I do not plot the standard E measure. Instead I plot the distribution ofdust mass relative to its t = 0 distribution. This produces extreme enhancement in the outer disk
due to the outward diffusion relative to the t = 0 distribution (though not relative to the evolving
gas distribution), and the mid-disk does show inward drift of the dust. However, this inward drift
does not correspond to a relative increase in solids at small AU. The effects of headwind dragalone appear sufficient to preferentially remove mass from the inner disk. Radial advection only
normalizes the dust distribution to the gas-disk evolution.

194t=7.5x104 yr

Enhancement

100.0

10.0

1.0
Measured E
Eprecip
0.1
0.1

1.0

10.0
100.0
R (AU)

1000.0Figure 6.18: Eprecip versus measured E at t = 7.5×104 years for a simulation in the MD,0 = 0.09M¯
disk model in which the 2 cm-sized particles were barred from reaching the inner disk edge.

6.5

Discussion and mm-size Grains

In all of my simulations, it is the rapid loss of the largest grain sizes that stymies significant
enhancement of the dust-to-gas ratio. Therefore, as in the mixing simulations of Chapters 4 & 5,
the transport processes for large grains at small AU have an important impact on the dust-solids
population as a whole. In Figure 6.18, I plot a simulated enhancement distribution at t = 7.5 × 10 4
years in the MD,0 = 0.09M¯ disk for a case wherein the 2 cm-sized grains are given an artificial
barrier to inward drift at small AU. In less that 105 years, they pile up so much mass in the innerdisk as to cross the precipitation threshold of Eprecip > ×10. Without such a barrier, however,mm–cm sized grains are rapidly lost from the disk, as demonstrated in the fiducial simulations.
This rapid loss of large grains presents problems on two fronts: that of observations of disks aroundother stars, and that of theories of planetesimal formation.
In terms of the observations, numerous observational studies of disks around other stars findevidence for rapid grain growth within disks (e.g., Rodmann et al. (2006); Kessler-Silacci et al.
(2006); Kwon et al. (2009); Ricci et al. (2010a)). Furthermore, observations in the (sub-)mm bands
see evidence for mm-sized grains, not just within disks, but out to 100 AU or farther within thosedisks (Testi et al., 2003; Ricci et al., 2010b). The cartoon grain-growth model I use to constrain
195the radial extent of simulation particles does not allow mm-sized grains to be initiated beyond 22
AU an any of the disk models I’ve run, and the inward flow from both accretion and headwind
drag certainly precludes any simulated mm grains from reaching 100 AU. Furthermore, the mixing
simulations run for Chapters 4 & 5 include simulations of particles experiencing outward-flowing
gas at the disk midplane, but less than 6% of mm-sized grains in that gas-flow case reach even 25
AU, and then only for a brief period of time, before the disk experiences substantial accretional
evolution and thinning.
That headwind drag tends to bar large grains from the outer disk, in contradiction with observations, has been noted and studied by several authors (e.g.,Weidenschilling (1977a); Takeuchi &Lin (2005); Brauer et al. (2007)). Brauer et al. (2007) specifically focus on attempting to solve theradial-drift problem by including more and more detailed physics into the transport calculations,
including dust settling and collective effects, and angular momentum exchange between a dust
sub-layer and the main gas disk. The authors find these effects in general insufficient to slow radial
drift of mm-sized grains to velocities compatible with their observed persistence in disks that are
generally 1 Myr old or older. Brauer et al. (2007) relate the problem of the survival of mm-sizedgrains in the outer disk to the problem of the survival of m-sized grains in the inner-main disk.
However, one point in this puzzel that my global-dust-distribution simulations emphasize is that
aerodynamic effects not only keep mm-sized grains from the outer disk, but lose them rapidly fromthe main and inner disks as well.
It is in the main and inner disks that loss of large grains is problematic for theories of planetesimal formation. In streaming-instability models of planetesimal formation, clumping and collapseof particle ensembles is not only strongly dependent on above-solar disk metallicities (Johansen,
Youdin, & Mac Low, 2009), but occurs specifically for particles that are marginally coupled to thegas motions (Johansen et al., 2007; Johansen, & Youdin, 2007; Bai & Stone, 2010a). This means
particles with normalized stopping times, τs , preferentially near 1 (though Bai & Stone (2010a)
report that grains as small as τs ∼ 10−2 still participate in the streaming instability), requiring
at least cm-sized grains at around 1 AU. Furthermore, chondrules (∼mm in size) make up a large
fraction of most meteorites (Cuzzi, Davis, & Dobrovolskis, 2003), implying that planetesimals in
the inner-main disk of the Solar Nebula were largely built from mid-to-large sized grains (see, e.g.,
Cuzzi, Hogan, & Bottke (2010) for a discussion). Therefore, planetesimal formation models using196

6×10620 microns

6×106Time (years)

5×106

20 microns 5×10

0.2 mm
2 mm

4×106

6

2 mm

3×106
2 cm

2 cm

2×106

2×106

20 cm

20 cm

1×106
2 m

0
−1

0.2 mm

4×106

3×106

1×106

0.2 mm02 m
12
log10 R (AU)

0.0

0
−1

4

3

0

1
2
log10R (AU)

3

4

1.5
0.5
1.0
2.0
Enhancement of dust/gas

Figure 6.19: τs = 1 grain-size contours (0.2 µm–2 m) overlying two maps of dust distribution for
two individual grain-size components. Pink contour highlights the contour matching the mapped
dust distribution. Component distributions also shown in Figure 6.1 and following the conventionsof that figure.

such grain sizes may be well justified. Of course, τs ∼ 1 requires smaller grain sizes in the outer,more tenuous parts of a disk. In the Epstein-drag regime,τs = ΩK tstop =s d ρd Ω K,ρg vtherm(6.27)

where sd and ρd are the dust-grain radius and internal density, ρg is the local gas density, andpvtherm is the thermal velocity of the gas particles. However, vtherm = cs 8/π, where cs is the sound¡¢ √speed, and, in a vertically-isothermal disk, ρg = Σg exp −z 2 /2Hg2 / 2πHg , where Hg = cs /ΩK .
Therefore, at the midplane of a vertically-isothermal disk, in the Epstein-drag regime,τs =πsd ρd.
2Σg(6.28)In Figure 6.19, I plot the grain-size contours matching τs = 1 in my evolving, fiducial disk model,
overlying the dust-enhancement distributions of both 20 µm and 0.2 mm-sized grains in my fiducial

197

simulations. From this, one can see that even in the outer disk, grains of all sizes are confined
inward of their τs = 1 contours by at least one and a half orders-of-magnitude, leaving very few
grains within the τs & 10−2 bounds for participation in the streaming instability. This is because
the headwind-drag barrier also operates most strongly near τs = 1. This grain-distribution behavior
is duplicated with minor variation in the Sc = 2 and Sc = 1/2 simulations. Grains of a given sizeare entirely confined within τs ≤ 10−1 , and enhanced final infall of the population occurs whenlocal τs contours at the outer edge of the distribution begin to evolve inward within the disk.
As with planetesimal formation via precipitation, simulations suggest that the streaminginstabillity method is also largely confined to certain regions of a disk by the local value of theradial pressure gradient. In Bai & Stone (2010b), the authors report simulations of particle clumping via the streaming-instability in which they vary their pressure-gradient term, Π η , between 0.025,
0.05, and 0.1. They find clumping requires τs & 10−2 for most particles and super-solar metallicity
when Πη = 0.05, and a monotonic increase in required metallicity for increasing Π η . Bai & Stone(2010b) define Πη ≡ ηδφ vK /cs , where ηδφ = (vK − vφ,g ) /vK . Therefore,Πη =(vK − vφ,g ).
cs(6.29)In Figure 6.20 panel-a, I plot Πη contours in red overlying the τs = 1 grain-size contours forthe fiducial-disk model. The spatial relationship between the Πη and τs parameter constraints
suggests, then, that the streaming-instabillity model mostly confines planetesimal formation fromthis mechanism to the inner-main regions of the disk, where cm-sized particles are required to allow
for marginal-coupling between the solids and the disk gas. This is doubly the case when paired withmy global-transport simulations suggesting that the outer disk will tend to be at least somewhat
depleted in solids relative to the gas, though again, super-solar metallicities and cm-sized grains,which my simulations are unable to supply simultaneously, are required as well.
The Bai & Stone (2010b) simulations are actually run considering a minimum-mass-style Solar
Nebula disk, with a steeper Σg profile and therefore steeper radial pressure gradient than that of myfiducial disk model. However, observations suggest that some disks have experienced grain-settlingand flattening (Miyake & Nakagawa, 1995), which will tend to rapidly cool a disk toward larger
radii. As discussed briefly in §5.6.2, this effect may produce shallower surface-density profiles and198

6×106a) fiducial model (q T=−1/2)

5×106

Time (years)

0.2 mm

4×10

6

0.25

3×106
0.1
2 cm

2×106

0.05
0.025

1×106

2 m

0
−1

8×106

20 cm

0

1
2
log10R (AU)34b) qT=−3/4

0.2 mm
2 mm

6×106Time (years)

2 mm

0.05

0.1

4×106
0.025

2×106

2 cm

20 cm
2 m

0
−1

0

1
2
log10R (AU)3

4

Figure 6.20: Πη contours (red) overlying τs = 1 grain-size contours (black) for the fiducial-disk
model and an evolving-disk model having a steeper temperature gradient and therefore a shallower
surface-density profile (see §5.6.2). Πη contours are for values of 0.025, 0.05, 0.1, 0.25, 0.5, and 1.
τs = 1 grain-size contours are spaced in decades 0.2 µm–2 m. Note that the second disk is colder
and therefore longer lived.

199

radial-pressure gradients within a disk, and I plot the Πη and τs grain-size contours for such a diskin panel-b of Figure 6.20. From this, one sees that while a steeper temperature gradient and colder
outer disk does expand the region of low Πη more favorable to streaming-instabillity planetesimalformation, it also pushes the τs = 1 grain-size contours outward so that large particles are likely stillrequired, even at larger distances, for the streaming-instabillity mechanism to operate. Because
headwind drag inspiral is weaker for shallower pressure gradients, my mixing simulations of §5.6.2
suggest that the dust population of a given grain size might not be confined so far inward of it’sτs = 1 contour in such a disk as in the fiducial-disk model. However, inspiral and loss of large
grains is likely to still be problematic, even in a shallow Σg -profile disk model.
Observations of disks around other stars suggest that growth of disk solids to at least tens-ofmicrons size is fairly rapid (signs of grain growth are observed for essentially all disks) (Kwon etal., 2009; Ricci et al., 2010a), and also that disks tend to maintain a small-dust population over alarge fraction of their lifetime (Bouwman et al., 2003; Dominik & Dullemond, 2008). Furthermore,
the fraction of stars observed to have circumstellar debris disks (with scattered-light levels arounda thousand times brighter than that of our own zodiacal dust and Kuiper belt) peaks at around
50% for 20 Myr old B and A stars (Currie, Plavchan, & Kenyon, 2008). These facts lend credence
to the idea that planetesimal formation is a fairly ubiquitous process that can form large bodies
early on and over a decent range of disk radii; collisional fragmentation of large bodies orbiting
within a disk is one of the most efficient mechanisms for maintaining a small-dust population.
However, my global-dust-transport simulations suggest that aerodynamic transport of dust
grains within a smooth, evolving disk model will equally ubiquitously lead to large-scale inspiral and loss of the dust population, directly counter to both observations and current theories ofplanetesimal formation. It is unlikely that aerodynamic processes of small grains within a tenuous fluid are so poorly understood as to have falsely represented this general trend in behavior
for the decades of disk study. Furthermore, the general 1D, fluid-dynamic picture of protoplanetary disks that is most often represented in models appears to be a reasonably good fit between
known disk-population properties, such as disk sizes, lifetimes, and accretion rates. Even the
mixing simulations I presented in Chapters 4 & 5 are qualitatively consistent with observed disk
compositions. Therefore, I propose that a 1D protoplanetary disk model is simply insufficient torepresent dust-gas interactions at the global-distribution and planetesimal-building level. Likely,
200

non-axisymmetric and local structures play an important role in providing locations where dust
can concentrate relative to the gas, and it is possible, furthermore, that such structures may be
integral to understanding the large-scale distribution and transport of large dust grains.

