Roman FFP Revolution: Two, Three, Many Plutos

At the time that a “Wide Field Imager in Space for Dark Energy and Planets” was proposed (Gould, 2009) to the 2010 Decadal Committee and was later adopted by the National Research Council¹¹1Astro2010: The Astronomy and Astrophysics Decadal Survey; New Worlds, New Horizons in Astronomy and Astrophysics; https://science.nasa.gov/astrophysics/resources/decadal-survey/astro2010-astronomy-and-astrophysics-decadal-survey as the Wide-Field Infrared Space Telescope (WFIRST), microlensing planets were being discovered at the rate of a few per year. In that context, the resulting homogeneous sample of $\cal{O}$(1000) microlensing planets, over the full range of masses, in the otherwise unreachable cold, outer regions of solar systems, would indeed be a “revolution” by completing the systematic census of exo-planets, which had been pioneered in the warm and hot regions by radial-velocity (RV) and transit studies, respectively. Moreover, in contrast to the ground-based detections, which delivered only planet-host mass-ratio ($q$) measurements, a substantial fraction of WFIRST planets would yield host-mass ($M_{\rm host}$) measurements, and thereby also planet-mass ($M_{\rm planet}=qM_{\rm host}$) measurements.

Fast forward 15 years, and these formerly “revolutionary” prospects have begun substantially merging into the mainstream. Based on a systematic analysis (Zang et al., 2024) of the first four years (2016-2019) of KMTNet (Kim et al., 2016) microlensing, there are already about 200 KMTNet planets in a homogeneous sample (2016-2024), and this will likely increase to about 300 by the time that WFIRST (now renamed Roman) is launched. As soon as adaptive optics (AO) are available on extremely large telescopes (ELTs), it will be possible to make mass measurements of the majority of hosts (and therefore planets) in the 2016-2019 KMTNet sample, and the planets detected in later years will also gradually become amenable to measurement as the source-lens separations continue to increase (Gould, 2022).

Certainly, Roman will detect several times more planets than KMT. Moreover, late-time AO observations will enable mass measurements for an even larger fraction of Roman planets than KMT planets, simply because its sources are systematically fainter²²2For example, according to Figure 9 of Gould (2022), about 20% of KMT planetary-microlensing sources are giants, which probably require source-lens separations of 10 FWHM ($140\,{\rm mas}$ for a 39m telescope) to allow for the 10 magnitude (factor 10000) contrast ratios that are needed to probe down most of the main sequence. The wait time for typical lens-source relative proper motions $\mu_{\rm rel}\sim 6\,{\rm mas}\,{\rm yr}^{-1}$ would be about 25 years. By contrast, only a tiny fraction of Roman planetary-microlensing sources will be giants.. However, in most respects, this will be an evolutionary, not revolutionary, development. The one remaining revolutionary element of the original WFIRST/Roman plan is its potential to probe the planet mass-ratio function well below what has been achieved from the ground.

On the other hand, the fading revolutionary potential of the original WFIRST/Roman program has been more than matched by the surging potential of its application to free-floating planets (FFPs) and wide-orbit planets (Yee & Gould, 2023).

Observationally, FFPs are short single-lens single-source (1L1S) microlensing events. These could indeed be unbound to any host, but may also be due to wide-orbit planets whose hosts are too far away to leave any trace on the event. As discussed by Gould (2016), FFP events can be resolved into moderately wide (hereafter: Wide), Kuiper, Oort, and Unbound objects using late-time high-resolution observations. As we will show, for the great majority of space-based FFP discoveries, these four categories can be distinguished within 10 years after the microlensing event, but at the time of discovery, there will be at most hints as to whether they are actually unbound (“free”) or they are bound. And most often, there will not even be hints. Hence, in the present work, we keep the nomenclature “FFP” as an observational classification of objects whose physical nature must still be determined on an event-by-event basis.

