# Probing Dust and Gas Properties Using Ringed Disks

Eve J. Lee Department of Physics and Trottier Space Institute, McGill
University, 3600 rue University, H3A 2T8 Montreal QC, Canada  Trottier
Institute for Research on Exoplanets (iREx), Université de Montréal, Canada

###### Abstract

How rapidly a planet grows in mass and how far they may park from the host
star depend sensitively on two non-dimensional parameters: Stokes number St
and turbulent $\alpha$. Yet, these parameters remain highly uncertain being
difficult or impossible to measure directly. Here, we demonstrate how the
ringed disks can be leveraged to obtain St and $\alpha$ separately by
constructing a simple toy model that combines dust radial equation of motion
under aerodynamic drag and coupling to gas motion with the measured
distribution of dust masses in Class 0/I disks. Focusing on known systems with
well-resolved dust rings, we find that the range of St and $\alpha$ that are
consistent with the measured properties of the rings are small:
$10^{-4}\lesssim{\rm St}\lesssim 10^{-2}$ and $10^{-5}\lesssim\alpha\lesssim
10^{-3}$. These low St and $\alpha$ ensure the observed rings are stable
against clumping. Even in one marginal case where the formation of bound
clumps is possible, further mass growth by pebble accretion is inhibited.
Furthermore, the derived low $\alpha$ is consistent with the nearly inviscid
regime where Type I migration can be prematurely halted. Our analysis predicts
minimal planet population beyond $\sim$10s of au where we observe dust rings
and significantly more vigorous planet formation inside $\sim$10 AU,
consistent with current exo-giant statistics. We close with discussions on the
implications of our results on small planet statistics at large orbital
distances.

##  1 Introduction

One of the biggest uncertainties in planet formation is their initial
conditions. Whether a planet becomes a gas giant or remain a gas-poor sub-
Neptune or a rocky super-Earth/Earth is sensitively determined by how rapidly
the underlying rocky core can grow (e.g., Lee, 2019; Lee et al., 2022b). The
initial growth of the planetary cores is expected to be governed by pebble
accretion (e.g., Ormel & Klahr, 2010; Lambrechts & Johansen, 2012; Ormel,
2017) whereby particles of Stokes number St $<1$ settle onto an accreting body
via aerodynamic drag. The growth rate by pebble accretion is then necessarily
sensitively determined by St, with smaller St slowing down the growth of solid
cores. In fact, under pebble accretion and the radial drift of solids in
protoplanetary disks (e.g., Weidenschilling, 1977), Lin et al. (2018) report
that a necessary (but not sufficient) condition for the rapid coagulation of
massive cores that can generate a gas giant is for the particle Stokes number
to be larger than the Shakura-Sunyaev turbulent parameter $\alpha$.

Large $\alpha$ (stronger diffusive turbulence) can puff up the dust disk
diluting the solid density and therefore slowing down pebble accretion. More
importantly, once $\alpha\lesssim 10^{-4}$–$10^{-3}$, disk-induced Type I
migration is expected to stall prematurely as the perturbed gas interior to
the planet’s orbit continue to pile up before it is diffused, driving the net
torque on the planet to zero (e.g., Rafikov, 2002; Li et al., 2009; Yu et al.,
2010; Fung & Chiang, 2017; Fung & Lee, 2018). It follows that the two non-
dimensional parameters St and $\alpha$ critically determine nearly all aspects
of the early stages of planet formation, from the mass growth of rocky cores
to the emplacement of the planet, which altogether determine the final
observed masses, radii, and orbital periods of exoplanets.

Yet, direct measurements of St and $\alpha$ are either impossible or
challenging. Under the Epstein drag regime, Stokes number is defined as

${\rm St}=\Omega\frac{\rho_{s}s}{\rho_{g}c_{s}}$ (1)

where $\Omega\equiv\sqrt{GM_{\star}/a^{3}}$ is the Keplerian orbital
frequency, $G$ is the gravitational constant, $M_{\star}$ is the mass of the
host star, $a$ is the orbital distance, $\rho_{s}$ is the material density of
the grain, $s$ is the size of the grain, $\rho_{g}$ is the volumetric mass
density of disk gas, $c_{s}\equiv\sqrt{kT/\mu m_{H}}$ is the sound speed, $k$
is the Boltzmann constant, $T$ is the midplane temperature, $\mu$ is the gas
mean molecular weight, and $m_{H}$ is the Hydrogen atomic mass. The wavelength
of observation that probes the dust thermal emission can be used as a proxy
for the grain size, albeit not exact. At large enough distances (beyond
$\sim$10 au), the disk midplane temperature can be approximated to be from
heating by stellar irradiation (e.g., Chiang & Goldreich, 1997; D’Alessio et
al., 1998). Gas measurements are limited and so $\rho_{g}$ remains largely
unknown or poorly constrained, and even when CO measurements can be made
(e.g., Zhang et al., 2021), conversion to the total gas mass depends on the
uncertain CO-H2 factor.

Measurement of $\alpha$ is equally challenging for protoplanetary disks given
that the turbulent dynamics is expected to be highly subsonic. Nevertheless,
reports of $\alpha$ based on the depth of the gap in dust emission (Pinte et
al., 2016) and non-thermal broadening of molecular lines (Flaherty et al.,
2015, 2017) find $\alpha\lesssim 10^{-3}$ over a wide range of vertical depth
and radial extent.

Gas kinematic data of sufficiently high resolution are expensive and often
require complex chemical modelling to disentagle. In this paper, we seek to
derive St and $\alpha$ with simpler and readily available dust emission
measurements of ringed disks. Ring-like substructures are common in radio
interferometric images of protoplanetary disks (e.g., ALMA Partnership et al.,
2015; Andrews et al., 2018). Modeling the gas pressure perturbation as a
Gaussian of width $w$, the observed dust ring can be interpreted as the
equilibrium between the aerodynamic drift that collects the particles at the
center of the pressure bump and the turbulent diffusion so that the width of
the dust ring is (Dullemond et al., 2018)

$w_{\rm d}=w\left[1+\left(\frac{{\rm Sc}{\rm
St}}{\alpha}\right)\right]^{-1/2}$ (2)

where ${\rm Sc}=1$ is the Schmidt number taken as 1 for simplicity (see also
Johansen & Klahr, 2005). Taking $w$ as the local gas disk scale height
$H\equiv c_{s}/\Omega$, resolved dust rings can be leveraged to obtain the
local ratio St/$\alpha$.

How can we break the degeneracy and separately determine St and $\alpha$?
Rosotti et al. (2020) computed gas rotation curves from 12CO emission for two
ringed disks HD 163296 and AS 209 to obtain a precise value of St/$\alpha$ and
placed an upper limit on $\alpha$ assuming the maximum St to be set by
fragmentation (i.e., any larger grains would be ground down by relative motion
between the grains set by their coupling to the gas turbulent motion). The
fragmentation velocity however is highly uncertain and depends on not only the
size of the grains but also their chemical composition and porosity (e.g.,
Musiolik & Wurm, 2019; Kimura et al., 2020). Here, we demonstrate how, in
addition to the dust ring width, dust ring mass can be used to solve for St
and $\alpha$ separately.

The paper is organized as follows. Section 2 outlines our toy model of
computing St and $\alpha$ and how we determine where or not a given dust ring
is able to nucleate planetary mass objects. Section 3 presents our results,
and their implications are summarized in Section 4. We conclude in Section 5.

##  2 Method

We build a simple toy model of the dust mass growth in local pressure traps
using the equation of motion of dust particle within a gaseous protoplanetary
disk. Combining this equation with the measured properties of the observed
rings, we break the degeneracy between $\alpha$ and ${\rm St}$.

