large mass ratio between pebbles and planetesimals, this process excites the
eccentricities of pebbles and damps the eccentricities of planetesimals.
Viscous stirring transfers angular momentum from planetesimals to pebbles,
raising $e$ for the pebbles.

Figure 15:  Snapshots of the eccentricity distribution of solids in a model
with $a_{0}$ = 75 au, $M_{0}$ = 45 $\rm M_{\oplus}$, and $f$ = 0 at five
epochs as listed in the legend. The large grey dot at $r$ = 1 cm and
$e=10^{-3}$ shows the starting point for a system composed only of pebbles.

Fig. 15 illustrates the impact of these processes for a calculation with
$m_{0}$ = 45 $\rm M_{\oplus}$ and $f$ = 0 at 75 au. Initially, all pebbles
have $e=10^{-3}$. During the first 1000 yr of evolution, collisions among
pebbles create 10 cm rocks and debris with $r\lesssim$ 1 mm. Damping reduces
the eccentricity of 1–100 $\rm\,\mu{\rm m}$ (3–10 cm) particles to $\sim
5-6\times 10^{-4}$ ($\sim 3\times 10^{-4}$). As pebbles continue to grow,
modest stirring by m-sized to km-sized planetesimals tries to raise $e$;
pebble damping continues to keep $e$ low.

This balance between the smallest and largest solids in the ring continues
until the runaway produces large protoplanets. At 10 Myr, several 0.25 $\rm
M_{\oplus}$ planets try to stir all of the lower mass solids. Solids with
$R\approx$ 100 m to 10 km have little mass; protoplanets effectively stir them
to larger $e$. Among smaller solids, damping maintains low $e$. At 100 Myr,
the smallest solids continue to resist viscous stirring by the largest
objects. Although 100 m and larger objects now have $e\sim$ 0.01–0.1, small
solids maintain $e\sim 2-4\times 10^{-3}$. Small fluctuations in $e$ for 0.1–1
m solids result from variations of the mass contained in each mass bin: bins
with smaller $e$ have more mass.

As the evolution proceeds to 1–10 Gyr, the cascade gradually reduces the
population of solids with $r\lesssim$ 1 m. With less mass among pebbles,
damping is less effective. For nearly all solids with $r\gtrsim$ 4 m,
$e\approx$ 0.1–0.2. Among the 1–4 m solids, damping maintains a small $e$. The
linear rise in the $e$ distribution from 1 m to 1 $\rm\,\mu{\rm m}$ results
from the smaller mass in these objects: damping maintains small $e$ for 1–4 m
objects but damping is less and less effective at smaller and smaller sizes.
By 10 Gyr, nearly all solids have roughly the same eccentricity, $e\approx$
0.3; the largest protoplanets have $e\approx$ 0.1.

Figure 16:  As in Fig. 15 for a system with $f=10^{-3}$. The two grey dots
indicate the initial $e$ for pebbles and planetesimals.

Fig. 16 shows snapshots of the eccentricity evolution for a massive ring with
planetesimals and pebbles. During the first Myr, growth is slow: mergers
produce a few objects with $r\approx$ 200–300 km and a swarm of smaller solids
with $r\lesssim$ 10 km. Collisional damping reduces $e$ by 10–20% for the
smaller solids; dynamical friction lowers $e$ by almost a factor of 100 for
the largest solids.

Over the next 30 Myr, the largest solids grow dramatically to nearly 10,000
km. With most of the mass, protoplanets stir themselves to larger and larger
$e$. Solids with $R\approx$ 0.1–10 km have little mass and are stirred to even
larger $e$. Despite having less mass, pebbles maintain smaller $e$ through
collisional damping. The combined effects of damping and stirring yield an
eccentricity distribution where the largest and the smallest solids have
$e\approx 2\times 10^{-3}$. Dynamical friction between pebbles and 100–300 km
solids keep the larger solids at very small $e\approx 10^{-4}$.

