A Stellar Mass Dependence of Structured Disks: A Possible Link with Exoplanet Demographics

Our sample consists of millimeter continuum fluxes from all known protoplanetary disk surveys from the literature: Ophiuchus (Cieza et al. 2019), Taurus (Akeson et al. 2019), Chamaeleon (Pascucci et al. 2016), Lupus (Ansdell et al. 2018), CrA (Cazzoletti et al. 2019), IC 348 (Ruíz-Rodríguez et al. 2018), Upper Sco (Barenfeld et al. 2016), and TW Hya (Rodriguez et al. 2015), and fluxes of disks in Cha and η Cha from A. Aguayo et al. (2021, in preparation). The main references for the disk surveys of each region are summarized in Table 1.

Table 1. Overview Regions

Region	$\langle d\rangle$	Age	N	Resolution	Resolution	References a
	(pc)	(Myr)		('')	(au)
Ophiuchus	140	1–2	163	0.2	28	1,2
Taurus	145	1–2	167	∼0.2–0.4	∼30–60	3, 4
Lupus	160	1–3	94	0.25	40	5, 6, 7
Cham I	180	1–6	85	0.6 b	108	8, 9, 7
CrA	150	1-5	39	0.32	39	10, 11
IC 348	250	2–3	59	0.8 b	200	12, 13
Cha	110	3-8	9	0.25	28	14, 15
TW Hya	56	7–13	4	∼1.0	∼60	16, 17, 18
η Cha	94	8–14	2	0.25	28	16, 15, 19
Upper Sco	145	10–12	72	0.34	49	20, 21, 22
Lower Cen Crux (LCC)	118	12–18	2	∼0.2	∼24	23, 24
Upper Cen Lupus (UCL)	140	14–18	4	∼0.2	∼28	23, 24

Notes.

^a References for age, disk survey, and stellar properties, respectively. (1) Wilking et al. (2008), (2) Cieza et al. (2019), (3) Kraus & Hillenbrand (2009), (4) Akeson et al. (2019), (5) Comerón (2008), (6) Ansdell et al. (2018), (7) Manara et al. (2018), (8) Luhman et al. (2007), (9) Pascucci et al. (2016), (10) Meyer & Wilking (2009), (11) Cazzoletti et al. (2019), (12) Luhman (2003), (13) Ruíz-Rodríguez et al. (2018), (14) Murphy et al. (2013), (15) Aguayo et al. (2021, in preparation), (16) Bell et al. (2015), (17) Rodriguez et al. (2015), (18) Venuti et al. (2019), (19) Rugel et al. (2018), (20) Esplin et al. (2018), (21) Barenfeld et al. (2016), (22) Manara et al. (2020), (23) Pecaut & Mamajek (2016), (24) Individual studies; see Table 3. ^b Although the beam size of the images is relatively large for Chamaeleon and IC 348, the performed uvmodelfit fitting by the authors results in recovered diameters as small as 011/20 au (Chamaeleon) and 021/52 au (IC 348) comparable to the resolution of other surveys.

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For most regions, we take the entire selection from the corresponding ALMA disk survey, which generally encompasses a complete sample of Class II objects as identified from previous Spitzer studies. Class II objects are generally considered as protoplanetary disks according to the Lada classification (Lada & Wilking 1984). Targets with a later spectral type than M6 (stellar mass ≤0.08 M_⊙) are excluded as the disk surveys are likely incomplete for these low-mass stars, and a statistical comparison becomes unreliable. For Ophiuchus, Upper Sco, Cha and η Cha, and Taurus, the sample selection was a bit more complicated due to the nature of the disk surveys as described below.

Ophiuchus—For Ophiuchus, Class I and F targets are included in our sample following the same approach as the corresponding ALMA survey (Cieza et al. 2019). The reason is that Ophiuchus suffers from high extinction by foreground clouds (e.g., McClure et al. 2010), which can make protoplanetary disks appear as Class I or Flat objects. Infrared color criteria such as the Lada classification are not a good indicator of evolutionary stage in that case. Excluding Class I and Flat sources from the survey would remove a large number of protoplanetary disks.

Upper Sco—The Upper Sco disk survey by Barenfeld et al. (2016) was selected using different color criteria than the Lada criteria, used by most of the other ALMA disk surveys, and it turns out that this sample contains a large number of Class III objects (Michel et al. 2021). In this work, we only consider the Class II objects from Barenfeld's survey for consistency with the other disk surveys.

Cha and η Cha—The samples of Cha and η Cha are semicomplete using currently available ALMA observations, with 8 of the 12 Class II objects in Cha as identified in Murphy et al. (2013) and 3 of the 6 Class II objects in η Cha as identified in Sicilia-Aguilar et al. (2009).

Taurus—For Taurus, we adopt the target list by Akeson et al. (2019), who constructed a full millimeter continuum sample of Class II objects based on, e.g., Andrews et al. (2013), Akeson & Jensen (2014), Ward-Duong et al. (2018), and their own work. All but Andrews et al. (2013) provide moderate-resolution ALMA observations (02–04), covering about 65% of their sample. The remaining 82 targets with millimeter data are based on low-resolution SMA fluxes from Andrews et al. (2013). However, 33 of these have been reobserved with ALMA at higher resolution in, e.g., Long et al. (2019; Francis & van der Marel 2020; and a few individual works, see Table 3) and 8 disks have been analyzed with visibility modeling by Tripathi et al. (2017), so we use these constraints. For the remaining SMA targets, 15 upper limits (above 3 mJy) are removed as they do not constrain at a similar level to the ALMA disk surveys. The remaining 26 targets do not have any information about their structure or size, but most of their fluxes are relatively low (up to 30 mJy). Although it is possible that some of these are structured or extended, the fraction is very small compared to the total number of Taurus targets, so they are included. Finally, we add transition disks AB Aur, MWC 758, CQ Tau, and GM Aur (Francis & van der Marel 2020), and MWC 480, CI Tau, and T Tau from Long et al. (2019) to the sample, which were not included by Akeson et al. (2019). The final sample consists of 167 targets in Taurus.

All targets have been compared with Gaia DR2. Targets that can be excluded as nonmembers based on their Gaia distance are excluded from our sample. Stellar properties are taken from the literature (see Section 3.2 for details).

The total sample consists of 700 disks. Table 1 provides an overview per region, including distance, age, and references for the disk surveys and stellar properties. The individual targets and their properties are listed in Table 3.