6.6

Conclusions

In this chapter, I have presented simulations of the global (re)distribution of a protoplanetarydisk dust population that include evolution of the gas-disk surface-density profile, aerodynamic
advection and diffusion of the dust-particle ensemble within the gas, and simulated grain-growthconstraints confining the appearance of the largest grain sizes to the inner and main regions of the
disk. I find that the global distribution of solids within a 1D evolving disk model follows a fairly
uniform evolutionary pattern across a range of disk-model parameter space for simulations varying
the initial-disk mass, MD,0 , the initial-disk compactness, Rd , the α-scaling of the disk viscosity,and the relative diffusivity of the particle ensemble, Sc. Specifically,* Large, mm–cm-sized particles are lost rapidly onto the the parent star (within 0.5 Myr).
* Growth of dust grains up to microns size and larger leads to the depletion of dust relative tothe gas toward the outer regions of the disk (beyond ∼100 AU).
* And, the dust-to-gas ratio within the main and inner disks remains near solar (near it’s initialvalue) for at least the first 40–60% of the disk lifetime.
In general, grains of a given size are confined to regions of the disk interior to at least τ s ≤ 10−1 andexperience accelerated infall as τs (∝ Σg ) contours near the outer boundary of that dust-population
distribution evolve inward.
In Youdin & Shu (2002), the authors present calculations of dust-to-gas enhancement factors
resulting from radial drift of dust grains within a static disk, finding a steady increase in dust-togas ratios at small disk radii with time. Such increases are not seen in the simulations presented
in this chapter for two important reasons. First, the inclusion of gas-disk accretional evolution
adds a bulk inward flow to both the gas and dust within the disk that is absent from the Youdin& Shu (2002) calculations. Also, as disk surface-density contours evolve inward, the infall of thedust grains is accelerated beyond the simple accretion+headwind-drag loss rates. Second, the201

simulations I present include loss of gas and dust past an inner-disk boundary set at R = 0.1 AU.
The Youdin & Shu (2002) calculations extend the disk all the way to R = 0. Without loss of dust
mass, their calculations must produce dust enhancements at small R.
Currently, local enhancement of the dust-to-gas surface-density ratio above solar metallicity is
considered an important component of most active theories of planetesimal formation. Electrostatic and chemical forces appear insufficient to build bodies via sticking of grains above roughly
centimeter sizes. However, in order for ensembles of small grains to collapse into large bodies gravitationally, the local dust density must exceed the Roche density, allowing collapsing solids to resist
tidal disruption due to stellar gravity. This Roche density is commonly 2–3 orders of magnitude
greater than typical midplane densities (Chiang & Youdin, 2010). While significant enhancement of
the dust density may be achieved through settling of the dust toward the midplane and turbulentconcentration mechanisms such as the streaming instability, these mechanisms have generally beenfound insufficient to reach Roche density within a disk of Solar metallicity. Therefore, some increasein the dust-to-gas surface-density ratios within the disk is also required.
To consider specific proposed planetesimal-formation scenarios: the results of my simulations
suggest that radial drift of particles within an azimuthally-symmetric, evolving disk scenario cannot
produce enhancements in the dust-to-gas ratio in the inner disk sufficient to meet requirements forlarge-scale collapse and planetesimal formation via the Youdin & Shu (2002)-precipitation mechanism, particularly not early enough in the disk lifetime to account for the formation of giant-planets
via the core-accretion scenario. The precipitation criteria depend on the local disk mass, temperature, and radial pressure gradient, and are most favorable to collapse in disks that are substantiallymore-massive and colder than a typical Solar Nebula analog. Planetesimal formation via particle
clumping and the streaming instability is less dependent on disk mass, but still requires aboveSolar local metallicites, as well as a substantial population of mm–cm-sized grains. My simulations
suggest that a 1D model of disk evolution and grain transport cannot provide substantial dust
enhancements and a large-grain population simultaneously, owing to the short lifetime of the later
within the disk.
If large-scale radial-drift concentration of solids within a disk is non-viable, it then seems likelythat local structure and local concentration of dust grains play an important role in providing theconditions necessary for planetesimal formation. Local disk structures that could prove important202

for dust concentration include, e.g., disk-opacity transitions, such as at the snow-line, spiral arms
produced within gravitaionally unstable disks, and vortices within the flow of disk gas. Also,
some simulations of magnetorotationally unstable disks suggest that MRI turbulence produces
local structure sufficiently long lived to concentrate dust particles (Johansen, Youdin, & Klahr,2009).
However, understanding the global dust distribution is important not only for identifying potential sites of enhanced planetesimal formation. The simulations I present are run within 1D disk
models assuming azimuthal symmetry and a smooth description of the protoplanetary gas disk.
Such models are qualitatively compatible with observational descriptions of global disk structureand evolution, as well as (demonstrated in my mixing simulations of Chapters 4 & 5) with trends between disk compositions and evolutionary parameters. However, as in previous studies of headwind
drag and inward grain migration, these simulations are in direct conflict with observations of large
grains at large distances in disks around other stars. Therefore, I conclude that a one-dimensional
picture of disks and grain-transport is insufficient to represent dust transport and distribution ona global scale, particularly in the regime of marginally-coupled particles. Non-axisymmetric disk
structure then also likely plays an important role in the long-distance transport of large dust grains,and should be studied with the radial transport/distribution of grains in mind.
Studies of global dust transport as a function of local structure may prove challenging. Numerically, local strucure is often preferentially captured within small-scale shearing-box-style simulations
often poorly compatible with the inclusion of disk evolution and accretional flows (which I haveshown are centrally important to understanding long-distance grain transport). However, advancesin computing resources and techniques continue to provide ever-richer details concerning disk-flow
dynamics and likely will do so on this front as well.
As with the mixing of dust species, future studies of large-scale grain distributions should also
include dust transport due to height-dependent grain migration and transport within alternative
disk-gas models, such as, e.g., disks that are flattened and colder in their outer regions, and disks
experiencing prolonged or episodic infall of material from their parent molecular clouds. However,
future disk observations will likely play a key role in improving our understanding of protoplanetarydisk physics. Three areas where further, better observations could prove invaluable include:203

1. Better spatial resolution and more samples of the youngest and forming disks.
Because age-dating evidence suggests that CAIs and iron-core meteorite parent bodies formed
around the same — very early — time in Solar Nebula history, the properties of disks as they
are forming are likely important for the mixing and distribution of high-temperature material,
the earliest growth of solids to larger sizes, and the first-stage opportunities of planetesimalformation. A spatially resolved, clear picture of the gas and dust distribution, as well as thedust size-distribution, in the youngest disks could provide great insight into disk properties
relevant to solids processing. Unfortunately, these early times in a disk history are only asmall fraction of the disk lifetime, and so such disks are rarer and often observered at greater
distances. Also, disks that are first forming are still embedded within their parent molecular
cloud and so obscured at many wavelengths (Eisner et al., 2005). However, the Atacama
Large Millimeter Array (ALMA) should provide great sensitivity and spatial resolution at
wavelengths important for the study protoplanetary disks, and the selected study of trace gas
species allows observers to probe some of the densest regions of molecular clouds. Therefore,
new and greater observations of the youngest stages of disk formation are hopefully on thehorizon.
2. Stronger constraints on the large-scale radial surface-density structure of thegas within a disk. While headwind-drag (apparently wrongly) predicts the rapid loss oflarge, mm-sized grains from the outer regions of a disk, that prediction is based on certain
assumptions concerning the disk gas distributions at those large distances. Alternative gas
distributions providing a much shallower radial pressure gradient in the outer disk would be
more likely to retain large grains, and, for example, a scenio of lower disk viscosity leading
to slower accretion and mass pile-up at large distances might also be more condusive to theformation of such large grains. Therefore, while the dust population provides most of the
optical depth within a disk and is therefore easier to probe than disk gas, more examples andbetter statistics about gas-mass distributions out to large distances would prove most useful.
Such information could either confirm the present conundrum of current dust-distribution
observations, or else allow us to rethink our understanding of disk structure in ways that are
more compatible with those observations.

204

3. Information about the turbulent velocities/properties of protoplanetary disk gas.
For decades, the viscosity driving disk evolution has been believed to be derived from turbulent motions within the disk gas (Shakura & Sunyaev, 1973). However, the source and natureof that turbulence, particularly within relatively small, cold protoplanetary disks, have remained elusive, and observations have only recently just reached the threshold of possibly
detecting turbulence within disks (M. Hughes et al., 2011). At present some of the challengesof observing turbulence include merely separating out turbulent line-widths from signaturesof infall of gas onto or outflow from the disk surface. Ideally, though, observations will
eventually provide us with not just detection of turbulent motion within disks, but spatial
information (radially and vertically within the disk structure) about the varying strength ofthat turbulence, to compare to current theories of turbulence generation within disks. The
nature and strength of disk turbulence is essentially the greatest source of uncertainty in any
calculations of disk structure, disk evolution, dust-gas aerodynamic interactions, and dusttransport within a disk. Resolving the nature of disk turbulence would vastly strengthen all
future calculations of solids processing within protoplanetary disks.
Note that while deep observations of the earliest times of disk formation could prove most valuable
for building an understanding of our own Solar system chronology, having a strong understanding of
the gas structure and dynamics of any observed disks would prove equally valuable. The apparent
ubiquity of debris disks and millimeter grains distributed out to large distances in protoplanetarydisks means that there remain problems to solve concerning disk structure and large solids withindisks of all sizes and ages.

205

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215

Appendix A

Derivation of Analytical Test Cases
for Diffusion
In this appendix, I follow Clarke & Pringle (1988) and expand the derivation for a set of analytical
test cases for radial diffusion of a contaminant within a gas disk. I demonstrate the fidelity of therandom-walk method in my particle-transport model in reproducing these solutions in §3.3.2 of themain thesis. Analytical solutions to the diffusion of a contaminant in a disk may be found for a setof idealized disk scenarios. These scenarios are not always physically meaningful — and so require
nonstandard equations with which to obtain them — but they may all be simulated numericallywith my code. Therefore, they are useful test cases. Second (§3.3.2), I present numerical simulationsof these test cases using my particle-transport model. All solutions and simulations presented below
use static-disk–surface-density profiles.
In Clarke & Pringle (1988), the authors solve for the time-dependent concentration, C, of acontaminant initiated in a ring at specified radius: C (t = 0) = C0 δ (R − R0 ), where C is the ratioof the local contaminant mass over the local disk-gas mass, R is the distance from the central star,and R0 is the initial position of the contaminant. The disk that they consider has a surface-density
profile given by Σg = Σ0 R−a , where Σ0 and a are constants; it is also a steady disk, so that fordisk viscosity, ν = ν0 Rb (b and ν0 constants), b = a, and the gas-accretion velocity may be written
as vR = −3ν/2R. For the following derivations, I also consider cases with zero gas velocity and I
relax the steady-disk assumption. I use:Σg = Σ0 R−a ,ν = ν0 R b ,216

vR = v0 Rb−1;

v0 = 0 or −

3ν0
.
2

(A.1)

Assuming azimuthal symmetry and that the gas and contaminant are vertically well-mixed, thediffusion equation in polar coordinates may be written (Gail, 2001)
·
µ
¶¸
∂
1 ∂
1 ∂
∂C
(Σg C) +(RΣg CvR ) −RDkg Σg= Pk (R, t) ,
∂t
R ∂R
R ∂R
∂R

(A.2)

where Dkg is the coefficient of diffusion of the contaminant within the gas, and Pk corresponds to
the rate of production of the contaminant per radial increment within the disk. Clarke & Pringle(1988) and the setup of my simulations assume zero production of the contaminant, P k (R, t) = 0,
and that the diffusivity scales with the disk viscosity, Dkg = ζν, where ζ is a constant. For myparticle-transport simulations using a static disk profile, Equation (A.2) is further constrained by∂Σg /∂t = 0, and so vR refers specifically to the prescribed dust-mean-radial velocity, irrespectiveof the (nonexistent) gas radial velocity. Equation (A.2) then becomes
µ
·
¶¸
∂C
∂C
1 ∂
1 ∂
Σg
RζνΣg
= 0.
+
(RΣg CvR ) −
∂t
R ∂R
R ∂R
∂R

(A.3)

Note that the standard approach (Clarke & Pringle, 1988; Gail, 2001) is to assume inwardlyaccreting–co-moving disk mass and contaminant (with vR ) and to then employ the equation for
mass-conservation,
∂Σg
1 ∂
+
(RΣg vR ) = 0.
∂t
R ∂R
The result is
·
µ
¶¸
∂C
∂C
1 ∂
∂C
Σg
+ Σ g vR
−
RζνΣg
= 0.
∂t
∂R R ∂R∂R
I use Equation (A.3) rather than the above equation in order to match the static-disk conditions
used in the simulations of §3.3.2. While my model is capable of evolving the disk surface-density
profile so that the usual conservation-of-mass equation would apply, analytically solving for themass distribution of the contaminant within an evolving disk profile if the disk is nonsteady (a 6= b)
is not tractable.

