it may have been more difficult to form multiple giant planets nearby, because
the inner one may have migrated inward and away from outer-growing
protoplanets. Ida et al. (2013) did not take account of secular perturbations
between giant planets in their Monte Carlo method for planet-planet scattering
either, which can also trigger orbital instabilities and is of course
automatically included in our N-body simulations. Furthermore, Ida et al.
(2013) explored a relatively narrow range of metallicities of
${\rm[Fe/H]}=[-0.2,\,0.2]$ as opposed to our ${\rm[Fe/H]}=[-0.5,\,0.5]$. Since
the number of giant planets per system is more sensitive to disc masses for
lower metallicities (e.g. see cases with ${\rm[Fe/H]}\lesssim 0.0$ in Figure
7), they may have generated a lower number of giant planets per system on the
average. Although, this effect may have been mitigated to an extent since they
covered a wide range of disc masses (spanning two orders of magnitude), and
the low metallicities can be compensated for by more massive discs. Also, it
is possible that the eccentricity damping effect was too strong in their
simulations, which prevented the occurrence of dynamical instabilities (see
next sub-section as well as Bitsch et al., 2020).

###  4.7 Comparison with Bitsch et al. (2019)

Here, we briefly compare our work with a similar N-body work on giant planet
formation by Bitsch et al. (2019). A direct comparison is not possible since
the details of their models are different from ours (e.g. equation of motion,
pebble and gas accretion models, pebble isolation mass). However, we list
differences in disc parameters below and compare their results with ours for
similar disc parameters. We find that there are both similarities and
differences.

Their gas accretion rate follows Hartmann et al. (1998) and Bitsch et al.
(2015), and it is the same as the formula adopted by Matsumura et al. (2017):

$\log\left(\frac{\dot{M}_{*,\,B19}}{M_{\odot}\,{\rm
yr}^{-1}}\right)=-8-\frac{7}{5}\log\left(\frac{t+10^{5}\,{\rm
yr}}{10^{6}\,{\rm yr}}\right).$ (52)

Although we did not use the formula for this paper, their accretion rate very
closely follows Disc 2 of our model (which corresponds to
$M_{d}=0.06\,M_{\odot}$ with $t_{{\rm diff}}=0.1\,$Myr and thus $\alpha_{{\rm
acc}}\sim 7.4\times 10^{-2}$). Since they evolve the disc from 2 Myr to 5 Myr,
their mass accretion changes from $\dot{M}_{*}\sim 3.8\times
10^{-9}\,M_{\odot}$ to $\sim 1\times 10^{-9}\,M_{\odot}$. They also adopted
$\alpha_{{\rm acc}}=5.4\times 10^{-3}$ for disc accretion and $\alpha_{{\rm
turb}}=5.4\times 10^{-4}$ and $10^{-4}$ for disc turbulence. Although the mass
evolution is similar to our Disc 2, their disc could be close to our Disc 6
(i.e. $M_{d}=0.02\,M_{\odot}$ with $t_{{\rm diff}}=1\,$Myr and thus
$\alpha_{{\rm acc}}\sim 7.4\times 10^{-3}$) since they started with an evolved
disc and assumed $\alpha_{{\rm acc}}=5.4\times 10^{-3}$. In our simulations,
both produce similar types of planets (see Figure 9 and a discussion below).

On the other hand, their pebble mass flux is

$\dot{M}_{F,\,B19}\propto Z^{7/3}\alpha^{-1}\dot{M}_{*,\,B19},$ (53)

where $Z=1\%$ is the metallicity. Although they assumed the solar metallicity
for their simulations, they scaled the pebble mass flux by the factor of
$S_{{\rm peb}}=1-10$, which corresponds to exploring the metallicities of
${\rm[Fe/H]}=0.0$ to $0.43$. Since our pebble mass flux is $\dot{M}_{F}\propto
Z^{2}\alpha^{-1}\dot{M}_{*}$, the dependence on each parameter is similar.

