Architecture of Planetary Systems Predicted from Protoplanetary Disks Observed with ALMA. I. Mass of the Possible Planets Embedded in the Dust Gap

In the standard core-accretion scenario, a giant planet forms out of a small solid core and grows via two-stage accretion processes as illustrated in Figure 1. At the first stage when its core mass is small, a planet mainly accretes solid materials from the surrounding disk. While it was believed that the accreting material at this stage is mainly km-size planetesimals (e.g., Safronov 1969), recent studies (e.g., Ormel & Klahr 2010; Ormel 2017) have suggested that the aerodynamically small particles with Stokes number between 10⁻³ and 1, so-called pebbles, may instead be the major source of accretion. In the following, we will use the terms “dust grains” and ”pebbles” interchangeably.

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As pebble accretion continues, the planet becomes more massive and starts to gravitationally interact with its surrounding disk with increasing strength. Several studies have suggested that the dust disk is more susceptible to gap-opening torque than the gas disk (e.g., Paardekooper & Mellema 2004; Muto & Inutsuka 2009; Rosotti et al. 2016). Before the planet is massive enough to generate a pressure maximum, it is possible to open a gap only in the dust disk through tidal torque and gas drag (Dipierro et al. 2016; Dipierro & Laibe 2017). For instance, Rosotti et al. (2016) showed that even a planet with 15 M_⊕ can open the dust gap without creating a pressure maximum. This result implies that dust gaps carved by planets above the minimum dust gap-opening mass can be possibly observed by ALMA which is sensitive to millimeter-sized dust particles. Once the dust gap is opened, the planet loses access to the fraction of pebbles sensitive to the ALMA band. Nevertheless, until the planet reaches the pebble isolation mass M_iso and generates a gas pressure maximum, some pebbles may still cross the planetary orbit depending on their Stokes number.

Pebble accretion is a self-terminating process that ends when the planetary mass reaches a certain mass threshold, namely, the pebble isolation mass (e.g., Lambrechts et al. 2014). At the pebble isolation mass, a planet is just massive enough to carve a shallow gas gap that generates a pressure maximum at the outer edge of the gap. Such a pressure maximum can trap all the pebbles with Stokes number >0.005 (Bitsch et al. 2018), a range that effectively covers those pebbles that are most favorable to pebble accretion. The planet has no access to the inflow of the pebbles afterward and the pebble accretion process is stopped. Since both the width and depth of the gas gap increase with the planetary mass, it is reasonable to expect that a relatively deep gas gap is formed once the planetary mass exceeds the pebble-isolation mass, and the dust gap is also strongly shaped by the gas gap due to the pressure maximum. Before reaching pebble isolation, it is reasonable to assume that a visible gap should be formed only in the dust disk because the gas gap is too shallow.

Three-dimensional hydrodynamical simulations by Bitsch et al. (2018) indicate the pebble isolation mass M_iso(R) at the location of R from the central star is written in terms of the stellar mass M_*, the aspect ratio of the disk h/R, the pressure gradient , and the viscosity of the gas α, as

Throughout the present paper, we consider locally isothermal disks with a temperature profile of T ∝ R^−1/2 and a surface density profile of Σ ∝ R⁻¹. We simply adopt the mid-plane pressure . Then we have η = − 2.75 and the second bracket in Equation (4) is 25/24. It is worth noting that M_iso obtained from 2D simulations is scaled down by a factor of 1.5 due to different treatments of the surface density profile (Bitsch et al. 2018, Appendix B). In this study, we nevertheless adopt M_iso,3D as the pebble isolation mass as we think it is more reliable to apply to real systems.

After reaching the pebble isolation mass, the gas envelope around the planet starts cooling, which triggers the runaway gas accretion that marks the beginning of the second stage. At this stage, the planet grows quickly via gas accretion and becomes a gas giant. As the planet continues to grow, it carves a deeper gap in the disk, and the gas accretion rate is slowed down. The accretion eventually stops when the PPD is dispersed due to both stellar accretion and photoevaporation.