217

Next, I combine Equations (A.1) & (A.3), giving
·
µ
¶¸
1 ∂C
v0 b−1 ∂Cv0 b−2
1+b−a ∂C
a−1 ∂+ R
+ (b − a) R C = R
ζR
.
ν0 ∂t
ν0
∂R
ν0
∂R
∂R

(A.4)

Note that for a steady disk (a = b), the third term on the left hand side disappears, producing thesame equation as Equation (2.2.2) in Clarke & Pringle (1988).
Following Clarke & Pringle (1988) I next separate variables by assuming¡¢
C (R, t) = Rp g (R) exp −ζν0 λ2 t ,(A.5)

where p is a constant, λ is the separation constant, and the spatial dependence, R p × g (R), follows
d2 g
R2 2
dR

µ
¶
v0
dg
2p + 1 + b − a −
+ R
dR
ζν0
¶
¸
· µ
v0
v0
λ2
+ g p p+b−a−− (b − a)+ g R2−b = 0 .
ζν0
ζν0
ζ

(A.6)

This equation looks like it could be a Bessel equation if I choose the correct value for p, but thelast term needs to go as R2 . Changing variables using kx = R1/q , where k is a constant andq = 2/ (2 − b), I have
d2 g
x2 2
dx

· µ
¶
¸
v0
dg q
2 2p + 1 + b − a −
−b
+ x
dx 2
ζν0
¶
¸
· µ
v0
λ2
v0
2
+ (a − b)+ gx2 q 2 2 = 0 .
+ gq p p + b − a −
ζν0
ζν0
ζk

1
Equation (A.7) is not valid for the case of b = 2, but for all other cases, when p =2
it yieldsµ¶|q|λ 1/q
g (R) = Jβ √ R,ζwhere Jβ (x) is the Bessel function of the first kind of order β =C (R, t) =

Z ∞

λ=0

¡
¢
exp −ν0 λ2 t Rp Jβ

218

µ

(A.7)

µ
¶
v0
a−b+
,
ζν0
(A.8)

|a − b − v0 /ζν0 |. Therefore,|2 − b|¶|q|λ 1/q√ R
A (λ) λdλ ,
ζ

(A.9)

where A (λ) is a function used to match the solution to the initial conditions.
Next, I use the initial conditions and insert the delta-function for C at t = 0 to write

C0 R−p δ (R − R0 ) =

Z ∞

Jβ

λ=0

µ

¶
|q|λ 1/q√ R
A (λ) λdλ .
ζ

(A.10)√
By using the variable transform s = |q|R1/q / ζ and keeping careful track of the signs of the limits
on the integral, I can apply a Hankel transform and write

A (λ) =Z ∞C0 R−p δ (R − R0 ) Jβ (sλ) s dsµ¶
|q|λ 1/q|q|C0 2q −1−pR0J β √ R0
.
ζ
ζ
s=0

=

(A.11)

Plugging Equation (A.11) into Equation (A.9) and solving yields:

C (R, t) =
q =
p =
β =

µ ¶
|q|C0 R01−b R p
2ζν0 t

R0

Iβ

Ã

1/q
q 2 R0 R1/q2ζν0 t2,(2 − b)
µ
¶
v0
1
a−b+
,
2
ζν0
|a − b − v0 /ζν0 |.
|2 − b|!



exp −q 2³

R02−b + R2−b4ζν0 t´,(A.12)

where Iβ (x) is the modified Bessel function of the first kind of order β. Note, solving the integral
in Equation (A.9) requires an identity. I have used the relation listed as Equation (3) in §2.12.39
of Prudnikov, Brychkov & Marichev (1986):
Z ∞
0

xe

−px2

" ¡
¢# µ ¶
b2 + c 2
1
bcJν (bx) Jν (cx) dx =
exp −
Iν
.
2p
4p
2p

For a = b and v0 = −3ν0 /2, Equation (A.12) reduces to Equation (3.2.4) in Clarke & Pringle(1988). In fact, if I had assumed the usual vR belonging to the gas with mass conservation, as in
Clarke & Pringle (1988), instead of using a static disk, I would have obtained the same solution
given in Equation (A.12) above, except that the order of the Bessel function would change to|a − b+v0 /ζν0 |β =. In this case, I would then also need to know the time-evolution of the|2 − b|219

nonsteady disk profile in order to convert Equation (A.12) into a contaminant-mass distribution
for a 6= b and nonzero vR .
The general solution in Equation (A.12) is valid for most combinations of a, b, and v 0 . However,
it is not valid when b = 2, nor in the simplest scenario for diffusion in R: when the mass distribution
is uniform in R (a = 1), the diffusivity is constant (b = 0), and there is zero background velocity(v0 = 0). But, in that simple-diffusion case, the solution is well known:"#
C0(R − R0 )2
C (R, t) = √exp −.
4ζν0 t4πζν0 t(A.13)

I do not solve for the general solution when b = 2. However, Clarke & Pringle (1988) present
the solution for the steady-disk case when a = b = 2:Ã!
[ln (R/R0 ) − v0 t]2C0−1/2(4πζν0 t)
exp −.
C (R, t) =R0
4ζν0 t(A.14)

Because this is a steady-disk case (a = b), the initial diffusion equation used to solve for C (R, t)
in Clarke & Pringle (1988) matches the diffusion equation I have used above for my static-disk
scenario, Equation (A.4). Note, however, that Equation (A.14) does not match Equation (3.1.3) in
Clarke & Pringle (1988) because of a typo in that paper, which accidentally presents the solution
for twice the background velocity, vR → 2vR . Also, when a = b = 2, the natural space to solve theequation is logarithmic space. This happens to be the space used for the width of my model-disk
grid cells, which is relavent to the only odd behavior that I find in my particle transport simulations(see §3.3.2).
My particle-transport simulations output the number of particles per radial grid space. Therefore, to compare the solutions provided by Equations (A.12), (A.13), & (A.14) to my simulations,
I next need to convert the analytic C (R, t) function into an expected fractional-mass distribution. The concentration can be expressed as the ratio of the contaminant-to-gas surface densities,
C = σ/Σg , where σ is the surface density of the contaminant. Because these test cases assume astatic-disk–surface-density profile, it is then relatively simple to solve for the fractional mass of thecontaminant in a given grid space, mi , where i denotes the grid space, which I will normalize by220mtot , the total mass of the contaminant in the disk at t = 0.

mtot = 2πZ ∞0RΣg C0 δ (R − R0 ) dR = 2πR0 Σg (R0 ) C0 .

I write:
µ

mi
mtot

¶

=

1
R0 Σg (R0 ) C0

Z Ri+1/2RΣg (R) C (R, t) dR ,(A.15)

R=Ri−1/2

where Ri−1/2 and Ri+1/2 are the inner and outer boundaries of the grid space, respectively. Combining Equations (A.12) & (A.15), and using a variable substitution, s = R 1/q , I can write thegeneral expression for the general-case expected mass distribution as
µ

mi
mtot

¶

"
#Z
µ 2
¶
µ
¶
Ri+1/2
q|q|R0γ1
q 2 R02−b
q s0 sq 2 s2
γ2
exp −
s Iβ
exp −
ds
2ζν0 t4ζν0 t2ζν0 t2ζν0 t
R=Ri−1/2
µ
¶
v0
1
a−b−
2
ζν0
(2 − a + v0 /ζν0 )(2 − b)

=

γ1 =
γ2 =

1/q

s0 = R 0 .

(A.16)For comparison with my simulations, I solve the integral numerically using a Simpson integrator
and the Numerical Recipes functions for the modified Bessel functions (Press et al., 1992).
Equation (A.16) represents the expected mass distributions for most cases of a, b, and v 0 , but
not all. For the simplest diffusion case (a = 1, b = 0, v0 = 0) with C (R, t) given in Equation (A.13),
the expected mass distribution is given by
µ

mi
mtot

¶

= (4πζν0 t)−1/2#(R − R0 )2
exp −
dR .
2ζν0 t
R=Ri=1/2

Z Ri+1/2"(A.17)

And for the case of a = b = 2 given in Equation (A.14) the expected mass distribution in a static
disk is given by
µ

mi
mtot

¶

= (4πζν0 t)−1/2#(Y − v0 t)2
exp −
dY ,
4ζν0 t
R=Ri−1/2

Z Ri+1/2"(A.18)

where Y = ln (R/R0 ), and again, I use a Simpson integrator to solve for the expected (mi /mtot )for these cases.

221

Appendix B

Notes on Numerical Techniques
To aid in the transparency of my models, in this chapter, I outline some of the numerical techniques
used within the code built for this thesis. While the techniques discussed here cannot be classified
as new research, they are important tools for implementing the research I have done.
In §B.1, I focus on the numerics of the disk-evolution model, detailing constraints imposed bynumerics, numerical conservation of mass and angular momentum, and the fidelity of the numericalmodel to analytic solutions of the disk-evolution equation. In §B.2, I discuss a range of simple
numerical techniques used ubiquitously throughout my code. And in §B.3, I discuss the iterativesolvers used to find radial-drift velocities and midplane disk temperatures within my model disks.
While iterative solving is simple in concept, it becomes more complex in practice, with each solver
tailored to the particular set of equations under consideration.

B.1

Time-Explicit Viscous-Disk Evolution

The material in this section is a supplement to §2.3.2, fully outlining the numeric considerations
pertinent to solving the disk-evolution equation including stability, convergence, and conserved
quantities, and elucidating choices and constraints for time-step sizes, grid-spacing, and the radialextent of the grid.

B.1.1

Numerical Setup and Fidelity

In setting up the numerical schemes for the disk-evolution code, I have followed Numerical Recipes(Press et al., 1992), which clearly outlines the most straight-foward, stable methods for approaching
the evolution of basic PDEs. For a diffusion equation, Press et al. (1992) §19.2 provides the single-

222

level, Forward-Time-Centered-Space (FTCS) scheme, using a finite-difference approximation of therelevant derivatives:
∂u
∂t
j+1
ui − uji
∆t

= D
= D

∂2u
,
2
∂x
" j
#
ui+1 − 2uji + uji−1
(∆x)2

.