The spatial resolution is relevant for the detectability of large-scale substructures such as gaps and rings at scales of tens of astronomical units, or in the absence of detectable substructure, to determine the disk dust size. Disk surveys have been performed at a range of spatial resolutions and are located at various distances. Table 1 lists the approximate spatial resolution for each region.

Most disk surveys have a spatial resolution of ∼50 au (25 au radius) or less, except for IC 348 and Chamaeleon, which have been observed at lower spatial resolution at ∼06–08 (>100 au; Pascucci et al. 2016; Ruíz-Rodríguez et al. 2018). However, both papers model the disk size in the uv plane using the CASA task uvmodelfit, which provides size estimates down to 02 diameter (∼40 au), which is comparable to the spatial resolution of other surveys and thus sufficient for our purposes.

Protoplanetary disks in the older Upper Centaurus Lupus and Lower Centaurus Crux (UCL/LCC) regions are not studied systematically in millimeter wavelengths, as the stellar population is thought to be much older than the other regions in our study, with a median age of ∼15 Myr (Pecaut & Mamajek 2016). A handful of exceptions are the famous, bright transition disks HD 142527, HD 135344B, PDS 70, and AK Sco in UCL; and HD 100546 and HD 100453 in LCC. Interestingly, the latter two are the only massive disks identified in an infrared study of A stars in Sco-Cen by Chen et al. (2012), with the third massive disk having an estimated dust mass of only 1.5 M_lunar. Also, AK Sco, HD 139614, and MP Mus have been identified as primordial disks in UCL and LCC (Preibisch & Mamajek 2008; Chen et al. 2011), with massive infrared disks in contrast with the bulk of the disk populations that have disk masses <2 M_lunar. For MP Mus and HD 139614, no high-resolution data are available, so it is unclear whether they are structured and marked as such.

In addition, the isolated structured disks HD 163296 (Isella et al. 2016) and HD 169142 (Fedele et al. 2017) are included in our sample. These disks are located just southeast of the Upper Sco region and have individual ages of 12 ± 3 and 7 ± 2 Myr, respectively (van der Marel et al. 2019). Millimeter-dust masses are derived from available ALMA fluxes (see Table 3). No other millimeter fluxes of UCL/LCC members have been derived to our knowledge, but considering the infrared studies mentioned above, it is unlikely that any other disks are in the >0.1 M_Earth regime. The UCL/LCC and isolated disks are thus only included in the age comparison in Figure 4 to demonstrate the survival of massive disks, but not in the stellar mass diagrams to avoid inclusion of highly incomplete samples of these older regions.

All disks are initially classified as either transition disk, ring disk, or "nonstructured" (no large-scale structure identified). A large-scale structure is defined as the presence of gaps at scales of 25 au or more, i.e., with a gap edge located at at least 25 au radius. The classification is based entirely on the literature classification. We have thus not reanalyzed the disk images ourselves, as gaps are often only revealed through visibility modeling. This means that transition and ring disks are classified as such as identified by the disk survey papers from Table 1, but we have also compared our list with transition disk surveys that have derived cavity sizes (Pinilla et al. 2018; van der Marel et al. 2018b; Francis & van der Marel 2020) and ring disk studies (Andrews et al. 2018; Long et al. 2019) for our classification. The distinction between a transition disk and ring disk is as follows: a ring disk is defined as a disk with one or multiple dust gaps throughout the disk, but with dust emission in the center, whereas a transition disk is defined as a disk with an inner cleared dust gap of at least 25 au in radius, without significant dust emission at the central location. The threshold of 25 au is chosen based on the largest gap widths seen in ring disks (Huang et al. 2018). Transition disks are only classified as such when the millimeter continuum image has confirmed the inner cavity, in contrast to classical SED-selected transition disks, which are known to be less well defined and often inconsistent in cavity size (e.g., Brown et al. 2009; van der Marel et al. 2018b). Transition disks with additional gaps in the outer disk are still classified as transition disks.

With this classification of large-scale structure, the total sample contains 42 transition disks and 37 ring disks out of 700 disks. When the incomplete regions UCL, LCC, and isolated sources are excluded, as there is no information about the nonstructured disks in their corresponding region, these numbers become 37 transition disks and 30 ring disks out of 692 disks: about 10% of the sample is thus structured for the given resolution.

This classification obviously underestimates the number of structured disks, as the majority of the disks have not been observed at sufficiently high angular resolution to detect large-scale gaps. However, dust evolution models have shown that pressure bumps are required to explain the presence of millimeter-dust grains at large orbital radii to limit the radial drift of millimeter grains (e.g., Weidenschilling 1977; Birnstiel et al. 2010; Pinilla et al. 2012b). In the absence of pressure bumps, radial drift would inevitably lead to compact, low-mass dust disks in a few Myr. This implies that large, extended dust disks can only exist when supported by pressure bumps, which could reveal itself as continuum large-scale substructure at high resolution. Therefore, the outer disk dust size can provide an alternative way to infer the likely presence of substructure in dust disks, which will be used in our subsequent classification of disks without observed structure. We make the explicit assumption here that all dust substructure is caused by pressure bumps and dust size is entirely regulated by dust evolution and radial drift.

For the "nonstructured disks" in our sample, we thus use the dust radius for further classification from a dust evolution argument. The dust radius of each disk R_size is estimated as half of the major axis or FWHM. For most disk studies in our sample, the FWHM of each disk is estimated using the CASA task imfit or uvmodelfit. For some studies, the effective radius R_eff,68 (the radius enclosing 68% of the total flux) is derived (Tripathi et al. 2017; Long et al. 2019), which is comparable to half of the FWHM. The R_eff,68 radii of the resolved disks in Lupus were recomputed, as Ansdell et al. (2018) had only derived R_eff,90 values. The dust disk outer radii are reported in Table 3. Although this is a rather simple approach, it is sufficient for the purpose of identifying extended and compact disks across this large sample for the purpose of our classification. For 148 targets, no flux was detected, and only an upper limit on the flux was derived: there is no information on R_size, but under the assumption of the size–luminosity relation (Tripathi et al. 2017), they can be considered compact.

In order to estimate a threshold for extended disks, we take a look at the distribution of structured disks and nonstructured disks in dust size and dust mass. Dust masses are recomputed with the new Gaia DR2 distances using the commonly used relations in Ansdell et al. (2016, 2018) following the assumptions of isothermal, optically thin emission of 20 K as demonstrated by Hildebrand (1983):

$$\begin{eqnarray}&&{M}_{\mathrm{dust}}=\displaystyle \frac{{F}_{\nu }{d}^{2}}{{\kappa }_{\nu }{B}_{\nu }\left({T}_{\mathrm{dust}}\right)},\end{eqnarray} \tag{ 1 }$$

where B_ν is the Planck function for a characteristic dust temperature T_dust, κ_ν the dust grain opacity, d the distance to the target in parsecs, and F_ν the millimeter flux, using the Gaia distances when available. Although the dust opacity may change as a function of disk age due to grain growth (Testi et al. 2014), we have chosen to follow the general assumptions in the literature of constant dust opacity across age for the purpose of this study.