Sumi et al. (2011) were the first to propose a large FFP population, a year too late to be considered by the Astro2010 report as part of the WFIRST mission. While Mróz et al. (2017) did not confirm the Sumi et al. (2011) Jupiter-mass FFP population, they did find evidence for a large FFP population of much lower mass based on the detection of six 1L1S events with short Einstein timescales, $t_{\rm E}<(1/3)\,$d. These were all point-source point-lens (PSPL) events. Subsequently, several studies led to the detection of nine additional FFPs, all finite-source point-lens (FSPL) events, which therefore yielded measurements of their Einstein radius, $\theta_{\rm E}\propto\sqrt{M}$ (Mróz et al., 2018, 2019, 2020a, 2020b; Ryu et al., 2021; Kim et al., 2021; Koshimoto et al., 2023; Jung et al., 2024), all with $\theta_{\rm E}\lesssim 9\,\mu$as.

While both Sumi et al. (2011) and Mróz et al. (2017) expressed the ensemble of their detections of short-$t_{\rm E}$ PSPL events in terms of simplified $\delta$-function FFP mass functions, Gould et al. (2022) used the four $\theta_{\rm E}<9\mu$as FSPL detections from their own study, combined with their study’s absence of FSPL events in the “Einstein Desert” ($9\,\mu{\rm as}\lesssim\theta_{\rm E}\lesssim 25\,\mu{\rm as}$), to derive a power-law FFP mass function. They showed that this mass function was consistent with the six PSPL events reported by Mróz et al. (2017). They also showed that if this, albeit crudely measured, power law were extended by 18 orders of magnitude, it was consistent with the previous detections of interstellar asteroids and comets.

Thus, Gould et al. (2022) both confirmed the earlier suggestions of Sumi et al. (2011) and Mróz et al. (2017) that there were substantially more FFPs than stars, but also tied these objects to the potentially vast population of very small planets, dwarf planets, and sub-planetary objects, either analogs of Kuiper-Belt and Oort-Cloud objects or potentially ejected from their solar systems.

Indeed, there is already important, if still suggestive, evidence for a very large population of sub-Earth-mass objects. Compared to most of the rest of the current sample of FFPs, OGLE-2016-BLG-1928 (Mróz et al., 2020b) has an unusually small $\theta_{\rm E}=0.84\,\mu$as. Considering that,

this could in principle be a few-Earth-mass FFP in the Galactic bulge, for which the lens-source relative parallax is usually in the range $\pi_{\rm rel}\lesssim 30\,\mu$as. However, by chance, this scenario can be virtually ruled out by the event’s high observed relative proper motion $\mu_{\rm rel}=10.6\,{\rm mas}\,{\rm yr}^{-1}$, and the measured vector proper motion of the source, which together are strongly inconsistent with bulge kinematics for the lens. See their Figure 3. For a typical disk $\pi_{\rm rel}\sim 60\,\mu$as, this object would have mass $M\sim 0.5\,M_{\oplus}$.

Moreover, one of the two FFPs found by Koshimoto et al. (2023) has $\theta_{\rm E}=0.90\,\mu$as. While, in contrast to OGLE-2016-BLG-1928, there are no constraints for this object on whether it resides in the disk or the bulge, the high-frequency (i.e., two!) of FFPs with $\theta_{\rm E}\lesssim 1\,\mu$as, despite the severe difficulty of detecting them, suggests an intrinsically high frequency of low-mass FFPs.

The physical origins of FFPs, whether each is ultimately identified as being bound or unbound, and whether it is of low or high mass, are likely tied together by a single history of planet formation and early dynamical evolution. Thus, detailed statistical studies of these objects, ultimately broken down into relatively Wide, Kuiper, Oort, and Unbound, will shed immense light on the process of planet formation and evolution. In particular, for the bound subsample, it will be possible to measure the masses and physical host-planet projected separations on an object-by-object basis, as was already discussed by Gould (2016), and which we will further discuss below. Hence, it will also be possible to study the differences in the mass functions of these different sub-populations, which will be critical input for theories of planet formation. Individual masses for genuinely free-floating objects will be more challenging, and we will also discuss these challenges.

However, the main point from the perspective of this introduction is that in the 15 years since the 2010 Decadal process, the issue of the FFP mass function (or mass functions at different levels of host separation) has emerged from absolutely nothing, to weak sister of the more recognized question of a bound-planet census, to intriguing question of the day, to a unique probe of planet formation and evolution that links planets with protoplanetary objects. Yet, as in most proto-revolutionary situations, understanding of the spectacular emergence of this new field and new questions has lagged dangerously behind actual developments.