One of the conclusions by Bitsch et al. (2019) was that formation of CJs was
possible for cores originating from 10 au with $\alpha_{{\rm turb}}=5.4\times
10^{-4}$ and those from 5 au with $\alpha_{{\rm turb}}=10^{-4}$. From their
Figures $4-6$, the formation of giant planets up to $\sim 300\,M_{\oplus}$ was
possible for both $\alpha_{{\rm turb}}$ cases above $S_{{\rm peb}}=2.5$ (which
corresponds to ${\rm[Fe/H]}\gtrsim 0.17$), while no giant planets formed for
the solar metallicity case. Our simulations show similar trends. In Disc 6
with $\alpha_{{\rm turb}}=10^{-4}$, giant planets primarily become low-mass
CJs with $\sim 100-400\,M_{\oplus}$ for metallicities ${\rm[Fe/H]}=0.3$ and
$0.5,$ while no giant planets form for ${\rm[Fe/H]}\leq 0.0$ (see Figure 9).
In Disc 2, the outcomes are similar except that planetary masses are lower
$\sim 100-300\,M_{\oplus,}$ and very low-mass giant planets can be formed with
${\rm[Fe/H]}=0.0$ (but not for lower metallicities). Out of formed giant
planets, those starting from $3-5\,$au typically become the closest-in CJs
with orbital radius beyond $\sim 1\,$au. For higher (lower) $\alpha_{{\rm
turb}}$, planets tend to migrate further (less) (see Section 3.2.1 and Figure
12) as indicated by Bitsch et al. (2019).

One major difference between Bitsch et al. (2019) and our work is that their
simulations are left with a number of giant planets with low eccentricities,
while we have successfully reproduced the eccentricity distribution of giant
planets (see Figure 5). The result may be surprising at a glance because the
number of giant planets that formed and survived in simulations by Bitsch et
al. (2019) is higher than in our simulations, and typically $\sim 5$ or more
from their Figures 6 and 7. We note that the higher number of surviving
planets is not likely due to the difference in the initial number of cores (60
for their simulations as opposed to 10 for ours). Jurić & Tremaine (2008)
showed that, even when there were 50 giant planets, the final number of
planets would be 2-3, as long as the dynamical instabilities occur. The
eccentricity distribution from such simulations with an initially high number
of giant planets agrees well with observations, and also with dynamical
instability simulations with lower number of giant planets (e.g. Jurić &
Tremaine, 2008; Chatterjee et al., 2008). Thus, the large number of surviving
giant planets in Bitsch et al. (2019) indicates that the dynamical instability
among giant planets was rare in their simulations. Indeed, even their long-
term evolution of 100 Myr did not lead to a dramatic increase in orbital
eccentricities (Bitsch et al., 2019).

While this manuscript was under revision, we became aware of the work by
Bitsch et al. (2020),in which they studied the effects of eccentricity and
inclination damping efficiencies on the eccentricity distribution of giant
planets. They parameterised the eccentricity and inclination damping
timescales as $\tau_{a}=K\,\tau_{e}=K\,\tau_{i}$ with $K=5,\,50,\,500$, and
$5000$, and found that the observed eccentricity distribution of giant planets
can be recovered for slow damping with $K\sim 5-50$. Since Bitsch et al.
(2019) adopted a faster damping with $K=100$, it may be the reason why they
did not obtain eccentric giants.

As seen in Figure 1, we also have $K\sim 100,$ though we managed to reproduce
the eccentricity distribution of giants. Since our eccentricity and
inclination damping prescriptions are similar to those by Bitsch et al.
(2019), it is possible that subtle differences in disc conditions changed the
dynamical outcomes of simulations. For example, the choice of $\alpha_{{\rm
acc}}=5.4\times 10^{-3}$ in Bitsch et al. (2019) may be inconsistent with the
time evolution of the stellar mass accretion rate they adopted (Equation 52).
The fit to this observed mass accretion rate requires a rather high viscosity
$\alpha_{{\rm acc}}\sim 0.01$ for the disc size of $10-100\,$au (Hartmann et
al., 1998) and even higher $\alpha_{{\rm acc}}$ for a larger disc. By assuming
the lower $\alpha_{{\rm acc}}$ for the same accretion rate, the estimated
surface mass density and thus the disc mass becomes higher, which leads to
more efficient eccentricity and inclination damping. In fact, we observed a
similar lack of dynamical instabilities in the no-migration simulations of
Matsumura et al. (2017) — we adopted the same stellar mass accretion equation
as Equation 52 and used $\alpha\leq 5\times 10^{-3}$ when $3-5$ giant planets
were formed (except for one case with 3 giant planets with $\alpha=0.01$). For
completeness, Matsumura et al. (2017) had $K\sim 100$ in the type I regime and
$K\sim 10$ in the type II regime, which should favour more eccentric systems.
Moreover, we had twice as many cores in a much narrower range of disc radii
($0.3-5\,$au) compared to the current runs. It is possible, however, since
these simulations did not include migration, that planets were separated too
far from one another to invoke dynamical instabilities within simulation
times. In that case, convergent migration may also play an important role in
determining the eccentricity and inclination distributions of planetary
systems.