###  2.1 Disk profile

All the well-characterized dust rings are located at wide orbits beyond
$\sim$10 AU where the disk heating is irradiation-dominated. Following
Dullemond et al. (2018), we adopt a simple flared irradiated disk profile for
the midplane temperature:

$T=\left(\frac{\phi L_{\star}}{8\pi a^{2}\sigma_{\rm sb}}\right)^{1/4}$ (3)

where $L_{\star}$ is the measured luminosity of the central star, $\sigma_{\rm
sb}$ is the Stefan-Boltzmann constant, and $\phi$ is the grazing angle of the
incident starlight which range from $\sim$0.1 to $\sim$0.4 between 10 and 200
AU according to equations 14b and 14d of (Chiang & Goldreich, 1997); see also
equation 1 and Figure 3 of Dullemond et al. (2001)—we take $\phi=0.25$.

The shape of the disk surface density profile is less certain owing to
uncertainty in the mechanism governing angular momentum loss/redistribution
and the difficulty of obtaining well-resolved gas line measurements for many
of the protoplanetary disks. We adopt $\Sigma_{g}\propto a^{-1}$ motivated by
the gas emission line measurements of TW Hya by Zhang et al. (2017) and the
MAPS survey (see Zhang et al., 2021, their Table 2).

###  2.2 Dust motion

The radial motion of a dust particle in the inertial frame can be written as

$\displaystyle v_{\rm drift}$ $\displaystyle=-\frac{3\nu}{2a}\frac{1}{1+{\rm
St}^{2}}-2v_{\rm hw}\frac{\rm St}{1+{\rm St}^{2}}$
$\displaystyle\sim-\frac{3}{2}\left(\frac{c_{s}}{v_{k}}\right)^{2}v_{k}\left(\alpha+\frac{2}{3}|\gamma|{\rm
St}\right)$ (4)

where the negative sign indicates inward motion (towards the central star),
$\nu\equiv\alpha c_{s}H$ is the kinematic viscosity of the gas, $\mu=2.3$ is
the local disk gas mean molecular weight, $v_{k}=\Omega a$ is the Keplerian
velocity, $v_{\rm hw}\equiv-(c_{s}^{2}/2v_{k})(\partial\log P/\partial\log
a)\equiv-(c_{s}^{2}/2v_{k})\gamma$ is the azimuthal headwind velocity, and $P$
is the local disk gas pressure. With our adopted disk temperature and density
profiles, $\gamma=-11/4$. We have taken the approximation of ${\rm St}\ll 1$
to write the second line, which we find to be valid a posteriori.

The first term in the bracket of equation 2.2 corresponds to the grain’s
coupling to the radial “advective” motion of the gas (while to be more
precisely, this is really a diffusive term for the gas but we assume $v_{\rm
drift}$ to be always evaluated sufficiently inside the characteristic radius
of the disk so that the gas motion under turbulent viscosity is inward rather
than outward) and the second term in the bracket is the aerodynamic drag term.
We have taken the drag term to also be inwardly-directed as is expected for a
typical smooth disk. We emphasize that equation 2.2 is intended to capture the
motion of the dust grains through the unperturbed region of the disk before
the grains are trapped.

We then posit that there is some local perturbation in the gas pressure,
without specifying the cause of it as it is irrelevant for our calculation.
The dust grains from orbits outside of this perturbation will drift in
following equation 2.2 and be trapped into an axisymmetric ring. To find an
expression for the amount of mass that is expected to be trapped within this
ring, we first empirically derive the radial profile of initial dust mass
budget using the mass and radius measurements of Class 0/I disks in the Orion
provided by Tobin et al. (2020), where both the ALMA (0.87 mm) and VLA (9 mm)
measurements are available. Following the procedure of Chachan et al. (2022),
we take the mass measurements from VLA (because disks are expected to be more
optically thin at longer wavelengths) and the size measurements of
ALMA—because disks are observed to be more compact at longer wavelengths which
could be physical but also could be an effect of wavelength-dependent optical
depths (see, e.g., Tazzari et al., 2016; Tripathi et al., 2018). Figure 1
illustrates the data and our best-fit initial dust mass profile using least-
squares fit of the form $M_{\rm dust}=M_{0}(a/{\rm 1\,AU})^{\beta}$ accounting
for the spread in the data in each bin. Fitting through the median $M_{\rm
dust}$ in each radial bin, we derive $M_{\rm dust}\propto a^{0.57\pm 0.08}$
which is steeper than what was reported by Lee et al. (2022a) but still agrees
within 1-$\sigma$. While in Figure 1, we display only the fit to the median
$M_{\rm dust}$, in our analysis, we explore the full range from 5th to 100th
percentiles of $M_{\rm dust}$ distribution, whereby $M_{0}$ and $\beta$ are
empirically fit to the $M_{\rm dust}$ values of a given percentile in each
radial bin.

Figure 1: Total dust mass vs. disk size for Class 0/I disks in the Orion star
forming region with both ALMA (0.87 mm) and VLA (9 mm) measurements, denoted
with grey circles. Vertical dotted lines delineate the boundaries of the 8
logarithmically spaced radial bins from the minimum to maximum disk sizes.
Median $M_{\rm dust}$ for each bin is at the center of the orange crosshair
with the lower and the upper tips of the crosshair indicating 16th and 84th
percentiles, respectively. Least-squares fit accounting for the spread in
$M_{\rm dust}$ in each bin (taken as the quadrature sum of the lower and upper
spread) are shown in blue line with the result of the fit annotated at the top
in blue text. Only the outer 6 radii bins are included in the fit as the inner
two bins contain less than 3 points.

We now write an expression for the total amount of dust that is expected to be
trapped within a ring at a radius $a_{\rm ring}$ at time $t$ which is simply
the amount of dust mass that exists between $a_{\rm ring}$ and some outer
radius $a_{0}>a_{\rm ring}$ from which the dust grains originate, corrected
for some trapping efficiency:

$M_{\rm ring}(t)=M_{0}\epsilon_{\rm trap}\left[\left(\frac{a_{0}(t_{\rm
age}-t)}{\rm 1AU}\right)^{\beta}-\left(\frac{a_{\rm ring}}{\rm
1AU}\right)^{\beta}\right]$ (5)

where $t_{\rm age}$ is the system age, and $\epsilon_{\rm trap}\leq 1$ is the
efficiency at which a given pressure perturbation can trap dust particles.
This efficiency is found to be $\sim$0.6–0.8 by Lee et al. (2022a) through
direct numerical simulations (see their Figure 4) but it is expected to vary
with time, the properties of the pressure bump and the ratio between ${\rm
St}$ and $\alpha$ (see, e.g., Zhu et al., 2012, their Section 4.4). Verifying
with systematic investigations in 2D and 3D hydrodynamic simulations is an
ongoing effort. For simplicity, we take a constant value $\epsilon_{\rm
trap}=0.6$.

In equation 5, we use $t_{\rm age}-t$ as the input time for $a_{0}$ to clarify
that we are going back in time to locate the origin site of dust grains: i.e.,
$a_{0}$ is defined by integrating equation 2.2 back in time $t$ with the
boundary condition of $a_{0}(t_{\rm age})=a_{\rm ring}$. From equation 3,
$c_{s}^{2}\propto T\propto a^{-1/2}$ and because $v_{k}\propto a^{-1/2}$,
$(c_{s}/v_{k})^{2}v_{k}$ in equation 2.2 is spatially constant. If we further
assume $\alpha$, $|\gamma|$, and ${\rm St}$ to be spatially constant:

$a_{0,{\rm St}}(t^{\prime})=a_{\rm
ring}+\frac{3}{2}\left(\frac{c_{s,r}}{v_{k,r}}\right)^{2}v_{k,r}\left(\alpha+\frac{2}{3}|\gamma|{\rm
St}\right)(t_{\rm age}-t^{\prime}),$ (6)

where the subscript ‘r’ clarifies $c_{s}$ and $v_{k}$ to be evaluated at the
location of the observed ring and the subscript ‘St’ specifies spatially
constant St. Note that with our definitions, $M_{\rm ring}(t=0)=0$ (no grains
have drifted in yet) and $M_{\rm ring}(t_{\rm age})$ is maximized (maximum
time elapsed).