In particular, at the moment, the final observation strategy is being formulated for Roman based primarily on bound-planet yield, and the only role of FFPs in this process is to verify that a relatively weak Level 1 requirement on FFPs can be met given whatever strategy is adopted in pursuit of bound planets.

By prioritizing revolutionary FFP science, this paper truly turns the entire process on its head.

One approach is to consider what Roman can achieve for FFPs according to its adopted strategy. There have been two major studies of the detection and characterization of FFPs by Roman, which individually and collectively provide valuable insights on the measurement process. Johnson et al. (2020) studied Roman detections of FFPs at a range of masses. For example, they considered an FFP mass function that is flat at $2\,{\rm dex}^{-1}$ (per star) for $M<5.2M_{\oplus}$ and falls with a $p=-0.73$ power law above that value. This was necessarily arbitrary because there were no published estimates of the FFP mass spectrum at that time. From the present standpoint, the fact that the mass function is flat at low masses will simplify the interpretation of their results.

Their Figure 5 and Table 2 show that there are 5.0 times more detections at $M=1\,M_{\oplus}$ (“Earth”) than at $M=0.1\,M_{\oplus}$ (“Mars”) despite the fact that their adopted mass function assigns equal frequency to each class. At first sight, this seems natural because smaller masses generate shorter and/or weaker perturbations, which are harder to detect. And their Figure 11 seems to confirm this by showing that the source stars for detected events are systematically brighter by about 2 magnitudes for the Mars-class than Earth-class objects, seemingly because (according to one’s first instinct) brighter source flux is required for the former. Table 2 similarly shows that $M=0.01\,M_{\oplus}$ (“Moon”) class objects are nearly impossible to detect unless they are extraordinarily numerous.

Johnson et al. (2022) investigate the seemingly intractable degeneracies in the basic microlensing parameters $(t_{0},u_{0},t_{\rm E},\rho,f_{s})$ in the large-$\rho$ limit, which generally applies to the lowest-mass FFPs. Here $(t_{0},u_{0},t_{\rm E})$ are the basic Paczyński (1986) parameters: the time of maximum, the impact parameter (scaled to $\theta_{\rm E}$), and the Einstein timescale $t_{\rm E}=\theta_{\rm E}/\mu_{\rm rel}$, while $\rho\equiv\theta_{*}/\theta_{\rm E}$ is the ratio of the angular source radius to the Einstein radius, and $f_{s}$ is the source flux. For example, in this limit, there is an almost perfect degeneracy between $f_{s}$ and $\rho$ because the observed excess flux is $\Delta F=2f_{s}/\rho^{2}$. There are several other degeneracies as well.

Here we build on these studies by taking the opposite approach: we ask what can be done to resolve the problems they identify, notably the poor sensitivity to low-mass FFPs, by altering the Roman strategy.

In this quest, we begin by ignoring any constraints arising from the “official goal” of Roman to “complete the census of planets”. Subsequently, recognizing that even the most successful revolutions must eventually come to terms with the “old order”, we ask what compromises can be made to reconcile these two somewhat conflicting goals.

For a primer on microlensing as it specifically relates to FFP microlensing events and mass measurements, see Appendix A.

In Section 2, we show analytically that two decades of low-mass FFP detections are being “killed off” by Roman’s low adopted survey cadence of $\Gamma=4\,{\rm hr}^{-1}$. In Section 3, we show how the critical Einstein-radius parameter, $\theta_{\rm E}$, can be measured for large-$\rho$ (low-mass) FFPs, despite the fact that very few will have source color measurements, which is usually considered a sine qua non for such measurements. In Section 4, we discuss two issues that are specifically related to bound FFPs, including that all FFP detections must be subjected to late-time, high-resolution imaging to determine whether the FFP is bound or Unbound. In Section 5, we show that measuring the microlens parallax, ${\mbox{\boldmath$\pi$}}_{\rm E}$, is extremely challenging for large-$\rho$ FFPs, although this only presents fundamental difficulties for the Unbound among them. In Section 6, we sketch the science that can be extracted from measuring the FFP mass functions for each of the four categories (Wide, Kuiper, Oort, Unbound). We show how these can be extracted from the Roman observations, augmented by late-time (e.g., 5–10 years later) high-resolution observations, and possibly microlens-parallax observations. In Section 7, we show that our proposed changes to the Roman observing strategy are beneficial to the remaining revolutionary aspects of the original Roman program, namely low-mass bound planets and wide-orbit planets. We describe various other benefits of the change. We also present strategy options that represent more of a compromise with the “old order”, although we do not advocate these.