##  5 Conclusion

For this paper, we studied the formation of planetary systems via pebble
accretion by using N-body simulations, and we investigated the effects of disc
parameters such as masses, dissipation timescales, and metallicities. This is
a continuation of Matsumura et al. (2017), in which we modified the N-body
code SyMBA (Duncan et al., 1998) to incorporate the pebble accretion model by
Ida et al. (2016), gas accretion, type I and type II migration, and
eccentricity and inclination damping. In this work, we updated the code as
detailed in Section 2 to take account of the recent development of the field,
and we also adopted a two-$\alpha$ disc model, where mass accretion and disc
turbulence have different drivers.

We find that the disc masses, dissipation timescales, and stellar
metallicities all affect the efficiency of planet formation. The effects of
each parameter can be summarised as follows (see Section 4.6):

  * •

Disc metallicities ${\rm[Fe/H]}$ affect the formation timescales of
protoplanetary cores, but they do not strongly affect the final planetary
masses.

  * •

Initial disc masses $M_{D,0}$ affect both core formation and gas accretion
timescales, and thus the final planetary masses.

  * •

Disc diffusion timescales $t_{{\rm diff}}$ set time limits on planet formation
in the disc, and thus affect the final planetary masses.

We identified two types of giant planet formation trends, depending on whether
planet formation is fast compared to the disc’s dispersal or not. When a
disc’s dissipation timescales are long in comparison to typical planet
formation timescales (Discs 4, 5, 7, and 8 in our simulations, formation-
dominated case), giant planets are massive ($\sim M_{J}$ or higher) and
distributed over a wide range of orbital radii (from $\sim 0.1\,$au to over
$100\,$au). On the other hand, when a disc’s dissipation timescales are
comparable to planet formation timescales (Discs 1, 2, 3, and 6, disc-
dissipation-limited case), giant planets tend to be low-mass ($\sim M_{J}$ or
lower) and CJs (with $a\gtrsim 1\,$au). The formation timescale depends both
on stellar metallicities and disc masses — the timescale is shorter for more
massive, more metal-rich discs. Therefore, protoplanetary cores tend to
migrate significantly before accreting gas to become giant planets in low
metallicity discs, while giant planets can form in situ in the outer part of
high-metallicity, massive discs. For low-mass, low-metallicity discs, giant
planet formation is difficult.

Our main findings are the following:

  * •

Differently from Matsumura et al. (2017), we successfully reproduced the
overall distribution trends of semi-major axes $a$, eccentricities $e$, and
planetary masses $M_{p}$ of extrasolar giant planets (see Section 3.1.1 and
Figure 5), though we tend to overproduce CJs compared to HJs and WJs. The
success of reproducing the $a-M_{p}$ distribution, especially for CJs, is
largely due to the new type II migration formula and the two-$\alpha$ disc
model, as proposed by Ida et al. (2018). The success in reproducing the $e$
distribution is likely due to a more self-consistent disc model, a higher
number of giant planets formed per system compared to Matsumura et al. (2017),
and not too efficient eccentricity and inclination damping in the disc (see
Section 4.7).

  * •

The overall occurrence rates of giant planets as a function of orbital periods
agree well with observed trends (see Section 4.1). The occurrence rates
increase with periods in the inner region, decrease in the outer region, and
peak at $\sim 1-10\,$au. The most abundant giant planets are CJs ($>50\%$),
and thus more than half the giant planets in our simulations stay near their
formation region.

  * •

As discussed in Section 4.2, our simulations naturally explain why HJs tend to
be alone (e.g. Wright et al., 2009; Steffen et al., 2012; Huang et al., 2016),
and also why the eccentricities of HJs are low around low-metallicity stars
and vary more widely around high-metallicity ones (e.g. Dawson & Murray-Clay,
2013; Shabram et al., 2016; Buchhave et al., 2018). The same trend is expected
for stellar obliquities of their host stars, and the current observations
support that (see Section 3.1.2).