The planetary formation picture illustrated in Figure 1 states that a planet opens a gap when it is massive enough. Since the properties of the gap are determined by the planet–disk interaction, one may infer the mass of the embedded planet from the parameters characterizing the gap morphology. Indeed, Kanagawa et al. (2016) performed a series of hydrodynamical simulations in the gas gap open case, and found the following empirical relation for the mass of the embedded planet:

where Δ_gap, h_gap, and R_gap are the width, scale-height, and location of the gap. The empirical formula, Equation (5), has been basically confirmed by Dong & Fung (2017).

It is important to note that Δ_gap in Equation (5) is defined for the gas profile in the original study. Therefore, while applying this mass prediction formula to those gaps observed in the dust continuum, we need to assume that the gas gap has already opened, and its width roughly matches that of the dust gap due to good gas–dust coupling.

Zhang et al. (2018) carried out gas–dust hydrodynamic simulations and investigated the scaling relation between the planet mass and the width of the gap as in Equation (5). Overall their fitting is consistent to Equation (5) ⁴ , though it deviates slightly due to the decoupling between gas and dust when St ≳ 10⁻². Due to this deviation, Equation (5) may overestimate the planet mass. To take this difference into account, we include an uncertainty factor for the estimated mass as will be discussed in Section 3.3. Also, as will be seen in Section 3.2, the estimated mass given by Equation (5) agrees with that estimated by Zhang et al. (2018) within this level of uncertainty.

Before reaching the pebble-isolation mass, no significant gas gap is expected to form and only a dust gap is visible as described in Section 3. Equation (5) is not relevant in order to estimate the mass of the embedded planet in such a dust-only gap regime. For instance, Figure 12 of Zhang et al. (2018) exhibits a large scatter of the range of small planetary mass (or small ), which may possibly indicate the transition of different scaling between the dust-only gap regime and gas gap regime that we assumed in this paper. In the dust-only gap regime, the width of the dust gap would be better described by the Hill radius scaling, instead of Equation (5).

The Hill radius model suggests that the gap width Δ_gap is proportional to the the planetary Hill radius R_H at the gap location R_gap. This relation can be rewritten for the planetary mass as

where C is the proportional constant and previous studies indicate that 4.5 < C < 7 (Rosotti et al. 2016). Following Lodato et al. (2019), we set C = 5.5 in this study.

While Equations (5) and (6) should connect to each other, the connection might not be smooth. Indeed, Rosotti et al. (2016) investigated both the dust-only gap case and relatively deep gas gap case and found the gap width can be nicely scaled by a single Hill radius (see Figure 16 of Rosotti et al. 2016). Although their scaling result does not explicitly depend on any gas parameters, it implies that Equation (6) may be valid even in the gas gap-opening regime for some specific setup. Investigation of the connection between the two scaling relations is beyond the scope of this paper and will be reserved for the future work.

The above argument indicates that Equation (5) is applicable only for a gaseous planet embedded in a disk with both dust and gas gaps, while Equation (6) may be more relevant for a low-mass planet with a dust gap alone. Unfortunately, the high-resolution disk substructure is available only for the dust component due to the difficulty of resolving the line emission from gas. Therefore, it is not easy to observationally distinguish between the two different predictions for the mass of embedded planets.

According to the planetary formation picture described in Section 3.1, however, the pebble isolation mass M_iso may be used as a reasonable threshold of the planetary mass, which is consistent with the presence or absence of the unobservable gas gap; if the predicted planetary mass is greater than M_iso, the gas gap is likely to be opened as well.

Equation (5) implies that M_p,gas > M_iso(R_gap) translates into the lower limit of (Δ_gap/R_gap):

Thus, if (Δ_gap/R_gap) of the observed dust gap satisfies the above inequality (7), the gap is likely expected to be associated with the gas gap of the similar value of . Thus Equation (5) provides a reasonably good mass estimate of the embedded planet.

Similarly, Equation (6) implies that M_p,dust < M_iso(R_gap) is rewritten as

If a dust gap satisfies the inequality (8), it is likely to be a dust-only gap, and Equation (6) may be preferred to predict the mass of the embedded planet.