As the collisional cascade continues, pebbles contain less and less mass.
Damping and dynamical friction are less effective; viscous stirring dominates.
Eccentricities for all solids grow from $10^{-3}$ to 0.1–0.3. At either end of
the distribution, $e$ drops by a factor of 2–3. For the large solids,
dynamical friction with the rest of the solids reduces $e$. The smallest
solids have just enough mass for damping to reduce and maintain the smaller
$e$.

These two examples illustrate how damping among pebbles fuels runaway growth
of the largest planetesimals. Without damping, pebbles have a factor of 10–100
larger eccentricity. Larger eccentricity lowers gravitational focusing factors
and reduces the growth rate of the largest planetesimals. Runaway growth is
delayed and is much weaker. Instead of reaching super-Earth masses on 1–10 Myr
time scales, systems with no damping would produce super-Earths on 100 Myr to
Gyr time scales. Delaying the runaway allows smaller planetesimals to accrete
more pebbles, reducing the dust luminosity to levels below observed systems at
10–100 Myr. Once the runaway begins, a delayed collisional cascade would
probably raise the dust luminosity to levels well above those observed at 1–10
Gyr. In this way, the lack of collisional damping would challenge our ability
to match observations with an evolving swarm of pebbles and planetesimals.

Table 1: Results for Coagulation Calculations at 36–54 auaa The columns list the initial mass $M_{0}$ for each calculation, the initial fraction $f$ of mass in large planetesimals, the ratio of the final mass in the grid $M_{f}$ to the initial mass, the time scale $t_{1k}$ to produce the first protoplanet with $r\geq$ 1000 km, the final radius $r_{max}$ of the largest planet, the final number $N_{L}$ of protoplanets with masses at least as large as 10% of the mass of the largest planet, the total mass $M_{L}$ in these large protoplanets, and the maximum relative dust luminosity $L_{d}/L_{\star}$ in units of $10^{-5}$.  $M_{0}$ ($\rm M_{\oplus}$) | $f$ | $M_{f}/M_{0}$ |  $t_{1k}$ (Myr) |  $r_{max}$ (km) | $N_{L}$ |  $M_{L}$ ($\rm M_{\oplus}$) | $L_{d,max}/L_{\star}$
---|---|---|---|---|---|---|---
0.01 | 1.0 | 1.000 |  $\cdots$ | 167.5 | 8017 | 0.01 |  $<$0.01
0.01 | 0.3 | 0.979 |  $\cdots$ | 257.0 | 98 |  $<$0.01 | 0.50
0.01 | 0.1 | 0.986 |  $\cdots$ | 192.8 | 869 |  $<$0.01 | 0.58
0.01 | $10^{-2}$ | 0.993 |  $\cdots$ | 295.1 | 75 |  $<$0.01 | 0.61
0.01 | 0.0 | 0.993 |  $\cdots$ | 453.9 | 5 |  $<$0.01 | 0.62
0.03 | 1.0 | 1.000 |  $\cdots$ | 199.5 | 23714 | 0.03 | 0.01
0.03 | 0.3 | 0.933 |  $\cdots$ | 639.7 | 9 |  $<$0.01 | 1.02
0.03 | 0.1 | 0.940 |  $\cdots$ | 638.3 | 15 |  $<$0.01 | 1.04
0.03 | $10^{-2}$ | 0.885 |  $\cdots$ | 540.8 | 178 | 0.01 | 1.04
0.03 | $10^{-3}$ | 0.545 | 7067.8 | 1049.5 | 17 | 0.01 | 1.05
0.03 | 0.0 | 0.962 |  $\cdots$ | 736.2 | 7 |  $<$0.01 | 1.05
0.10 | 1.0 | 1.000 |  $\cdots$ | 206.1 | 78705 | 0.10 | 0.01