Therefore, for the most compact form of the disk-evolution equation (2.17), one can write:jΣj+1
g,i − Σg,i∆t

3 1
= ³ ´34
Xij"(νΣg X)ji+x − 2 (νΣg X)ji + (νΣg X)ji−1
(∆X)2

#

,

(B.1)

(assuming constant ∆X grid spacing) where i and j are the spatial and temporal indices, respectively.
If, on the other hand, one wanted to integrate the expanded form of the equation, with the
diffusive and advective components separate, one would need a different scheme for the advective
component as FTCS is not stable for the advection equation (Press et al., 1992). The simplest
approach for advection provided in Press et al. (1992) (§19.1) is called upwind differencing, which
directly considers the direction of the flow of advection. In this scheme:
∂u
∂t
j+1
ui − uji∆tj+1
ui − uji
∆t

∂u
:
∂x
uj − uji−1= −vij i, vij > 0∆x
uj − uji= −vij i+1, vij < 0∆x
= −v

so one could write the advective term in Equation (2.18) as
99 ∂(νΣg ) →
2R ∂R
2Rij

"

(νΣg )ji+1 − (νΣg )ji∆R#(assuming constant ∆R grid spacing).
For each of the diffusive- and advective-equation schemes there is a stability criterion (taken

223

from Press et al. (1992)). For the diffusive scheme it is
2D∆t
≤ 1,
(∆x)2

(B.2)

so that the scheme of Equation (B.1) requires∆t ≤ (∆X)

2 2X

3 ν

2

2
−→ ∆t ≤ (∆X)2 ∗ min
3

µ

Xi2
νi

¶

.

(B.3)

The stability criterion for the advective scheme is given by the Courant condition (material may
not flow beyond the adjacent grid cells in a given time step):
|v|∆t
≤ 1.
∆x

(B.4)

In my disk-evolution model for numerical integration, I use only the most compact form of
the disk-evolution equation. The scheme outlined by Equations (B.1) & (B.3), using a constantgrid spacing, ∆X, has been used by several authors (e.g., Hueso & Guillot (2005); Alexander &Armitage (2007)), and I compare it to my preferred scheme below. In my preferred scheme, I do notuse constant grid-spacing, but instead space grid points logarithmically in R (and X), as outlinedin §2.3.2. Very small grid-cells at small R means that this scheme requires more time-steps and soruns more slowly than the traditional-∆X scheme, but it also requires fewer grid-spaces to reachconvergence and is more conservative (as discussed in §B.1.2). Also, uneven grid-spacing requires amodification to Equations (B.1) & (B.3) in order to be properly applied; they must be expanded toconsider the finite-differencing between individual grid cells separately. This gives (Equations (2.24)& (2.25) )jΣj+1
g,i − Σg,i

∆t

∆t

=
≤

"
#
(νΣg X)ji+1 − (νΣg X)ji(νΣg X)ji − (νΣg X)ji−1
3 1 1
+
,
4 Xi3 ∆Xi
∆Xi+1/2∆Xi−1/2
Ã!
∆Xi ∆Xi−1/2 Xi2
2
min
,
3
νi

(B.5)
(B.6)

where ∆Xi = Xi+1/2 − Xi−1/2 is the width of grid cell i and ∆Xi+1/2 = Xi+1 − Xi is the distancefrom grid point i to i + 1.

224

140

100

Sigmag (g cm−2)

analytic
ngrid=200
ngrid=399

120
80
60
40
20
0

(numeric−analytic)/analytic

a1)

0

2

4

6
8 10
R (AU)

12

constant Delta X
0.2
0.1
0.0
−0.1
b1)

−0.2

0

2

4

6
8 10 12 14
R (AU)

analytic
ngrid=200
ngrid=399

a2)

120
100
80

time (yr):
103
4x103
104
2x104
4x104
7x104

60
40
20
0

14

(numeric−analytic)/analytic

Sigmag (g cm−2)

140

0

2

4

6
8 10
R (AU)

1214

log−spaced Delta X i
0.2
0.1
0.0
−0.1
b2)

−0.2

0

2

4

6
8 10 12 14
R (AU)

Figure B.1: Comparison between analytic and numerical solutions for disk evolution of an initialdelta-function in Σg at R0 = 5 AU. Red/Orange simulations run in the constant ∆X numerical
scheme, and Blue/Green simulations run in the log-spaced ∆Xi scheme. Top panels-a1 and a2 plot surface-density contours for the analytic solution and the numeric results using two grid
resolutions (grid from 0.1–100 AU). Bottom panels-b1 and -b2 plot the deviation between the
numeric and analytic results. Note that the magnitude of the plotted surface density is dependanton the width of the R0 grid cell, defining the t = 0 disk mass, and that the ngrid = 200 Σg curves
are plotted at half-magnitude for ease of comparison with the rest.
In Figure B.1, I plot a comparison of disk evolution for the two numeric schemes outlined above(constant ∆X and log-spacing ∆Xi ) versus the analytic solution for the viscous spreading of a ringof disk material (given in §2.3.1). These examples are run at two spatial resolutions of the grid:
ngrid = 200 and ngrid = 399, extending from 0.1 to 100 AU. While these simulations are not a perfectmatch for the analytic solution, they are a still a good match, where inspection shows noticeable
deviation only at the inner-disk edge at late times. Panels-b1 and -b2 plot the deviation between
the numeric and analytic results, and it is clear both that higher spatial resolution does lead to acloser fit with the analytic solution and that the log-grid-spacing scheme has better convergence
for a given ngrid in the main part of the disk than does the constant ∆X scheme. However, some of225

the deviation appears intrinsic to all of the numerical simulations. This intrinsic deviation is due
both to a finite inner-grid boundary (causing loss of material that otherwise continues to diffuse in
the analytic solution) and to the fact that a discreet grid where Σg is constant across the width ofa grid cell simply cannot completely replicate the delta-function initial condition.

B.1.2

Conservation of Mass and Angular momentum, and the Grid-Spacing
SchemeIn this section, I discuss the link between the choice of grid and grid spacing, and the numericalconservation of mass and angular momentum in an evolving model disk. In the model disk, thesurface density is considered constant across the width of a grid cell, and I consider two forms forcalculating the mass and (Keplerian) angular momentum of a cell. An approximate formulation:

M1,i = 2πRi ∆Ri Σg,i ,p
1/2
L1,i =
GM? M1,i Ri ,

(B.7)

´
³
2
2
Σg,i ,− Ri−1/2
M2,i = π Ri+1/2
³
´
p
4π
5/2
5/2
L2,i =
Ri+1/2 − Ri−1/2 Σg,i ,
GM?
5

(B.8)

and a more precise formulation:

where M1,i and M2,i are two expressions for the mass contained in grid-cell i, L1,i and L2,i are two
expressions for the angular momentum, ∆Ri is the width of grid-cell i, and Ri−1/2 and Ri+1/2 are
the inner and outer grid-cell boundaries. The angular-momentum expressions also assume Keplerian velocities for the disk, which pressure-support of the gas tells us is not completely accurate.
However, the disk-evolution equation was also derived assuming Ωg = ΩK , so calculating angularmomentum conservation this way is consistent with investigating numerical fidelity to the baseequation.
In order to track mass and angular-momentum conservation in an evolving disk, one must also
account for the quantities lost past the inner and outer grid-space boundaries. This can be done
easily by creating dummy cells at the inner and outer boundaries, setting the surface-density inthese dummy cells equal to the bounding Σg used in the code, and tracking the mass and angular226

Figure B.2: Diagram of Σg evolution using two dummy grid cells.

momentum deposited into them at each time-step.
Finally, dummy cells are also a useful way to calculate a numerical-scheme’s potential to deviate
from conservation. Consider a grid cell that starts out containing a finite amount of disk mass,
while the (dummy) cells to either side have Σg = 0. After one numerical time-step, all three cells
will have finite Σg , as depicted in Figure B.2, and, if the scheme is conservative, should also have
j+1
j+1
Mij ≡ Mi,init = Mi,fin ≡ Mdummy,i−1+ Mij+1 + Mdummy,i+1,
j+1
Lji ≡ Li,init = Li,fin ≡ Lj+1
+ Lj+1dummy,i−1 + Li
dummy,i+1 .

Therefore, the potential numerical deviation from mass and angular-momentum conservation canbe calculated for a grid cell asδMi = Mi,fin − Mi,init ,δLi = Li,fin − Li,init .

(B.9)

Performing these calculations with dummy cells for the two disk-evolution grid-spacing schemes,
produces for constant ∆X:

δM1,i = 0 ,δL1,i = 0 ,227δM2,i =¢¡3π ∆t∆X 3 3Xi2 − ∆X 2
¢2 νi Σg,i,0 ,¡ 2
2Xi
X − ∆X 2i

δL2,i ≈
and for log-spaced ∆Xi :3π ∆t∆X 3¡¢ νi Σg,i,0 ,2 Xi2 − ∆X 2

δM1,i = 0 ,δL1,i = 0 ,δM2,i = 0 ,δL2,i = 0 .

This method, therefore, predicts that both disk-evolution grid-spacing schemes are conservative tofirst order, but that only the log-spaced ∆Xi scheme is conservative in the more precise formulation.
To verify that this is so, Figure B.3 plots the various mass and angular-momentum quantities andtheir sum as a function of time for three evolving-disk scenarios using each of the disk-evolution
grid-spacing schemes. Of particular interest, are the heavy lines in the figure, which plot a total
accounting of all the mass (or angular momentum) in the disk, adding the mass within the disk,
the net mass which has been lost inward past the grid, the net mass which has been lost outward
past the grid, and substracting and estimated excess deviated mass. This curve is flat for all cases,
showingthat this calculation accounts for all of the real or numerically added mass of the systems.
Finally, the curves summing the excess-deviated mass (angular momentum) for the constant ∆X
scheme, do show this value to be fairly low, and further investigation reveals that this scheme does
deviate less from conservation for a higher grid-space resolution. However, a higher grid-spaceresolution is also needed for the result to reach convergence, as suggested by the difference in diskevolution produced by the two schemes, particularly when there is significant mass to process in
the inner disk.
Finally, consider the fact that Figure B.3 shows a non-negligible amount of mass being lost
past the outer-grid boundary (except, of course, where the outer boundary is the mass source).
Of course, it is preferable that, in most disk simulations, such loss have only a negligible effecton the overall disk evolution. However, early simulations with MD,0 = 0.05M¯ disks, ngrid = 500
228

a1)

10
5
0
2×105 4×105 6×105
Time (years)

8×105

15
b1)

10
5
0
0

2×105 4×105 6×105
Time (years)

8×105

200
100
0
−100
c1)

−200

0

2×105 4×105 6×105 8×105 1×106
Time (years)

Total l(L) (MJupiter AU1/2)

Total Mass (MJupiter)0

Total Mass (MJupiter)

Total l(L) (MJupiter AU1/2) Total l(L) (MJupiter AU1/2)

Total Mass (MJupiter)

15

80
60

a2)

40
20
0
0

2×105 4×105 6×105
Time (years)

8×105

80
b2)

60

Disk Total
lost past Rinlost past RoutDeviated total
Sum − deviated

40
20
0
0

2×105 4×105 6×105
Time (years)

8×105

50

0

−50
c2)

−100

0

2×105 4×105 6×105 8×105 1×106
Time (years)√
Figure B.3: Plots of the total mass and normalized angular momentum (l = L/ GM? ) in threedifferent evolving-disk scenarios using the constant ∆X (red curves) and log-spaced ∆X i (bluecurves) numerical schemes. Curves plotted include the total mass (angular momentum) on thedisk at a given time, the total lost inward past Rin , the total lost outward past Rout , the total
excess mass (angular momentum) from deviation (δM or δL) (for the constant ∆X scheme only),
and the total+lost-inward+lost-outward−deviated-excess values. Panels-a1 and -a2 show a disk
with constant Σg ν (full disk) at t = 0 and an outer boundary of Σg = 0.1 g cm−2 . Panels-b1
and -b2 show a full disk with a (silly) inner boundary of Σg = 105 g cm−2 . And panels-c1 and-c2 show a scenario where the disk is initial empty and is fed entirely by the Σg = 0.1 g cm−2
outer-boundary condition.