The above argument offers a possibility to distinguish between the presence and absence of the gas gap with a similar profile of the observed dust gap. In reality, however, the mass predictions suffer from significant uncertainties for multiple reasons. The major uncertainty comes from the gap width Δ_gap. Rosotti et al. (2016) found that the gap profile in their simulations changes in a time-dependent fashion and the observed gap width would vary accordingly, which leads to a factor of 2–3 uncertainty for the planetary mass prediction. Also Equation (5) derived by Kanagawa et al. (2016) depends on the coupling strength between dust and gas components in the disk, which would affect its overall normalization. Furthermore, the definitions of the inner and outer boundaries of the gap are not unique, and vary among different papers. Finally, the host stars in PPDs are generally young, and their luminosities and masses cannot be well constrained from the stellar evolution model. Therefore, we have to keep in mind such uncertainties in interpreting inequalities (7) and (8).

Due to various uncertainties of the observed dust profiles, we introduce fudge factors in the planetary mass predictions. Specifically, we impose the following criteria (hereafter “C1” and “C2”) for the dust-only gap and for the gas gap, respectively:

where F₁ and F₂ are the fudge factors. Substituting Equations (3), (5), and (6), inequalities (9) and (10) are rewritten as the relations between Δ_gap/R_gap and h_gap/R_gap:

According to the two criteria C1 and C2, the observed gaps are classified into four groups on h_gap/R_gap—Δ_gap/R_gap plane, as schematically illustrated in Figure 2 and summarized in Table 2. While the relevant choice of the fudge factors is not clear, we adopt F₁ = F₂ = 2 below, just for simplicity.

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On the basis of the classification, we can interpret the nature of the gap. Group I implies that the observed dust gap is not associated with a gas gap, and Equation (6) is preferred for the planetary mass estimate. Group II, however, implies that the disk has a gas gap as well, and Equation (5) is a reasonable estimate for the embedded gas planet. Gaps in group III are consistent with both interpretations and we cannot distinguish between the two possibilities M_p,dust and M_p,gas, unless the presence or absence of the gas gap is clarified observationally.

Finally, gaps in group IV are inconsistent with either interpretation. Such gaps may be of nonplanetary origin, or it may simply point to a possible inaccuracy or limitation of the mass prediction models that we adopted here. In this context, it is interesting to note that there is no gap classified as group IV in our sample. More details of the interpretation of the groups can be found in Appendix A, where we show how the gap width varies with planetary mass at three different locations of a disk.

According to C1 and C2, we classify the gaps into groups I to IV as shown in Figure 3. The red and green regions correspond to group I (dust gap only) and group II (gas gap opened) gaps, respectively, and the overlapping region (light brown) bounded by Equations (9) and (10) is group III in which both interpretations are possible. We adopt the fiducial value of the viscosity, α = 10⁻³. The relations of M_p,dust = M_iso and M_p,gas = M_iso are plotted in red and green lines, respectively. The quoted error bars of each symbol along the y-axis is estimated from the 1σ uncertainties of R_gap and Δ_gap listed in previous studies. If they are not available, we do not plot the error bars.

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The result shows that out of 55 gaps, there are 21 gaps in group I, 19 gaps in group II, and 15 gaps in group III, i.e., more than two-thirds (73%) of the gaps are classified as either dust-only gaps or gas gaps according to our criteria. The majority (52%) of the gaps in single-gap disks are classified as dust-only gaps, which are also the dominant group in the multigap disks. It should be noted that when the predicted planetary mass is close to M_iso, the gap classification becomes sensitive to the adopted model of the temperature and aspect ratio. Only one gap is below the dashed line Δ_gap = h_gap, which is the minimum dust gap width for the Stokes number St = 1. If this gap is opened by a planet, the coupling between the dust and gas should be weak with St > 1 (Dipierro & Laibe 2017; Lodato et al. 2019). While we give mass predictions to this gap as well, it may be of nonplanetary origin.