Figure 1 shows the disk size R_size (∼R_eff,68) versus disk dust mass for all detected dust disks in our sample. This plot is equivalent to the size–luminosity relation (Andrews et al. 2010; Tripathi et al. 2017; Hendler et al. 2020), showing that lower continuum fluxes generally correspond to smaller dust disks. Although the sizes for the nonstructured disks are not derived uniformly, the plot immediately demonstrates that transition and ring disks are large and massive compared to the bulk of the disk population. Transition disks have been known to be at the high end of the disk mass distribution (Owen & Clarke 2012; van der Marel et al. 2018b) and also ring disks are generally found in the largest, most massive disks (Andrews et al. 2018), as expected from the presence of pressure bumps (Pinilla et al. 2020). Interestingly, also Long et al. (2019) found a clear distinction in high-resolution observations between compact disks without substructure around single stars with R_eff,68 radii ≲40 au and disks with substructures with R_eff,68 radii ≳40 au for an unbiased disk survey in Taurus. This motivates the use of a 40 au threshold for the presence of large-scale substructure.

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Although obviously detection biases exist (see Discussion), the disk dust size provides an approximate tool to identify disks with large-scale substructure when the resolution is too low to identify gaps in the dust distribution. We set the threshold above which disks are considered "Extended" (due to substructure) at 40 au and include this in our disk classification. In our sample of nonstructured disks, we classify 40 targets or ∼6% as "Extended." All nonstructured disks with sizes <40 au are classified as "Compact."

Stellar masses have been derived using Gaia-corrected luminosities and isochrones for the bulk of our sample; only for Ophiuchus do stellar masses remain unconstrained for 93% of that survey due to the high optical extinction, which prevents proper determination of the stellar properties. For CrA and IC 348 the stellar masses have been derived using an average distance, as individual Gaia distances were limited (Ruíz-Rodríguez et al. 2018; Cazzoletti et al. 2019). As the initial sample selection excluded spectral types outside the B9-M6 range, the final stellar mass distribution ranges from 0.09 to 3 M_⊙ and is shown in Figure 2. This figure, showing all disks for which the stellar mass has been constrained, represents 73% of our entire sample. The sample is dominated by subsolar masses, as expected from the stellar mass distributions in star-forming regions, but there is also a relatively large amount of intermediate-mass stars, due to the additions of bright transition Herbig disks. The sample is thus not fully representative for the initial mass function but contains a broad range of targets.

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Age ranges for each region have been derived from evolutionary diagrams of the full population of each region (see references in Table 1). The disk sample is grouped into three categories: young (1–3 Myr), intermediate (3–5 Myr), and old age (∼8–15 Myr) using these values, where the group was based on the mean age of each association. Due to the uncertain nature of the age of CrA (see discussion in Cazzoletti et al. 2019) and Chamaeleon, they are included in the intermediate age regime. All regions are indicated in Table 2. LCC and UCL have estimated median ages of ∼15 Myr, but the individual age estimates of the eight disks in our sample are systematically lower (4–12 Myr; Garufi et al. 2018) and likely located in the younger, ∼10 Myr regimes of the UCL/LCC complex (Pecaut & Mamajek 2016, Figure 9). These disks are thus included in the old age category.

Table 2.  M_dust−M_* Regression Fit Parameters

Group	α (intercept)	β (slope)	δ (scatter)	Regions
Young (1–3 Myr)	0.6 ± 0.1	1.1 ± 0.2	0.3 ± 0.04	Oph, Tau, Lup
Intermediate (3–5 Myr)	0.4 ± 0.1	0.7 ± 0.2	0.4 ± 0.05	Cham, IC 348, CrA, Cha
Old (∼10 Myr)	0.5 ± 0.2	1.9 ± 0.3	0.3 ± 0.07	Upp Sco, TW Hya, η Cha,UCC, UCL, Isolated
Structured	1.3 ± 0.1	1.0 ± 0.2	0.1 ± 0.02	all

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In this work we compare the different disk morphologies as a function of age and stellar mass to learn more about the disk evolution of structured and nonstructured disks and comparison with exoplanet demographics. The results are presented in Figures 3–11.

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No single exoplanet survey spans the same range of stellar mass as the protoplanetary disk sample described above and also has enough sensitivity to detect planets over a wide range of planet masses and semimajor axes. Thus, we construct a demographical model from radial-velocity-detected giant planets by combining the planet mass and semimajor axis distribution around solar-mass stars from Mayor et al. (2011) and Fernandes et al. (2019) with the stellar mass dependence from Johnson et al. (2010). The latter derived occurrence rates of Jupiter-mass planets around three samples of stars: low-mass M dwarfs, solar-mass stars, and higher-mass stars that have evolved off the main sequence. While the stellar mass of this sample of "retired A stars"' is somewhat uncertain, the identification of these stars as significantly more massive than solar (M_⋆ > 1.5 M_⊙, and thus in our highest-mass bin) appears secure (Ghezzi et al. 2018; Malla et al. 2020).

To estimate the fraction of stars with an exoplanet massive enough to open up a wide cavity (transition disk) or an annular gap (ringed or extended disk), we start with the occurrence rate distribution of giant planets around Sun-like stars measured by Mayor et al. (2011) because that survey extends into the Neptune-mass regime. Figure 8 shows the inverse cumulative distribution, e.g., the integrated planet occurrence rate of planets more massive than a certain value. The left axis shows how this translates to the fraction of disks with an exoplanet above that mass. This is essentially a renormalization of the y-axis that assumes there is one giant planet per planetary system and that exoplanets occur only around stars that also have disks, which is here assumed to be 50%. The latter assumption compensates for the typical infrared disk lifetime of a few Myr (Mamajek 2009) and the inclusion of young and old star-forming regions in our disk sample. We note that the infrared disk lifetime ("presence of small grains") is not equivalent to the presence of a massive millimeter-dust disk ("presence of large grains") as these dust populations evolve separately (Sellek et al. 2020).