    * –

In low-metallicity discs, HJs tend to form in situ: protoplanetary cores
migrate to the inner disc and accrete gas there. This is because planet
formation is slower in the low-metallicity discs, which leads to greater
migration of a protoplanetary core before it reaches the PIM and starts
accreting a significant gas to become a giant planet. Since the low-
metallicity discs tend to form just 1-2 giant planets, HJs tend to be alone
and on nearly circular and coplanar orbits.

    * –

In high-metallicity discs, HJs can be formed either via tidal circularisation
of highly eccentric orbits or via a migration scenario (including in situ
formation; see Section 3.1.2). The higher metallicity discs tend to produce a
number of giant planets that are prone to dynamical instabilities. A HJ could
be formed from a WJ/CJ as its eccentric orbit is circularised. Alternatively,
HJs could be first formed in situ (i.e. via core migration followed by gas
accretion) or via migration, along with WJs and CJs. The dynamical
instabilities in such systems often remove either HJs and/or WJs, leaving
either (i) only CJs, or (ii) HJs with CJs. HJs formed in high-metallicity
discs have a wider variety of eccentricities and inclinations and also tend to
be alone.

  * •

If an SCJ is formed, as a giant planet grows within $\sim 20\,$au and then
gets scattered outward, we expect that such an SCJ (1) was born in a high-
metallicity disc ($\left[{\rm Fe/H}\right]\gtrsim 0.0$), (2) has an eccentric
orbit, and (3) tends to be accompanied by another giant planet ( $\sim
80\,\%$) (see Section 3.1.3).

  * •

Most warm Jupiters ($0.1\,{\rm au}\lesssim a\lesssim 1\,$au) are formed in the
formation-dominated discs (i.e. Discs 4, 5, 7, and 8 in our simulations). In
other words, in our simulations, it is difficult to form WJs in rapidly
dissipating, low-mass and/or low-metallicity discs.

  * •

CJs tend to be formed in high-mass and/or high-metallicity discs, where the
planet formation timescale is comparable to or shorter than the disc
dissipation timescale.

Finally, there are still several issues that need to be resolved/explored in
our work. Most importantly, type I migration is still too fast and we tend to
lose SEs. For example, type I migration can be slowed in the inner disc region
if we fully adopt the wind-driven disc, as in Ogihara et al. (2018). Resolving
the migration issue is also important when choosing a more appropriate gas
accretion formula, which would provide more accurate planetary compositions
(see Section 4.5). Furthermore, when $\alpha_{{\rm turb}}\ll\alpha_{{\rm
acc}}$ as we assumed, the gap depth may also be affected by the wind-driven
accretion.

###### Acknowledgements.

We thank Man Hoi Lee and Eduard Vorobyov for useful discussions and an
anonymous referee for detailed comments. SM is grateful to Aurora Sicilia-
Aguilar for valuable discussions and also for kindly sharing her data from
Sicilia-Aguilar et al. (2010). SM would also like to thank the Earth-Life
Science Institute at Tokyo Institute of Technology for its hospitality, where
part of this work has been done. SM is partly supported by the STFC grant
number ST/S000399/1 (The Planet-Disk Connection: Accretion, Disk Structure,
and Planet Formation). RB acknowledges financial assistance from the Japan
Society for the Promotion of Science (JSPS) Shingakujutsu Kobo (JP19H05071).
SI is supported by MEXT Kakenhi 18H05438.