The distribution of the gaps from both single-gap disks and multigap disks shows that gaps with small aspect ratios (and thus at the inner region) are more likely to be gaps that are opened in both gas and dust disks. In the outer region where the aspect ratio is larger, the gaps are more likely to be dust alone. This trend is more clearly seen in the case of multigap disks (bottom panel of Figure 3), which may be attributed to the presence of planet–planet interaction in a multiplanetary system. We will discuss this difference further in Section 5.2.

We estimate planetary masses in the observed disk gaps according to our gap classification, assuming that planets are fully responsible for creating the gap substructures, and gaps and planets are in one-to-one correspondence. The mass prediction result is summarized in Table 3. In the case of group III, we list both M_p,dust and M_p,gas.

4.2.1. Case with Fiducial α = 10⁻³

The predicted planetary mass ranges from 1.6M_⊕ to 4.8M_J , but with significant diversities. Figure 4 plots the distribution of the planets, M_p /M_* against the semimajor axis for α = 10⁻³. We also plot the observed exoplanet population (extracted from exoplanet.eu as of 2021 March) with confirmed planetary mass, semimajor axis, and stellar mass for comparison. We specifically label several widely separated giant planets observed by direct imaging including HR 8799, β Pictoris, and PDS 70, which would be relevant to compare with our prediction.

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Figure 4 indicates that our predicted planets are located in a fairly distinct region relative to that of the observed population. It may imply that these observed planets have experienced significant inward orbital migration before the disk dispersal due to disk–planet interaction (Zhang et al. 2018; Lodato et al. 2019). Furthermore, their mass should have increased via accretion during the migration. A possible model of such evolutionary tracks is introduced by Tanaka et al. (2020), which suggests the following relation between mass and semimajor axis:

where is a constant. Just for reference, we plot examples of the tracks with the range of values that match our predicted planets. While a fraction of observed planets is consistent with the evolutionary tracks, a majority of them is far outside of the region that the simply model predicts. At this point, the reason of the discrepancy is not clear, and more detailed study of their evolution is necessary, which we plan to show elsewhere.

The architecture of the planets predicted from the multigap systems is shown in Figure 5. Mass predictions based on M_p,dust and M_p,gas are plotted in circles and squares, respectively. The filled (open) symbols indicate that they are consistent (inconsistent) with the criteria C1 and C2. The pebble isolation mass is also indicated by a straight line. As mentioned before, the planetary mass of most disks generally decreases against R, except for the cases of DL Tau, HD 163296, HL Tau, and V1094 SCO: the masses of the three planets in the DL Tau system increase with R; for the rest of the systems, at least one pair of planets exists that does not follow the trend. We found planets that violate this trend are always gas gap planets. If an outer planet forms first and later approaches the inner planet via inward migration, it may start runaway gas accretion earlier than the inner planet and quench its growth by shielding the gas flow. This implies that the architecture of a multiplanetary system is sensitive to its initial configuration.

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Figure 5 also plots the locations of the outer 2:1 resonance period for the inside gas gap planets (including marginal ones in group III) in dashed vertical lines. In many systems (i.e., AA Tau, AS 209, GY 91, Elias 24, DL Tau, GO Tau, HD 163296, and V1094 SCO) the gas gap planet has a dust gap planet close to its 2:1 period ratio. It may be due to the fact that the dust gap planet is likely to form at the 2:1 resonance place of the gas gap (or relatively large) planet. Alternatively, such a configuration can be the consequence of the combination of planetary migration and resonance trap. These mass orders may be related to alternative formation scenarios of planets, which will be discussed in the next section.

4.2.2.  α Dependence

So far we have adopted the fiducial α value 10⁻³. In fact, however, both M_p,gas and M_iso depend on α, and different values of α affect both classification criteria and the predicted mass. To investigate the α dependence, we calculate the planetary masses when α = 10⁻⁴ and 10⁻² in addition to the fiducial value and list the results in Table 3. We also present the gap classification results in Appendix B. When the α is lower, C2 is less likely to be satisfied as M_p,gas is scaled by α^1/2, so more gaps are classified as dust-only gaps (group I). As α increases, in contrast, M_p,gas increases and M_p,dust/M_iso decreases, therefore both C1 and C2 are easier to be satisfied, and the number of indistinguishable gaps (group III) increases.