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Only sufficiently massive giant planets are able to open up a wide enough cavity to create a transition disk. We can therefore estimate the expected fraction of transition disks from the number of stars that have an exoplanet more massive than a certain mass threshold, M_trans. The red area shows that the estimated fraction of cavity-opening planets based on a threshold mass of M_trans = 1 M_Jup, which is 16.3% for the sample of Sun-like stars.

Planets that are not massive enough to create a transition disk but massive enough to create gaps with pressure bumps at its edges in the disk would create ringed dust disks in high-resolution observations. We estimate the fraction of gapped disks from the number of stars that have an exoplanet more massive than M_ring but less massive than M_trans. The blue area shows the estimated fraction of disks with a planet more massive than M_ring = 0.2 M_Jup is 16.5% and the black area the fraction of disks with a planet <0.2 M_Jup or no planet.

Now we extend the demographic model to stars of lower and higher masses to compare with Figure 6. We add a stellar mass dependence to the parametric planet population model of Fernandes et al. (2019):

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}f({M}_{p},{M}_{\star })}{{{dM}}_{p}{{dM}}_{\star }}={{CM}}_{p}^{a}{M}_{\star }^{b},\end{eqnarray} \tag{ 3 }$$

where a = − 0.45 from Fernandes et al. (2019), b = 1 from Johnson et al. (2010), and C a normalization constant such that ${\int }_{{M}_{p}\gt {M}_{J}}f({M}_{p},{M}_{\odot }){{dM}}_{p}=6.2 \%$ again following Fernandes et al. (2019). We note that this functional form implicitly assumes that the stellar mass dependence measured for Jovian-mass planets also applies to lower-mass planets, though this has not been observationally constrained. We will test this assumption below.

We omit the semimajor axis dependence because the bulk of the giant planet are located near the peak at 2–3 au, and the population model contains only a small number of planets outside of the detection limits of  10 au. The dotted line in Figure 8 shows that this parametric occurrence rate model provides a good match to the occurrence rates from Mayor et al. (2011) around solar-mass stars. We therefore do not expect that exoplanets at large separations but below the detection limits of direct imaging surveys would provide a significant contribution to the total number of giant planets.

Finally, we calculate the probability that each star in the disk sample has a cavity or a gap based on its stellar mass and the occurrence of exoplanets above the mass thresholds M_trans and M_ring. Figure 9 shows the predicted number of transition and gapped disks for the same stellar mass bins as Figure 6, assuming a cavity-opening mass of 1 Jupiter mass and a gap-opening mass of 0.2 Jupiter mass. Figure 9 shows a similar trend as the morphologies in Figure 6: Super-Jovian planets (>1 M_Jup, red) are much more common around higher-mass stars than lower-mass stars, but the predicted trend for the number of transition disks is weaker than the one observed. The predicted fraction of structured disks (with either a gap or cavity) matches the observed trend using M_ring = 0.2 M_Jup (blue).

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To get a better match to the observed fractions of transition disks, there are two possibilities: varying the threshold mass for a planet to open a cavity or adjusting the assumed scaling of planet occurrence rate with stellar mass from Equation (3).

First, we allow the threshold planet masses M_trans and M_ring to vary with stellar mass. We parameterize this dependence as a power law in stellar mass, ${M}_{\mathrm{trans}}={M}_{p}{\left({M}_{\star }/{M}_{\odot }\right)}^{q}$. We then do a maximum likelihood estimate using emcee (Foreman-Mackey et al. 2013), where we vary M_p and q to match the observed fraction of transition disks at each stellar mass. The likelihood is calculated using binomial statistics as in Johnson et al. (2010). Figure 10, left panel, shows the estimated mass thresholds for a planet opening a gap (blue) or cavity (red). The binned fractions of the best-fit model are also presented in Figure 11. The maximum likelihood estimation (MLE) fit has a steep anticorrelation between the threshold planet mass and stellar mass,

$$\begin{eqnarray*}&&{M}_{p,\mathrm{trans}}={1.16}_{-0.24}^{+0.30},{q}_{\mathrm{trans}}=-{1.36}_{-0.33}^{+0.35},\end{eqnarray*}$$

which provides a better match to the transition disk fraction in the lowest and highest stellar mass bins. The threshold mass for a ringed disk is consistent with having no stellar mass dependence:

$$\begin{eqnarray*}&&{M}_{p,\mathrm{ring}}={0.21}_{-0.03}^{+0.04},{q}_{\mathrm{ring}}={0.12}_{-0.21}^{+0.21}.\end{eqnarray*}$$

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It is not clear why M dwarfs would require more massive planets to open a cavity. If cavities are indeed caused by eccentric companions as suggested by Muley et al. (2019) and van der Marel et al. (2021), the planet/star ratio required for eccentricity is constant, and thus lower-mass stars would require lower-mass planets. Therefore, we also explore a second scenario next where we keep the threshold mass fixed but fit for the stellar mass dependence of the planet occurrence rate, p.

The right panel of Figure 10 shows the estimated stellar mass dependence of the giant planet occurrence rate that would match the fractions of structured disks. The occurrence rate of gap-opening planets (b = 0.9 ± 0.1) is consistent with the linear stellar mass dependence from Johnson et al. (2010) as previously assumed. The occurrence rate of cavity-opening planets would have to depend more strongly on stellar mass (b = 1.9 ± 0.3) than is observed for giant exoplanets. The binned fractions are statistically indistinguishable from those of the other scenario presented in Figure 11. To discriminate between the two scenarios, a larger exoplanet sample is needed, in particular one probing the planet mass distribution around M dwarfs, including sub-Jovian planets.

The occurrence rates of transiting planets smaller than Neptune, commonly referred to as super-Earths, are known to be anticorrelated with stellar mass (Howard et al. 2012; Mulders et al. 2015) and increase from spectral types F through M in the Kepler survey. Recently, this increase in planet occurrence rate has been shown to correspond to an increase in the fraction of stars with planetary systems as well (Yang et al. 2020; He et al. 2021). Thus, it is not likely that these planets cause or form directly in the observed disk gaps and cavities, which are positively correlated with stellar mass. Instead, we explore the opposite hypothesis that these structures, and the associated giant planets, inhibit the formation or inward migration of the super-Earths that are detectable with Kepler. This would give rise to an anticorrelation between the fraction of disks with gaps and the fraction of stars with transiting exoplanets.