## References

  * Adachi et al. (1976) Adachi, I., Hayashi, C., & Nakazawa, K. 1976, Progress of Theoretical Physics, 56, 1756
  * Ali-Dib et al. (2020) Ali-Dib, M., Cumming, A., & Lin, D. N. C. 2020, MNRAS, 494, 2440
  * Andrews et al. (2018) Andrews, S. M., Huang, J., Pérez, L. M., et al. 2018, ApJL, 869, L41
  * Andrews & Williams (2007) Andrews, S. M. & Williams, J. P. 2007, ApJ, 659, 705
  * Andrews et al. (2010) Andrews, S. M., Wilner, D. J., Hughes, A. M., Qi, C., & Dullemond, C. P. 2010, ApJ, 723, 1241
  * Ataiee et al. (2018) Ataiee, S., Baruteau, C., Alibert, Y., & Benz, W. 2018, A&A, 615, A110
  * Bai (2016) Bai, X.-N. 2016, ApJ, 821, 80
  * Bai (2017) Bai, X.-N. 2017, ApJ, 845, 75
  * Bai & Stone (2013) Bai, X.-N. & Stone, J. M. 2013, ApJ, 769, 76
  * Balbus & Hawley (1998) Balbus, S. A. & Hawley, J. F. 1998, Reviews of Modern Physics, 70, 1
  * Baron et al. (2019) Baron, F., Lafrenière, D., Artigau, É., et al. 2019, AJ, 158, 187
  * Batygin et al. (2016) Batygin, K., Bodenheimer, P. H., & Laughlin, G. P. 2016, ApJ, 829, 114
  * Birnstiel et al. (2012) Birnstiel, T., Klahr, H., & Ercolano, B. 2012, A&A, 539, A148
  * Bitsch et al. (2019) Bitsch, B., Izidoro, A., Johansen, A., et al. 2019, A&A, 623, A88
  * Bitsch et al. (2015) Bitsch, B., Lambrechts, M., & Johansen, A. 2015, A&A, 582, A112
  * Bitsch et al. (2018) Bitsch, B., Morbidelli, A., Johansen, A., et al. 2018, A&A, 612, A30
  * Bitsch et al. (2020) Bitsch, B., Trifonov, T., & Izidoro, A. 2020, A&A, 643, A66
  * Boley et al. (2016) Boley, A. C., Granados Contreras, A. P., & Gladman, B. 2016, ApJL, 817, L17
  * Bowler (2016) Bowler, B. P. 2016, PASP, 128, 102001
  * Bowler et al. (2020) Bowler, B. P., Blunt, S. C., & Nielsen, E. L. 2020, AJ, 159, 63
  * Brasser et al. (2017) Brasser, R., Bitsch, B., & Matsumura, S. 2017, AJ, 153, 222
  * Brasser et al. (2018) Brasser, R., Matsumura, S., Muto, T., & Ida, S. 2018, ApJL, 864, L8
  * Brittain et al. (2019) Brittain, S. D., Najita, J. R., & Carr, J. S. 2019, ApJ, 883, 37
  * Brittain et al. (2020) Brittain, S. D., Najita, J. R., Dong, R., & Zhu, Z. 2020, ApJ, 895, 48
  * Bryan et al. (2016) Bryan, M. L., Knutson, H. A., Howard, A. W., et al. 2016, ApJ, 821, 89
  * Buchhave et al. (2018) Buchhave, L. A., Bitsch, B., Johansen, A., et al. 2018, ApJ, 856, 37
  * Casassus & Pérez (2019) Casassus, S. & Pérez, S. 2019, ApJL, 883, L41
  * Chambers (2016) Chambers, J. E. 2016, ApJ, 825, 63
  * Chatterjee et al. (2008) Chatterjee, S., Ford, E. B., Matsumura, S., & Rasio, F. A. 2008, ApJ, 686, 580
  * Chiang & Laughlin (2013) Chiang, E. & Laughlin, G. 2013, MNRAS, 431, 3444
  * Coleman & Nelson (2014) Coleman, G. A. L. & Nelson, R. P. 2014, MNRAS, 445, 479
  * Coleman & Nelson (2016a) Coleman, G. A. L. & Nelson, R. P. 2016a, MNRAS, 460, 2779
  * Coleman & Nelson (2016b) Coleman, G. A. L. & Nelson, R. P. 2016b, MNRAS, 457, 2480
  * Cumming et al. (2008) Cumming, A., Butler, R. P., Marcy, G. W., et al. 2008, PASP, 120, 531