To illustrate the α-dependence, we choose Elias 24 as an example. Figure 6 shows how the mass changes for the gaps in the Elias 24 system by varying α. For gap 1 (innermost gap), when 5 × 10⁻⁵ < α < 2 × 10⁻³, M_p,dust/2 < M_iso is not satisfied, while 2M_p,gas > M_iso is satisfied. Therefore, only the gas gap interpretation is valid and the gap is in group II. In the range 2 × 10⁻³ ≤ α, both dust gap and gas gap interpretations are valid, therefore the gap is in group III. However, for gap 2 (outer gap), the condition M_iso > 2M_p,gas holds for all ranges of α, therefore the gap cannot be interpreted as a gas gap and is always in group I. Note that the original viscosity range in Bitsch et al. (2018) is 10⁻⁴ ≤ α ≤ 10⁻². Outside of this range, the prediction of M_iso becomes an extrapolation of Bitsch et al. (2018) that may not be accurate.

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We note that while M_p,dust defined by Equation (6) does not exhibit any explicit dependence on α, it should depend on α in reality. We discuss this issue in Section 5.

Our sample disks are largely identical to those adopted by Zhang et al. (2018) and Lodato et al. (2019), thus we compare the results to see the agreement. Zhang et al. (2018) carried out gas–dust two-fluid hydrodynamical simulations and developed a more sophisticated method to derive the gas gap width by considering three different dust models (DSD1, “1mm”, DSD2). Compared with their study, our study simply assumes that the gas gap width matches that of the dust gap due to strong gas–dust coupling, which is not necessarily valid if the coupling between dust and gas is weak. In principle, if the dust grains are decoupled from the gas (Zhu et al. 2012; Weber et al.2018), the dust gap should be wider than its associated gas gap. Therefore, the planetary masses predicted by the well-coupled DSD1 model of Zhang et al. (2018) are supposed to be comparable to ours, while those predicted by the less-coupled DSD2 model should be smaller.

At α = 10⁻³, the majority of the predictions given by the DSD1 model from Zhang et al. (2018) are similar to ours within a factor of two (it is the same level of the uncertainly that we used), such as the first planet of Elias 24 (45%), Elias 27 (31%), and GW Lup (21%). However, Zhang et al. (2018) gave significantly larger predictions than ours for the inner or wide gaps: for instance, the predicted masses of the innermost gaps in HD 143006 and SR 4 differ by nearly a factor of five and even the masses given by the “DSD2” model are larger than ours. This discrepancy may be attributed to the difference in measurement of the gap width and computation time between Kanagawa et al. (2016) and Zhang et al. (2018). The predictions of Zhang et al. (2018) are based on fitting of gaps at 1000 planetary orbits, while the result of Kanagawa et al. (2016) is based on the stationary shape of gaps in long-term simulations. Also the computational time required to reach the stationary shape is longer for a wider gap (see Figure 14 of Kanagawa et al. 2017). If the gap width does not reach the stationary value within 1000 orbits, Zhang et al. (2018) may give a larger planet mass as compared with our predictions. In the case of the innermost gap of HD 143006, the stellar age (∼4 Myr) corresponds to ∼55,000 orbits at the location of the innermost gap (21 au), and hence the simulation time could be insufficient for the gap to reach its stationary width.

We also compare our results with those of Lodato et al. (2019). When a gap is classified as dust-only (group I), we adopt the same Hill radius scaling relation as Lodato et al. (2019), and thus our predicted planetary mass for group I gaps most agrees with those from Lodato et al. (2019) within a 50% uncertainty level, such as RY Tau (35%) and the second planet of GO Tau (48%). This deviation is mainly due to the different stellar masses and gap widths adopted. Nevertheless, the planetary mass predicted from group II gaps in our study is significantly smaller than that in their study. For example, Lodato et al. (2019) predicted the mass of the first gap of CI Tau to be 15.7M_J , while the same planet is predicted to be 0.85M_J in our fiducial case with α = 10⁻³. Apart from the different gap width data we adopted, it is because M_p,gas given by Equation (5) scales with , while M_p,dust based on the Hill radius relation scales with (also see Appendix A). Physically, it means that in the gas gap regime the dominant planet–gas interaction can create a wider gap than that expected by the Hill radius argument. Therefore, the planetary mass in this regime may be overestimated if Equation (6) is used. It should be noted that in most cases, the width of the gas gap given by Equation (5) is wider than that of the dust gap predicted by Equation (6).