There is some theoretical motivation for this hypothesis, motivated by the lack of super-Earths in the solar system and linking this to the presence of Jupiter. For example, Izidoro et al. (2015) hypothesized that Jupiter may have prevented the inward migration of super-Earths. Along similar lines, Lambrechts et al. (2019) found that super-Earth formation is dependent on the pebble flux and that Jupiter may have starved the inner solar system from pebbles. Giant planets such as Neptune and Uranus are expected to limit pebble drift as well. Such mechanisms, if they also operate in the disks with observed cavities, would reduce the occurrence rates of close-in super-Earths. In this section, we test if the anticorrelation between close-in super-Earths and disk structures predicted by such mechanisms could be consistent with the Kepler planet occurrence rates.

We calculate the fraction of stars with close-in planetary systems, F, from the Kepler planet occurrence rates, η, and an estimate for the number of planets per planetary system, $\bar{n}$,

$$\begin{eqnarray*}&&F=\bar{n}\,\eta .\end{eqnarray*}$$

We calculate the planet occurrence rate, η, for different stellar mass bins as described in Mulders et al. (2018), using the Kepler DR25 planet candidate catalog (Thompson et al. 2018) and survey completeness (Burke & Catanzarite 2017). The highest stellar mass bin (>1.5 M_⊙) does not contain enough stars and planets to calculate an occurrence rate for sub-Neptunes. The fraction of stars with planets, $\bar{n}$, is in the range of 2–6 based on forward modeling of planet populations (e.g., Zhu et al. 2018; Mulders et al. 2018; He et al. 2019). Here we will assume $\bar{n}=4.5$, mainly such that the fraction of stars with planets, F, does not exceed 1. Figure 12 shows the fraction of stars with Kepler planetary systems and the predicted fraction of structured disks, equal to 1 − F. The model underpredicts the number of structured disks for M dwarfs but overpredicts that same number for the more massive stars.

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Planet occurrence rates for single stars are underestimated by the presence of binaries in the Kepler sample. Moe & Kratter (2019) showed that accounting for binaries can explain up to half of the known stellar mass dependence for sub-Neptunes. Because the disk sample is biased toward single stars due to the infrared selection criteria (Kraus et al. 2012 and Section 6.5), we apply this binary correction in Figure 13, where we also adjust the number of planets per system, $\bar{n}$, to 6.5 to make sure that the fraction of M dwarfs with planets does not increase above 100%. The number of stars with close-in planetary systems now roughly matches the observed incidence of disks without gaps or cavities in each of the stellar mass bins. This comparison shows that the elevated occurrence rates of transiting planets around M dwarfs may be connected to those stars having fewer disk structures to impede the drift of pebbles or the migration of super-Earths.

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The similarities between the disk demographic fractions and exoplanet mass bin fractions as a function of stellar mass (Figures 6 and 9) suggest a connection between the observed gaps in disks and recently formed planets, where the most massive planets (∼Jupiter mass) are responsible for the carving of the widest gaps in transition disks, moderately massive planets (∼Neptune–Jupiter mass) are responsible for the structures seen in ring disks, whereas lower-mass planets below about a Neptune mass do not affect the disk structure and the disk dust structure is set by radial drift. The fact that both structured disk fraction and massive planet fraction show a strong positive correlation with stellar mass provides the most convincing evidence for a connection here. As massive stars are generally surrounded by more massive disks (Figure 3), it is not surprising that they are also forming more massive planets.

The majority of giant exoplanets are found at smaller orbital radii than the ALMA spatial resolution limit and thus could not be responsible for creating the gaps in protoplanetary disks if they formed at their current locations. However, it is very likely that these planets have migrated inwards (Type I/II migration) during the lifetime of the disk due to the torques between the disk and planet (Paardekooper et al. 2010; Lodato et al. 2019), and we consider the current location of the exoplanets as less relevant for our conclusions. The potential low efficiency of planet formation at the large gap radii is further discussed in Section 6.3.

The observed connection between disk morphology and giant planet fraction is consistent with pressure bumps at the edge of planet gaps, where the dust gets trapped and concentrated, resulting in rings, gaps, and cavities (Pinilla et al. 2012a). Planet gap edges require a minimum planet mass to have sufficiently strong pressure bumps to trap the millimeter dust at large radii. This threshold was initially set at 20 M_Earth (∼1.2 M_Nep) for a small set of planet–disk interaction models at α = 10⁻³ for a 1 M_⊙ star (Rosotti et al. 2016), but later works show that the threshold depends on, e.g., stellar mass, scale height (location in the disk) and viscosity (Sinclair et al. 2020) and the minimum planet mass encompasses a range of values in the super-Earth regime with a mass of ∼10–20M_Earth, or ∼0.6–1.2 M_Nep. Interestingly, Sinclair et al. (2020) show that subsolar stars require somewhat higher-mass planets to create a pressure bump compared to Sun-like stars. We do not see evidence of such a reverse stellar mass dependence in the estimated fraction of ring disks though.

Our results suggest that with a higher threshold of ∼0.2 M_Jup or ∼60 M_Earth, there are already enough exoplanets to account for the observed number of disks with gaps. This allows for the possibility that not all planets create observable disk structures, for example, because some planets are located too close to their host stars for their gaps to be observable. On the other hand, many ring disks contain multiple gaps, which are likely caused by multiple planets, whereas our exoplanet demographic model assumes a single giant planet per star. If multiple planets per disk are required to create the observed structures, that implies that the minimum planet mass must be lower. Furthermore, gaps in the inner 10 au of the disk would remain undetectable due to high optical depth and lower scale height (Andrews 2020). An exact determination of the planet mass range responsible for ring disks is beyond the scope of this study, but even the current simple approach strongly suggests that there must be a minimum value in the (super-)Neptune mass range for clearing gaps, which is consistent with the exploration studies in planet–disk interaction simulations (Rosotti et al. 2016; Sinclair et al. 2020).

Second, large transition disk cavities are likely caused by the most massive giant planets, around a Jupiter mass and above, where the required minimum planet mass decreases with stellar mass in order to be consistent with the high transition disk fractions in the highest stellar mass bins. van der Marel et al. (2021) proposed a scenario where transition disks must harbor planets that are so massive that their orbits become eccentric, clearing a much wider gap than a single planet would be capable of (D'Angelo et al. 2006; Muley et al. 2019). This threshold, however, depends directly on the stellar mass, and the requirement for eccentricity is set at q > 0.003 (Kley & Dirksen 2006), implying a minimum planet mass requirement that instead increases with stellar mass and is a factor of a few higher than what is derived in our planet mass threshold fit (Figure 11).