  * Dawson & Murray-Clay (2013) Dawson, R. I. & Murray-Clay, R. A. 2013, ApJL, 767, L24
  * Duffell & MacFadyen (2013) Duffell, P. C. & MacFadyen, A. I. 2013, ApJ, 769, 41
  * Duncan et al. (1998) Duncan, M. J., Levison, H. F., & Lee, M. H. 1998, AJ, 116, 2067
  * Fabrycky & Tremaine (2007) Fabrycky, D. & Tremaine, S. 2007, ApJ, 669, 1298
  * Fendyke & Nelson (2014) Fendyke, S. M. & Nelson, R. P. 2014, MNRAS, 437, 96
  * Fernandes et al. (2019) Fernandes, R. B., Mulders, G. D., Pascucci, I., Mordasini, C., & Emsenhuber, A. 2019, ApJ, 874, 81
  * Ford & Rasio (2008) Ford, E. B. & Rasio, F. A. 2008, ApJ, 686, 621
  * Forgan & Rice (2011) Forgan, D. & Rice, K. 2011, MNRAS, 417, 1928
  * Fortney & Nettelmann (2010) Fortney, J. J. & Nettelmann, N. 2010, SSRv, 152, 423
  * Fung et al. (2014) Fung, J., Shi, J.-M., & Chiang, E. 2014, ApJ, 782, 88
  * Hall et al. (2017) Hall, C., Forgan, D., & Rice, K. 2017, MNRAS, 470, 2517
  * Hartmann et al. (1998) Hartmann, L., Calvet, N., Gullbring, E., & D’Alessio, P. 1998, ApJ, 495, 385
  * Helled et al. (2020) Helled, R., Nettelmann, N., & Guillot, T. 2020, SSRv, 216, 38
  * Huang et al. (2016) Huang, C., Wu, Y., & Triaud, A. H. M. J. 2016, ApJ, 825, 98
  * Huang et al. (2018) Huang, J., Andrews, S. M., Dullemond, C. P., et al. 2018, ApJL, 869, L42
  * Hueso & Guillot (2005) Hueso, R. & Guillot, T. 2005, A&A, 442, 703
  * Hut (1981) Hut, P. 1981, A&A, 99, 126
  * Ida et al. (2016) Ida, S., Guillot, T., & Morbidelli, A. 2016, A&A, 591, A72
  * Ida & Lin (2004) Ida, S. & Lin, D. N. C. 2004, ApJ, 604, 388
  * Ida & Lin (2008) Ida, S. & Lin, D. N. C. 2008, ApJ, 673, 487
  * Ida et al. (2013) Ida, S., Lin, D. N. C., & Nagasawa, M. 2013, ApJ, 775, 42
  * Ida et al. (2020) Ida, S., Muto, T., Matsumura, S., & Brasser, R. 2020, MNRAS, 494, 5666
  * Ida et al. (2018) Ida, S., Tanaka, H., Johansen, A., Kanagawa, K. D., & Tanigawa, T. 2018, ApJ, 864, 77
  * Ida et al. (2019) Ida, S., Yamamura, T., & Okuzumi, S. 2019, A&A, 624, A28
  * Ikoma et al. (2000) Ikoma, M., Nakazawa, K., & Emori, H. 2000, ApJ, 537, 1013
  * Izidoro et al. (2019) Izidoro, A., Bitsch, B., Raymond, S. N., et al. 2019, arXiv e-prints, arXiv:1902.08772
  * Johansen et al. (2019) Johansen, A., Ida, S., & Brasser, R. 2019, A&A, 622, A202
  * Johansen & Lambrechts (2017) Johansen, A. & Lambrechts, M. 2017, Annual Review of Earth and Planetary Sciences, 45, 359
  * Johansen et al. (2015) Johansen, A., Mac Low, M.-M., Lacerda, P., & Bizzarro, M. 2015, Science Advances, 1, 1500109
  * Johansen et al. (2007) Johansen, A., Oishi, J. S., Mac Low, M.-M., et al. 2007, Nature, 448, 1022
  * Johansen et al. (2009) Johansen, A., Youdin, A., & Mac Low, M.-M. 2009, ApJL, 704, L75
  * Jurić & Tremaine (2008) Jurić, M. & Tremaine, S. 2008, ApJ, 686, 603
  * Kanagawa et al. (2018) Kanagawa, K. D., Tanaka, H., & Szuszkiewicz, E. 2018, ApJ, 861, 140
  * Keppler et al. (2018) Keppler, M., Benisty, M., Müller, A., et al. 2018, A&A, 617, A44
