The mass function (MF) of isolated objects measured by microlensing consists of both a stellar and a planetary component. We compare the microlensing MFs of A. Gould et al. and T. Sumi et al. to other measurements of the MF. The abundance of brown dwarfs from the tail of the T. Sumi et al. stellar MF is consistent with measurements from the local solar neighborhood. Microlensing free-floating planets (
Over the past 15-20 years, several techniques (transits, radial velocities, microlensing, direct imaging, and coherent structures in circumstellar disks) have provided insights into the frequency and mass distribution of planetary mass objects at distances ranging from a few stellar radii to 10-100 au from a host star (e.g., Poleski et al. 2021;Pearce et al. 2022;Bae et al. 2023;Currie et al. 2023;Lagrange et al. 2023;Lissauer et al. 2023;Weiss et al. 2023Weiss et al. , 2024)). Data from microlensing, radial velocities, and transits demonstrate that planets with masses comparable to Saturn and Jupiter are significantly less common than planets with masses comparable to Earth and Neptune. Although direct imaging studies has only detected planets more massive than Jupiter beyond ∼ 10 au, the derived frequencies of these worlds are reasonably consistent with results from microlensing and radial velocities at similar separations from their host stars.
The declining frequency of planets with increasing mass mirrors a similar phenomenon among stars (e.g., Salpeter 1955;Miller & Scalo 1979;Chabrier 2005;Kroupa et al. 2013;Kirkpatrick et al. 2024). From the most massive O-type stars to the lowest mass stars, the frequency is a declining function of mass. For stars more massive than ∼ 0.1-0.2 M ⊙ , all derivations of the initial mass function (IMF) agree reasonably well. Among the lowest mass hydrogen-burning stars and lower mass brown dwarfs, however, different formulations of the IMF diverge. jyee@cfa.harvard.edu The vertical gray lines are drawn at 1 M Sat and 0.08 M Sun (the hydrogen burning limit). Black horizontal lines indicate the approximate mass ranges of "Stars", "Brown Dwarfs", and "Planets."
The lines are straight for masses that clearly belong to that population; when they extend to the edge of the plot, arrows indicate that the range extends beyond that point. The lines are wavy and end with an arrow in regions where the extreme limit is unknown or ambiguous. The upper and lower limits of the planetary and stellar MFs, respectively, and the location of the minimum between them are particularly uncertain, leading to ambiguity in the nature of brown dwarfs, both individually and as a population.
The schematic in Figure 1 illlustrates these two populations. In most mass regimes, one mass function (MF) or the other dominates; assigning most stars and planets to one of the two populations is straightforward. However, the minimum (maximum) mass of the stellar (planetary) MF and the slope near the limit are unknown. For example, isolated cores of molecular gas with estimated masses < 30 M Jup , where M Jup is the mass of Jupiter, have been observed in young star-forming regions (e.g., Pearson et al. 2021;Damian et al. 2023); planets with estimated masses 10 M Jup are observed embedded in disks or coplanar configurations (e.g., Marois et al. 2008;Lagrange et al. 2009;Marois et al. 2010;Lagrange et al. 2019;Hinkley et al. 2023). Where the MFs overlap, there is ambiguity in assigning an object to one population or the other based solely on mass (e.g., Burrows et al. 1997;Schlaufman 2018;Kirkpatrick et al. 2024, and references therein).
We assume that different formation mechanisms shape the stellar and planetary MFs. Current theories for the formation of planets and stars differ in significant ways. Stars grow at the center of a collapsing molecular cloud core; planets grow within a disk of material orbiting a newly-formed star (e.g. Shu et al. 1987;Drazkowska et al. 2023;Pineda et al. 2023). Regardless of mass, companions that form via binary star formation processes would be part of the stellar MF whereas those that form in a disk would be part of the planetary MF. Because objects referred observationally as brown dwarfs span masses from the hydrogen burning limit to an unknown lower limit, they may be drawn from either the stellar MF or the planetary MF, and so, formed by either mechanism. The nature of objects near the minimum between the MFs (those with mass between ∼ 3 and ∼ 15 M Jup ) is particularly ambiguous. Improving our understanding of the frequency of objects in this mass range may yield better constraints on the lowest (highest) masses generated by star (planet) formation processes.
To investigate the mass range where the stellar and planetary mass functions converge, we consider the microlensing free-floating planet (µFFP) population (Mróz et al. 2020a;Ryu et al. 2021;Gould et al. 2022;Koshimoto et al. 2023;Sumi et al. 2023). The spatial scale of a microlensing event is characterized by the Einstein radius, which is ∝ M 1/2 , where M is the lens mass. The time it takes the source to travel one Einstein radius sets the timescale of the event. Thus, lenses with masses 1 M Jup produce events with short timescales or small Einstein radii. Microlensing measures the MF by fitting timescale or Einstein radius distributions for single lens objects. We can decompose this MF into separate populations of stars and planets as illustrated in Figure 1. The properties of the µFFP population are inferred by measuring deviations from the extrapolation of the stellar MF to planetary masses. In the region of overlap, objects from both populations may have the same masses, and hence, the same timescales and Einstein radii. Because the microlensing MF spans masses from planets through stars, it provides limits on both the low-mass end of the stellar population and the high-mass end of the planet population.
Although the microlensing MF is measured for single lenses, those lenses could have companions (µFFPs could have host stars or stellar lenses could have secondaries or primaries) that are simply too far away to create a microlensing signal. Hence, the term µFFP refers to objects detected by microlensing from the planetary component of the population of single lenses. Whether or not they are truly free-floating is ambiguous. The populations represented by the microlensing MFs comprise both isolated objects and those on wide orbits ( 10 au for µFFPs).