0.10 | 0.3 | 0.828 | 8851.2 | 1047.1 | 8 |  $<$0.01 | 2.15
0.10 | 0.1 | 0.625 | 3143.6 | 1538.2 | 2 | 0.01 | 2.01
0.10 | $10^{-2}$ | 0.495 | 5601.9 | 1039.9 | 37 | 0.01 | 1.63
0.10 | $10^{-3}$ | 0.312 | 2293.1 | 1207.8 | 40 | 0.02 | 2.37
0.10 | 0.0 | 0.468 | 1402.8 | 1367.7 | 17 | 0.02 | 2.05
0.30 | 1.0 | 1.000 |  $\cdots$ | 434.5 | 1683 | 0.02 | 0.03
0.30 | 0.3 | 0.774 | 1453.5 | 1552.4 | 10 | 0.01 | 4.95
0.30 | 0.1 | 0.587 | 820.4 | 2113.5 | 9 | 0.03 | 4.30
0.30 | $10^{-2}$ | 0.319 | 659.9 | 2387.8 | 3 | 0.02 | 5.16
0.30 | $10^{-3}$ | 0.254 | 550.4 | 1836.5 | 12 | 0.03 | 6.34
0.30 | $10^{-4}$ | 0.341 | 95.4 | 2138.0 | 15 | 0.04 | 8.32
0.30 | 0.0 | 0.752 | 317.4 | 1625.5 | 5 | 0.01 | 2.51
1.00 | 1.0 | 0.998 |  $\cdots$ | 517.6 | 6295 | 0.17 | 0.20
1.00 | 0.3 | 0.750 | 427.6 | 2128.1 | 11 | 0.05 | 13.30
1.00 | 0.1 | 0.547 | 229.0 | 3076.1 | 9 | 0.09 | 10.57
1.00 | $10^{-2}$ | 0.272 | 149.8 | 3639.2 | 5 | 0.12 | 17.22
1.00 | $10^{-3}$ | 0.217 | 129.5 | 2393.3 | 6 | 0.04 | 20.04
1.00 | $10^{-4}$ | 0.186 | 30.6 | 3191.5 | 2 | 0.04 | 26.61
1.00 | 0.0 | 0.267 | 39.4 | 2055.9 | 10 | 0.04 | 7.57
3.00 | 1.0 | 0.993 | 4818.2 | 2157.7 | 49 | 0.11 | 0.66
3.00 | 0.3 | 0.676 | 98.7 | 3243.4 | 3 | 0.06 | 32.21
3.00 | 0.1 | 0.520 | 87.5 | 4385.3 | 4 | 0.16 | 29.65
3.00 | $10^{-2}$ | 0.234 | 66.6 | 5407.5 | 5 | 0.28 | 65.61
3.00 | $10^{-3}$ | 0.168 | 51.9 | 5211.9 | 2 | 0.16 | 65.92
3.00 | $10^{-4}$ | 0.136 | 19.6 | 2747.9 | 46 | 0.22 | 86.50
3.00 | $10^{-5}$ | 0.094 | 0.6 | 3396.3 | 17 | 0.27 | 110.41
3.00 | 0.0 | 0.118 | 3.1 | 4335.1 | 6 | 0.15 | 44.16
10.00 | 1.0 | 0.962 | 716.9 | 4375.2 | 22 | 0.66 | 2.18
10.00 | 0.3 | 0.561 | 19.8 | 4168.7 | 20 | 0.50 | 119.40
10.00 | 0.1 | 0.468 | 24.9 | 4405.5 | 13 | 0.48 | 132.74
10.00 | $10^{-2}$ | 0.211 | 14.5 | 6280.6 | 5 | 0.58 | 154.17
10.00 | $10^{-3}$ | 0.143 | 15.8 | 6683.4 | 6 | 0.85 | 350.75
10.00 | $10^{-4}$ | 0.140 | 6.2 | 3784.4 | 22 | 0.28 | 239.88
10.00 | $10^{-5}$ | 0.091 | 0.3 | 3097.4 | 75 | 0.81 | 340.41
10.00 | 0.0 | 0.047 | 1.0 | 3810.7 | 4 | 0.23 | 127.94
Table 2: Results for Coagulation Calculations at 60–90 auaa The columns list the initial mass $M_{0}$ for each calculation, the initial fraction $f$ of mass in large planetesimals, the ratio of the final mass in the grid $M_{f}$ to the initial mass, the time scale $t_{1k}$ to produce the first protoplanet with $r\geq$ 1000 km, the final radius $r_{max}$ of the largest planet, the final number $N_{L}$ of protoplanets with masses at least as large as 10% of the mass of the largest planet, the total mass $M_{L}$ in these large protoplanets, and the maximum relative dust luminosity $L_{d}/L_{\star}$ in units of $10^{-5}$.  $M_{0}$ ($\rm M_{\oplus}$) | $f$ | $M_{f}/M_{0}$ |  $t_{1k}$ (Myr) |  $r_{max}$ (km) | $N_{L}$ |  $M_{L}$ ($\rm M_{\oplus}$) | $L_{d,max}/L_{\star}$
---|---|---|---|---|---|---|---
0.01 | 1.0 | 1.000 |  $\cdots$ | 152.4 | 8035 | 0.01 |  $<$0.01