  * Kikuchi et al. (2014) Kikuchi, A., Higuchi, A., & Ida, S. 2014, ApJ, 797, 1
  * Kokubo & Ida (2002) Kokubo, E. & Ida, S. 2002, ApJ, 581, 666
  * Kratter et al. (2010) Kratter, K. M., Murray-Clay, R. A., & Youdin, A. N. 2010, ApJ, 710, 1375
  * Kretke & Levison (2014) Kretke, K. A. & Levison, H. F. 2014, AJ, 148, 109
  * Lambrechts & Johansen (2012) Lambrechts, M. & Johansen, A. 2012, A&A, 544, A32
  * Lambrechts & Johansen (2014) Lambrechts, M. & Johansen, A. 2014, A&A, 572, A107
  * Lambrechts et al. (2014) Lambrechts, M., Johansen, A., & Morbidelli, A. 2014, A&A, 572, A35
  * Lambrechts et al. (2019) Lambrechts, M., Morbidelli, A., Jacobson, S. A., et al. 2019, A&A, 627, A83
  * Levison et al. (2015) Levison, H. F., Kretke, K. A., & Duncan, M. J. 2015, Nature, 524, 322
  * Lin et al. (1996) Lin, D. N. C., Bodenheimer, P., & Richardson, D. C. 1996, Nature, 380, 606
  * Lin & Papaloizou (1986) Lin, D. N. C. & Papaloizou, J. 1986, ApJ, 309, 846
  * Lin & Papaloizou (1993) Lin, D. N. C. & Papaloizou, J. C. B. 1993, Protostars and Planets III (The University of Arizona Press)
  * Lodato et al. (2019) Lodato, G., Dipierro, G., Ragusa, E., et al. 2019, MNRAS, 486, 453
  * Long et al. (2018) Long, F., Pinilla, P., Herczeg, G. J., et al. 2018, ApJ, 869, 17
  * Lynden-Bell & Pringle (1974) Lynden-Bell, D. & Pringle, J. E. 1974, MNRAS, 168, 603
  * Marzari & Weidenschilling (2002) Marzari, F. & Weidenschilling, S. J. 2002, Icarus, 156, 570
  * Matsumura et al. (2017) Matsumura, S., Brasser, R., & Ida, S. 2017, A&A, 607, A67
  * Matsumura et al. (2010) Matsumura, S., Peale, S. J., & Rasio, F. A. 2010, ApJ, 725, 1995
  * Mayor et al. (2011) Mayor, M., Marmier, M., Lovis, C., et al. 2011, ArXiv e-prints [arXiv:1109.2497]
  * Morbidelli & Nesvorny (2012) Morbidelli, A. & Nesvorny, D. 2012, A&A, 546, A18
  * Mordasini et al. (2009) Mordasini, C., Alibert, Y., & Benz, W. 2009, A&A, 501, 1139
  * Mordasini et al. (2012) Mordasini, C., Alibert, Y., Benz, W., Klahr, H., & Henning, T. 2012, A&A, 541, A97
  * Mustill et al. (2015) Mustill, A. J., Davies, M. B., & Johansen, A. 2015, ApJ, 808, 14
  * Muto et al. (2011) Muto, T., Takeuchi, T., & Ida, S. 2011, ApJ, 737, 37
  * Nagasawa & Ida (2011) Nagasawa, M. & Ida, S. 2011, ApJ, 742, 72
  * Nagasawa et al. (2008) Nagasawa, M., Ida, S., & Bessho, T. 2008, ApJ, 678, 498
  * Naoz (2016) Naoz, S. 2016, ARA&A, 54, 441
  * Naoz et al. (2011) Naoz, S., Farr, W. M., Lithwick, Y., Rasio, F. A., & Teyssandier, J. 2011, Nature, 473, 187
  * Nielsen et al. (2019) Nielsen, E. L., De Rosa, R. J., Macintosh, B., et al. 2019, AJ, 158, 13
  * Ogihara et al. (2010) Ogihara, M., Duncan, M. J., & Ida, S. 2010, ApJ, 721, 1184
  * Ogihara & Hori (2020) Ogihara, M. & Hori, Y. 2020, ApJ, 892, 124
  * Ogihara et al. (2018) Ogihara, M., Kokubo, E., Suzuki, T. K., & Morbidelli, A. 2018, A&A, 615, A63
  * Ogihara et al. (2017) Ogihara, M., Kokubo, E., Suzuki, T. K., Morbidelli, A., & Crida, A. 2017, A&A, 608, A74