Under our assumption that the two MFs represent two different formation models, the low-mass end of the microlensing stellar MF can be compared with the mass function of the lowest mass brown dwarfs derived from surveys of the nearest stellar objects (e.g., Kirkpatrick et al. 2024). It may also include objects found through radial velocity and direct imaging planet searches if those objects formed through binary star formation processes rather than in a disk. The planetary microlensing MF, i.e., the µFFP population, can be compared to the wide-orbit planet populations derived from radial velocity and direct imaging observations under the assumption that those companions formed in a disk. µFFPs also test constraints on the population of Jupiter-mass planets inferred from studies of debris disks and protoplanetary disks. Clanton & Gaudi (2017) also tried to reconcile wide-orbit planets with the µFFP population. Sumi et al. (2011) had inferred a large population of free-floating Jupiters based on an excess of short-timescale microlensing events. Clanton & Gaudi (2017) tested whether the observations could be explained by the bound planet population derived in Clanton & Gaudi (2016) from microlensing (Gould et al. 2010;Sumi et al. 2010), radial velocity (Montet et al. 2014), and direct imaging (Lafrenière et al. 2007;Bowler et al. 2015) measurements of the planet frequency. The constraints
available at that time from radial velocity and direct imaging were relatively weak and limited to planets somewhat more massive than Jupiter. Since then, the measurements of the µFFP populations have been updated several times. Mróz et al. (2017) found no evidence for a large population of free-floating Jupiters in their timescale distribution, although they did find tentative evidence of a population of free-floating super-Earths. Gould et al. (2022) made an independent measurement based on the Einstein radius distribution and constrained both the frequency and power-law index of the µFFP population. Sumi et al. (2023) combined timescale and Einstein radius distributions to measure those properties for a larger sample of events. In addition, several techniques (radial velocities, microlensing, direct imaging, and coherent structures in circumstellar disks) have provided new insights into the frequency and mass distribution of planetary mass objects at 10-100 au distances from a host star (e.g., Poleski et al. 2021;Pearce et al. 2022;Bae et al. 2023;Currie et al. 2023;Lagrange et al. 2023).
We compare the MFs inferred from µFFPs (Gould et al. 2022;Sumi et al. 2023) to the wide-orbit planet populations inferred from other techniques. We begin with a description of our methodology and then discuss the constraints on µFFPs and brown dwarfs from the microlensing perspective. After comparing these constraints to constraints from direct imaging and radial velocity surveys for massive planets, we discuss indirect constraints from structures observed in debris and protoplanetary disks. We conclude with a brief summary.
As shown in Figure 1, we assume the observed MF of objects can be decomposed into two components: a stellar MF and a planetary MF. Stars are objects from the stellar MF. Planets are any objects that come from the planetary MF. Observationally, objects with masses ≥ 0.08 M ⊙ (i.e., above the hydrogen burning limit) or cores from star formation regions are clearly stars, and objects that clearly formed in a disk or those with masses 1M Sat , where M Sat is the mass of Saturn, can be unambiguously defined as planets. Objects in the mass ranges 1M Sat M < 0.08 M ⊙ whose formation mode is unclear can be divided into "stellar brown dwarfs" (those from the low-mass tail of the stellar mass function) and "massive planets" (from the high-mass tail of the planetary mass function), although the exact classification of individual objects is unclear.
For our quantitative comparisons, we quote the planet (or object) frequencies as the number of planets (objects) per 100 stars (following the conventions of Fulton et al. 2021). In this context, we define stars to be objects from the stellar MF in the range 0.08 M ⊙ ≤ M * ≤ 8 M ⊙ . That is, we assume that neither brown dwarfs (M < 0.08 M ⊙ ) nor high-mass stars (M * > 8 M ⊙ ) form a significant number of planets. We use the Chabrier (2005) MF as our reference stellar MF. We restrict our comparisons to objects with semi-major axis a ≥ 10 au and focus on objects with masses m p (or m p sin i) ≤ 13 M Jup , where i is the orbital inclination. We use M * for the masses of host stars, m p for masses of the companions, and M for masses of objects in a mass function. We also include some comparisons for more massive objects if the comparison study calculates frequencies in bins beyond the 13 M Jup limit.
For studies that quote planet frequencies for a particular spectral type or types rather than a host mass range, we use Pecaut & Mamajek (2013) to derive assumed mass ranges for a given spectral type. Table 1 lists the fraction of stars in each spectral type bin from the Chabrier (2005) MF. For studies that cover host stars from only part of the range, we consider either the scenario that those hosts are representative of all stars or re-weight the frequencies by the fraction of 0.08 M ⊙ ≤ M * ≤ 8 M ⊙ stars represented by those hosts.
For radial velocity and direct imaging, the distinction between massive planets and stellar brown dwarfs is unclear, especially for objects with masses m p ∼ 3 -15 M Jup . For radial velocity studies, it may be possible to separate massive planets and stellar brown dwarf companions based on mass ratio. Because ground-based direct imaging studies are still limited to detecting companions with masses > 1 M Jup , the origin of many companions is ambiguous. The exceptions are companions that lie within a disk (Bowler et al. 2025;Close et al. 2025;Zhou et al. 2025) or with additional coplanar objects (e.g., the HR 8799 system; Marois et al. 2008Marois et al. , 2010) ) or both (e.g., β Pic b and c; Lagrange et al. 2009Lagrange et al. , 2019)); in these cases, the objects probably formed in a disk and are, therefore, part of the planetary MF.