229

surface density (g/cm 2)

t=0 years
t=3.75x106
t=5.5x106
t=5.7x106
t=5.75x106
t=5.85x106
t=5.95x106
t=6.1x106 (n=581)

102
101
100
10−1
10−2
10−3
10−4
10−50.1

1.0
10.0 100.0 1000.0
orbital radius (AU)Figure B.4: Comparison of disk evolution between two grid extensions: Solid lines: n grid = 500,
Rout = 2000 AU; Dashed lines: ngrid = 581, Rout = 9981.4 AU. MD,0 = 0.05M¯ , Rd = 10 AU,
α = 10−2 .

and R = 2000 AU outer-grid boundaries suggested that the Σg profile was being influenced by thelocation of the outer boundary. Therefore, I ran a comparison simulation with and extra 81 grid
points, extending the grid out to R = 9981.4 AU, and the results are plotted in Figure B.4. This
plot suggests that loss of disk mass past the nearer outer-grid boundary is significant enough tosubstantially alter the disk evolution near the end of the disk lifetime, causing the disk to thin anddissipate more rapidly than it otherwise would. In light of this result, I have chosen a fiducial gridfor the simulations of this thesis that uses ngrid = 600 and is extended out to R = 15,000 AU.

B.2

Some Simple Numerical Tools

In this section, I discuss some simple numerical tools that each have multiple applications within
the code used for this thesis. These tools are simple enough that there is nothing particular about
my use of them, and my purpose for discussing them here is simply to clearly specify the ways they
are applied within my code.

230B.2.1

Interpolation: Trajectories and Probabilities

Everything used to calculate particle trajectories in my particle-transport model is defined at eachdiscreet grid point of my model-disk grid, and the code outputs the grid-binned particle distribution
in order to retain smaller output-file sizes. However, the particles within the simulation exist along
the R continuum, and not just at discreet locations. Therefore, it is often necessary to interpolatetransport values between two grid points in order to calculate trajectories. This is especially so forthe code discussed in §3.2.1 that directly integrates particle trajectories via force-balance. In thatcode, each of the terms in Equations (3.10) & (3.11) are linearly interpolated between grid points
to give values for specific rd at a given t. However, this direct-integration code is used only to verify
the fidelity of my primary transport code. In my primary transport code I use interpolation in twoplaces:* For 1st -order (linear) interpolation of particle trajectories according to gridded values of radial
dust velocities (§3, Equation (3.1) ).
* For 2nd -order (parabolic) interpolation of the probability that a particle will (diffusively)random-walk inward versus outward for a given time-step, ∆t (§3.3.1).
First-order interpolation, to find a local value between two grid points, simply assumes a linear
function for that value between points (a and b):y (R) = y (Ra ) + (R − Ra )[y (Ra ) − y (Rb )].
(Ra − Rb )(B.10)In order to use interpolation to calculate particle trajectories in my code, I use Equation (B.10) to
calculate the local radial-drift velocity, vsrd,0 , and and turbulent-stepping velocity, vturb,0 (for the
primary particle time-step, ∆t) at R0 . If these velocities will carry the particle beyond the next
forward grid point, Ri , or ∆t > ∆tnext ≡ (Ri − R0 ) /vtotal , then I first advance the particle to thatgrid point following Equation (3.1), then use the local advection velocity at that point, v srd,i tocalculate the particle motion for the rest of the time step. To complete the trajectory step this way,
I do not adjust to vturb,i at the grid point, because the diffusion-stepping properties are calculated
for the whole ∆t step at the starting point, R0 . Therefore, for a trajectory step completely within231the bounds of two grid points

Rfin = R0 + ∆t (vsrd,0 + vturb,0 ) ,and for a trajectory-step that crosses grid-point i

Ri = R0 + ∆tnext (vsrd,0 + vturb,0 ) ,Rfin = Ri + (∆t − ∆tnext ) (vsrd,i + vturb,0 ) .
I use this simple, first-order interpolation rather than a higher-order scheme because a) lower-orderschemes are less computationally expensive, b) a second-order scheme cannot be employed between
grid-points with oppositely-directed velocities, and c) as shown in §3.2.3, this first-order scheme
reproduces the direct-integrated trajectories to sufficient accuracy.
To calculate the probability of diffusively stepping inward (versus outward) for a given time-stepat each grid point, I do use second-order interpolation, this time in ∆t. Second-order interpolation
assumes a parabolic function, requires three points (a, b, and c), and in generaly (x) = Ax2 + Bx + C ,
A =
B =
C =

(B.11)(ya − yb ) / (xa − xb ) − (yb − yc ) / (xb − xc ),(xa − xc )
¡
¢
¡
¢
¡
¢
−ya x2b − x2c + yb x2a − x2c − yc x2a − x2b,(xa − xb ) (xb − xc ) (xa − xc )ya xb xc (xb − xc ) − yb xa xc (xa − xc ) + yc xa xb (xa − xb ).
(xa − xb ) (xb − xc ) (xa − xc )For calculating pin (∆t), these equations are somewhat simplified because I calculate pin at eachgrid point for ∆tevolv and ∆tevolv /2 (where ∆tevolv is the global constraining time-step of the diskevolution model), and I know pin (∆t = 0) = 1/2. Thenpin (∆t) = A∆t2 + B∆t + C ,A =B =2pin,e − 4pin,he + 1,∆t2evolv−pin,e + 4pin,he − 3/2,∆tevolv

232(B.12)

1
,
2

C =

where pin,e ≡ pin (∆tevolv ) and pin,he ≡ pin (∆tevolv /2). This equation is used the find pin (∆t) forthe bounding grid points of a particle’s starting location, and then linear interpolation is used tofind pin for the precise location.

B.2.2

Time Derivatives, Radial Gradients, and q-values

Outside of the disk-evolution equation discussed in §2.3.2 and §B.1, a few other model calculations,particularly equations for the various gas velocities (§2.6), require knowing the gradient of somedisk property or properties. In general, for time derivatives, I use a first-order approximation:
µ

∂y
∂t

¶

=
j

yj+1 − yj,∆t(B.13)

where j is the time-stepping index, and y is the property-of-interest. For spatial derivatives, I
generally use a second-order approximation:
µ

∂y
∂R

¶

=
i

·
¸
1 (yi+1 − yi ) (yi − yi−1 )+,2 ∆Ri+1/2∆Ri−1/2(B.14)

where i is the grid-space index, and ∆Ri+1/2 = Ri+1 − Ri . This allows the calculation for thederivative at a given grid point, rather than half-way between two grid points. (Note that thez-derivatives needed for vr,merid (Equations (2.56)–(2.59) ) are calculated analytically within thecontext of a vertically-isothermal disk structure.)
However, many of the spatial derivatives needed to calculate gas velocities come in the form ofµ¶R ∂y. This is a special case of R-derivatives that has the added numeric pitfall of diverging aty ∂Rgrid points where y → 0. For a simple power-law function in R
y = y0 R q ,
µ
¶
R ∂y
q =
,
y ∂R
where q is a constant. Therefore, for these special-case derivatives in the gas-velocity calculations,
I calculate a separate grid-vector of what I term ’local q-values’, that are simply an extension of233

the second-order spatial derivative above:¸·
Ri (yi+1 − yi ) (yi − yi−1 )qy,i =+,
2yi ∆Ri+1/2∆Ri−1/2(B.15)and the subroutine for calculating qy,i sets qy,i = 0 where yi = 0 to avoid local divergences.

B.2.3

Large Summing Sets

One of the trickiest of the most basic numerical operations is summing, particularly of large setswith component values that range over orders of magnitude. This is simply because of the finite
accuracy of even double-precision numbers that cannot record a change to the base value that is
too many orders-of-magnitude less than the whole. The places that I have had to be wary of thisin my code are in the summing up of mass, both of gas and dust.
The simplest way to correct for errors in summing is to add a loop within the summing thatcauses the code to first add up subsets of the data and then to add those subsets together:n/nsetMtot =

X

Mi0 ,

i0 =1

Mi0

=

0
nX
set i

Mi

(B.16)

i=(i0 −1)nset +1

where Mtot is the total mass sought, Mi is the mass in, e.g., grid-cell i, n is the total number ofcells being summed, and nset is the number of cells in a subset. For a very large set, one mightneed to use multiple nested loops, but in the code used for this thesis, I generally use a single set,
as shown here.
Using a scheme like this is important if I want to calculate the total gas mass of my model
disk. My disks are on a grid with n = ngrid = 600. The inner-most cells are small, due to thelogarithmic spacing of grid points, so each contain small amounts of mass. While the outer-mostcells are large, the disk surface-density drops off exponentially at the outer edge, so those, too,
contain small amounts of mass. Therefore, summing the mass from either direction will lead to thesituation of trying to add small-mass values to a large summed total. For this type of summing, I
tend to use a subset with nset = 20. It is possible that this still does not capture gas added fromthe very outer edge of the disk at thousands of AU, but such mass is negligible for these purposes.
234For the dust-mass–allocation scheme used to assign masses to each representative particle in
the Chapter 6 global-distribution runs, it is important to use subsets to learn both how much dust
mass is currently in a zone-of-interest, and how many particles of each size. For these sets, I use
nset = 200, because of the large number of particles in the simulation and that could be in givenzone (from a few hundred to several thousand). For these, I also sum up the dust mass in each size
bin separately, before adding all of that together. For a large range in particle size, there are several
orders-of-magnitude difference in the mass allocated between the bins of largest and smallest size,
and therefore a wide range in masses assigned to each individual particle.

B.3

Instances of Itterative Solving

There are two important aspects of the disk and particle-transport models that do not have direct,
analytic expressions and must be solved for iteratively. These are the the steady-radial-drift velocities of the particles vsrd , and the midplane disk temperature in the models using an evolving,energy-balanced temperature profile. Solving for vsrd is simpler than for the midplane temperature,
because vsrd requires only two equations and two unknowns (vr,d and vφ,d ), whereas the temperature requires three equations and three unknowns (Tmid , κR , and ν). Both cases also require sometrickiness to obtain a full solution set, but in the most general cases encountered in the disk arereasonably well-behaved.

B.3.1

Itteration Strategies and the Early Model for Finding Epstein-drag Velocities

There are two, simple numerical techniques used in my code for iterative solvers. The first is aloop/walk method diagramed in Figuere B.5. This method involves picking a starting-guess solution(x0 ), and then alternating back and forth between the equations to be solved (e.g., y = f (x) andx = g (y)) until convergence to a solution of sufficient accuracy. However, unless I want only toinvestigate the direction that this solution carries me in parameter space, my code does not use
this method with a pure loop, where* y0 = f (x0 )* x1 = g (y0 )235

Figure B.5: Diagrams for iterative-solving with a loop/walk method. Blue and teal curves represent two equations to be solved, the orange arrow represents the initial guess, and red arrows
represent the motion through parameter space of the solver routine. Direct-looping can lead to
non-convergence as shown in panel-a1, and loop/walking in some directions may be non-tractableor lead to divergence, as shown in panels-b1 and -c1.

* y1 = f (x1 )* x2 = g (y1 )* etc...
As shown in panel-a1 of Figure B.5, there are cases where a pure loop will not converge, or will
be too slow to converge to be numerically useful. Therefore, I modify the loop by only taking ahalf-step in one of the parameter dimensions, as shown in panel-a2, and prescribed by* y0 = f (x0 )236

* x1 = g (y0 )* y1 = f (x1 ) → y1a = 0.5 (y0 + y1 )* x2 = g (y1a )* y2 = f (x2 ) → y2a = 0.5 (y1a + y2 )* etc...
With this method, the solver may converge to a centralized solution much more quickly. However,as shown in panels-b1 and -c1 of Figure B.5, even this half-step method will not work at all if
loop/walking causes the solutions to diverge rather than converge. When solving for v srd caused
by Epstein drag, this problem can be circumvented by inverting the two equations (to produce
x = f 0 (y) and y = g 0 (x)) and loop/walking in the opposite direction, as shown in panels-b2 and-c2, with:* y0 = g 0 (x0 )* x1 = f 0 (y0 )* etc...
However, panel-c2 also shows the primary drawback of the loop/walk method, which is that it cansometimes be quite slow to converge, with poor constraints on the accuracy of that convergence.
The second iterative-solver method, which is much faster to converge, is the bisection method,
diagramed in Figure B.6. This method requires two initial guesses for the solution (x 0a and x0b ), one
on either side of that solution. At each guess, the direction to the other curve and the magnitudeof the separation are calculated. As long as the solution lies between the two guesses, the next
guess is made half-way between them, the direction and magnitude to the other curve calculated,
and the distance halved again. It goes as:* y0a = f (x0a ) ; y0b = f (x0b )* ∆x0a = g (y0a ) − x0a ; ∆x0b = g (y0b ) = x0b* for ∆x0a ∆x0b < 0:237

Figure B.6: Diagram for iterative-solving with a bisection method. Blue and teal curves represent
the equations to be solved. Dotted lines represent guesses with arrows pointing to the magnitude
and direction of separation at each guess. Guesses moving inward from dark to light toward
convergence (sufficiently small separation between curves).