Regarding $\mu_{\rm rel}$, Equation (4) is scaled to a typical value for microlensing fields, the prospective Roman fields in particular. One can, in principle consider only the slower events, e.g., $\mu_{\rm rel}=3\,{\rm mas}\,{\rm yr}^{-1}$, at which the events will have 6 points (keeping the other parameters the same).

To gain analytic understanding, we consider the ideal case of bulge-bulge lensing with the distributions of the sources and lenses each characterized by a 2-dimensional isotropic Gaussian with $\sigma=3\,{\rm mas}\,{\rm yr}^{-1}$. Keeping in mind that the event rate is proportional to $\mu_{\rm rel}$, the distribution of event proper motions is $f(\mu)d\mu\propto\mu^{2}d\mu\exp(-\mu^{2}/4\sigma^{2})$, or to simplify the algebra,

It is immediately clear from this formula that only a fraction

of the distribution will have $\mu_{\rm rel}<\sigma=3\,{\rm mas}\,{\rm yr}^{-1}$. Figure 1 shows the full cumulative distribution in the upper panel.

We also consider the case of disk lenses (and bulge sources). For this purpose, we adopt $D_{l}=6\,{\rm kpc}$, $D_{s}=8.2\,{\rm kpc}$, a disk rotation speed of $v_{\rm rot}=240\,{\rm km}\,{\rm s}^{-1}$, an asymmetric drift of $v_{\rm asym}=25\,{\rm km}\,{\rm s}^{-1}$, bulge proper motion dispersions $(3.2,2.8)\,{\rm mas}\,{\rm yr}^{-1}$ in the $(l,b)$ directions, disk velocity dispersions of $(64,41){\rm km}\,{\rm s}^{-1}$, and solar motion $(+12,+7){\rm km}\,{\rm s}^{-1}$. The lower panel of Figure 1 shows the resulting cumulative distribution for the disk.

We return to a discussion of this figure in Section 2.3.

From the form of Equation (4), it is clear that by doubling $\theta_{*}$ one can also double $N_{\rm exp}$, which would bring it to the required six 3-$\sigma$ points for the fiducial parameters of this equation. Of course, the cost of relying on such bigger (solar-type) source stars is that they are much rarer than the early M-dwarfs that are used to scale the relation. Indeed, the main point of conducting microlensing from space and in the infrared is to access these much more numerous stars.

We can understand the role of $\theta_{*}$ as a continuous variable as follows. Because we are treating $D_{s}$ as constant, and because $R\simeq R_{\odot}(M/M_{\odot})$ on the main sequence (below the turnoff), we have $\theta_{*}\propto M_{s}$. The cross section for events in the $\rho\gg 1$ regime that we are investigating scales as $d\Gamma_{\rm events}/d\log M_{s}\propto\theta_{*}(M_{s})(M_{s}dN/dM_{s})\propto M_{s}^{2}dN/dM_{s}$. That is, $d\Gamma_{\rm events}/d\log M_{s}\propto(M_{s}/M_{\odot})^{\alpha+2}$ for the case that the mass function (of sources) is described by a power law, $\alpha$.

Based on Hubble Space Telescope (HST) optical counts of bulge stars by Calamida et al. (2015), we adopt a broken power law, with break point $M_{\rm br}=0.56\,M_{\odot}$, and powers $\alpha_{\rm high}=-2.41$ and $\alpha_{\rm low}=-1.25$, respectively above and below the break. Thus, $(\alpha+2)$ changes sign ($-0.41$ to $+0.75$) at the break, implying that on a log-$M$ plot, there is a peak at early M-dwarfs. In Figure 2, we express this rate in terms of $\theta_{*}$, by first employing the above approximations, i.e., $\theta_{*}=R_{s}/D_{s}=R_{\odot}/D_{s}(M_{s}/M_{\odot})=0.58\,\mu{\rm as}(M_{s}/M_{\odot})$. We express this relation in terms of $\theta_{*}$ (rather than $\log\theta_{*}$) because it is more familiar.