It is possible that this threshold holds and only some of the transition disks are caused by individual planetary companions, whereas the remainder are caused by close stellar companions, i.e., circumbinary disks (e.g., Harris et al. 2012) or multiple Jovian planets (Dodson-Robinson & Salyk 2011; Close 2020). A binary companion can clear a cavity of 2–5 times the binary separation (Artymowicz & Lubow 1994; Hirsh et al. 2020), so for transition disk cavities of ∼25–100 au, we are interested in binary companions at ∼10–50 au separation. This is further discussed in Section 6.5. When multiple Jovian planets are responsible for the wide transition disk cavities, this would shift our planet mass threshold by a factor of a few, but this does not change our major conclusions.

The two observed evolutionary pathways in Figure 4 are consistent with our main picture of the disk dust evolution process dominating the disk dust size and morphology: whereas structured disks maintain their dust mass due to the presence of pressure bumps (and thus planets), the bulk of the disks (compact) decrease their dust mass over time as they do not contain sufficiently strong pressure bumps to prevent radial inward drift (Birnstiel et al. 2010; Pinilla et al. 2012b). Pinilla et al. (2020) demonstrated that disks without pressure bumps drop in dust mass by a factor of 3 in 5 Myr, whereas disks with pressure bumps generally maintain their dust mass. The latter is only true for disks around high-mass stars as enhanced dust growth to boulders in disks around low-mass stars also decreases the observed dust mass. By including the stellar mass dependence of the presence of substructure (pressure bumps), these models are consistent with the observed trends between dust mass and stellar mass in younger and older regions (Figure 7 in Pinilla et al. 2020) and the two evolutionary pathways observed in Figure 4 in this work.

Figure 6 demonstrates that compact disks are more common around low-mass stars, which also form the majority of any stellar population: hence the majority of disks drops in dust mass over time, which is a general observation in disk dust mass survey comparisons (Ansdell et al. 2017; Cieza et al. 2019). The overall drop in dust mass was initially interpreted as disk dissipation, but this appeared inconsistent with the high accretion rates in the older Upper Sco region (Manara et al. 2020). A modeling approach with both low viscosity (thus slow decrease in accretion rates) and radial drift explains this combined process of disk evolution (Sellek et al. 2020). Our results are compatible with this scenario, with the addition of the structured disks, which are unaffected by radial drift. As this is only about 17% of the total disk population (including the extended disks) and particularly biased toward the highest stellar masses, these structured disks do not have a very pronounced effect on the overall disk evolution: the compact disks dominate the statistics. This scenario is also consistent with the size–luminosity relation derived by Tripathi et al. (2017), which is overall quadratic consistent with a grain size distribution dominated by radial drift rather than fragmentation (Rosotti et al. 2020), but with a larger scatter at larger radii, suggesting that a fraction of the disks is consistent with a grain size distribution set by fragmentation (and thus pressure bumps) instead. Overall, the presence of giant planets (above the threshold to create pressure bumps) thus sets the evolutionary path of the dust mass and size of the disk.

A second aspect of disk dust mass evolution is that the dust mass decreases faster for lower-mass stars (Figure 3): this is also called the steepening of the M_dust−M_* relation (Ansdell et al. 2017). Our results naturally explain this as structured disks are much less common around low-mass stars compared to higher-mass stars (Figure 6), which can be understood immediately as giant planets are also much less common around low-mass stars (Figure 9). Interestingly, Pinilla et al. (2020) already predicted that strong pressure bumps were required to explain the steepening of the M_dust−M_* relation. However, they introduced pressure bumps equally in all stellar mass bins. Strong traps were indeed required for the high stellar masses, but for the low stellar masses, the M_dust value dropped regardless of the presence of pressure bumps (Figure 7 in Pinilla et al. 2020): for the model with an unperturbed density profile, the decrease is due to drift and for the model with pressure bumps due to growth up to boulders. Thus, one can conclude that these dust evolution models, when the presence of pressure bumps was introduced with respect to their relative presence in their stellar mass bins, would indeed reproduce our results.

The dust disk evolution is thus not uniquely set by its stellar mass but by its capability to form giant planets. As intermediate-mass stars have higher-mass disks and thus a higher likelihood to form giant planets, they are more likely to have structured disks, so one can still conclude that the majority of intermediate-mass stars are expected to have longer-lived disks. Interestingly, Ribas et al. (2015) concluded that disks around intermediate-mass stars evolve faster than low-mass stars. However, this can be understood as they considered transition disks as evolved disks rather than primordial, as they only considered the infrared fluxes in their work. A drop in infrared flux can be caused by the presence of a large cavity, rather than disk evolution. The millimeter fluxes of transition disks are high, and these disks should thus be seen as primordial.

Another interesting aspect about the disks around intermediate-mass stars is the origin of the disks without substructure. Why were these disks incapable of forming giant planets? Inspection of their dust masses reveals that their dust masses are generally somewhat lower than the structured disks. More importantly, more than half of these disks are known wide binaries in the 30–300 au separation range, which might affect the planet-formation efficiency in these disks (Harris et al. 2012). Furthermore, we notice that five of these disks are so-called Group II disks in Herbig studies (Meeus et al. 2001). In contrast to Group I disks (mostly transition disks), Group II disks are fainter in the infrared and do not show signs of an inner cavity (Maaskant et al. 2013). Their millimeter properties are not well constrained. Based on a large range of properties, Garufi et al. (2017) proposed that Group I and Group II disks evolve through different evolutionary pathways. A similar suggestion was proposed by van der Marel et al. (2018b) for T Tauri stars in Lupus with and without cavities. This suggestion is consistent with our proposed scenario of disk evolution with and without pressure bumps.

A final question is what happens at the end of the lifetime of the disk. According to viscous evolution, the disk accretion rate drops with time (Hartmann et al. 1998) until eventually photoevaporation takes over and clears the gas in the disk (Clarke et al. 2001; Alexander et al. 2014, and references therein). At this stage, all pressure bumps will be removed and the dust grains are expected to largely dissipate with the gas. However, the dust that has grown to planetesimal sizes will no longer be affected by the presence or absence of gas, as they are completely decoupled. A ring of planetesimals formed in a pressure bump in the primordial disk would thus remain after most of the disk is dissipated, with a fraction of the original dust mass remaining, dominated by the large solid bodies. Such a planetesimal ring is seen in cold debris disks, where a collisional cascade caused by stirring of the planetesimals results in smaller, observable dust grains (Wyatt 2008; Matthews et al. 2014, and references therein). Debris disks generally display large dust rings or belts at tens or even hundreds of astronomical units (Matrà et al. 2018), similar in appearance to but much fainter than transition disks. How common are these cold debris disks? According to the Herschel DEBRIS survey, the detection rate of debris disks, when corrected for completeness in particular to correct for detection biases toward more luminous stars, increases with stellar mass (Thureau et al. 2014; Sibthorpe et al. 2018). In contrast, warm debris disks with dust near the habitable zone show no correlation with spectral type in the Large Binocular Telescope Interferometer results (Ertel et al. 2020), but structures at these scales in protoplanetary disks cannot be resolved by ALMA.