* x1a = 0.5 (x0a + x0b )* y1a = f (x1a )* ∆x1a = g (y1a − x1a )* for ∆x1a ∆x0a < 0:
* x1b = x0a ; ∆x1b = ∆x0a → x2a = 0.5 (x1a + x1b )* etc...
* else for ∆x1a ∆x0b < 0:
* x1b = x0b ; ∆x1b = ∆x0b → x2a = 0.5 (x1a + x1b )* etc...
This method always converges as fast as possible, and convergence is judged directly by when theseparation between the two curves is sufficiently small. It never requires that the equations-to-besolved are invertible, but it does require knowing ahead of time the general region of parameter
space in which the solution exists.
While the loop/walk method is the most computationally expensive of these two methods, aslong as f (x) and g (y) are invertible, it can, however, be applied relatively blindly and still usually
produce stable, reliable results. Therefore, the earlier simulations of particle transport (presented
238

Figure B.7: Diagram of iterative solving by loop/walk method (panel-a) and bisection method(panel-b) for a case in Epstein-drag vsrd when there are no solutions. One of the vsrd equations has
an asymptote at vK /2 with curves open in these directions when vφ,g < vK /2. Black dotted lines
denote asymptotes. Left of the black axis line vφ,d < 0, non-viable solution space.

in Chapters 4 & 5) use Epstein-drag vsrd values derived from a loop/walk-only iterative solver, with
Equation-set (3.24):´³
2 − v2
3 vsφdK

vsrd = vr,g +,CR Rρg vthermµ¶ µ¶
CR Rrhog vtherm
CR Rρg vtherm
vsrd vKvsφd =
vφ,g +/ vsrd +,3
2
3(B.17)(B.18)and inverse equations:
CR Rρg vtherm(vsrd − vr,g ) ,3(vsφd − vφ,g )1= − CR Rρg vtherm.
3(vsφd − vK /2)

vsφd =
vsrd

r

2 +
vK

(B.19)(B.20)

Using a loop/walk-only method with these equations is almost always well-behaved. However,
because one of the vsrd equations is hyperbolic, there is one case, sometimes appearing, for which
no steady vsrd solution exists. This is diagramed in Figure B.7. Note, that neither the loop/walkmethod, nor the bisection method can handle this no-solution situation on their own. A better
iterative solver does not simply switch to a faster solving method, but instead takes the input
equations into account and is carefully designed to search for solutions where they are likely to befound. Such a map for finding the full solution set to the vsrd equations is outlined in §B.3.2.
239

B.3.2The Mapped Approach to Iteratively Solving for Steady Radial Drift
Velocities

Here I present the schematic for finding the full solution set for vsrd under the influence of Epstein
drag using the equations (B.17)–(B.20) given above and the schematic map (in v srd versus vsφd
space) shown in Figure B.8. Due to the nature of hyperbolic equations, solutions for v srd arebroken into two regimes.
The assymptotes for the hyperbolic exist at

vsφd = vφH =vK
2(B.21)

vsrd = vrH = −vdrag ,(B.22)

where vdrag = CR Rρg vtherm /3 and is so-named because it scales the Epstein-drag force and has theunits of velocity. While in general vφ,g < vK , it is almost always greater than vK /2, and therefore
solving for vsrd usually falls into the simpler of the two regimes. In this vφ,g > vK /2 regime, there isalways a solution (A) intersecting the upper-RH branch of the hyperbolic equation, and sometimes
vphi,g > vK

vsrd

vsrd

vphi,g < vK
C

A
D
A

vsphidvsphidE
G

B

F

Figure B.8: Schematic for the vsrd solutions with Epstein drag (Equations (B.17)–(B.20) ). Note,curves have been exaggerated for clarity. Dashed lines mark the assmpytotes of the hyperbolic
equation (green curves). Green x’s mark where the hyperbolic equation crosses the zero axes
(vsφd = vφ,g for vsrd = 0) and blue x’s mark where the parabolic equation (purple curve) intersects
the vsφd = 0 axis.
240

also a solution (B) for the lower-LH branch as well. The hyperbolic intersects the v sφd = 0 axis at

vH0 = −

2vφ,g vdrag.
vK(B.23)

2
vK
.
vdrag

(B.24)The parabolic intersects the axis at

vP0 = vr,g −Solution (B) exists if vH0 ≥ vP0 . Finally, both solutions (A) and (B) can be found with thebisection method, bisecting in vsrd — the parameter of most interest. Solution (B) must exist at
vP0 ≤ vsrd,(B) ≤ vH0 , and so the bisect bounds are already defined. If solution (B) exists, then one
bisect bound for solution (A) should be placed at vsrd = 0. The other bound must be searched
for both toward lower vsrd approaching (but not crossing) the hyperbolic assymptote, and toward
greater vsrd , since it cannot be known before hand in which direction the solution will lie. If,
however, solution (B) does not exist, then a bound for (A) can be placed at v P0 and the othersearched toward larger vsrd . In the case that both (B) exists and there are two solutions to chose
from, I choose the solution closest to (vr,g , vφ,g ), almost always solution (A).
Searching for vsrd solutions in the vφ,g < vK /2 regime is rarer (but not negligibly so, as thegas may orbit very slowly just interior to a gap or at the very outer disk edge), and also more
complicated, and there may be no solution at all. If in this regime vH0 < vP0 , then there is generally
one solution (C) on the upper-LH branch of the hyperbolic that can be found using the same bisect
method as for (A) but with the upper-bound already given by where the parabolic crosses thevK /2 assmpytote. However, it is conceivable that vH0 < vP0 harbors three solutions (C1,C2,C3), as
depicted in Figure B.9. If three solutions do exist, then the bisect search will locate either (C1) or(C3). Therefore, solution one must search for a possible (C2) using the loop-walk method - starting
just above or below the found solution (based on the requested accuracy for convergence), and thenloop/walk searching in both directions (which is one reason why the invertible equations for the
Epstein-drag case are very nice). (Note that this search will fail incorrectly if two of the solutions
are with the error-margine of each other.) Moving toward larger values will not provide (C2) if theloop/walk goes past vsφd = vK /2. Moving to small values will not provide (C2) if the loop/walk

241

vsrd

vphi,g < vK

vsphidC3

C2
C1

Figure B.9: Map of the solution-space for finding vsrd due to Epstein drag when there are threepossible solutions (in the vφ,g < vK /2 regime). Same conventions as in Figure B.8.

goes below vsrd = vP0 . Therefore, this search method has clear failure criteria. If, however, (C2) isfound, then a third solution must also exist, either between (C2) and vP 0 or (C2) and where theparabolic crosses vK /2. This provides clear bounds for a bisect method. However, numerically, the(C2) bound must be placed at least one error-margine away. If these later two solutions are within
an error-margin of each other, then they will have to be treated as a single solution.
Next, in the more difficut vφ,g < vK /2 regime, when vH0 > vP0 , there are either 0 or 2 solutions(either both on the upper-LH branch of the hyperbolic (D and E) or both on the lower-RH branch(F and G)). Fortunately, there are clear criteria for determining whether there are no solutionsin this regime, and if that is the case, my code simply sets the radial-drift velocity equal to theassymptote vrH = −vdrag . The schematic for seeking the first solution in this regime is shown in
Figure B.10. The first step is to start at the point where the parabolic crosses the v K /2 assymptote
and loop/walk toward smaller values. If the walk crosses below vH0 , then solutions (D) and (E)do not exist. Otherwise, loop/walking will locate solution (D). Solution (E) may then be foundwith bisection, setting on bound at vH0 , and searching for the other at vsrd < vsrdD . If solutions(D) and (E) do not exist, next is to loop/walk toward larger values from v P0 . If the walk crosses

242

vsrd

vsrd
vsphidDGvsphidF

E

Figure B.10: A map seeking vsrd solutions in the case that vφ,g < vK /2 and vH0 > vP0 . Looping
forward from two limits will either find the first of two solutions, (D) or (F), or will identify thecriteria for zero solutions. Plot uses same conventions as in Figure B.8.
above the assymptote vrH = −vdrag , then solutions (F) and (G) do not exist (and there are no
solutions). Otherwise, the loop/walk will reach solution (F), and solution (G) can be found by
bisection. Setting the bisection bounds for (G) are a bit tricky, however, as they must be > v srdF
and < vrH and not actually at either.
As complex as these schematics may seem, there are really quite clear paths to reaching all
obtainable solutions. Making a map to solve for vsrd in the case of Stokes drag would be harder.
This is primarily for two reasons. First, because the vsrd equations for Stokes drag are made more
complex by being spread over three regimes in Re (see §3.2.1), and the boundaries of those regimes
depend on one’s position in both vsrd and vsφd parameter space. And second, the Stokes-dragequations cannot be easily inverted, making the direction loop-walking fixed and even some of
the axis intersection bounds troublesome to locate. One possible approach would be to solve theequations in each of the thee Re regimes separately, and then sort out which solutions were viable.
However, each regime would still have its individual challenges.

243B.3.3

To Solve for the Disk Temperature at the Midplane by Balancing Energy
FluxHere, I present my schematic for finding an energy-balanced midplane-temperature solution (see§2.4). Because of the number and complexity of equations involved, my code does not account forfinding the full solution set. Instead, I have taken a simplified approach generally appropriate to
my protoplanetary-disk models, wherein there are usually only 1 or 3 possible solutions, as depicted
in Figure B.11.
The primary equations to itterate to find an energy-balanced midplane temperature are (see§2.4)
4
σB Tmid

=

1 2
R νΣg
2

µ

∂Ωg
∂R

¶2 µ

3
1
τR +
8
2τP

¶

¡ 4
¢
4
+ σB Tirr+ Tcloud,κR = κi ρai T bi ; with 8 regimes for κi , ai , and bi ,
s
kB T
1
ν = α
,
ΩK µmHalso remembering that

τR

Σg ΩK
= √
2π
= κ R Σg

τP
τR

≈ 1

ρg,mid

r

µmH
kB T

(B.25)

This set of equations is not invertible, so, in the case of itteratively loop/walking to find a solution
in the space shown in Figure B.11, motion toward the energy-balance (blue) curve must be for agiven κR (and ν) and is left-right only, while motion toward the Bell-Lin opacity (green) curve
must be for a given T and is up-down only.
To begin looking for an energy-balanced temperature solution, I first mark a minimum guessed
temperature, Tmin , taken from the energy-balanced curve due to external luminosity only (ν = 0).
This then also gives a first guess for ν and κR . With this guessed minimum temperature, I next
consider the resultant κR (T ) curve and each of the intersection points between the different opacity
regimes. For each intersection point, I consider the (left-right) direction to the local energy-balance
temperature, TEB , and flag each point for which Tmin < Tpoint < TEB . (The change in the κR (T )244

log(kappaR)

A
A

A
C

B
C
B
C

log(T)

Figure B.11: Schematic for iteratively solving for an energy-balanced midplane disk temperatureplotted in κR versus T space. However, the solution must actually be found in three parameters,
including the disk viscosity, ν. Green curves plot the Rossalind-mean opacities κR obtained fromBell & Lin (1994) for a higher (dark) and lower (light) gas density, ρ g,mid ∝ T −1/2 . Blue curves
plot instances of the energy-balance equation (2.44) (for lower (darker) to higher (lighter) disk
viscosity, να ∝ T ).
slope at intersection points means that there is a chance for multiple crossings of the energy-balanceand opacity curves under that condition.) In Figure B.11, only the left-most set of blue curves would
have no flagged intersection points (so I would need to look for one solution (A) only).
Finally, I first seek a solution to the set using the simple loop/walk method of itteration startingat Tmin . For each guessed T , I advance both κR and ν before seeking the next T . This loop/walkmethod then produces either solution (A) or (C). If the solution is at T greater than any of the
flagged points, then it is solution (C) and I assume this is the only solution (as in the right-mostblue curve of Figure B.11). Otherwise it is solution (A) and, if there are flagged points, I next
search for solutions (B) and (C).
Solution (C) is at T greater than any of the flagged points, so the greatest flagged T marks
one of the bounds for a bisection search. The other bound must be searched for toward larger T .
Solution (B) can only be found using bisection, but the only clear bounds are that T A < TB < TC .
Therefore, the first bound must be guessed between TA and TC , and a second bound searched left
and right toward those solutions. And (B), of course, cannot be found separately if it is within an
accuracy margin of one of the other solutions.
245

Last, if there are multiple solutions found, my code picks the temperature solution closest to
the the previously calcuated temperatre (or Tmin if this is the first calculation).