0.01 | 0.5 | 0.995 |  $\cdots$ | 153.8 | 4121 | 0.01 | 0.12
0.01 | 0.1 | 0.998 |  $\cdots$ | 153.1 | 914 |  $<$0.01 | 0.21
0.01 | $10^{-2}$ | 0.998 |  $\cdots$ | 195.9 | 81 |  $<$0.01 | 0.23
0.01 | 0.0 | 1.000 |  $\cdots$ | 333.4 | 2 |  $<$0.01 | 0.23
0.10 | 1.0 | 1.000 |  $\cdots$ | 157.8 | 79616 | 0.10 |  $<$0.01
0.10 | 0.5 | 0.959 |  $\cdots$ | 358.9 | 272 |  $<$0.01 | 1.20
0.10 | 0.1 | 0.973 |  $\cdots$ | 474.2 | 49 |  $<$0.01 | 1.49
0.10 | $10^{-2}$ | 0.979 |  $\cdots$ | 421.7 | 411 | 0.01 | 1.51
0.10 | $10^{-3}$ | 0.916 |  $\cdots$ | 901.6 | 43 | 0.01 | 1.51
0.10 | 0.0 | 0.984 |  $\cdots$ | 772.7 | 2 |  $<$0.01 | 1.51
0.50 | 1.0 | 1.000 |  $\cdots$ | 209.4 | 433511 | 0.05 |  $<$0.01
0.50 | 0.5 | 0.883 |  $\cdots$ | 863.0 | 56 | 0.01 | 4.46
0.50 | 0.1 | 0.820 | 4135.2 | 1548.8 | 7 | 0.01 | 4.06
0.50 | $10^{-2}$ | 0.361 | 3203.7 | 2249.1 | 5 | 0.03 | 3.79
0.50 | $10^{-3}$ | 0.277 | 2164.2 | 1927.5 | 3 | 0.01 | 3.82
0.50 | $10^{-4}$ | 0.207 | 685.6 | 2393.3 | 8 | 0.03 | 5.51
0.50 | 0.0 | 0.962 | 1174.9 | 1188.5 | 6 | 0.01 | 2.65
1.50 | 1.0 | 1.000 |  $\cdots$ | 304.8 | 54576 | 0.29 | 0.07
1.50 | 0.5 | 0.804 | 2807.4 | 1745.8 | 21 | 0.02 | 10.54
1.50 | 0.1 | 0.659 | 2629.0 | 2118.4 | 5 | 0.03 | 8.79
1.50 | $10^{-2}$ | 0.217 | 716.6 | 3372.9 | 2 | 0.04 | 9.33
1.50 | $10^{-3}$ | 0.194 | 838.4 | 3475.4 | 1 | 0.04 | 10.79
1.50 | $10^{-4}$ | 0.180 | 601.2 | 2306.7 | 36 | 0.09 | 15.67
1.50 | $10^{-5}$ | 0.585 | 3.7 | 2685.3 | 6 | 0.06 | 11.04
1.50 | 0.0 | 1.000 |  $\cdots$ | 304.8 | 54576 | 0.29 | 0.07
5.00 | 1.0 | 0.998 |  $\cdots$ | 691.8 | 6209 | 0.34 | 0.33
5.00 | 0.5 | 0.743 | 401.8 | 1892.3 | 133 | 0.20 | 25.41
5.00 | 0.1 | 0.521 | 272.1 | 4405.5 | 7 | 0.20 | 20.51
5.00 | $10^{-2}$ | 0.229 | 245.7 | 5533.5 | 4 | 0.46 | 31.05
5.00 | $10^{-3}$ | 0.153 | 185.7 | 5546.3 | 2 | 0.22 | 50.47
5.00 | $10^{-4}$ | 0.106 | 73.5 | 2831.4 | 86 | 0.33 | 43.75
5.00 | $10^{-5}$ | 0.081 | 1.4 | 3507.5 | 22 | 0.27 | 69.34
5.00 | 0.0 | 0.256 | 10.1 | 3499.5 | 8 | 0.13 | 20.65
15.00 | 1.0 | 0.995 | 2753.5 | 3404.1 | 11 | 0.27 | 1.14
15.00 | 0.5 | 0.671 | 95.1 | 3715.4 | 10 | 0.25 | 62.95
15.00 | 0.1 | 0.459 | 79.8 | 5081.6 | 5 | 0.22 | 76.74
15.00 | $10^{-2}$ | 0.180 | 79.3 | 7979.9 | 4 | 0.92 | 137.72
15.00 | $10^{-3}$ | 0.116 | 69.2 | 6745.3 | 7 | 0.98 | 233.88
15.00 | $10^{-4}$ | 0.099 | 29.8 | 3435.6 | 45 | 0.40 | 153.82
15.00 | $10^{-5}$ | 0.076 | 0.7 | 3396.3 | 110 | 1.11 | 163.68
15.00 | 1.0 | 0.152 | 4.7 | 4285.5 | 11 | 0.34 | 70.63
45.00 | 1.0 | 0.979 | 853.7 | 4909.1 | 18 | 1.53 | 3.37
45.00 | 0.5 | 0.667 | 28.9 | 5382.7 | 16 | 0.97 | 195.88
45.00 | 0.1 | 0.962 |  $\cdots$ | 861.0 | 3289 | 0.35 | 163.31
45.00 | $10^{-2}$ | 0.986 | 35.7 | 1253.1 | 7 | 0.01 | 103.04
45.00 | $10^{-3}$ | 0.100 | 19.0 | 9397.2 | 7 | 2.23 | 457.09
45.00 | $10^{-4}$ | 0.085 | 10.1 | 12302.7 | 2 | 2.82 | 941.89
45.00 | $10^{-5}$ | 0.083 | 0.3 | 3265.9 | 279 | 3.36 | 613.76
45.00 | 0.0 | 0.025 | 1.5 | 7079.5 | 5 | 0.86 | 309.03