We will compare the population of objects from radial velocity and direct imaging planet searches to the µFFP population. If there are more radial velocity and direct imaging companions than µFFPs, some of the radial velocity and direct imagining companions would have to come from the stellar MF. We can then infer the minimum fraction of such companions should be classified as stellar brown dwarfs (i.e., should be assigned to the stellar MF) rather than massive planets. Having fewer radial velocity and direct imaging companions than µFFPs places a constraint on the fraction of µFFPs that are bound vs. free-floating.
To generate the structures observed in protoplanetary and debris disks, embedded or nearby planets are one option among many (Vorobyov et al. 2020;Friebe et al. 2022;Smallwood et al. 2023; Stuber t E = θ E µ rel = 0.68 d M M Jup 1/2 π rel 0.016 mas 1/2 µ rel 6 mas/yr -1
(2) where κ = 8.14 M -1 ⊙ mas, π rel ≡ au(D -1 L -D -1 S ), D L is the distance to the lens, D S is the distance to the source, and µ rel is the lens-source relative proper motion. Microlensing experiments monitor the brightness of millions of stars to look for this particular light curve morphology (Gaudi 2012).
Equations 1 and 2 indicate that a planetary mass object will produce a microlensing event with a short timescale and will have a small θ E (if it is measurable). Figure 2 shows example light curves based on the µFFP event OGLE-2016-BLG-1540 (Mróz et al. 2018). Because the event was caused by a single body, it is a point-lens (PL) event, which had t E = 0.32 day, θ E = 9.2 µas, and a Neptunemass lens. The blue curve (PSPL) shows the light curve assuming a point source (PS). The orange curve (FSPL) includes the finite source (FS) effect (i.e., accounting for the physical extent of the source star), which is one way to measure θ E (Yoo et al. 2004).
A microlensing event that appears as a single lens event due to a planetary-mass object may lack a microlensing signature due to the host star (Han et al. 2005). Given a detected planet, the probability of detecting a host star decreases rapidly with the orbital separation of the planet. For a planet that appeared to be separated from its host star by 6.6 θ E (analogous to a Neptune-like orbit), there is only a ∼ 20% probability of detecting the microlensing signature of the host star (Gould 2016;Clanton & Gaudi 2017). For individual µFFP candidates, the microlensing light curve is usually only sufficient to rule out host stars within ∼ 10 au (Mróz et al. 2018(Mróz et al. , 2020a;;Ryu et al. 2021), making it ambiguous whether any given µFFP candidate is truly free-floating or merely on a wide orbit.
High-resolution follow-up observations can be used to search for potential host stars, but such searches are still in the early stages. For example, Mróz et al. (2024) searched for hosts for five µFFP candidates, but were only able to rule out the most massive ∼10-35% of possible host stars.
Even without knowing whether or not there are host stars for individual objects, the µFFP population measures the combined sum of the wide-orbit planet population and the population of freefloating planets. For example, if all µFFPs are wide orbit planets, there cannot be any true freefloating planets and vice versa. Based on published limits on host stars for individual events, we set 10 au as the lower bound of the semi-major axis range probed by µFFPs.
Equations 1 and 2 also show that while short t E or small θ E events are strong candidates for having planetary mass lenses, the mass of any given object is ambiguous. Hence, the properties of the µFFP population are inferred by measuring statistically significant excesses to the t E or θ E distributions after accounting for the stellar population. The first µFFP population studies measured or constrained excesses to the t E distribution (Sumi et al. 2011;Mróz et al. 2017). The studies discussed in this work have measured the θ E distribution, which has fewer unknowns but is restricted to a smaller sample of events. Relatively few objects with small θ E have been detected (Mróz et al. 2018(Mróz et al. , 2019(Mróz et al. , 2020b,a;,a;Kim et al. 2021;Ryu et al. 2021;Koshimoto et al. 2023;Jung et al. 2024).
Recently, both the Korea Microlensing Telescope Network (KMTNet; Kim et al. 2016) and Microlensing Observations in Astrophysics (MOA; Bond et al. 2004) surveys have made measurements of the µFFP MF using samples that include events with measured θ E .
Gould et al. ( 2022) measured a power-law distribution for µFFPs from a statistical sample of 30 events with measured θ E , including 4 detections in the "planetary" regime ( 10 µas in their case).
In particular, they observed an absence of objects with 9 µas < θ E < 26 µas, which they dubbed the "Einstein Gap." Because of this gap, they suggested that the MF could be comprised of two separate MF, one below the gap for "planets" and one above the gap for "stars." For the planetary component, they derive
with p likely to be in the range 0.9 < p < 1.2; p ≤ 0.6 is ruled out. Gould et al. (2022) showed that the µFFP population is consistent with the short timescale events from Mróz et al. (2017) and with the bound microlensing planet population from Poleski et al. (2021), who measured the occurrence rate of planets from 5 -15 au to be 1.4 +0.9 -0.6 per star with a power law index of p ∼ 1.0. Gould et al. (2022) also found consistency between their population and limits on the population of 'Oumuamua-like objects and the fraction of Uranus and Neptune-like objects that would be detected as free-floating vs. bound planets. They concluded that their results would be consistent with the larger mass planets being bound and as the mass decreases, an increasing fraction of planets could be free-floating.
Sumi et al. ( 2023) conducted a joint statistical analysis of events from the MOA survey (Koshimoto et al. 2023), some with measured θ E and others with only t E . They derived a similar power-law mass function to Gould et al. (2022):
Figure 3 compares the MFs from Gould et al. (2022) and Sumi et al. (2023). Both studies quote their MFs in planets per star. Sumi et al. (2023) defines stars as objects from their stellar MF with 3×10 -4 M ⊙ < M < 8 M ⊙ . We renormalize the Sumi et al. (2023) MF to have equal numbers of stars in this range as in the Chabrier (2005) MF. Gould et al. (2022) does not specify a particular mass range as corresponding to stars, so we use the same range as for Sumi et al. (2023) to renormalize the MF. Because stellar MFs are steep at both low and high masses, the normalization is relatively insensitive to the choices of the upper and lower limits. Using 0.013 M ⊙ as the lower limit or 100 M ⊙ as the upper limit changes the normalization by < 1%.