The initial drift of dust grains into the pressure bump is considered to occur
in a smooth disk so we maintain the assumption of spatially constant $\alpha$
and $|\gamma|$. Assuming a constant ${\rm St}$ is less certain as this number
depends on the disk gas surface density and temperature profile even for a
fixed grain size. We now derive $a_{0}(t)$ assuming a fixed grain size. Again,
since we are in the outer disk, the gas mean free path is relatively large and
the mm-cm sized grains are most likely in Epstein drag regime,111Using the
minimum mass extrasolar nebula (Chiang & Laughlin, 2013) and the gas mean free
path $\lambda=4\times 10^{-9}/\rho_{g}$ where $\rho_{g}$ is the volumetric
mass density of disk gas, $\lambda\gtrsim 200$ cm beyond 10 AU. so from
equation 1,

${\rm St}=\frac{\rho_{s}s}{\Sigma_{g}}$ (7)

where $\Sigma_{g}=\rho_{g}H$ is the surface density of the disk gas. Given
that $\Sigma_{g}\propto a^{-1}$, ${\rm St}\propto a$ for a fixed $s$.
Integrating equation (2.2) back in time with ${\rm St}\propto a$ with the same
boundary condition $a_{0}(t_{\rm age})=a_{\rm ring}$:

$\displaystyle\frac{a_{0,s}(t^{\prime})}{a_{\rm ring}}=$
$\displaystyle-\frac{3}{2}\frac{\alpha}{|\gamma|{\rm St_{r}}}$
$\displaystyle+\left(1+\frac{3}{2}\frac{\alpha}{|\gamma|{\rm
St_{r}}}\right){\rm Exp}\left[\frac{c_{s,r}^{2}}{v_{k,r}}|\gamma|{\rm
St_{r}}\frac{(t_{\rm age}-t^{\prime})}{a_{\rm ring}}\right],$ (8)

where the subscript ‘s’ denotes spatially constant grain size.

When a system features two rings, we account for potential reduction of the
dust masses that can reach the inner ring owing to the partial filtering by
the outer ring as follows. First, we check if $a_{0}$ of the inner ring is
within $a_{\rm ring}$ of the outer ring, and if so, the initial dust sourcing
the inner ring is unaffected by the outer ring. If the condition is not met
however, we account for the filtering by the outer ring by re-computing the
effective $M_{\rm 0,new}$ for the inner ring:

$M_{\rm 0,orig}\left(\frac{a_{\rm 0,out}}{1\,{\rm AU}}\right)^{\beta}-M_{\rm
ring,out}=M_{\rm 0,new}\left(\frac{a_{\rm 0,out}}{1\,{\rm AU}}\right)^{\beta}$
(9)

where $M_{\rm 0,orig}$ is the original unaffected $M_{0}$, $a_{\rm 0,out}$ is
the origin site of dust grains that source the outer ring, and $M_{\rm
ring,out}$ is the measured dust mass of the outer ring. We then recompute St
and $\alpha$ of the inner ring using the new $M_{\rm 0,new}$.

Table 1: Well-resolved dust rings in the DSHARP survey drawn from Dullemond et
al. (2018).

Source | Name | Age | $M_{\star}$ | $L_{\star}$ | $a_{\rm ring}$ | $w_{d}$ | $M_{\rm ring}$ | (St$/\alpha)_{\rm min}$ | (St$/\alpha)_{\rm max}$ | (St$/\alpha)_{\rm rot}$
---|---|---|---|---|---|---|---|---|---|---
|  | (log (yr)) | ($M_{\odot}$) | ($L_{\odot}$) | (au) | (au) | ($M_{\oplus}$) |  |  |
AS 209 | B74 | 6.0 | 0.83 | 1.41 | 74.2 | 3.38 | 31.5 | 4.54 | 7.14 | 5.56
AS 209 | B120 | 6.0 | 0.83 | 1.41 | 120.4 | 4.11 | 69.8 | 6.67 | 9.09 | 7.69
Elias 24 | B77 | 5.3 | 0.78 | 6 | 76.7 | 4.57 | 40.8 | 3.75 | 12.99 | 0
HD 163296 | B67 | 6.9 | 2.04 | 17 | 67.7 | 6.84 | 56.0 | 3.85 | 5.00 | 4.35
HD 163296 | B100 | 6.9 | 2.04 | 17 | 100.0 | 4.67 | 43.6 | 20.0 | 33.3 | 25.0
GW Lup | B85 | 6.3 | 0.46 | 0.33 | 85.6 | 4.8 | 37.0 | 0 | 3.23 | 0
HD 143006 | B41 | 6.6 | 1.78 | 3.8 | 41.0 | 3.9 | 9.9 | 0 | 5.56 | 0
HD 143006 | B65 | 6.6 | 1.78 | 3.8 | 65.2 | 7.31 | 25.6 | 0 | 0.91 | 0

Note. — Column 1: name of the source. Column 2: name of the ring. Column 3:
system age taken from Table 1 of Andrews et al. (2018), with the exception of
HD 163296 for which we take the value quoted in the original reference
(Fairlamb et al., 2015). Column 4: mass of the host star. Column 5: luminosity
of the host star. Column 6: measured orbital distance of the ring. Column 7:
underlying width of the ring after deconvolution. Column 8: measured dust mass
of the ring including the correction for the optical depth ($M^{\rm true}_{\rm
d}$ of Dullemond et al. 2018). Column 9: minimum inferred value of St$/\alpha$
(inverse of column 12 of Dullemond et al. 2018, their Table 3). Column 10:
maximum inferred value of St$/\alpha$ (inverse of column 11 of Dullemond et
al. 2018, their Table 3). Column 11: measured St$/\alpha$ with gas rotation
curves from Rosotti et al. (2020). For rings re-analyzed by Rosotti et al.
(2020), we replace the values of columns 9 and 10 with the lower and upper
1-$\sigma$ error limits based on the measurements of Rosotti et al. (2020).
The corresponding ratio ${\rm St}/\alpha$ is recorded as zero when it is not
available. For rings without gas rotation curve measurements, we update
(St/$\alpha$)min with our $\phi=0.25$ as compared to $\phi=0.02$ adopted by
Dullemond et al. (2018) which increases the gas disk scale height by $\sim$37%
and boosts the derived St/$\alpha$ by order unity factors. For GW Lup B85,
this correction drives minimum St/$\alpha$ to be slightly larger than the
maximum St/$\alpha$ and so we adopt the maximum value to be the true value and
set the minimum value to zero. For HD 143006 rings, the new gas disk scale
heights remain smaller than $w_{d}$ and so (St/$\alpha$)min are recorded as
zero.