We have extended the plot to the full main sequence ($1\gtrsim M/M_{\odot}\gtrsim 0.1$) for clarity, noting that while the above S/N relation only applies in a more limited range ($0.9\gtrsim M/M_{\odot}\gtrsim 0.2$), this relation does not play a direct role in the current discussion. Figure 2 singles out the cumulative distribution up to fiducial value of $\theta_{*}=0.3\,\mu$as, as well as for two other values, whose significance will be made clear in Section 2.3.

The only other scaling variable that can be changed in Equation (4) is the observing cadence, which is currently being set for Roman at the indicated scaling value, $\Gamma=4\,{\rm hr}^{-1}$.

Of course, it requires no special insight to realize that by doubling $\Gamma$, one also doubles $N_{\rm exp}$, albeit at the cost of halving the number of fields (and so, the total area) that can be observed. However, making use of the results in Sections 2.1 and 2.2, we are now in a position to understand the impact of such doubling on the rate of FFP detection in the large-$\rho$ (i.e., low-$M$) limit.

From Equation (4), we see that one can, in principle, reach the same adopted threshold for FFP detections, $N=6$, by either halving $\mu_{\rm rel}$ or doubling $\Gamma$. However, from Figure 1, we see that by doing the first, we cut the fraction of the cumulative $\mu_{\rm rel}$ distribution for bulge lenses from 43% (red) to 8% (blue), i.e., by a factor of 5.3. While doubling $\Gamma$ comes at the cost of halving the number fields, there is still an overall net increase in large-$\rho$ FFP detections of a factor 2.6. The corresponding numbers for the disk cumulative distribution are 47% (red), 14% (blue), and factors 3.3 and 1.7.

Motivated by this insight, one might consider increasing $\Gamma$ by a further factor of 1.5 to $\Gamma=12\,{\rm hr}^{-1}$, which would allow one to capture 79% (green) of the bulge-lens $\mu_{\rm rel}$ distribution, i.e., a further increase by a factor 1.8. Again, this would come at a price of reducing the number of fields by a factor 2/3, implying a net improvement of a factor 1.2. This factor is quite minor, and such a change would come at significant cost to other aspects of the experiment. A virtually identical argument applies to the disk-lens proper-motion distribution.

Figure 2 allows us to make a similar evaluation for the trade offs between changes in $\Gamma$ and $\theta_{*}$. Comparing the red and blue lines, one sees that restricting the mass (or luminosity, or $\theta_{*}$) function to stars with $\theta_{*}>0.5\,\mu$as (which would by itself not quite achieve the required doubling of $N_{\rm exp}$), would reduce the available cumulative $\theta_{*}$ distribution function by almost a factor of 5. This is similar to the case for the bulge-lens $\mu_{\rm rel}$ distribution that was just discussed.

As in that case, we can also ask about the impact of a further increase of $\Gamma$ by a factor of 1.5, which would drive the minimum source radius down to $\theta_{*}=0.2\,\mu$as. This is shown in green. The nominal improvement is a factor of 1.56, which would be almost exactly canceled by the loss of area due to higher $\Gamma$. In fact the range of “improvement” $0.3<\theta_{*}/\mu{\rm as}<0.2$, is actually pushing the FFP detections into a regime in which the assumptions underlying the S/N calculation start to break down, mainly because blending becomes a much more serious issue. Hence, there would actually be a net loss of FFP detections, even ignoring the negative impact on other aspects of the experiment of such a further increase in $\Gamma$.

If the Roman cadence remains at $\Gamma=4\,{\rm hr}^{-1}$, as derived by optimizing the bound-planet detections, then every large-$\rho$ event that is selected according to the criterion of six 3-$\sigma$ points will have a product $\beta\theta_{*}/\mu_{\rm rel}$ that is at least twice as big as that given by the fiducial parameters given in Equation (4). That is, the prefactor in this equation is 3.0, so the product of the remaining factors must be $\geq 2$ to achieve 6 points. Ignoring the narrow range available from the final term $(\beta)$, this implies some combination higher $\theta_{*}$ and lower $\mu_{\rm rel}$. To properly account for this, we should allow $\mu_{\rm rel}$ and $\theta_{*}$ to vary simultaneously, rather than holding the other fixed as in Figures 1 and 2. We therefore find the cumulative distribution of the product of the $\mu_{\rm rel}$ and $\theta_{*}$ factors from Equation (4).