The stellar MF of Sumi et al. (2023) consists of a 3-part power law and differs from Chabrier (2005) below ∼ 1 M ⊙ . Figure 3 shows that Sumi et al. (2023) predicts fewer M dwarfs and significantly more stellar brown dwarfs than Chabrier (2005), but the MFs agree well for higher-mass stars. These features are quantified in spectral-type bins in Table 1. For comparison, we also plot the IMF from Kirkpatrick et al. (2024)
Attempts to detect exoplanets with direct imaging benefit from increasingly sophisticated and sensitive AO cameras and high quality software (cf. Currie et al. 2023). However, direct imaging has only been sensitive to the very highest mass end of the "planetary" mass regime: m p 1 to a few M Jup . In microlensing populations, the planetary and stellar populations overlap at these masses (Fig. 3). We can test whether the direct imaging very low-mass companion population is consistent with being planetary by seeing whether it can be explained by the µFFP population or whether it also requires contributions of objects from the stellar population. 2019) analyze GPI data for 300 stars, which yielded 6 companions with masses ≤ 13 M Jup and 3 with masses 13 M Jup < m p < 0.08 M ⊙ . The host stars range in mass from 0.2 M ⊙ to 5 M ⊙ . Nielsen et al. (2019) fit their population with a power law:
For the full host sample, they derive α = -2.3 +0.8 -0.7 . This power-law index (α ≡ -(p + 1)) is consistent with the values for the µFFP population (p =0.7-1.4; Sumi et al. 2023). Nielsen et al. (2019) infer raw frequencies of 2.1-5.4 (68% confidence interval) 5-13 M Jup companions per 100 stars with 0.2 M ⊙ < M * < 5 M ⊙ . For comparison, Gould et al. (2022) calculate the total frequency of 5-13 M Jup objects as 2.0 per 100 stars from a combination of massive planets and stellar brown dwarfs. From Sumi et al. (2023), we infer a total frequency of 8.1 objects per 100 stars with
10 1 10 0 10 1 10 2 m p (M Jup ) 10 0 10 2 Objs / 100 Stars 5M Jup 13M Jup 13M Jup 80M Jup Nielsen+19 Fulton+21 M * [1.5, 5] [0.2, 5] [0.63, 1.58] 10 1 10 0 10 1 10 2 m p (M Jup ) 10 0 10 2 Objs / 100 Stars 1M Jup 75M Jup Vigan+21 M * [1.68, 3] [0.58, 1.68] [0.3, 0.58] 10 1 10 0 10 1 10 2 m p (M Jup ) 10 0 10 2 Objs / 100 Stars 1M Jup 20M Jup Fulton+21 M * [0.63, 1.58] 2021). There are more direct imaging and radial velocity companions than µFFPs, suggesting that some fraction of the imaging and radial velocity companions come from the tail of the stellar MF.
the majority coming from the tail of the stellar population (see Table 2). As illustrated in Figure 4, if we assume the results for 0.2 M ⊙ to 5 M ⊙ stars are representative of all stars, the Nielsen et al. (2019) frequencies are only consistent with the microlensing population if most of the companions are drawn from the stellar brown dwarf population.
However, stars from 0.2 M ⊙ to 5 M ⊙ account for only ∼ 65% of the stellar population. If we assume that no stars outside of this mass range have 5-13 M Jup companions, the frequency is reduced to 2.3 +1.2 -0.9 objects per 100 stars. This result is still in tension with the hypothesis that all of these companions are planets.
One way to reconcile these frequencies would be to use the lower limit from Nielsen et al. ( 2019) from the 95% confidence interval: 1.1 objects per 100 stars. Combined with the correction factor for the fraction of hosts and the uncertainties in the µFFP population, this approach may be sufficient to bring the measurements into agreement. But, it also requires that all of the µFFPs in this mass range are bound, wide-orbit planets.
All of the m p ≤ 13 M Jup companions in Nielsen et al. (2019) orbit stars with M * > 1.5M ⊙ (∼ 40% of their sample), so they also analyze this subsample of stars. They derive a similar power-law index to the full sample and a frequency of 5-13 M J companions at 10-100 au of 9 +5 -4 objects per 100 stars with 1.5 M ⊙ < M * < 5 M ⊙ . Such a large frequency of planets, if it extended to host stars of all masses, is inconsistent with the µFFP population. However, 1.5-5 M ⊙ stars account for only about ∼ 6% of the stellar population. Assuming no other stars have companions of this mass, which is supported by the lack of detections around lower-mass stars, the frequency from Nielsen et al. (2019) becomes 0.5 +0.3 -0.2 objects per 100 stars. This frequency is in good agreement with the µFFP population, so all of the companions could be drawn from the planetary MF, but it still requires all µFFPs to be bound planets. Nielsen et al. (2019) also measure the frequency of companions in the 13-80 M Jup range (the commonly used mass range for observationally defining brown dwarfs). After scaling according to the host-star population (0.2-5.0 M ⊙ ), the frequency of such companions is 0.5 +0.5 -0.3 objects per 100 stars. These values are roughly consistent with the Gould et al. (2022) and Sumi et al. (2023) µFFP populations (0.2 and 0.1 planets per 100 stars, respectively), so these companions could be drawn from the planetary MF and formed in a disk. The total number of microlensing objects in this mass range, including brown dwarfs from the tail of the stellar MF, is ∼ 30 objects per 100 stars. Because this value vastly exceeds the frequency of companions in the Nielsen et al. (2019) study, it suggests that the vast majority of such microlensing objects cannot be bound to a star, which is consistent with the hypothesis that they are part of the tail of the stellar MF.