Table 2: Limits on St and $\alpha$ for dust rings in Table 1

Source | Name | $f(>M_{\rm dust})_{Q}$ | ${\rm St}_{Q}$ | $\alpha_{Q}$ | $f(>M_{\rm dust})_{M}$ | ${\rm St}_{M}$ | $\alpha_{M}$
---|---|---|---|---|---|---|---
|  | (percentile) |  |  | (percentile) |  |
AS 209 | B74 | 72.54 | 4.50E-04 | 8.10E-05 | 30.91 | 2.03E-03 | 3.66E-04
AS 209 | B120 | 72.54 | 1.22E-03 | 1.59E-04 | 30.91 | 5.01E-03 | 6.51E-04
Elias 24 | B77 | 100 | 9.21E-04 | 2.46E-04 | 15.56 | 1.60E-02 | 4.26E-03
HD 163296 | B67 | 29.48 | 3.54E-04 | 8.14E-05 | 27.07 | 4.73E-04 | 1.09E-04
HD 163296 | B100 | 29.48 | 3.54E-04 | 1.42E-05 | 27.07 | 4.56E-04 | 1.82E-05
GW Lup | B85 | 34.45 | 9.31E-04 | 2.88E-04 | 5 | 2.62E-03 | 8.12E-04
HD 143006 | B41 | 34.05 | 9.62E-05 | 1.73E-05 | 5 | 3.08E-04 | 5.55E-05
HD 143006 | B65 | 34.05 | 2.30E-04 | 2.53E-04 | 5 | 6.72E-04 | 7.38E-04

Note. — Column 1: name of the source. Column 2: name of the ring. Column 3:
maximum $M_{\rm dust}$ percentile for gravitational stability ($Q_{T}\geq 1$).
Column 4: minimum St for $Q_{T}\geq 1$. Column 5: corresponding $\alpha$ for
StQ. Column 6: minimum $M_{\rm dust}$ percentile to explain ring dust mass.
Column 7: maximum St. Column 8: corresponding $\alpha$ for maximum St.

###  2.3 Solving for St and $\alpha$

Substituting equations (6) or (8) into equation (5) and equating it to the
measured total dust mass in an observed ring $M_{\rm ring}$ at the measured
ring location $a_{\rm ring}$ and at the estimated age of the host system
$t_{\rm age}$, we can solve for a quantity that is a combination of $\alpha$
and ${\rm St}$ which can be separated by using the measured ${\rm St}/\alpha$
from, e.g., Dullemond et al. (2018); see Table 1 for the list of DSHARP disks
with well-resolved rings on which we focus our analysis. In case of spatially
constant ${\rm St}$,

$\alpha+\frac{2}{3}|\gamma|{\rm St}=\frac{\left(\frac{M_{\rm
ring}}{M_{0}\epsilon_{\rm trap}}+\left(\frac{a_{\rm ring}}{{\rm
1AU}}\right)^{\beta}\right)^{1/\beta}-\frac{a_{\rm out}}{\rm
1AU}}{\frac{3}{2}\left(\frac{c_{s,r}}{v_{k,r}}\right)^{2}v_{k,r}t_{\rm
age}/{\rm 1AU}}.$ (10)

All the quantities on the right hand side are either measured or specified in
our model setup and so is $\gamma$ so we can calculate the left hand side and
combine it with the measured ${\rm St}/\alpha$ to separately determine
$\alpha$ and St.

In case of spatially constant $s$, we can directly solve for ${\rm St_{\rm
r}}$:

$\displaystyle{\rm St}_{\rm r}=$
$\displaystyle\left(\frac{v_{k,r}}{c_{s,r}^{2}}\right)\frac{a_{\rm
ring}}{t_{\rm age}|\gamma|}$
$\displaystyle\times\log\left(\frac{\left(\frac{M_{\rm
ring}}{M_{0}\epsilon_{\rm trap}}+\left(\frac{a_{\rm ring}}{{\rm
1AU}}\right)^{\beta}\right)^{1/\beta}+\frac{3}{2}\frac{\alpha}{|\gamma|{\rm
St}_{r}}a_{\rm ring}}{\left(\frac{a_{\rm ring}}{\rm
1AU}\right)+\frac{3}{2}\frac{\alpha}{|\gamma|{\rm St_{r}}}\left(\frac{a_{\rm
ring}}{\rm 1AU}\right)}\right)$

where all values on the right hand side are known or measured (and the $\log$
here is natural logarithm). We can then compute $\alpha$ using the measured
${\rm St}/\alpha$.

Above analysis implicitly assumes the $\alpha$ that parametrizes the diffusion
of dust mass in the pressure bump (equation 2) is equivalent to that
parametrizes the angular momentum transfer of the disk gas (equation 2.2). Our
approach here is valid as long as the effective values of the two
parametrization are similar, confirmation of which would depend on the exact
source of both the pressure bump and the redistribution of disk gas angular
momentum. We defer such confirmation to future numerical studies.

We illustrate in Figure 2 the overall trend in the inferred St and $\alpha$
for a range of ${\rm St}/\alpha$ and $M_{\rm ring}$, showing Elias 24 B77 as
an example case. The Stokes number is largely insensitive to the ratio ${\rm
St}/\alpha$ and more of a function of $M_{\rm ring}$ except when
$\alpha\gtrsim{\rm St}$. Larger ring mass translates into larger St because
larger amount of mass needs to drift into a given ring location for a fixed
total dust budget (equivalent behavior is observed when $M_{\rm dust}$ is
lower for a fixed ring mass). The St-$M_{\rm ring}$ relation flattens out at
lower ${\rm St}/\alpha$ because the dust radial motion becomes more dominated
by their coupling to the gas advection.

Compared to the inferred St, the turbulent $\alpha$ varies with both ${\rm
St}/\alpha$ and $M_{\rm ring}$, increasing overall with lower ${\rm
St}/\alpha$ and higher $M_{\rm ring}$, with the $\alpha$-$M_{\rm ring}$
relation strengthening (and $\alpha-{\rm St}/\alpha$ relation weakening) at
lower ${\rm St}/\alpha$. The overall increase in $\alpha$ with more massive
ring can be understood as due to the need to drift more dust into a ring
location, identical to the cause for the positive St-$M_{\rm ring}$ relation.
The rise in $\alpha$ for lower St/$\alpha$ down to St/$\alpha\sim 1$ appears
because for a fixed St, lower St/$\alpha$ implies higher $\alpha$. Below
St/$\alpha\lesssim 1$, the $\alpha-M_{\rm ring}$ relation strengthens because
dust coupling to the gas advection dominates dust drift (the same effect
reducing St-$M_{\rm ring}$ relation).

Comparing the two columns of Figure 2, the aforementioned observed behaviors
as well as the magnitude of St and $\alpha$ appear generally identical between
constant St and constant grain size models, demonstrating the robustness of
our calculation to the assumptions of microphysics of dust grains. Even though
the physics of dust-gas dynamics is controlled by St, the images from ALMA
observations are sensitive to grain size and so for the remainder of the
paper, we will focus our attention on our constant grain size models.

Figure 2: Contour map of St (top row) and $\alpha$ (bottom row) as a function
of ${\rm St}/\alpha$ and the ring mass for a system parameter following Elias
24 B77 (see Table 1) with median $M_{\rm dust}$ (see Figure 1). Whether we
assume constant St (left column) or constant grain size (right column) makes
negligible difference in the calculated St and $\alpha$. The purple dashed
lines in the top row draw the St that corresponds to Toomre $Q=1$ such that
larger St would be consistent with local gravitational stability (see Section
2.4). The purple dashed line in the bottom row delineates $\alpha=10^{-4}$
below which the disk is in the strongly inviscid limit. The measured
St/$\alpha$ and $M_{\rm ring}$ for Elias 24 B77 are indicated with black
circles with the upper and lower limits connected by a line.

###  2.4 Placing limits on St and $\alpha$

Following the calculations outlined in Sections 2.2 and 2.3, we find that the
derived St and $\alpha$ vary most sensitively to $M_{0}$ and $\beta$, which
are controlled by the chosen percentile of $M_{\rm dust}$ distribution (see
Figure 1). Here, we describe how we constrain the lower and upper limits on
reasonable $M_{\rm dust}$ percentiles, which in turn correspond to upper and
lower limits, respectively, on the derived St and $\alpha$.