Tables of Results. To facilitate comparisons between these calculations and
those in other studies, Tables 1–2 summarize results for each calculation. The
first two columns list $M_{0}$ and $f$. In the third column the ratio of the
final mass $M_{f}$ to $M_{0}$ provides a measure of the mass ejected in the
form of 1 $\rm\,\mu{\rm m}$ and smaller particles. In energetic (weak)
cascades, $M_{f}/M_{0}$ is small (large). Rings with small $f$ lose much more
material than those with large $f$. The next columns measure the ability of
collisional growth to concentrate solids into massive planets. The variable
$t_{1k}$ in column 4 quantifies the time scale for the growth of Pluto-mass
planets; $r_{max}$ in column 5 lists the radius of the largest protoplanet at
10 Gyr. Rings that generate super-Earths (Plutos) have large (small) $r_{max}$
and small (large) $t_{1k}$. The next two columns quantify the amount of mass
in the largest objects at the end of each calculation: $N_{L}$ ($M_{L}$) is
the number (mass) of objects with masses no less than 10% of the mass of the
largest object. Sometimes, the calculation produces 1–3 large protoplanets;
others with similar amounts of mass in large objects generate many much
smaller protoplanets. These two quantities allow a comparison of systems with
similar initial masses but very different $r_{max}$. Finally, the last column
quantifies the maximum dust luminosity throughout the calculation. As noted in
the main text, more massive rings have more luminous debris disks. This column
quantifies that statement.