246

Appendix C

Glossary of Physical Constants andSymbolic Variables

SPECIAL FUNCTIONS
SymbolFunction

Iβ

Modified Bessel function of the First Kind oforder βJβBessel function of the First Kind of order βδ (x − x0 )Dirac-delta function

Example Use
Equations (2.21) & (A.12)
Equation (A.8)
Equations (2.20), before (A.1)

Table C.1: Table of Special Functions

247

PHYSICAL CONSTANTS
Symbol

Value

Name

AU

1.4959787066 × 1013 cm

astronomical unitc2.99792458 × 1010 cm s−1

speed of light

Equation (3.2)

G

6.6742 × 10−8 cm3 g−1 s−2

the gravitational constantEquation (2.2)

kB

1.3806505x10−16 ergs K−1

Boltzmann’s constantEquation (2.4)

L¯

3.846 × 1033 ergs s−1

solar luminosity

M¯

1.988 × 1033 g

Solar mass

MEarth

5.9723 × 1027 g

Earth massmH

1.67262171 × 10−24 gproton mass

Equation (2.4)αB

2.6−13 cm3 s−1

recombination coefficient of
atomic Hydrogen

Equation (2.31)

σ13.6eV

6.3×10−18 cm2

absorbing cross-section for
EUV ionizing radiation

after Equation (2.32)

σB

5.6704×10−5 ergs s−1 cm−2
K−4

Stefan-Boltzman constantEquation (2.38)

σH2

2.4 × 10−15 cm2collisional cross-section ofmolecular Hydrogenπ3.1415926....

pi
Table C.2: Table of Physicals Constants

248

Example UseSYMBOLIC VARIABLES
SymbolCommon Units

Description

Example Usea–(negative) power-law constantfor power-law disk surface density

Equation (A.1)

A

s−1

local rate of shearing in thedisk

Equation (2.15)

A (λ)

***

initial-condition fitting function for contaminant-diffusion
distribution

Equation (A.9)

ai–power-law for ρg in the Bell &Lin (1994) opacity parameterization

Equation (2.47)

aph–fitting constant for EUV photoevaporation

Equation (2.33)

Aph–fitting constant for EUV photoevaporation

Equation (2.31)b–power-law constant for powerlaw disk viscosity

Equation (A.1)

bi–power-law for T in the Bell &Lin (1994) opacity parameterization

Equation (2.47)

Bph–fitting constant for EUV photoevaporation

Equation (2.30)

Table C.3: Table of Symbolic Variables

249

SymbolCommon Units

Description

Example Use

bq T

–

power-law for time-scaling for
qT in the TPR fit to disk temperature

bTAU

–

power-law for time-scaling for
TAU in the TPR fit to disk
temperature

Equation (2.50), Table 2.1–

power-law of time-scaling of
TAU or qT for TPR fit to disk
temperature

Equations (2.50) & (2.51)

C–

concentration(localcontaminant-to-gas
mass
ratio) of diffusing contaminant

Equation (A.2)

C0

***

scaling constant for t = 0 contaminant concentration

before Equation (A.1)

bx

Equation (2.50), Table 2.1& (A.10)
C1,ph–fitting constant for EUV photoevaporation

Equation (2.31)

C2,ph

–

fitting constant for EUV photoevaporation

Equation (2.35)

CD–

coefficient of Stokes dragEquation (3.14)

CN–

the normalized concentrationof a contaminant in a region(the outer disk in this thesis)

Equation (4.2)

CR

cm2 g−1

surface-area-to-mass ratio of adust particle

Equation (3.10)

CR,maxv

cm2 g−1

CR that gives fastest inward
drift from headwind dragEquation (3.23)

CR,steadycm2 g−1

CR that balances drag for astable orbit

Equation (3.22)

Table C.3 continued: Symbolic Variables

250

SymbolCommon Units

Description

Example Usecs

km s−1

local sound speed

Equation (2.4)

cs,ionz

km s−1

sound speed of ionized gas

Equation (2.29)

Dg

cm2 s−1

diffusivity of the gas

Equation (3.30)

Dkg

cm2 s−1

diffusivity of a contaminant

Equation (A.2)

Dp

cm2 s−1

diffusivity of the particle ensemble

Equation (3.29)

Dp,CDC1993

cm2 s−1

particle-ensemble diffusivity
according to Cuzzi, Dobrovolskis, & Champney (1993)

Equation (3.34)

Dp,YL2007

cm2 s−1

particle-ensemble
diffusivity according to Youdin &Lithwick (2007)

Equation (3.33)

Dph–fitting constant for EUV photoevaporation

Equation (2.30)

E–

local enhancement factor ofthe dust-to-gas ratio

Equation (6.5)

Eprecip–

enhancement required for collapse/precipitation

Equation (6.19)

0
Eprecip–

enhancement required for precipitation using Ri0c

Equation (6.25)

EQp–

enhancement required for direct collapse of particle layer

Equation (6.21)

0
EQ
p

–

enhancement required for direct collapse of particle layer

Equation (6.26)

dFν

g s−2

viscous force per unit length

Equation (2.15)

Fcloudergs s−1 cm−2

energy-flux of molecular-cloudillumination

Equations (2.37) & (2.40)

Table C.3 continued: Symbolic Variables

251

SymbolCommon Units

Description

Fcool

ergs s−1 cm−2

energy-flux of disk coolingFDg cm s−2

force of gas dragFD,rg cm s−2

radial component of the gasdrag force

Equation (3.7)

FD,φg cm s−2

azimuthal component of thegas-drag force

Equation (3.8)

fph–smoothing function for the
direct-photoevaporation wind

Equations (2.33) & (2.34)

Fph

g cm s−2

force of photophoresis

Equation (3.3)

Fradg cm s−2

force of radiation pressure

Equation (3.2)

Fstarergs s−1 cm−2

energy-flux of stellar illumination

Equations (2.37) & (2.39)

Fvisc

ergs s−1 cm−2

energy-flux of disk viscousheating

Equations (2.37) & (2.42)

g (R)

***

R-dependence ofseparated C (R, t)Gνg cm2 s−2

local viscous torque

h (ψ)

–

H0

Example Use
Equations (2.37) & (2.38)
below Equation (3.8)

variable-

Equation (A.5)
Equations (2.14) & (2.15)

gravitational scaling for critical particle layer

Equation (6.20)

scaling constant for power-law
disk scale-height

Equation (2.10)

HgAU

local disk-gas scale-heightHpAUparticle scale-height in thedisk

Equation (3.4)

Hp,c

cm

height of critical particle layer

Equation (6.20)

Equations (2.3) & (2.4)

Table C.3 continued: Symbolic Variables

252

SymbolCommon Units

Description

Example Use–

index variable of disk-model
grid cells

Equation (2.23)

i0–

index variable contracted gridspace: one i0 corresponds toseveral i

Equation (6.10)

I

ergs s−1 cm−2

intensity of incident light

Equation (3.2)

j–index variable for
evolution time steps

J1–

assymetry parameter for calculating photophoresis

Equation (3.3)

k***change-of-variables constantEquation (A.7)

Kdiff

g cm−2 s−1/2

scaling constant for the
diffuse-photoevaporation wind

Equations (2.30) & (2.31)

Kdir

g cm−1/2 s−1/2

scaling constant for directwind photoevaporation

Equations (2.33) & (2.35)

kthergs s−1 cm−1 K−1

l (L)g cm1/2L?

L¯ithermalgrain

conductivity

disk-

of

Equation (2.24)

anormalized
angularmomentum measure

Equation (3.3)

Figure (B.3)

stellar luminosity

Equation (2.39)

L1,ig cm2 s−1

less-precise
momentumgrid-cell i

Equation (B.7)

L2,ig cm2 s−1

more-precise
angularmomentum calculation in
grid-cell i

Equation (B.8)

mgmass of a (growing) dust grain

Equation (3.35)

angularcalculation in

Table C.3 continued: Symbolic Variables

253

Description

SymbolCommon Units

Example Use

m1 , m2g

massesoftwocolliding/coagulaing dust grains

before Equation (3.35)

M1,i

g

less-precise mass calculation in
grid-cell i

Equation (B.7)

M2,i

g

more-precise mass calculationin grid-cell i

Equation (B.8)

M?

M¯

mass of the central star

Equation (2.2)

ṀM¯ yr−1

local accretion rate

Equation (2.19)

Ṁ0

M¯ yr−1

initial disk accretion rate

Equation (2.1)

mdgmass of a dust particle

Equation (3.7)

MD,0

M¯

initial disk mass

Equation (6.4)

Mdummygmass viscously transfered into
a dummy grid-cell with nomass initially in it

Mgg

disk-gas mass

Equation (3.32)

Mg,i

g

gas mass in grid-cell i

Equation (6.10)mGE

ggas-equivalent mass of a dust
particle

after Equation (6.5)

MGEg(source-) gas-equivalent massof dust particles

Equation (6.5)

MGE,i

g

gas-equivalent mass of dust
particles in grid-cell i

Equation (6.10)

migcontaminant mass in grid-celliEquation (A.15)

mtot

g

total mass of diffusing contaminant

Equation (A.15)

before Equation (B.9)

Table C.3 continued: Symbolic Variables

254

SymbolCommon Units

Description

Example Use

n

cm−3

gas particle number density

ngrid

–

number of disk-model radialgrid cells

Equation (2.23)

np# cm−3

number density of particles

Equation (6.6)

P

g cm−1 s−2

local gas pressure

Equation (2.2)

p–power-law–index
variablechangefor
diffusion-of-acontaminant solution

Equation (A.5)

pin–probability of a particle diffusively stepping radially inward

Equation (3.32)

Pkg cm−2 s−1

production rate of a contaminant

Equation (A.2)

pout

–

probability of a particle diffusively stepping radially outward

Equation (3.32)

q–change-of-variables power-law
constant

Equation (A.7)

qH–

disk scale-height power-law
constant in R

Equation (2.10)

Qp–

Q-collapse criterion of particle
layer

Equation (6.20)

qs–

grain-size distribution powerlaw index

Equation (6.6)

qT–

disk temperature power-law
constant in R

Equation (2.10)

qT,0

–

qT at t = 0

Equation (2.50), Table 2.1

qT,∞

–

qT at t = ∞Equation (2.50), Table 2.1

after Equation (2.2)

Table C.3 continued: Symbolic Variables

255

SymbolCommon Units

Description

Example Useqν–

disk viscosity power-law scaling in R

Equation (2.10)qΣ–

local power-law constant forthe disk surface density

Equation (2.54)

RAU

radial distance from the central star

Equation (2.1)

R0AU

initial radial position of a diffusing quantity

Equation 2.20

rdcm

dust radial distance from thestar

Equation (3.1)

RdAU

initial-disk exponential fall-off
radiusEquation (2.1)

Re–

Reynold’s number

Equation (3.12)

Redge

AU

outer edge of photoevaporative gap

Equation (2.32)

Rfin

cm

radial position of a particle atthe end of a time-step§3.3.2Rg

AUradius where an ionized (hot)gas is gravitationally unboundfrom the star

Equation (2.29)

Rgap

AU

location where EUV photoevaporative gap opens

before Equation (2.32)