First, we choose which St/$\alpha$ to adopt. When the rotation curve
measurements are available (AS 209 and HD 163296 rings), the width of the gas
pressure bump can be more precisely determined so we always use
(St/$\alpha$)rot. In the absence of such measurements, we have a choice
between the minimum and maximum St/$\alpha$. The minimum value is evaluated
using equation 2 by taking $w=H$ whereas the maximum value is derived by
setting $w$ as the separation between rings in a given system or the distance
between the dust ring to the nearest minimum (see Table 3 caption of Dullemond
et al. 2018 for more detail). We favor the minimum value where available
because it is more self-consistent with the setup of our model.

Next, we establish that with lower $M_{\rm dust}$ and at fixed St/$\alpha$,
the derived St is higher, as seen in Figure A.1 in the Appendix, and it
follows that $\alpha$ is also higher. Such a trend arises because in disks
lighter with dust, larger radial extent ($a>a_{\rm ring}$) of dust must be
sourced to account for a given $M_{\rm ring}$ and for given system age,
necessitating more rapid inward drift of dust.

We now proceed to establish limits on $M_{\rm dust}$ percentiles. The lower
limit is given by when the corresponding origin site of the dust $a_{0}$ for
either the inner or the outer ring exceeds 300 AU, which we consider to be
unphysically large—this is the maximum size among the Class 0/I disks shown in
Figure 1. Any smaller $M_{\rm dust}$ is considered incompatibly low with the
measured disk conditions. In cases where $M_{\rm dust}$ remains consistent
with a given ring dust mass down to 5th percentiles, we take the 5th
percentile as the lower limit. For all such cases, St appears flat for
$\lesssim$10th percentile so we set this 5th percentile as the absolute lower
limit (see Figure A.1).

The upper limit on $M_{\rm dust}$ percentile is given by the requirement for
the underlying gas disk to be gravitationally stable, which should locally
satisfy

$Q_{T}\equiv\frac{c_{s}\Omega}{\pi G\Sigma_{g}}=\frac{c_{s}\Omega}{\pi
G\rho_{s}s}{\rm St}>1$ (12)

where the second equality derives from equation 7. For our analysis, we adopt
$\rho_{s}=1\,{\rm g\,cm^{-3}}$ and $s=\lambda/2\pi$ where $\lambda=1.25$mm is
the wavelength of ALMA observations (Andrews et al., 2018). It follows that
the local St for a given ring has a lower limit:

${\rm St}>\frac{\pi G\rho_{s}s}{c_{s}\Omega}$ (13)

while keeping in mind the uncertainties in $\rho_{s}$ and $s$ likely result in
at least order unity variations. In one case (Elias 24 B77; see Figure A.1),
we find that even at 100th percentile of $M_{\rm dust}$, $Q_{T}>1$ and so we
take 100 as the upper limit percentile.

For systems with more than one ring, we ensure the same $M_{\rm dust}$
percentile is used within a given system. Among the rings in a system, we take
the smallest $M_{\rm dust}$ percentile for local gravitational stability
(i.e., the global upper limit on $M_{\rm dust}$) and the largest $M_{\rm
dust}$ percentile for dust mass budget (i.e., the global lower limit on
$M_{\rm dust}$). Table 2 summarizes the derived limits on St and $\alpha$.

Figure 3: Derived St (top row) and $\alpha$ (bottom row) for individual rings
from Table 1. Filled circles represent minimum St for local gravitational
stability and corresponding $\alpha$ whereas the unfilled circles represent
maximum St that satisfies the disk mass budget and its corresponding $\alpha$.
In general, all the rings indicate low St $\lesssim 10^{-2}$ and low
$\alpha\lesssim 10^{-3}$, consistent with being approximately inviscid. Figure
4: Estimated local gas surface density $\Sigma_{g}$ for derived minimum St for
gravitational stability (StQ in Table 2; filled circle) and maximum St to be
consistent with ring dust mass (StM in Table 2; open circle) for each ring.
Where available, we indicate with blue squares the estimated $\Sigma_{g}$ from
Powell et al. (2019, AS 209 B74 and B120; HD 163296 B67 and B100) and with
magenta squares the measured $\Sigma_{g}$ from Booth et al. (2019, HD 163296
B67 and B100) where for the latter, we take the values in Table 2 of Rosotti
et al. (2020).

##  3 Results

As illustrated in Figure 3, all the rings studied in this paper feature low St
$\lesssim 10^{-2}$ and low $\alpha\lesssim 10^{-3}$. The derived range in both
St and $\alpha$ are within order unity with the exception of Elias 24 B77
because we can find solutions for this particular ring over the full range of
$M_{\rm dust}$ (see Figure A.1). As we will discuss more in Section 4, such
low St would slow down much of the effects of dust-gas dynamics including
clumping and core coagulation. The low values of $\alpha$ we obtain are
consistent with molecular line measurements of Flaherty et al. (2017) and
suggest the disks to be nearly inviscid, at least at the large orbital
distances ($\gtrsim$10 AU) where we observe dust rings.

In Figure 4, we compare the estimated range of $\Sigma_{g}$ based on our
derived St to what is reported in the literature. For AS 209 and HD 163296
rings, Powell et al. (2019) derive local $\Sigma_{g}$ assuming the local
particle drift timescale to be equal to the system age (see also Powell et
al., 2017). We find that their $\Sigma_{g}$ falls well within the range
expected from our derivation for AS 209 B74 and B120. For HD 163296 rings,
their $\Sigma_{g}$ are lower than even our lower limit. For this disk, a more
direct measurement of gas disk exists with 13C17O (J=3-2) line that is
expected to be optically thin down to $\sim$50 AU (see Booth et al., 2019,
their Figure 2d). Using the disk model of Booth et al. (2019), Rosotti et al.
(2020) quote $\Sigma_{g}=68\,{\rm g\,cm^{-2}}$ and $57\,{\rm g\,cm^{-2}}$ for
B67 and B100 which we show in Figure 4.222Reading Figure 3e Model 2 of Booth
et al. (2019) and converting the line intensity to gas surface density
following the procedure of Carney et al. (2019) with spectral information from
Müller et al. (2005) and 13C17O to H2 abundance ratio $5.39\times 10^{-10}$,
we can recover $\Sigma_{g}$ that are close but higher than what are quoted in
Rosotti et al. (2020) by $\sim$20–40% likely due to estimating the measured
13C17O line intensity by eye. These more directly measured $\Sigma_{g}$ are at
the maximum value allowed for gravitational stability. In case of B67, the
measured $\Sigma_{g}$ is even slightly higher; the difference however is less
than a factor of 2 which can be easily absorbed into the uncertainties in the
grain bulk density $\rho_{s}$, grain size $s$, and also the age of the system.

We close this section with a comment on the difference between Powell et al.
(2017) (and by extension Powell et al. 2019) and our model. For the specific
case of HD 163296, while the adopted stellar parameters are different, we do
not think this is the main cause of the different $\Sigma_{g}$. Powell et al.
(2019) use 36$L_{\odot}$ and 2.3$M_{\odot}$ as compared to our 17$L_{\odot}$
and 2.04$M_{\odot}$. Combined with their choice of temperature profile (see
their equations 3 and 4), their local headwind velocity is smaller than ours
by $\sim$20%. They also adopt younger age estimate for the system 5 Myr which
is $\sim$37% shorter than the age we use. These differences, however, are
washed out by their adopting $\rho_{s}=2\,{\rm g\,cm^{-3}}$ instead of our
$\rho_{s}=1\,{\rm g\,cm^{-3}}$ (c.f. equation 1 of Powell et al. 2019).