Among the published numerical calculations of pebble accretion, only Shannon
et al. (2015) consider the 1–10 Gyr evolution of a swarm of pebbles and
planetesimals. Starting with $M_{0}$ = 0.01 $\rm M_{\oplus}$ in a ring at
42–48 au with $f=10^{-3}$ for pebbles and km-sized planetesimals, they follow
the growth of larger solids with a single annulus coagulation calculation.
With a much smaller initial eccentricity $e_{0}=10^{-6}$ and binding energy
for solids ($Q_{D}^{\star}$, eq. A3), large objects in these models grow more
slowly and fail to reach $r_{max}$ $\gtrsim$ 1000 km in 10 Gyr. in contrast,
For the initial conditions considered here, results are similar: $r_{max}$
$\approx$ 1000 km at 1–2 Gyr and $\sim$ 2000 km at 10 Gyr.

Growth Time as a Function of Initial Mass. To conclude this sub-section, Fig.
17 plots values for $t_{1k}$ in Tables 1–2 as a function of $M_{0}$ for rings
at 45 au (lower panel) and 75 au (upper panel). The data show clear relations
between $t_{1k}$ and $M_{0}$. For calculations with the same $f$, the time
scale to form Pluto-mass protoplanets scales inversely with the initial mass
of solids, $t_{1k}\propto M_{0}^{-1}$. Among the full ensemble, there are
three trends. Systems with $f$ = 1 ($f\lesssim 10^{-5}$) have the longest
(shortest) formation times, with a three order of magnitude difference
independent of $M_{0}$. Midway between, systems with $10^{-4}\lesssim
f\lesssim 0.5$ have a factor of $\sim$ 30 shorter (longer) formation time than
those with $f$ = 1 ($f\lesssim 10^{-5}$). The short formation times with small
$f$ is a hallmark of pebble accretion (Goldreich et al., 2004; Rafikov, 2005).

The relation for the slow growth rates for systems with $f$ = 1 is
straightforward to derive. In any swarm of solids, the growth time for a solid
of radius $r$ is $t_{c}\propto(rP/\Sigma)[1+(v_{esc}/v)^{2}]^{-1}$, where $P$
is the orbital period, $\Sigma$ is the surface density of solids, $v_{esc}$ is
the escape velocity of the pair of colliding solids, and $v$ is their
collision velocity (e.g., Safronov, 1969; Lissauer, 1987; Wetherill & Stewart,
1993; Goldreich et al., 2004; Rafikov, 2005; Kenyon & Bromley, 2008). When $f$
= 1, $v_{esc}\approx v$; $t_{c}\propto rP/\Sigma$. Compared to the prediction
in eq. 1 derived from scaling results in Kenyon & Bromley (2008), the growth
time scales listed in Tables 1–2 are a factor of two smaller. Analysis of each
calculation suggests a simple explanation: in the ring-like geometries
considered here, planetesimals are concentrated in the middle of the ring.
Collisions with other planetesimals outside the central annuli tend to
concentrate additional material in these annuli. This additional material
increases the surface density and lowers the growth time. In the full suite of
calculations, the typical increase in $\Sigma$ is a factor of two, accounting
for the factor of two smaller growth time.

Figure 17:  Evolution time $t_{1k}$ to form objects with $r_{max}$ $\gtrsim$
1000 km for each calculation in Table 1 (lower panel) and Table 2 (upper
panel). Filled circles indicate results for $f$ = 1 (black), 0.3 and 0.5
(purple), 0.1 (blue), 0.01 (dark green), 0.001 (light green), 0.0001 (orange),
0.00001 (orange-red), and 0 (maroon). Dashed lines show the relation
$t_{1k}\propto M_{0}^{-1}$ discussed in the text.