  * Oka et al. (2011) Oka, A., Nakamoto, T., & Ida, S. 2011, ApJ, 738, 141
  * Ormel & Klahr (2010) Ormel, C. W. & Klahr, H. H. 2010, A&A, 520, A43
  * Ormel & Kobayashi (2012) Ormel, C. W. & Kobayashi, H. 2012, ApJ, 747, 115
  * Ormel et al. (2015a) Ormel, C. W., Kuiper, R., & Shi, J.-M. 2015a, MNRAS, 446, 1026
  * Ormel & Liu (2018) Ormel, C. W. & Liu, B. 2018, A&A, 615, A178
  * Ormel et al. (2015b) Ormel, C. W., Shi, J.-M., & Kuiper, R. 2015b, MNRAS, 447, 3512
  * Paardekooper et al. (2011) Paardekooper, S.-J., Baruteau, C., & Kley, W. 2011, MNRAS, 410, 293
  * Pfalzner et al. (2014) Pfalzner, S., Steinhausen, M., & Menten, K. 2014, ApJL, 793, L34
  * Santerne et al. (2016) Santerne, A., Moutou, C., Tsantaki, M., et al. 2016, A&A, 587, A64
  * Sato et al. (2016) Sato, T., Okuzumi, S., & Ida, S. 2016, A&A, 589, A15
  * Schlaufman (2018) Schlaufman, K. C. 2018, ApJ, 853, 37
  * Shabram et al. (2016) Shabram, M., Demory, B.-O., Cisewski, J., Ford, E. B., & Rogers, L. 2016, ApJ, 820, 93
  * Shakura & Sunyaev (1973) Shakura, N. I. & Sunyaev, R. A. 1973, A&A, 24, 337
  * Sicilia-Aguilar et al. (2010) Sicilia-Aguilar, A., Henning, T., & Hartmann, L. W. 2010, ApJ, 710, 597
  * Simon et al. (2016) Simon, J. B., Armitage, P. J., Li, R., & Youdin, A. N. 2016, ApJ, 822, 55
  * Stamatellos & Whitworth (2008) Stamatellos, D. & Whitworth, A. P. 2008, A&A, 480, 879
  * Steffen et al. (2012) Steffen, J. H., Ragozzine, D., Fabrycky, D. C., et al. 2012, Proceedings of the National Academy of Science, 109, 7982
  * Suzuki et al. (2016) Suzuki, T. K., Ogihara, M., Morbidelli, A. r., Crida, A., & Guillot, T. 2016, A&A, 596, A74
  * Tanaka & Ward (2004) Tanaka, H. & Ward, W. R. 2004, ApJ, 602, 388
  * Tanigawa & Tanaka (2016) Tanigawa, T. & Tanaka, H. 2016, ApJ, 823, 48
  * Thommes et al. (2008) Thommes, E. W., Matsumura, S., & Rasio, F. A. 2008, Science, 321, 814
  * Thorngren et al. (2016) Thorngren, D. P., Fortney, J. J., Murray-Clay, R. A., & Lopez, E. D. 2016, ApJ, 831, 64
  * Tsukagoshi et al. (2019) Tsukagoshi, T., Muto, T., Nomura, H., et al. 2019, ApJL, 878, L8
  * Tychoniec et al. (2018) Tychoniec, Ł., Tobin, J. J., Karska, A., et al. 2018, ApJS, 238, 19
  * Vicente & Alves (2005) Vicente, S. M. & Alves, J. 2005, A&A, 441, 195
  * Weidenschilling (1977) Weidenschilling, S. J. 1977, MNRAS, 180, 57
  * Wimarsson et al. (2020) Wimarsson, J., Liu, B., & Ogihara, M. 2020, MNRAS, 496, 3314
  * Winn (2018) Winn, J. N. 2018, Planet Occurrence: Doppler and Transit Surveys, 195
  * Wright et al. (2009) Wright, J. T., Upadhyay, S., Marcy, G. W., et al. 2009, ApJ, 693, 1084
  * Wu & Lithwick (2011) Wu, Y. & Lithwick, Y. 2011, ApJ, 735, 109
  * Yasui et al. (2010) Yasui, C., Kobayashi, N., Tokunaga, A. T., Saito, M., & Tokoku, C. 2010, ApJL, 723, L113
  * Youdin & Goodman (2005) Youdin, A. N. & Goodman, J. 2005, ApJ, 620, 459
  * Zhang et al. (2016) Zhang, K., Bergin, E. A., Blake, G. A., et al. 2016, ApJL, 818, L16