Functionally, the calculation of Powell et al. (2017) is equivalent to
removing the coupling to gas advection term in $v_{\rm drift}$ (equation 2.2),
integrating it to some prescribed outer edge of the disk $r_{\rm edge}$ for a
chosen grain size over the system age to solve for St:

${\rm St}\sim\frac{r_{\rm edge}}{t_{\rm age}\frac{c_{s}^{2}}{v_{k}}|\gamma|}$
(14)

and then plugging this St into equation 7 to solve for $\Sigma_{g}$. In this
sense, our approach is more complete as we account for the dust coupling to
inward gas flow which in turn requires us to solve for St and $\alpha$ that
are also consistent with the measured dust mass within the dust ring in
addition to the location of the dust ring. We conclude that the difference
between our result and that of Powell et al. (2019) mostly arises from our
more complete treatment of dust motion and our introduction of extra
constraint from dust mass within rings. We are further encouraged by the
greater agreement between our derived $\Sigma_{g}$ and the direct gas
measurements in HD 163296.

##  4 Discussion

While the overconcentration of dust in the observed rings potentially provides
ideal sites of planet formation, the fact that we see these rings over the
system age imply that they must be stable against vigorous formation of stable
clumps and planetary mass objects. In this section, we discuss the implication
of our derived St and $\alpha$ on the survival of rings and planet formation
at wide orbital separations.

###  4.1 Rings are Stable to Clumping by Streaming Instability

Figure 5: Estimated local dust-to-gas mass ratio for minimum St (filled
circles) and maximum St (unfilled circles) for each ring. Plotted in red
circles are the critical dust-to-gas mass ratio for triggering streaming
instability taking into account extra gas turbulence (e.g., unknown source
generating the dust trap) in addition to what would be generated from dust-gas
interaction. All rings studied are expected to be stable against streaming
instability.

One oft-invoked mechanism to produce dust clumps is streaming instability (SI;
e.g., Youdin & Goodman, 2005) which arises from the non-zero drift of solids
with respect to gas and the dust feedback on the motion of the gas. Key
criteria for clump formation by SI is to have the local dust-to-gas mass ratio
$\Sigma_{\rm ring}/\Sigma_{g}$ and St high enough.333Clumping is inherently
nonlinear and so while (linear) SI always exists, we are interested in the
conditions that allow for SI growth timescale that is shorter than the system
age and more critically that are explicitly found to trigger clump formation.
We first compute the dust-to-gas ratio for the rings studied in this paper by
converting $M_{\rm ring}$ of each ring into $\Sigma_{\rm ring}$ following
(c.f. equation 8 of Dullemond et al. 2018)

$\Sigma_{\rm ring}\equiv\frac{M_{\rm ring}}{(2\pi)^{3/2}a_{\rm ring}w_{d}},$
(15)

and $\Sigma_{g}$ follows from Figure 4.

Li & Youdin (2021) provide an empirical fit to their numerical calculations of
the critical dust-to-gas mass ratio for clumping by SI. We adopt their
equation 14 which takes into account extra turbulence in addition to that by
dust-gas interaction that drives the instability to be consistent with our
definition of dust ring (equation 2). We emphasize that for our derived St and
$\alpha$ the additional turbulence term that sets the dust disk scale height
($H_{\rm p,\alpha}$ in Li & Youdin 2021) dominates over that expected from
pure dust-gas interaction (their $H_{\rm p,\eta}$). Figure 5 demonstrates that
for all the rings we study, the derived dust-to-gas mass ratio is always
smaller than that required for clumping by SI.

It is not surprising that accounting for the extra turbulence impedes SI. Dust
feedback onto gas is most effective when the relative local concentration of
dust is already comparable to that of gas. Higher level of turbulence puffs up
the solid disk diluting the dust concentration in the midplane, which in turn
would lessen the effect of dust feedback. In fact, Chen & Lin (2020) find that
when $\alpha\gtrsim{\rm St}^{1.5}$, the growth timescale by SI becomes
untenably long (defined as $\sim$106 times local orbital time; see also
Umurhan et al. 2020) and we verify that this inequality is met for all the
rings we study with the exception of Elias 24 B77 at its maximum St and
corresponding $\alpha$ (but even for this case, $\alpha/{\rm St}^{1.5}\sim
0.6$).

Faced with the difficulties with clump formation by SI in turbulent rings, Liu
& Bai (2023) identify dust-driven Rossby wave instability that may lead to
clumping. Although intriguing, we cannot verify whether these new
instabilities would be active in the rings we study because Liu & Bai (2023)
do not report results for sufficiently low dust-to-gas mass ratio. We
therefore conclude that the rings studied here are stable against clumping by
SI but proceed with a case where initial clumps may nevertheless emerge by
other dust-driven instabilities.

###  4.2 Dust Clumps in Rings are Unstable

Figure 6: The ratio of ring surface density (equation 15) to the minimum
density required for nucleation of bound clump (equation 16) vs. the radial
location for each ring. Plotting scheme is identical to that of Figure 3. None
of the rings are able to nucleate stable clumps.

If rings were to create planetary cores, the initial dust clumps must be
gravitationally bound against particle diffusion (Klahr et al., 2018) and
stable against tidal shear (Gerbig et al., 2020). Clump formation that meets
both requirements is possible when

$\Sigma_{\rm ring}>\left(\frac{M_{\star}v_{\rm rms}^{2}}{\pi^{2}Ga_{\rm
ring}^{3}}\right)^{1/2}=\left(\frac{M_{\star}}{\pi^{2}Ga_{\rm
ring}^{3}}\right)^{1/2}c_{s}\left(\frac{\alpha}{1+{\rm St}}\right)^{1/2}$ (16)

(c.f. equation 40 of Lee et al. 2022a; note that they had a typo on the power-
law index of $a_{\rm ring}$, equivalent to their $R_{f}$, which we correct
here), where $\Sigma_{\rm ring}$ is the solid surface density of the dust
ring, and $v_{\rm rms}$ is the velocity dispersion of particles within the
ring which we take as $v_{\rm rms}=c_{s}\sqrt{\alpha/(1+{\rm St})}$ (Youdin &
Lithwick, 2007) assuming the particle random velocities to be coupled to gas
turbulence.

As shown in Figure 6, rings that are at wider orbital separations are more
likely to nucleate bound dust clumps, owing to the minimum surface density
scaling as $\propto c_{s}a_{\rm ring}^{-3/2}$ (physically, the tidal shear
effect diminishes farther out). Nevertheless, none of the rings studied in
this paper are dense enough to nucleate a stable bound clump. Repeating the
analysis with using (St/$\alpha$)max uniformly so that the derived $\alpha$ is
smaller (and therefore the minimum required $\Sigma_{\rm ring}$ is small; c.f.
equation 16) recovers only AS 209 B120 to be able to nucleate a stable bound
clump and only at the lowest St (1.21$\times 10^{-3}$) and corresponding
$\alpha=1.33\times 10^{-4}$.

###  4.3 Mass growth by pebble accretion

Focusing on the most optimistic scenario of a bound clump formation in AS 209
B120 as identified in the previous section, we now evaluate how massive such a
clump may grow. First, we evaluate the minimum mass of a bound planetesimal
clump following equation 41 of Lee et al. (2022a):

$M_{\rm pl}>\frac{1}{9}M_{\star}\left(\frac{c_{s}^{2}}{3\pi G\Sigma_{\rm
ring}a_{\rm ring}}\right)^{3}\left(\frac{\alpha}{1+{\rm St}}\right)^{3},$ (17)

which evaluates to 2.2$\times 10^{-6}M_{\oplus}$ and will serve as the minimum
initial mass.