In Figure 15 we plot the cold debris disk fraction for our stellar mass bins. There is again a similarity with the structured disk fractions in Figure 6, in particular the strong increase with stellar mass, although the fractions do not match exactly. This leads to the hypothesis that the detected cold debris disks are the final outcome of the primordial structured disks rather than a general representation of protoplanetary disk remnants, as proposed by Michel et al. (2021). As the majority of protoplanetary disks are affected by radial drift, no cold dust or planetesimals would remain present at large radii. Although stirring is required for the collisional cascade to generate the observable dust debris disks to begin with, it is not unlikely that continuous stirring would occur in a system with at least one giant planet and a surrounding planetesimal belt. This scenario thus suggests that all debris disks must host massive planets, as they are thought to be responsible for clearing the large gaps in the primordial phase. A tentative correlation has indeed been found between the presence of giant planets at wide orbits and bright debris disks (Meshkat et al. 2017), but Yelverton et al. (2020) showed that a correlation between debris disks and close-in planets does not exist. Perhaps the planet mass and detectability in the relevant orbital range can solve this discrepancy, but the connection between debris disks, giant planets, and structured disks thus currently remains debatable.

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The proposed scenario suggests that giant planets must form in the first Myr in order to create the structures that are observed in protoplanetary disks ranging in age from 1 to 10 Myr old. These planets would then have to migrate inward to 1–10 au, where they are detectable as giant exoplanets around main-sequence stars in radial velocity surveys. This very early formation is consistent with recent insights into the formation of the solar system. Jupiter must have grown massive enough early on to create a gap that separates different reservoirs of material in the solar nebula with different isotopic compositions (Kruijer et al. 2017). Jupiter subsequently migrated inward through the disk to ∼1.5 au (Walsh et al. 2011), though the proposed outward migration afterwards may have been unique for the solar system and does not have to be common for giant exoplanets.

The anticorrelation between Kepler planets and structured protoplanetary disks (Section 5.2) suggests that the pressure bumps at large disk radii suppress the formation of close-in super-Earths. A plausible mechanism to achieve this is if these planets form through pebble accretion. Lambrechts et al. (2019) show that the formation of super-Earths is controlled by the magnitude of the flow of pebbles into the inner disk. A reduction in this flow by disk structures would prevent detectable super-Earths from forming. Thus, the high planet occurrence around M dwarfs (e.g., Mulders et al. 2015) could thus be directly connected to the observed lower incidence of structured disks around low-mass stars. This scenario also suggests that close-in planetary systems form preferentially in the small, low-mass dust disks that are harder to resolve spatially with ALMA.

Our evolutionary scenario follows along the lines of the separation of super-Earth and terrestrial planet-formation pathways as regulated by the pebble flux (Lambrechts et al. 2019), as the pebble flux would be naturally reduced by the presence of giant planets and pressure bumps, leading to the formation of terrestrial planets only. Our solar system does not contain super-Earths but several giant planets and small terrestrial planets, and would thus at some point have been a structured disk with rings and gaps where radial drift and pebble flux is limited, according to this scenario. The large-scale structured disks proposed in our study of 40 au and above would be too far out to be explained as limited pebble drift by only Jupiter within this scenario, but Neptune and Uranus are sufficiently far out and would limit pebble drift as well. The Kuiper Belt might be a remnant of the pebbles from a large-scale structured disk.

Our planet-formation scenario requires a specific timing of the formation of different types of planets: Giant planets form first, super-Earths form later. This timeline is consistent with the measured properties of Kepler planets. The low density of many of these planets suggests they must have formed while disk gas was still present. In addition, the presence of a radius valley suggests that even planets that are currently rocky (<1.5R_Earth, Rogers 2015) initially formed with a gaseous envelope (e.g., Fulton et al. 2017; Owen & Wu 2017; Rogers & Owen 2021)—and hence, before the gas in the disk had dissipated. Because sub-Neptunes have accreted only a small fraction of their total mass in gas, this suggests that formation happens toward the end of the disk lifetime when the disk gas is already depleted (e.g., Dawson et al. 2016; Lee & Chiang 2016).

The hypothesis that disk structures induced by giant planets suppress the formation of close-in super-Earths would affect the exoplanet population in multiple ways. First, it implies that super-Earths and cold giant planets would likely not occur in the same systems. This is slightly at odds with multiple studies that indicate correlations between those planets (Bryan et al. 2019; Herman et al. 2019) though this is not always found (Barbato et al. 2018). This hypothesis also provides an explanation for why the planet occurrence rates of sub-Neptunes follow different scaling relations with stellar mass and metallicity as (cold) giant planets (see Mulders 2018 for a review on this topic), though we note that binarity may also play a role in explaining the observed (anti)correlations with stellar properties (e.g., Moe & Kratter 2019; Kutra & Wu 2020).

The question remains how giant planets can be present at the large gap radii of tens of astronomical units observed in protoplanetary disks when core accretion is generally not efficient at these radii (e.g., Helled et al. 2014 and references therein). One possibility is that planets form quickly in the very inner part of the disk and then initially migrate outwards (Type II migration) to the position they keep during the protoplanetary disk phase (Paardekooper & Papaloizou 2009). It is also possible that they form quickly through gravitational instability (Boss 1997) in the young, compact circumstellar disk. Planet-formation models in the conditions of embedded, compact disks are required to test these scenarios.

In this work we have shown that the observed fraction of structured disks is positively correlated with stellar mass, that such a correlation matches the occurrence rates of giant exoplanets, and that enough exoplanets of sufficient mass to open disk gaps are available to explain the observed structures. However, correlation does not automatically imply causation, and alternative explanations are possible.

One source of possible bias in our study is the ability to detect substructure with ALMA in disks of different sizes and brightnesses (hence dust mass). In this work only large-scale substructure (>25 au) is considered as a tracer of pressure bumps caused by giant planets. The concentration of observed transition and ringed disks at high mass and radius may be a result of the preference of high-resolution observation ALMA studies of the brightest and most extended disks, where gaps are more easily resolved. Disk gaps in disks that are smaller or fainter may fall below the detection limits. If these detection biases are indeed significant, the large fraction of transition disks around high-mass stars is not a result of those disks having giant planets but of those disks being larger (and/or more massive). In that scenario all disks may have substructure, but only the large-scale substructure leading to larger disk sizes is identified here to lead to a classification as a structured disk.