Vigan et al. ( 2021) analyze SPHERE results for 105 stars and are sensitive to 1-75 M Jup companions over the semi-major axis range a = 5-300 au. They conclude that the predicted fraction of the population due to massive planets vs. stellar brown dwarfs is a function of stellar mass: a large fraction of substellar companions are drawn from the planetary MF for BA stars, while for M stars, most companions are drawn from the stellar MF. Vigan et al. (2021) report their results as the frequency of systems, which represents a lower limit on the frequency of companions, if stars can host multiple planets or companions (as in the case of HR8799).
As shown in Figure 4, Vigan et al. (2021) derive the frequencies of at least one companion with mass 1 M Jup ≤ m p ≤ 75 M Jup as 23.0 +13.5 -9.7 (BA stars), 5.8 +4.7 -2.8 (FGK stars), and 12.6 +12.9 -7.1 (M stars) per 100 stars.
To obtain the total companion frequency, we re-weight each spectral-type bin based on the fraction of the stellar population it corresponds to (Table 1). The mass ranges for the M and BA bins are truncated because Vigan et al. (2021) state their sample is limited to host stars with 0.3
After correcting for these population fractions and assuming no stars outside the range 0.3 M ⊙ ≤ M * ≤ 3 M ⊙ have companions, the Vigan et al. ( 2021) results imply a total frequency of 2.0 +2.7
-1.4
companions with mass 1 M Jup ≤ m p ≤ 75 M Jup per 100 stars. These values are comparable to the expected µFFP population from microlensing: 2.6 planets per 100 stars (Gould et al. 2022) and 3.6 planets per 100 stars (Sumi et al. 2023). See Table 2. However, as with Nielsen et al. ( 2019), if we assume that all of these companions come from the planetary MF and correspond to the µFFPs, then all of the µFFPs must be bound. If we consider the Vigan et al. (2021) sample to be representative of all stars (instead of excluding companions around stars with M * < 0.3 M ⊙ or M * > 3 M ⊙ ), the frequencies increase by a factor of 2. Similar to microlensing studies, Vigan et al. (2021) consider their direct imaging detections as a mix of massive planets and stellar brown dwarfs. We scale the parameterized population models from Vigan et al. (2021) by the host star population fraction and assume that no stars with M * < 0.3 M ⊙ or M * > 3 M ⊙ have companions. The total frequencies are ≥ 0.5 +3.4 -0.3 massive planets per 100 stars and ≥ 2.0 +2.7 -1.4 stellar brown dwarfs per 100 stars. Even with the factor of 2 correction, these frequencies are below the limits inferred from µFFPs and the microlensing stellar MF, respectively. Because the frequency derived from the low-mass tail of the microlensing stellar MF is much larger than the Vigan et al. ( 2021) stellar brown dwarf companion frequency, this result implies that a significant fraction of the microlensing objects either do not have host stars or orbit stars with masses outside the range 0.3
The one caveat is that Vigan et al. ( 2021) assumed fixed power-law MFs with β = -1.31 (where β ≡ -(p + 1)) for the planetary MF, which is shallower than allowed for the µFFP population (Gould et al. 2022). A steeper power-law would decrease the number of larger mass objects in the planetary component of the Vigan et al. (2021) population, but the number of observed objects and total frequencies remain fixed. This alternative implies a larger frequency of stellar brown dwarfs and smaller frequency of planets than considered above; however, the results remain consistent with the microlensing populations.
Radial velocity planet searches offer another technique to probe the wide-orbit companion population via direct detections. Several recent radial velocity measurements of the population of companions with masses corresponding to giant planets "beyond the ice line" (Fernandes et al. 2019;Wittenmyer et al. 2020;Fulton et al. 2021) indicate that the frequency of gas giants appears to increase with a out to 3-10 au and then drop. Of these, only Fulton et al. (2021) claims sensitivity to planets beyond 10 au, so we focus our comparison on those results. Fulton et al. (2021) analyze a sample of 719 stars containing 178 planetary-mass companions from the California Legacy Survey (Rosenthal et al. 2021). Based on their figure 9, we estimate the frequency of 1 -20 M Jup companions in orbits from 10-50 au as 6.3 +3.0 -1.9 objects per 100 stars (median and 68% confidence interval), and the frequency of 5 M Jup -13 M Jup companions in orbits from 10-100 au as 3.5 +2.3 -1.6 objects per 100 stars. These values are shown in Figure 4 compared to the values for the µFFP population from Table 2. For these companion mass ranges, the µFFP frequencies lie at the extreme lower edge of the probability distributions in figure 9 from Fulton et al. (2021), indicating significant tension between the measurements. Fulton et al. (2021) describe the sample as consisting of FGKM stars, corresponding to most stars; renormalizing the values to account for stars with M * > 1.68 M ⊙ has minimal effect on the totals. However, the sample has low completeness beyond 10 au. Because of Kepler's third law, the completeness could be correlated with host mass. To test this possibility, we use the data from Rosenthal et al. (2021) to calculate the total duration of observations, ∆t, for each star. We divide ∆t by the period, P , at 10 au for that star to determine the fraction of the orbital period that has been observed. Figure 5 shows that there is indeed a strong correlation between this fraction and the host mass.
Figure 5 also highlights the five companions in the sample with a ≥ 10 au and m ≤ 13 M Jup . The minimum value of ∆t/(P @ 10 au) is ∼ 0.6. If we assume the sample is only complete for stars above this value, then the range of host masses probed by Fulton et al. (2021) for wide-orbit planetary-mass companions is approximately -0.2 ≤ log M * /M ⊙ ≤ 0.2 or 0.63 M ⊙ ≤ M ⋆ ≤ 1.58 M ⊙ . Stars in this range account for only 17% of all stars.