Further mass growth of a clump ensues by accreting solids whose accretion rate
is generally written as

$\dot{M}_{\rm core}=2\Sigma_{\rm ring}R_{\rm acc}v_{\rm acc}\times{\rm
min}(1,R_{\rm acc}/H_{\rm sol})$ (18)

where particles that enter within $R_{\rm acc}$ of the protocore at speeds of
$v_{\rm acc}$ (measured as the relative speed between the protocore and
particles) will be accreted, and $H_{\rm sol}\equiv H\sqrt{\alpha/(\alpha+{\rm
St})}$ is the solid disk scale height (Youdin & Lithwick, 2007).

We consider the theory of pebble accretion to determine $\dot{M}_{\rm core}$
(e.g., Ormel & Klahr, 2010). The core-particle relative velocity $v_{\rm acc}$
is a combination of headwind $v_{\rm hw}$, shear $(3/2)R_{\rm acc}\Omega$, and
turbulent random velocity $v_{\rm rms}$ (see, e.g, Lin et al. 2018, their
equation 14; see also Ormel & Liu 2018 and Rosenthal et al. 2018). In
calculating $v_{\rm hw}$, we account for the deceleration of particles in the
pressure trap. Direct numerical simulations find that the peaks of dust
density often occur at the radial center of the pressure bump (e.g., Lee et
al., 2022a) and so we evaluate $v_{\rm hw}$ at the maximum size of the clump
min($w_{d}$,$H_{\rm sol}$) away from the center of the pressure bump. The
width of the gas pressure bump for AS 209 B120 is precisely determined using
gas rotation curves and so we follow equation 2 of Rosotti et al. (2020) and
derive

$\displaystyle v_{\rm hw,bump}$
$\displaystyle=-\left(\frac{c_{s}^{2}}{2v_{k}}\right)\gamma_{\rm bump}$
$\displaystyle=\left(\frac{c_{s}^{2}}{2v_{k}}\right){\rm
Exp}\left(-\frac{1}{2}\frac{\alpha}{\rm St}\right),$ (19)

where $w_{d}<H_{\rm sol}$ for AS 209 B120 and $v_{\rm hw,bump}\sim 980\,{\rm
cm\,s^{-1}}$ is $\sim$34% smaller than $v_{\rm hw}$ in the smooth background
disk (c.f. Appendix A of Rosotti et al. 2020). Because $v_{\rm hw,bump}>v_{\rm
rms}\sim 335\,{\rm cm\,s^{-1}}$, we only need to determine whether the
accretion is headwind or shear-dominated. Pebble accretion is expected to be
headwind rather than shear-dominated when the mass of the accreting core is
small (c.f. equation 7.9 and 7.10 of Ormel 2017; see also equation 24 of
Lambrechts & Johansen 2012):

$M_{\rm core}<\frac{1}{8}\frac{v_{\rm hw,bump}^{3}}{G\Omega}{\rm St}^{-1}.$
(20)

This maximum mass for headwind-dominated accretion is $\sim$1.8$M_{\oplus}$
for AS 209 B120 and we will show that a dust clump would never exceed this
mass. We therefore safely assume $v_{\rm acc}\sim v_{\rm hw,bump}$ at all
times.

If pebble accretion were to occur, the particle stopping time must be shorter
than the encounter time (${\rm St}/\Omega<R_{\rm acc}/v_{\rm acc}$; the
settling regime, see Ormel & Klahr 2010). We first assume this to be true to
compute $R_{\rm acc}$ and verify a posteriori that the condition is indeed
satisfied. Under the settling condition,

$v_{\rm acc}=4\frac{GM_{\rm core}}{R_{\rm acc}^{2}}\frac{\rm St}{\Omega}$ (21)

so that the particles that accrete reach a terminal velocity during the
encounter with the core. Substituting $v_{\rm acc}=v_{\rm hw,bump}$ to solve
for $R_{\rm acc}$, the settling condition is satisfied when

$M_{\rm core}>\frac{1}{4}\frac{v_{\rm hw,bump}^{3}}{G\Omega}{\rm St}$ (22)

which is larger by a factor of 2 compared to equation 7.11 of Ormel (2017)
because of different assumptions of order unity factors in the settling
condition and in equation 21. For AS 209 B120, this minimum mass for pebble
accretion is $\sim$5.23$\times 10^{-6}M_{\oplus}$ which is larger than the
minimum initial clump mass (equation 17) so we use equation 22 as the initial
core mass for pebble accretion.

In equation 18, the accretion proceeds in a three-dimensional fashion when
$R_{\rm acc}<H_{\rm sol}$:

$M_{\rm core}<\frac{1}{4}\frac{v_{\rm hw,bump}\Omega H^{2}}{G{\rm
St}}\left(\frac{\alpha}{\alpha+{\rm St}}\right).$ (23)

Headwind-dominated accretion will always be in three-dimensional regime if the
maximum mass in equation 20 is smaller than that in equation 23 which
translates to

$\frac{1}{2}\left(\frac{v_{\rm
hw,bump}}{c_{s}}\right)^{2}<\frac{\alpha}{\alpha+{\rm St}},$ (24)

which is satisfied for AS 209 B120.

The accretion rate then follows

$\displaystyle\dot{M}_{\rm core}$ $\displaystyle=\frac{8\Sigma_{\rm
ring}GM_{\rm core}{\rm St}}{c_{s}}\left(\frac{\alpha+{\rm
St}}{\alpha}\right)^{1/2}$
$\displaystyle=\left(\frac{2}{\pi}\right)^{3/2}\left(\frac{v_{k}}{c_{s}}\right)^{2}\frac{M_{\rm
ring}(t)}{M_{\star}}M_{\rm core}\Omega\frac{H}{w_{d}}{\rm
St}\left(\frac{\alpha+{\rm St}}{\alpha}\right)^{1/2},$ (25)

and

$M_{\rm core}(t)=M_{\rm pl,0}\,{\rm Exp}\left[F(M_{\star},a_{\rm
ring},\alpha,{\rm St})\int^{t}_{0}M_{\rm ring}(t^{\prime})dt^{\prime}\right]$
(26)

where $M_{\rm pl,0}$ is set by either equation 17 or equation 22 whichever is
larger, $F(M_{\star},a_{\rm ring},\alpha,{\rm St})$ encapsulates all the other
time-invariant quantities in equation 25, and $M_{\rm ring}(t^{\prime})$
follows from equation 5.

For constant particle size, taking $a_{0}$ from equation 8:

$\displaystyle\int^{t}_{0}M_{\rm
ring}(t^{\prime})dt^{\prime}=M_{0}\epsilon_{\rm trap}\times$
$\displaystyle\left[-\frac{(A+De^{Ct^{\prime}})^{1+\beta}\,_{2}F_{1}(1,1+\beta,2+\beta,1+\frac{D}{A}e^{Ct^{\prime}})}{AC(1+\beta)}-a_{\rm
ring}^{\beta}t^{\prime}\right]^{t}_{0}$ (27)

where

$A\equiv-\frac{3}{2}\frac{\alpha}{|\gamma_{\rm bump}|{\rm St}_{r}}a_{\rm
ring},$ (28) $D\equiv a_{\rm ring}-A,$ (29)
$C\equiv\frac{c_{s,r}^{2}}{v_{k,r}}|\gamma_{\rm bump}|{\rm St_{r}}a_{\rm
ring}^{-1},$ (30)

and ${}_{2}F_{1}$ is the hypergeometric function.

Figure 7 demonstrates minimal core growth within AS 209 B120 (pink line). The
low St, $\alpha$, and the wide orbital separation all conspire to render core
growth prohibitively sluggish. We therefore conclude that the observed rings
are stable against clump formation or even if it nucleates dust clumps, they
are expected to be destroyed (primarily by tidal shear) and even in a marginal
case where a bound clump can form, such clumps would not have enough time to
grow in mass, ensuring the survival of the rings as they are observed.