Disk size is known to be strongly correlated with dust mass (Tripathi et al. 2017) with no strong residual dependence on stellar mass (Andrews et al. 2018; Hendler et al. 2020). Thus, in this scenario, transition disks would appear more frequently in ALMA observations of more massive disks because these disks are also larger. Giant planets would also preferentially form in these more massive disks, but would not have to be the direct cause of the observed gaps. For example, if giant exoplanets form closer to their current locations, any gaps they cause would be located interior to the resolution of ALMA and thus not be detected. In particular, our threshold of 40 au disk size to label a disk as "Extended" (which is added to the ring disk fraction in the comparison with exoplanets) is chosen from the known transition and ring disks and thus possibly caused by this detectability bias. A lower threshold value would result in a larger fraction of extended disks and thus a larger range of planets capable of opening a gap within our interpretation. Furthermore, if giant planets were present in compact, low-mass disks at smaller orbital radii (either through formation or migration), their gaps would remain undetected in our classification. In principle, the fraction of ring/extended disks should be considered a lower limit, which means that the minimum planet mass responsible would be an upper limit.

On the other hand, the Taurus disk survey by Long et al. (2019) was relatively unbiased and does show a clear distinction between compact disks without and extended disks with substructure around 40 au, suggesting that there is a true separation between these two disk populations. Furthermore, if a significant number of compact disks were hiding small-scale substructure caused by giant planets, there are insufficient giant planets in the current exoplanet population, in particular around low-mass stars. The stellar mass dependence of the giant exoplanets is thus an argument against this possibility. We emphasize that this conclusion relies on the assumption that all dust substructures are caused by giant planets and that the occurrence rates of giant planets (in particular around low-mass stars) are not underestimated.

On a final note, the extended disks are considered exclusively as ring disks in our analysis, whereas some of these disks might be transitional as well, which poses some additional uncertainty on the derived transition disk fraction, but considering the number of extended disks, this is a marginal effect.

A scenario where giant planets preferentially form from the observed substructure is also possible. This is a chicken and egg situation that can only be resolved by directly observing forming planets in disks. The detection of multiple planets embedded in disks with substructure (e.g., Haffert et al. 2019; Pinte et al. 2019) provides a promising avenue to understand if the statistical relations between (exo)planets and disks identified in this paper are also borne out at the level of individual protoplanetary disks.

Binarity may play an important role in the comparison of disks and exoplanets as well, as binaries can either prevent the formation of a disk, limit its lifetime, or truncate the millimeter-dust disk either from the outside or inside, i.e., a circumbinary disk (Harris et al. 2012). Binarity is not well constrained for young disk surveys and its influence remains difficult to ascertain. We will just summarize the main statistics of binaries in this section and comment on their role within this study, in particular the role of circumbinary disks in explaining transition disks. For example, the HD 142527 transition disk was found to host a substellar companion (Biller et al. 2012; Close et al. 2014). However, for several transition disks, stringent upper limits on possible companions exist, excluding this as the main mechanism (Andrews et al. 2011; van der Marel et al. 2021). However, some transition disks in our sample may still be explained by circumbinarity. Rather than studying individual limits we approach this from a statistical perspective of the occurrence rate of binarity as a function of stellar mass.

Binarity of stars is known to increase with stellar mass (Raghavan et al. 2010), and the binarity fraction has been computed explicitly for two different separation regimes: close (<10 au) and wide (<100 au) separation by Moe & Kratter (2019, Figure 1) for companions down to 80 M_Jup (brown-dwarf limit). Using these curves, we compute the fraction of close <10 au binaries and wide binaries (10–100 au) for each stellar mass bin. The fraction of very wide binaries (>100 au) is not quantified, but such wide separations are unlikely to influence the disk structure (Harris et al. 2012). Figure 16 shows these binarity fractions.

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Close binarity increases by more than a factor of 3 from the lowest to the highest stellar mass bin, but the fraction of 10–100 au separation binaries increases by less than a factor of 2. We note that the occurrence rate of brown dwarfs (13–80 M_Jup) at 10–100 au is much lower than stellar and super-Jovian companions (brown-dwarf desert): its occurrence is estimated as ∼0.8% for all stars (Nielsen et al. 2019), but as this is not separated for different stellar mass bins, this fraction cannot be included in Figure 16.

Although the binary separation range is somewhat larger than the expected range for clearing transition disk cavities, these fractions may provide a possible explanation for the low occurrence rate of super-Jovians compared to the number of transition disks, as the fraction of wide stellar companions (for all stars) is also approximately 15%. An exact comparison is not possible due to the different ranges in separation and the lack of proper statistics on binaries truncating the outer rather than the inner disk (Harris et al. 2012). Also, it is unclear why the overabundance of transition disks is not seen in the lower stellar mass bins, where binarity is decreased, but not down to zero. Likely the disk mass and size distribution with respect to the stellar mass play a role here: if disks around low-mass stars are smaller, binaries at 10–100 au separation are less likely to truncate them internally rather than externally.

Binarity may also play a role in the disk fraction: the number of young stars that actually show an indication of a disk. The infrared disk fraction is known to drop with age (Hernández et al. 2007; Mamajek 2009), although perhaps not as quickly as previously estimated (Michel et al. 2021). ALMA disk surveys are biased as they have preselected young stars with infrared excess, i.e., a 100% disk fraction, whereas even the youngest star-forming regions have disk fractions of only 60%–80%, which drops to 20%–40% in the older regions. This deficit in young star-forming regions might be explained by multiplicity, as the tidal influence of close stellar companions within ∼40 au either prevent the formation or limit the lifetime of a disk (Kraus et al. 2012). This implies that the ALMA disk surveys are intrinsically biased toward single stars and binary stars with larger separations.

On the other hand, exoplanet surveys often have a similar bias toward single stars as known binaries are excluded from the radial velocity target star samples. In transit surveys such as Kepler, however, the binarity of target stars is not always known but can be statistically corrected for (e.g., Moe & Kratter 2019). In our work we have assumed an average disk fraction of 50% in the comparison with exoplanet demographics, but it is possible that this is an underestimate due to the bias in each sample. In that case, the number of gap-opening planets in disks would be a factor of 2 lower, which could be corrected for by adjusting the minimum planet masses downward by a factor of $\sqrt{2}$.