We would like to point out that theoretically, it is possible that the dust gap is wider than the gas gap; the dust gap width, Δ_gap in Equation (6), is independent of the gas parameters, while the gas gap width, Δ_gap in Equation (5), is scaled with viscosity as α^−1/4, and with scale height as (h/R)^−3/4. Therefore, for a given planetary mass M_p > M_iso, it is always possible to find a set of gas disk parameters (i.e., large enough α, h/R) with which the dust gap width by Equation (6) is wider than that of the gas gap by Equation (5). In such cases, M_p,dust is larger than M_iso, but the edge of the dust gap may be separate from the gas pressure bump. Thus, it might be appropriate to still use Equation (6) to estimate the planet mass.

Having said this, however, the parameter ranges for such cases (roughly α > 0.01 and h/R > 0.1) are fairly different from those commonly adopted by previous studies. Moreover, the dust grains have to be decoupled from the gas; if the dust is strongly coupled to the gas, the width of the dust gap should well match that of the gas gap. In this case, the dust gap width expected by Equation (6) is an overestimate. In the intermediate regime between the strongly coupled and the decoupled cases, the dust gap width roughly agrees with the gas gap width, and Equation (5) is still a good approximation in the gas gap regime.

Our classification of the observed gaps has shown gaps at the inner region of the disk are mostly gas gaps (group II), while those at the outer region are mostly dust gaps (group I). The cores at the inner region of the disk undergo faster accretion due to high surface densities of both pebbles and gas, so they more likely become gas giants that can open wider gas gaps. However, planets at the outer region of the disk undergo slow accretion because of insufficient material and runaway gas accretion is also harder to trigger because the larger aspect ratio leads to larger pebble isolation mass. In effect, it is harder for planets to open gaps in both gas and dust profiles at the outer region.

The abundance of dust gaps in the outer region also implies that the outer planets are, in fact, small in general, suggesting these distant planets are not those formed via gravitational instability, which should be much more massive and open gas gaps instead (Mayer et al. 2002). However, it remains unclear how the cores of the outer planets form at such large distances from the star since the formation of the solid cores via collision at the outer region is very inefficient due to low density and temperature. It is also possible that cores are not formed in situ: they may form at the inner dust-rich region first then migrate outward and inward again after growing up (e.g., Paardekooper & Mellema 2006; Paardekooper et al. 2011).

The configurations of the multiplanetary systems shown by Figure 5 suggest in general that the planetary mass decreases with radius. In single-planetary systems, the planet can migrate freely inward and grow quickly at the inner region. In multiplanetary systems, if a gas giant is formed first at the interior, the strong 2:1 orbital resonance will prevent the outer planet from migrating further inward, therefore in this case, the inner planet always grows faster than the outer planets and the mass order is retained.

However, there are four disks out of this trend, namely DL Tau, HD 163296, HL Tau, and V1094 SCO. For these four disks, it is possible that the outer planet is initially larger than the inner planet. As a result, it migrates to the current position and starts gas accretion earlier, which shields the inner planets from accessing accretion materials and therefore quenches its growth (also see Wang et al. 2020). Therefore, the fate of an individual planet strongly depends on its birth place and time relative to other planets. While investigating the evolution of a multiplanetary system, it is important to consider the coupling between planets, as the planet–planet interaction can effectively shape the orbital and mass configuration.