Our finding of minimal planet formation in dust rings differs from that of Lau
et al. (2022) and Jiang & Ormel (2023) who argue for rapid planet formation in
dust rings (but see Morbidelli, 2020). First, both studies assume stable
planetesimal formation in rings: Lau et al. (2022) use a prescribed activation
function that is independent of local turbulence while Jiang & Ormel (2023)
set up their rings so that the local dust-to-gas ratio is already one. We find
that such prescribed conditions are not realized in observed rings nor would
the clumps be stable if formed. Second, Lau et al. (2022) adopt a distribution
of St that samples high values $\sim$0.1 which dominates the dust mass (see
their Figure 3) and would dramatically accelerate the rate of pebble
accretion. We find that such high St are not likely in the rings we study.
Jiang & Ormel (2023) on the other hand explore low St $\sim$10-3 with default
$\alpha=10^{-3}$ so at face value, we would expect planet growth to be even
more impeded under their condition. The reduction by higher $\alpha$ is
dwarfed by their choosing dense $\Sigma_{\rm ring}$ which would exponentially
accelerate the rate of pebble accretion (see equation 25). From their equation
34, the dust surface density of AS 209 B120 would correspond to $\sim$1 g cm-2
which is about an order of magnitude larger than the observed value.

Figure 7: Mass growth of a dust clump within AS 209 B120 (pink line). Shown in
other lines are the rescaled mass growth if the ring were to be placed at
closer-in distances with the same $M_{\rm ring}$, $w_{d}/a_{\rm ring}$, St,
and $\alpha$. Planet formation is prohibitively slow at the measured location
of the ring but accelerates dramatically inside 10 au, potentially engulfing
all the ring masses if placed at 3 au.

###  4.4 Implication on planet formation

One caveat to our conclusion above is that we ignore the possibility of
creating multiple clumps. Without direct numerical simulation, it is not clear
how many bound clumps may form. Nevertheless, we can estimate the number using
the maximum size of the clump as the dust scale height:

$\displaystyle N_{\rm cl}$ $\displaystyle\sim\frac{2\pi a_{\rm ring}}{H_{\rm
sol}}$ $\displaystyle\sim\frac{2\pi a_{\rm ring}}{H}\sqrt{\frac{\alpha+{\rm
St}}{\alpha}}$ (31)

which is $\sim$170 for AS 209 B120. Unless all the clumps are initially about
half an Earth mass—which is more massive than typically quoted masses of
initial planetesimals (e.g., Cuzzi et al., 2010; Gerbig & Li, 2023)—the total
amount of mass locked within these clumps would be negligible compared to the
total ring dust mass, also consistent with the existence of rings.

By contrast, if we compute the same analysis with the ring placed at closer-in
distances, planet formation becomes much more efficient, so much so that the
entire ring can be engulfed into a planet inside 10 au (see Figure 7). In this
experiment, we keep $M_{\rm ring}$, $w_{d}/a_{\rm ring}$, St, and $\alpha$
fixed and scale all other quantities with $a$, including $\phi\propto a^{2/7}$
(Chiang & Goldreich, 1997) so that the accelerated growth at shorter $a$ is
due to the shorter dynamical time (see also Chambers, 2021).444At these
shorter distances, pebble accretion becomes shear-dominated but remain in
three-dimensional regime so that $\dot{M}_{\rm core}$ and $M_{\rm core}(t)$
still follow equations 25 and 26.

This strong sensitivity to the radial location of the ring has an interesting
implication on the connection between disk and exoplanet observations. In a
recent review, Bae et al. (2023) find that disks with substructures tend to be
of high disk mass (see their Figure 3, panel a), whereby the mass is obtained
from the measured mm-flux assuming optically thin limit. Given that larger
disks tend to also be luminous (e.g., Tripathi et al., 2017; Zormpas et al.,
2022), we may deduce that disks with substructures tend to be large. As
cautioned by Bae et al. (2023), the observed lack of rings in lighter
(smaller) disks may be an observational bias as such disks lack the required
high resolution data, but the results presented in this paper would suggest
that this may be physical. It may be that even if such rings existed, they may
quickly have transformed into planetary mass objects and therefore no longer
observable. Such a prospect is also consistent with current exoplanet
demographics. Direct imaging surveys report a decline in the population of gas
giants beyond 10 au (e.g., Nielsen et al., 2019), which is also consistent
with the long baseline radial velocity surveys (e.g., Fulton et al., 2021)
that show a peak in the giant occurrence rate at 1–10 au. Our calculation
demonstrates that the initial growth of planets is difficult beyond $\sim$10
au even within dust rings where we see concentration of mass. By contrast,
inside $\sim$10 au, formation of massive cores becomes precipitously more
likely.

Our calculation further demonstrates that at $\sim$10 au and beyond, rocky
bodies can at best be a Mars mass individually (see orange curve in Figure 7).
While planets are often invoked as the source of the rings (e.g., Teague et
al., 2021), conclusive evidence remains elusive for many of the observed
ringed disks (Speedie & Dong, 2022). Even if the first generation planets did
indeed create some of these rings, such rings are expected to be stirred up
(e.g., Bi et al., 2021, 2023), hindering both planetesimal formation and
pebble accretion and therefore the generation of second population of planets.
As long as the rings sampled in this study are a good representation of
typical planet-forming disks, our results would predict that, like the gas
giants, we would see a similar decline in the population of smaller Neptunes
and super-Earths beyond $\sim$10 au, to be confirmed by the Roman Space
Telescope.

##  5 Conclusion

By coupling the measured properties of dust rings in nearby bright T Tauri
disks with Class 0/I disk using dust radial equation of motion, we have
separately determined local St and $\alpha$ in ringed disks at large orbital
distances. We find that both parameters are low ($10^{-4}\lesssim{\rm
St}\lesssim 10^{-2}$; $10^{-5}\lesssim\alpha\lesssim 10^{-3}$) consistent with
inviscid disks where Type I migration would be prematurely halted. We further
find that given such low St, the estimated local $\Sigma_{g}$ is high so that
the dust-to-gas ratio is too small to trigger streaming instability for the
corresponding $\alpha$. Even if initial clumping may be possible by other
dust-driven instabilities, we find that the rings are not dense enough to
nucleate clumps that are stable against both turbulent diffusion and tidal
shear. In one marginal exception, at the lowest St and $\alpha$, AS 209 B120
may be able to create bound clumps but at such wide orbital distances,
combined with low St and $\alpha$, core growth by pebble accretion is minute.
Our analysis is consistent with the fact that the rings are observable over
the system age.

An interesting implication of our analysis is that beyond $\sim$10 au, we may
see a decline in the planet population even for Neptune-class objects, similar
to the already observed decline in wide-orbit gas giants, insofar as the
ringed disks being typical planet-forming disks. Such a prediction should be
testable by space-based microlensing studies such as the Nancy Grace Roman
Space Telescope.

The anonymous referee provided an encouraging and insightful report that
helped improve the manuscript. I thank Wlad Lyra and Wenrui Xu for their
questions at the Extreme Solar Systems V which motivated some of the
supporting calculations and Ruobing Dong for useful discussions. E.J.L.
gratefully acknowledges support by NSERC, by FRQNT, by the Trottier Space
Institute, and by the William Dawson Scholarship from McGill University.

##  Appendix A Limits on Stokes number for each system

We illustrate in Figure A.1 the range of allowable St for each dust rings over
5th and 100th percentile of initial disk mass as shown in Figure 1.

Figure A.1: Derived St as a function of initial disk dust mass percentiles.
For each system, the minimum St required for local gravitational stability is
drawn in horizontal dashed line.

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