Although the smaller growth times for systems with $f<$ 1 are simple to
understand, deriving a quantitative relation is a challenge. In a system of
pebbles and planetesimals, damping reduces collision velocities. With $v\ll
v_{esc}$, $t_{c}\propto(rP/\Sigma)(v/v_{esc})^{2}$. To evaluate
$(v/v_{esc}^{2}$, there are two regimes (e.g., Goldreich et al., 2004;
Rafikov, 2005). In the ‘dispersion regime’, collision velocities exceed the
Hill velocity, $v_{H}=\Omega R_{H}$, where $\Omega$ is the angular velocity.
Analytic results then yield $(v/v_{esc})^{2}\propto f$. In the ‘shear regime’,
$v<v_{H}$; then $(v/v_{esc})^{2}\propto f^{1/2}$.

These theories predict the sense of the results in Tables 1–2, but not the
clustering of $t_{1k}$ for $10^{-4}\lesssim f\lesssim 0.5$ and the second
cluster of results for smaller $f$. When $f$ is roughly zero, growth is in the
shear regime and very rapid as summarized in the main text. Larger $f$ places
growth more in the dispersion regime, where growth is somewhat slower. Our
results suggest that the boundary between rapid and extremely rapid growth is
$f\approx 10^{-5}-10^{-4}$. However, current theory is insufficient to isolate
this boundary. At the same time, theory suggests that the growth time should
scale with $f$ for intermediate $f$ between 0 and 1 rather than a common
growth time for a range of $f$. Because relating $t_{1k}$ to the initial
properties of rings of solids is not a central goal of this project, we leave
a detailed investigation of this issue to a separate study.

##  Appendix D COMPARISONS WITH PREVIOUS CALCULATIONS

To illustrate the impact of differences between the calculations in this paper
and those in Kenyon & Bromley (2010), we consider several examples. In Kenyon
& Bromley (2010), solids with radii $r\gtrsim$ 1 m evolve within 64 annuli
extending from an inner radius of 30 au to an outer radius of 150 au. Solids
interact with a gaseous disk; gas drag circularizes orbits while radial drift
removes smaller particles from the grid. Initially, the gas-to-solid ratio is
roughly 1:100; during the evolutionary sequence, the gas mass declines
exponentially with an e-folding time scale of 10 Myr. Although Kenyon &
Bromley (2010) consider a broad range of initial conditions, here we focus on
calculations starting with a mono-disperse set of solids and a total mass
roughly comparable with the minimum mass solar nebula (solid and gas mass).

In the calculations for this paper, the minimum size of solids is 1
$\rm\,\mu{\rm m}$ instead of 1 m. Interactions with gas are ignored. The
smaller size allows collisional damping to play a role during the collisional
cascade (e.g., Kenyon & Bromley, 2015, 2016b, 2016a, see also Figs. 15–16).
When damping is important, the mass in 0.1 mm to 10 cm particles may grow with
time and allow a second phase of runaway growth for the largest particles.
Thus, some solids reach larger masses than in calculations without the small
particles. The lack of interactions with gas tends to slow the growth of the
largest particles when their initial sizes are 1–10 km.

Figure 18:  Evolution of the largest protoplanets (lower panel) and the
relative disk luminosity $L_{d}/L_{\star}$ (upper panel) in disks of solids at
30–150 au. Dotted curves show results from Kenyon & Bromley (2010); solid
curves follow growth with 1 $\rm\,\mu{\rm m}$ and larger particles but without
gas drag. In the lower panel, the legend indicates the initial radius for
swarms of monodisperse solids; dashed lines show the masses of Pluto (lower),
Mars (middle), and the Earth (upper). Systems with 1–10 km objects evolve
faster when gas is included in the calculations but reach roughly the same
maximum $L_{d}/L_{\star}$. Addition of small particles in the new calculations
allows planets to reach larger radii.