Because the prefactor in Equation (4) is 3.0, while the detection criterion is $N=6$, $\zeta>2$ is required under the present Roman strategy, but $\zeta>1$ would suffice for $\Gamma=8\,{\rm hr}^{-1}$. To evaluate the cumulative distributions for the bulge (black) and disk (magenta) cases, we draw from the range $0.58\,\mu{\rm as}>\theta_{*}>0.25\,\mu{\rm as}$, as well as from the full proper-motion distribution.

The blue and red lines in Figure 3 highlight the cumulative distributions for the cases of $\Gamma=4\,{\rm hr}^{-1}$ ($\zeta=2$) and $\Gamma=8\,{\rm hr}^{-1}$ ($\zeta=1$), respectively. The ratios of the two are 3.63 and 2.74 for the bulge and disk respectively. For FSPL events, bulge lenses are 2.5 times more frequent than disk lenses (Figure 9 of Gould et al. 2022). Weighting the two ratios by this factor, we obtain a net improvement of a factor 3.38.

As mentioned above, this improvement must be divided by 2 because there will be half as many fields.

Finally, there will be some additional detections because $\Delta\chi^{2}$ will double, which will push some very-low-mass FFPs above the threshold, e.g., $\Delta\chi^{2}>300$, as adopted by Johnson et al. (2020). For example, Equation (6) in its current form predicts $\Delta\chi^{2}=150$, for $\theta_{\rm E}=0.067\,\mu$as. If $\Gamma$ were doubled, then $\Delta\chi^{2}=300$, and it would cross the threshold of detection. A lens of mass $M=M_{\rm Pluto}$ could then be detected provided that $\pi_{\rm rel}>\theta_{\rm E}^{2}/\kappa M_{\rm Pluto}=84\,\mu$as, which corresponds to $D_{l}\lesssim 4.8\,{\rm kpc}$. Hence, depending on whether Plutos are common (about which we presently have only the barest indication from our own Solar System), there could be many additional detections of FFPs from this class.

We adopt a net improvement of a factor 2 in low-mass FFP detections.

One might also consider other cadences than the two shown in Figure 3. To avoid cluttering this figure, we present these results in tabular form in Table 1. The final column in this table is a figure of merit, which takes account of both the added FFP detections due to higher $\Gamma$ and the reduced area. However, it does not take account of the extra (or reduced) FFP detections due to higher (or lower) $\Gamma$.

The usual method to measure $\theta_{\rm E}$ is to measure $\rho$ and then to determine $\theta_{*}$ using the method of Yoo et al. (2004). In this method, one measures source flux and color from fitting the light curve, measures its offset from the clump in these variables, and then uses tabulated color/surface-brightness relations to determine $\theta_{*}$. Finally, one calculates $\theta_{\rm E}=\theta_{*}/\rho$.

The challenges arise because each of these steps is, individually, either difficult or impossible for Roman large-$\rho$ FFPs. Hence, carrying out all of them would appear hopeless.

The first problem is that very few, if any, Roman large-$\rho$ FFPs will have a color measurement. These events have a total duration $\Delta t_{\rm event}\leq 2t_{*}=2\theta_{*}/\mu_{\rm rel}=0.9\,{\rm hr}(\theta_{*}/0.3\,\mu{\rm as})/(\mu_{\rm rel}/6\,{\rm mas}\,{\rm yr}^{-1})$. So, first, because the alternate-band observations are taken only twice per day, the chance is small that these will occur during the time that the source is magnified. This issue is well recognized.

However, what seems to be less recognized is that if the second-band observations are taken during the event, they will, in the overwhelming majority of cases, prevent the light curve from being properly monitored in the primary band. This is because the secondary-band exposures are much longer, so that to cycle through all the targeted fields requires about 50 min. Hence, the main impact of secondary-band observations on FFPs will not be to measure their colors but to prevent the detection of about 8% of otherwise detectable events.