Ri–

Richardson number

before Equation (6.15)

Ric–

critical Richardson number

Equation (6.15)

Ri0c

–

critical Richardson number
scaled to E following Lee et al.
(2010)

Equation (6.24)

Ri

AU or cmradial position of grid-cell i

Equation (2.23)

Table C.3 continued: Symbolic Variables

256

Description

SymbolCommon Units

Example Use

Ri+1/2cm

radial-position of the outerboundary of grid-cell i

Equation (B.8)

Rin

AU

(steady) inner edge of gas disk

Equation (2.1)

Rmax

AU

outer-most disk-grid-cell position

Equation (2.23)

Rmin

AU

inner-most disk-grid-cell position

Equation (2.23)

Rout

AU

∼ outer observable edge of agas disk§2.5

Rstartcm

radial position of a particle atthe beginning of a time-step§3.3.2Rthin

AU

inner edge of photevaporative
gap

Equation (2.32)

s***spatialchange-of-variables
for
contaminant-diffusion
distribution

Equation (A.11)

s0

***

s for R0 for contaminantdiffusion distribution

Equation (A.16)

s (ψ)

–

gravitational scaling for collapse criterion

Equation (6.15)

sbmax

cm

maximum grain radius of agiven grain-size bin

Equation (6.8)

sbmin

cm

minimum grain radius of agiven grain-size bin

Equation (6.8)

Sc–

the Schmidt number: ratio of
disk viscosity to the gas diffusivity

before Equation (3.33)

sd

cm

radius of a dust particle

Table C.3 continued: Symbolic Variables

257

Equation (3.2)

Description

SymbolCommon Units

Example Use

sd1 , sd2

cm

radii
of
two
colliding/coagulating dust grains

Equation (3.36)

smax

cm

maximum grain-radius for the
dust-size distribution

Equation (6.7)

smin

cm

minimum grain-radius for the
dust-size distribution

Equation (6.7)

St–

the Stokes number: ratio of
the gas-drag stopping to theeddy turn-over times

Equation (3.34)

t

years

time

Equation (2.1)

t0

yrtime (relative to t = 0 of thedisk evolution) that particles
are initiated in the transport
simulations

§5.3

teddy

s

time-scale for one eddy turnover

before Equation (3.33)

torbitsorbital period

before Equation (3.23)

tstop

s

gas-drag stopping time of aparticle

Equations (3.4) & (3.5)

tTAU

years

time-scaling for TAU in the
TPR fit to disk temperature

Equation (2.50), Table 2.1

t qT

years

time-scaling for qT in the TPR
fit to disk temperature

Equation (2.50), Table 2.1

tx

years

time-scaling for TAU or qT in
the evolving-power-law fit ofdisk temperature

Equations (2.50) & (2.52)

TK

local disk temperature

Equation (2.4)

T0

***

scaling constant for power-law
disk temperature

Equation (2.10)

Table C.3 continued: Symbolic Variables

258

SymbolCommon Units

Description

Example Use

TAUKdisk temperature at 1 AU

TAU,0

K

TAU at t = 0

Equation (2.50), Table 2.1

TAU,∞

K

TAU at t = ∞Equation (2.50), Table 2.1

TboundKboundary temperature between opacity regimes in the
Bell & Lin (1994) parameterization

after Equation (2.47)

TcloudKtemperature of the (molecular
cloud) disk environment

Equation (2.40)

TeK

effective radiative temperatureof the disk

Equation (2.38)

Tirr

K

effective temperaturestellar irradiation

Equation (2.39)

Tmid

K

midplane disk temperaturev

cm s−1

background contaminant radial velocity§3.3.2v0***

scaling-constant for power-law
radial velocity

Equation (A.1)

vK

cm s−1

Keplerian velocity

Equation (3.5)

vR

cm s−1

1D disk-model or contaminantradial velocity

Equations (2.13) & (A.1)

vr,acccm s−1

vr,g in the accretion-flow case

Equation (2.55)

vr,dcm s−1

radial dust velocity

Equation (3.5)

vr,dragcm s−1dust radial velocity induced bygas drag

Equation (3.28)

vr,driftcm s−1dust radial-drift velocity fromheadwind dragEquation (3.25)

Equation (2.49)

fromEquation (2.44)

Table C.3 continued: Symbolic Variables

259

SymbolCommon Units

Description

Example Usevr,g

cm s−1

radial gas velocity

Equation (2.56)

vr,merid

cm s−1

vr,g in the midplane-flow case

Equation (2.57)

vsettlecm s−1settling (toward the midplane)
velocity of a dust particle in agas disk

Equation (3.37)

vsrd

cm s−1dust steady mean-radial velocity

Equations (3.1) & (3.24)

vsφdcm s−1steady azimuthal dust velocity

Equation (3.24)

vthermcm s−1

thermal velocity of the gas
particles

Equation (3.10)

vturb

cm s−1dust turbulent random-walkvelocity

Equations (3.1) & (3.29)vz,gcm s−1

vertical (midplane normal) gas
velocity

Equations (2.56) & (2.58)vφ,dcm s−1

azimuthal (orbital) dust velocity

Equation (3.5)

vφ,gcm s−1

azimuthal (orbital) velocity ofthe disk gas

after Equation (2.54)

x***spatial change-of-variables forcontaminant diffusion

Equation (A.7)

x0

K or –TAU or qT at t = 0

Equation (2.50)

x∞

K or –TAU or qT at t = ∞Equation (2.50)

X

cm1/2

spatial variable used in the
disk-evolution equation

Equation (2.17)

Xi√√
AU or cmX-spatial position of grid-celliEquation (2.23)

Table C.3 continued: Symbolic Variables

260

SymbolCommon Units

Description

Example UseY***spatial change-of-variables forcontaminant-distribution solutionz

cm

height above the disk midplane

Equation (2.2)

Z–

disk metallicity (dust-to-gas
surface-density ratio)

Equation (3.38)

Z¯–

solar/initial dust-to-gas ratio(metallicity)

Equation (6.13)

Z0–

ratio of all-condensibles-to-gas(base metallicity)

Equation (6.13)

Zrel

–

relative fraction of all condensibles solid at a given disk temperature

Equation (6.13)

α–alpha-scaling for disk viscosity

Equation (2.8)αc–power-law constant used in solution for analytic diffusion

Equation (2.21)

β–order of the modified Bessel
function solution to diffusive
evolution

Equation (2.21)

χph–scaling constant for Φdir

Equation (2.36)

χR–

radial position scaled by Rg

Equation (2.30)

g cm2 s−1

extra angular-momentum created (or destroyed) in one numeric viscous-evolution timestep from grid-cell i

Equation (B.9)

δLi

Table C.3 continued: Symbolic Variables

261

Equation (A.18)

SymbolCommon Units

Description

Example UseδMigextra mass created (or destroyed) in one numericviscous-evolution
time-stepfrom grid-cell i∆Radvectcm

displacement of particle due togas-drag advection

∆Ri

cm

width of grid-cell i

Equation (B.7)

∆Rrms

cm

root-mean-square of particle
diffusive translation

Equation (3.31)cm

displacement of particle due
to diffusive random-walk stepping§3.3.2∆RturbEquation (B.9)§3.3.2∆t

years or s

size of various time-steps∆tevolv

years or stime-step of disk-model viscous evolution

after Equation (3.32)∆v

cm s−1collisional velocity for dustgrain coagulation

Equation (3.35)∆vB

cm s−1collisional velocity for dust coagulation via Brownian motion

Equation (3.39)

∆Xi

√
cm

width in X of grid-cell i

Equation (2.24)∆Xi+1/2√cm

distance in X between gridcells i and i + 1

Equation (2.24)²–

radiative efficiency of the diskgasEquation (2.38)²ρ–local-dust-to-gas
density ratio

Equation (3.27)

Equations (2.24) & (2.25)

volume-Table C.3 continued: Symbolic Variables

262

SymbolCommon Units

Description

Example Use–

dust-to-gas density ratio at the
midplaneEquation (6.22)

φ–incidence angle between disk
surface and stellar illumination

Equation (2.39)

Φphotons s−1

EUV ionization flux

Equation (2.32)

Φdiffphotons s−1

ionization flux for diffuse-windphotoevaporation

Equation (2.30)

Φdirphotons s−1

EUV ionization flux for directwind photoevaporation

Equations (2.33) & (2.36)–

power-lawconstant
contaminant-distribution
solution

in

γ1

–

power-lawconstant
contaminant-distribution
solution

in

γ2

η

g cm−1 s−1

kinematic viscosity of the gas

Equation (3.13)

ηδφ–normalized difference between
Keplerian and gas orbital velocities

Equation (3.27)

κi

cm2 g−1

local opacity scaling in the
Bell & Lin (1994) parameterization

Equation (2.47)

κR

cm2 g−1

Rossalind mean opacity

Equation (2.45)

λ–separation constant for separation of variables

Equation (A.5)

λmf

cm

mean-free path of a gas particle

below Equation (3.20)²ρ,0Equation (A.16)

Table C.3 continued: Symbolic Variables

263

Equation (A.16)

SymbolCommon Units

Description

Example Useµ–average gas-particle mass

Equation (2.4)

µionz

–

mH -scaled mass of an ionizedgas particle

Equation (2.31)

νcm2 s−1ν0***scaling constant for power-law
viscosity

Equation (A.1)

Ωg

s−1

local gas orbital/azimuthal velocity

Equation (2.14)

ΩK

s−1

Keplerian azimuthal angularvelocity

Equation (2.54)ΩK,mids−1local Keplerian velocity at the
midplaneEquation (2.4)Πη–

radial-pressure-gradient
rameterEquation (6.29)

ρdg cm−3dust-particle internal density

ρgg cm−3

local disk gas density

Equations (2.2) & (2.3)

ρg,midg cm−3

midplane gas density

Equations (2.3) & (2.6)ρpg cm−3

local dust-particle
densityρp,sd

g cm−3volume density of particles ina given sd size bin

Equation (6.8)σ

g cm−2surface density of diffusing
contaminant

before Equation (A.15)

Σ0

***

scaling constant for power-law
disk surface-density

Equation (A.1)

local disk viscosity

Equations (2.1) & (2.8)

pa-

Equation (3.25)

volumeEquation (6.7)

Table C.3 continued: Symbolic Variables

264

SymbolCommon Units

Description

Example Use

σc

cm2

collisional cross-section
dust-grain coagulationfor

σcrosscm2

collisional cross-section of agas particle

below Equation (3.11)

Σ̇diffuseg cm−2 s−1

loss rate of Σg due to diffusewind EUV photoevaporation

Equation (2.30)

Σ̇directg cm−2 s−1

loss rate of Σg due to directwind EUV photoevaporation

Equation (2.33)

Σgg cm−2

local disk-gas surface density

Equation (2.1)

Σg,0

***

initial scaling for a deltafunction in surface density

Equation (2.20)

Σjig cm−2

disk surface density in grid-celli at time-step j

Equation (2.28)

Σpg cm−2surface density of dust solids

Equation (6.12)

Σp,cg cm−3

critical particle surface density
for collapse/precipitation

Equation (6.15)

Σviscg cm−2the disk surface-density that
results from viscous evolution

Equation (2.28)

Σ̇windg cm−2 s−1

loss-rate of disk surface density due to photoevaporation

Equation (2.27)

Equations (3.35) & (3.36)

τe–the normalized eddy time

τP–

Plank optical depth

τR–

Rossalind mean optical depth

Equations (2.44) & (2.45)

τs–normalized gas-drag stoppingtime

Equation (3.27)

ψ–gravitational scaling constantEquation (6.17)

before Equation (3.33)
Equation (2.44)

Table C.3 continued: Symbolic Variables

265

Symbolζζ∆tCommon Units

Description

Example Use–

scale-constant between gas diffusivity and disk viscosity

after Equation (A.2)–

scaling of disk-evolution timesteps relative to the stability
criterion

Equation (2.26)

Table C.3 continued: Symbolic Variables
Units marked as:–
These variables are unitless.
***These variables have units that may change depending on the parameters of
their equations, or units that are important only within their given equations.

266