We also found that in half of our sample disks, the outermost gas gap is accompanied by a dust gap planet close to its outer 2:1 mean motion resonance. The resonant trap is one of the possible explanations: the dust gap planet migrates inward faster via type I migration, while the migration speed of the inner gas gap planet is slower due to the gap-opening effect (Kanagawa et al. 2018). Once the outer planet approaches the inner planet, it will be captured due to resonance and start co-migration (e.g., Mustill & Wyatt 2011; Deck & Batygin 2015). An alternative explanation is that the density of large pebbles or planetesimals may be enriched at the resonance sites (e.g., Wyatt 2003), therefore, a planet is preferred to form near the resonance. Further investigations are required to discuss the possibilities of the respective scenario.

Finally, the evolution tracks on Figure 4 do not pass through the region of the observed hot Jupiters. This implies that those gas gap-opening planets in our predicted population are unlikely to be the origin of the hot Jupiters, at least before the dispersal of the disk. It is still possible for low-mass planets to become hot Jupiters, but it depends on whether an inner gas giant exists in the disk that can stop the inward migration of a low-mass planet via resonance capture. Alternatively, if the configuration of the planetary system is dynamically unstable, hot Jupiters can be formed via later-stage scattering followed by tidal circularization (e.g., Nagasawa et al. 2008). Therefore, the stability of the planetary systems is crucially important to determine the final configurations, and that will be covered in our second paper.

There are a few caveats when applying our model. First, we assume that there is a one-to-one correspondence between the gap and the planet, which may not be true for all the gaps. It is possible that two close-in planets can create a wide common gap (e.g., Cimerman et al. 2018) that can be misinterpreted as a single gap created by a massive gas giant. However, the possibility for the common gap interpretation may be restricted to the wide gaps with high Δ_gap/R_gap, such as the first gaps of DS Tau and HD 143006 systems. For those narrow gaps with Δ_gap/R_gap ∼ 0.1 in our sample, they are unlikely to be common gaps, because otherwise each planet is too small to create even a dust gap. Moreover, a single planet can generate multiple gaps and rings in a very low viscosity disk (Bae et al. 2017; Dong et al. 2017), meaning we may overestimate the number of planets. We need to have better knowledge about the gap structure in these contexts and more observational details about the gap morphology to identify these cases.

Second, while applying Equation (5), we always assume the perfect dust-gas coupling by equating the width of the dust gap to that of the gas gap. The degree of gas–dust coupling depends on the Stokes number, which is a function of the dust grain size and gas density (e.g., Takeuchi & Lin 2005). In the well-coupled situation, our results are similar to Zhang et al. (2018); nevertheless, when the dust is strongly decoupled from the gas, our prediction may overestimate the planetary mass even though we have considered a relatively large uncertainty range in our estimate.

Third, though we neglect α-dependence in M_p,dust given by Equation (6), M_p,dust could depend on α through coupling between the gas and dust grains. In the strong coupling limit, the dust particles would depend on α similar to the gas, i.e., ∝ α^1/4 indicated by Equation (5). If the dust grains are only weakly coupled to the gas, however, the α-dependence should vanish for large St (in the limit α/St → 0) as indicated by Dipierro & Laibe (2017).

The dust particles visible in the ALMA band fall within these two limits, and therefore the α-dependence of M_p,dust should be small and does change our current result. Moreover, M_p,dust should depend on the Stokes number St as well as α. In a weakly coupled dust regime, Dipierro & Laibe (2017) suggest that the dust gap width only weakly depends on St^1/4, so the planetary mass can still be fairly well constrained even if an order of magnitude variance of St is taken into account. For the intermediate regime, the limitation of adopting Equation (6) may be kept in mind, and we note here that a more robust prediction of the planetary mass should consider both α dependence and St estimation carefully.

Moreover, our classification of gap depends on the pebble isolation mass that is sensitive to the aspect ratio, so the grouping result is subject to change when the temperature profile deviates from that given by Equation (1). Apart from the aforementioned factors, the predicted planetary mass also depends on properties of the dust grains, gas density, viscosity, and the way to measure gap widths. Hence, we should stress that our predicted mass can deviate by a factor of two or larger in some extreme cases. Our results can be improved if we can more reliably obtain the width of the gas gap with considerations of detailed dust models, temperature profiles, and gas density profiles.