Applying this correction factor to the Fulton et al. (2021) frequencies, yields 1.1 +0.5 -0.3 objects with masses 1 M Jup ≤ m p ≤ 20 M Jup in orbits from 10-50 au per 100 stars and 0.6 +0.4 -0.3 objects with masses 5 M Jup ≤ m p ≤ 13 M Jup in orbits from 10-100 au per 100 stars. These values are consistent with the µFFP population. For the 5 M Jup ≤ m p ≤ 13 M Jup bin, if all such objects are from the planetary MF, all µFFPs in this mass range must be bound. Also, no technique should detect planets of this mass around stars with masses outside the range 0.63 M ⊙ ≤ M * ≤ 1.58 M ⊙ because they should not exist. If they do exist, a significant fraction of the Fulton et al. (2021) objects are stellar brown dwarfs rather than massive planets.
5.1. Debris Disks: Pearce et al. (2022) to those planets with semi-major axes a ≥ 10 au (158 planets with a maximum semi-major axis of 300 au). Planet masses range from 0.012 M Jup to 7 M Jup with a median planet mass of 0.4 M Jup .
We also show the cumulative distributions in spectral type bins (M/K/G/F/Massive) according to the ranges given in Table 1 and the minimum and maximum host masses in the sample. The host stars in Pearce et al. (2022) have masses ranging from 0.22 M ⊙ to 3.7 M ⊙ but are heavily biased toward stars with masses exceeding 0.8 M ⊙ ; the sample contains only four M dwarfs. The inferred planet minimum mass distribution is a strong function of host mass. However, after shifting all the distributions by the median planet mass (bottom panels), a Kolmogorov-Smirnov test shows no significant differences between them.
The top panel of Figure 7 compares various power-laws to the Pearce et al. ( 2022) cumulative distribution. The power-laws are normalized to match the Pearce et al. (2022) distribution at log(m p /M Jup ) = -0.7 and 0.0 (0.2 M Jup and 1 M Jup ). A power-law index of p = 1.0, as suggested by the µFFP population, is clearly too steep for the inferred debris disk population. This difference could be resolved if there is a large population of small objects found as µFFPs that are not associated with a debris disk structure.
Table 3 lists planet frequencies for the Pearce et al. (2022) population. We count the number of planets (N pl ) in a given planet mass (m p ) and spectral type (SpT) bin and calculate the fraction of planets (frac) of this type. We weight this frequency by the fraction of the stellar population "Pop Frac." represented by the range of host masses in that bin (min M * , max M * ) to derive the predicted number of planets per 100 stars ("All"). This estimate would be the total planet frequency if we assume the sample of stars with debris disks is representative of the planet population around all stars. We also consider an additional weight to account for the debris disk frequency, assuming DD Frac. = 10%, 25%, and 50% for M, FGK, and AB stars, respectively (Lestrade et al. 2009). The resulting values are given in the "DD-only" column and are the minimum frequencies assuming that stars without debris disks have no planets. We sum the values to obtain the total predicted number of planets in various mass bins, which are given in Table 4.
The binned planet frequencies from Pearce et al. (2022) are compared to the µFFP population from Sumi et al. (2023) in Figure 6. For the mass ranges 1 M Jup ≤ m p ≤ 13 M Jup and 1 M Sat ≤ m p ≤ 1 M Jup , the frequency is 3.5 and 8.1 planets per 100 stars, respectively, for the Pearce et al. (2022) sample. These values are similar to the values from the µFFP populations (see also Table 2). However, they require that all of the µFFPs in this range are bound planets. This requirement can be relaxed if only stars with debris disks have planets of this type.
The Pearce et al. (2022) masses are lower limits on the masses of the planets. Because the estimates of the Pearce et al. (2022) planet frequency for m p > M Sat are already roughly equivalent to the µFFP population, it is difficult for them to be significantly more massive than the minimum values. As an extreme example, we predict 16.2 total planets with 1 M Earth ≤ m p ≤ 13 M Jup per 100 stars from Pearce et al. (2022). If all of them actually had masses between 1 M Jup and 13 M Jup , that would vastly exceed the allowable totals from the µFFP population. We estimate that 25% of the Pearce et al. (2022) planets with minimum masses < 1 M Jup can have true masses > 1 M Jup , even including the assumption that all planets in the 1 M Jup ≤ m p ≤ 13 M Jup bin have true masses > 13 M Jup . Thus, the µFFP population can place upper limits on the masses of the planets predicted from debris disk physics. The alternative is that the inner edges of the disks are sculpted by multiple planets. Pearce et al. (2022) conclude that the minimum masses for multiple, equal mass planets to maintain the inner edge of the disk are generally at least an order of magnitude smaller than for a single planet (see Figure 8), and the majority of the planets have masses m p < 1 M Sat . Pearce et al. (2022) do not calculate the total number of planets required for each disk, so we cannot directly compare the frequencies or the resulting cumulative distributions with the microlensing results. Depending on how many planets are required, this scenario may be more compatible with the microlensing results.