The lower panel of Fig. 18 compares results for calculations starting with
only 1 km, 10 km, or 100 km objects. At the start of each calculation,
planetesimals grow slowly. Over 1–3 Myr of evolution, gas drag gradually
results in larger gravitational focusing factors for the largest objects.
Despite the loss of collisional debris from radial drift, 1–10 km objects
reach Pluto masses in 30–100 Myr. After this spurt, gravitational interactions
between the swarm of Plutos and smaller planetesimals initiates a collisional
cascade, which grinds the leftovers into smaller and smaller objects. The loss
of material slows growth considerably. The largest objects approach the mass
of Mars but rarely grow larger.

When gas drag is not included, 1–10 km objects grow more slowly. It takes them
about three times longer to reach the mass of Pluto. Once they initiate the
collisional cascade, a modest amount of mass collects in mm- and cm-sized
objects which are stronger and more resistant to the cascade than m-sized and
larger solids. This reservoir allows Pluto-mass objects to grow beyond the
mass of Mars and reach the mass of the Earth. The time scale to reach Earth-
mass is $\sim$ 1–2 Gyr.

Gaseous disks have little impact on the collisional evolution of 100 km
planetesimals. On time scales of 100–300 Myr, these objects reach masses
comparable to Charon, Pluto’s binary partner. At this point, collisions
initiate a modest collisional cascade. In calculations with no sub-meter-sized
particles, the cascade effectively removes debris from the grid. Although
growth continues, the largest objects reach roughly the mass of Mars on 5–10
Gyr time scales. When the small particles are included, mass removal is less
efficient. Mars (Earth) mass planets then form on 1–2 Gyr (5–7 Gyr) time
scales.

Despite differences in approach, the evolution of the dust luminosity is
similar in the two sets of calculations (Fig. 18, upper panel). In Kenyon &
Bromley (2010), a second calculation uses the loss rate of sub-meter-sized
particles from the main coagulation calculation to derive the time evolution
of 1 $\rm\,\mu{\rm m}$ to 1 m particles; the surface area (and thus
reprocessed stellar luminosity) of these particles is therefore completely
distinct from the evolution of the largest particles in the grid. In the
calculations for this paper, the smallest particles evolve together with the
largest particles. This approach should provide a better measure of the
evolution of the dust luminosity.

The Figure illustrates that the dust luminosity follows the growth of the
largest objects in each set of calculations. early on, the largest particles
grow slowly; collisions generate little debris. In turn, the dust luminosity
is fairly small. As planetesimals grow more rapidly, collisions produce
copious amounts of dust; the dust luminosity rises in step with the growth of
the largest planetesimals. As the collisional cascade proceeds, there is less
and less solid mass in the grid. Growth slows; debris production declines; the
dust luminosity begins to fade.

For each pair of calculations, the maximum dust luminosity correlates well
with the initial sizes of planetesimals; $L_{d}/L_{\star}\approx 10^{-3}$ for
1 km planetesimals, $L_{d}/L_{\star}\approx 3\times 10^{-4}$ for 10 km
planetesimals, and $L_{d}/L_{\star}\approx 10^{-4}$ for 100 km planetesimals,
The timing of this peak is earlier for systems with gas drag and 1–10 km
planetesimals and is independent of gas drag for 100 km planeteimals.
Independent of the approach, all of the calculations have the same dust
luminosity at 10 Gyr, $L_{d}/L_{\star}$$\sim 10^{-4}$.

Although we do not compare these evolutionary tracks with the data in Fig. 3
for clarity, a simple comparison shows that the evolution of full disks
provides a poor match to the data. At 100 Myr, these models have
$L_{d}/L_{\star}$ $\sim 10^{-5}-10^{-3}$, as in observed systems. With
$L_{d}/L_{\star}$ $\approx 10^{-4}$ at 10 Gyr, all disk models decline much
more slowly than the observations where $L_{d}/L_{\star}$ $\sim$ $10^{-5}$ for
most sources. As discussed in the main text, rings of solids evolve in a
similar way as the observed systems.

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