As summarized in Najita et al. (2022), recent ALMA observations divide protostellar disks around solar-type stars into two general classes (see also ALMA Partnership et al. 2015;Avenhaus et al. 2018;Huang et al. 2018;Long et al. 2019;Cieza et al. 2019;van der Marel & Mulders 2021;Bae et al. 2023, and references therein). The youngest solar-type stars all have a compact disk with a radius of 20-30 au. Roughly 25% of these young stars have thin rings of gas and dust at larger radii. Rings have typical widths, δa/a ≈ 0.05-0.10, and lie at distances a ≈ 40-200 au from the host star (Hughes et al. 2018;Bae et al. 2023;Michel et al. 2023;Huang et al. 2024, and references therein). For robust ALMA detections, the rings are massive, 5-10 M ⊕ of solids in the form of mm-sized to cm-sized pebbles. There are also rings in transitional disk systems, where the compact disk has an inner hole (e.g., Bae et al. 2023, and references therein). A small sample of protostellar disks have spirals or crescents (van der Marel et al. 2021;Bae et al. 2023).
The gaps seen in ALMA disks probe the planet population at a much younger age (see also Andrews et al. 2016;Sheehan et al. 2020;Teague et al. 2021). In this section, we consider four works that calculate properties for planets required to open the observed gaps in protoplanetary disks: Zhang et al. (2018) 2023) (Section 5.2.4). We discuss our conclusions in Section 5.2.5.
For the four studies discussed below, we calculate the expected number of planets per 100 stars for each study. Each sample spans a range of stellar masses. However, the total number of stars in each sample is small, and there does not appear to be a strong correlation between inferred planet mass and host mass. Hence, we do not re-weight the sample by spectral type/stellar mass bin, but instead assume the frequency is independent of stellar mass. In addition, for consistency with previous calculations, we limit the proposed planets to those in gaps at R gap ≥ 10 au. The calculations of the planet frequencies are given in Table 5 and shown in Figure 6, and the cumulative distributions of the inferred planet populations are shown in Figure 9. 5.2.1. Zhang et al. (2018) Zhang et al. ( 2018) took a sample of 14 protoplanetary disks from the DSHARP survey (Andrews et al. 2018) containing a total of 19 gaps. Using hydrodynamical simulations under different assumptions for the disk turbulent viscosity coefficient, α, they map the properties of the gaps to planet masses (for the legend in Figure 9, we use A ≡ log α.). In this sample, there are 40-50 planets per 100 stars with a ≥ 10 au between 1 M Sat and 13 M Jup regardless of the assumed value of α. This frequency is well in excess of the number allowed by the µFFP population in this mass range: 7.5 (Gould et al. 2022) and11.7 (Sumi et al. 2023) planets per 100 stars. 5.2.2. Lodato et al. (2019) Instead of using detailed hydrodynamical simulations, Lodato et al. (2019) use a simple assumption that the gap width scales with the Hill radius of a planet. They calculate the masses of the hypothetical planets existing in several, non-overlapping, samples of observed proto-planetary disks: their own sample of disks from Taurus, the DSHARP sample from Zhang et al. (2018), and a compilation of gaps from Bae et al. (2018). Table 5, lists the number of planets per star for each of their samples and also for the combination. As with the Zhang et al. (2018) sample, the number of planets per star between 1 M Sat and 13 M Jup is well in excess of the number allowed by the µFFP populations. (2019) should apply (the "dust" regime). However, planets above this mass are likely to open gaps in the gas disk and are therefore governed by the empirical relation of Kanagawa et al. (2016) (the "gas" regime). That relation generally implies more massive planets than the Hill radius relation for the same gap width.
For each dust gap in their sample, Wang et al. (2021) classify it according to which regime applies or whether it is ambiguous. For our calculations of the number of planets per star, we perform two sets of calculations for each value of α. First, we assume that the ambiguous gaps are all in the "dust" regime, and then we assume that they are all in the "gas" regime. Thus, the labels in Figure 9 and Table 5 refer to the distinction only for the gaps in the ambiguous regime, rather than completely separating the populations by the assumed underlying physics. For the A ≡ log α = -4 regime, there are no ambiguous gaps, so we give only one set of values.
Regardless of the choice for ambiguous planets, there is significant tension with the µFFP population. The microlensing results allow for 2.4 (Gould et al. 2022) and3.4 Sumi et al. (2023) large gas giants (1 M Jup ≤ m p ≤ 13 M Jup ) and 5.1 and 8.2 medium gas giants (1 M Sat ≤ m p ≤ 1 M Jup ) per 100 stars, respectively. Even with Poisson statistics and regardless of the assumed physics, the inferred gap populations from Wang et al. (2021) imply too many large planets relative to the µFFP population. 5.2.4. Zhang et al. (2023) Zhang et al. ( 2023) calculate the inferred mass of planets for gaps in disks in Taurus using the framework of Zhang et al. (2018). In our calculations of the planet frequencies, in contrast to Zhang et al. (2023), we do not exclude any of the proposed planets from the sample, so our numbers may differ from those reported in the original paper. For this sample, the results are consistent with the µFFP population, but only under the assumption that the maximum grain size is α max = 1 cm. Smaller α max leads to larger planets and strong tension with the µFFP population.
The populations of planets inferred from ALMA gaps prefer a much shallower power-law index (p ∼ 0.5) than the µFFP population (Figure 9). There are also significantly more giant planets predicted by ALMA gaps than are compatible with the µFFP population. Unlike for direct imaging or radial velocity planets, because these inferred planets are embedded in a protoplanetary disk, they cannot be explained by assigning them to the stellar mass function.
There are several ways this tension might be resolved. First, we have assumed that the inferred ALMA planet populations are representative of the planet population as a whole. Lodato et al. 2019 argues the most significant bias in the Taurus sample is against massive disks (which are likely to have additional massive planets), which supports that assumption. On the other hand, they argue that only 35% of young stars have disks with gaps. This factor of ∼ 3 would reduce, but not entirely resolve, the tension with the microlensing results for certain physical assumptions. However, it also requires that planets with a ≥ 10 au and m p > M Sat are only found around stars with disks with gaps.
Another explanation could be that as the systems evolve, the planets migrate away from where they are inferred to form in the protoplanetary disk. However, the migration cannot be outward because those planets would still be detected as part of the µFFP population, even if they are ejected from their host systems. Inward migration would likely require that a significant fraction are absorbed by the star because the number of giant planets at a < 10 au has been measured by multiple studies to be ≪ 1 per star (Fernandes et al. 2019;Suzuki et al. 2016).
Finally, this tension could suggest problems with the assumed physics of gap formation by planets in these studies. Perhaps there is not a 1-to-1 correspondence between planets and gaps. For example, a single planet may explain multiple gaps in the disk around AS 209 (Fedele et al. 2018). Alternatively, theories about the physical mechanisms controlling gap formation could be incomplete. Regardless, the tension indicates that the µFFP population has a significant ability to constrain these theories and assumptions.
6. CONCLUSIONS
We have investigated the nature of planets and brown dwarfs in the mass regime for which the stellar and planetary mass functions (MFs) overlap. The MF of isolated objects (single lenses) identified with microlensing (Gould et al. 2022;Sumi et al. 2023) can be decomposed into two separate components: a planetary (µFFP) component and a stellar component. The µFFP (microlensing freefloating planet) population comprises all planets outside 10 au, including both bound and ejected free-floating planets. Comparing this population to various measurements of the wide-orbit planet population yields constraints on the relative proportions of planets from the high-mass tail of the planetary MF versus brown dwarfs from the low-mass tail of the stellar MF, and so, the lowest or highest mass objects formed in each case. These comparisons also constrain the fraction of µFFPs that are actually bound.
We showed that the microlensing stellar MF from Sumi et al. (2023) has significantly more objects with M < 0.1 M ⊙ than the Chabrier (2005) MF, but is similar to the local IMF derived by Kirkpatrick et al. (2024) (see Figure 3). The Sumi et al. (2023) MFs imply that objects with M 1 M Jup are dominated by objects from the low-mass tail of the stellar mass function, with relatively few objects coming from the high-mass end of planet formation. The conclusions are similar using the Chabrier (2005) stellar MF and the Gould et al. (2022) µFFP MF, but the transition point shifts to M 6 M Jup .
We compared the microlensing MFs with the populations of companions with masses m p > 1 M Jup and semi-major axes a ≥ 10 au observed by direct imaging (Nielsen et al. 2019;Vigan et al. 2021, Section 4.1) and radial velocity (Fulton et al. 2021, Section 4.2). The frequency of these companions is roughly equal to the frequency in the µFFP population. This result implies that, as hypothesized by Gould et al. (2022), all µFFPs with these masses could be bound planets for which there is no measurable microlensing signal from the host star. Furthermore, unless some of the direct imaging and radial velocity companions are drawn from the stellar MF, these planets can only exist around more massive stars; i.e., there cannot be additional undetected companions (e.g., with masses between ∼ 3 and ∼ 15 M Jup ) around stars outside the host mass ranges probed by those studies (see text). Finally, the microlensing objects from the stellar MF in this mass range should not have host stars, unless the hosts would be undetectable with existing radial velocity or direct imaging surveys.
Hence, based on these conclusions, we predict that if high-resolution imaging is taken for planetary-mass microlenses with M > 1 M Jup , there must be at least as many host stars with masses 1 M ⊙ as the number of such candidates predicted to belong to the planetary, rather than stellar, component of the microlensing MF at those masses. Alternatively, some fraction of the direct imaging and radial velocity companions are actually brown dwarfs from the tail of the stellar MF, which formed through the binary star formation process.
We also conclude that the number of planets with minimum masses > 1 M Sat inferred from debris disks assuming a single planet maintains the inner edge of the disk (Pearce et al. 2022, Section 5.1) is roughly equal to the number of such planets in the µFFP MF. This result agrees with our analysis comparing the µFFP population with radial velocity and direct imaging populations but extending to lower masses: either all µFFPs must be bound or stars without debris disks tend not to host planets with masses > 1 M Sat and orbits with a ≥ 10 au.
The inferred debris disk planet population extends below 1 M Sat . The power-law index for these planets has a much shallower slope than the µFFP MF. The µFFP population can easily accommodate this population of smaller planets and additional contributions from smaller planets, including those around stars without debris disks or ejected planets. At the same time, the planet masses from Pearce et al. (2022) are lower limits, but if they were much larger, they would violate the constraints from the higher mass µFFP population. Hence, the µFFP population constrains the masses of the population of planets sculpting debris disks or suggests that they are scuplted by multiple, smaller planets, an alternative hypothesis also investigated by Pearce et al. (2022).
In addition, the µFFP population is not consistent with the hypothesis that every ALMA gap contains a planet (Zhang et al. 2018;Lodato et al. 2019;Wang et al. 2021;Zhang et al. 2023, Section 5.2). Even if we account for the possibility that only stars with massive disks host such planets, the required numbers of m p > 1 M Sat are in tension with the µFFP population. Because the µFFP population constrains all objects with a ≥ 10 au, migration can only resolve this tension if the planets migrate inward. In that case, a large fraction would have to be absorbed by their stars due to the constraints on the frequency of giant planets with a < 10 au. More plausibly, there is not a 1-to-1 correspondence between ALMA gaps and planets. Instead, these results favor a scenario in which a single planet induces multiple gaps.
These comparisons demonstrate the power of microlensing measurements of the mass function to constrain the wide-orbit planet population and the physics governing structures in circumstellar disks. Stronger comparisons would be possible with larger samples of direct imaging and radial velocity planets, which would allow more precise frequency measurements as a function of companion mass, and a better understanding of the range of host star masses that are probed by direct imaging and ALMA disk studies.