Second, as mentioned in Section 1.3, Johnson et al. (2022) show that these events display a strong degeneracy between $f_{s}$ and $\rho$, meaning that in most cases, neither can be measured separately. Rather, what is measured is the parameter combination $f_{s}/\rho^{2}$. In particular, in the approximation of no limb darkening, the excess flux as the lens is transiting the source is just $\Delta f=2f_{s}/\rho^{2}$. In discussing this, Johnson et al. (2022) point back to the fact that Mróz et al. (2020a) had already shown that this degeneracy is actually the key to measuring $\theta_{\rm E}$, provided that the source color (hence surface brightness, $S_{H}$) is known. Then we can write $\Delta f=(2/\rho^{2})\times(\pi\theta_{*}^{2}S_{H})=2\pi\theta_{\rm E}^{2}S_{H}$. That is, $\theta_{E}^{2}=\Delta f/2\pi S_{H}$. Because $\Delta f$ and $S_{H}$ are empirically determined quantities, $\theta_{\rm E}$ can be robustly measured even if $f_{s}$ and $\rho$ are not separately measured.

The problem is that, as noted by Johnson et al. (2022), Roman will yield very few color measurements for large-$\rho$ FFPs. Indeed, we should say “essentially zero”.

The solution to this seemingly intractable problem of measuring $\theta_{\rm E}$ (as opposed to either of the above two problems, considered individually) comes in three parts. First, $\theta_{\rm E}$ measurements will span two decades, i.e., $0.1\,\mu{\rm as}\lesssim\theta_{\rm E}\lesssim 10\,\mu{\rm as}$. Hence, we can easily tolerate 10% (0.04 dex) errors in typical individual $\theta_{\rm E}$ measurements and even several tens of percent in some subset of cases. Second, one can estimate the surface brightness of the source to within 20% if its $H$-band luminosity is known exactly. Third, errors in the inferred $\theta_{\rm E}$ scale only as the sixth-root of errors in the luminosity. In brief, adequate estimates of surface brightness can be made without the customary source-color measurement.

For stars on or near the zero-age main sequence (ZAMS), their mass-radius and mass-$L_{H}$ relations are determined by their chemical composition. Together, these algebraically predict the surface brightness, $S_{H}$, as a function of $L_{H}$. From isochrone models, we know that in the $H$-band, the rms scatter in this relation is less than 20% over the range of bulge metallicities³³3 At fixed luminosity on the main-sequence, surface brightness $S_{H}$ only varies by $\lesssim 10\%$ for metallicities [Fe/H]$\geq 0$. It is higher by $\sim 15\%$ for [Fe/H]$=-0.5$, and somewhat higher than that at yet lower [Fe/H]. Thus, considering the distribution of metallicities of microlensed bulge stars as measured by Bensby et al. (2017), we find that the rms error made by adopting the [Fe/H]$=0$ surface brightness, would be $\sigma(\ln S_{H})=10.3\%$. However, a more precise evaluation, which would account for $\alpha/$Fe variation, should be undertaken before applying this method.. Because $\theta_{\rm E}=\sqrt{\Delta f/2\pi S_{H}}$, such 20% errors in $S_{H}$ lead to only 10% errors in $\theta_{\rm E}$.

This reasoning does break down for source stars $M_{s}\gtrsim 0.9\,M_{\odot}$, corresponding to $\theta_{*}\gtrsim 0.53\,\mu$as in Figure 2, because these stars have moved off the ZAMS by different amounts depending on their age. However, first, we can see from Figure 2 that these stars account for a small fraction of large-$\rho$ FFP events. Second, the stars themselves are both bright and sparse (in Roman data), so they will be only weakly blended in the great majority of cases (unless the FFP has a bright host, which can be determined from late-time high-resolution data). Therefore, their color (hence, surface brightness) can be well estimated from their well-measured color at baseline.

Finally, the luminosity $L_{H}$ can be estimated using the relation $L_{H}=f_{s}\times 4\pi D_{s}^{2}\times 10^{0.4A_{H}}$, where $f_{s}$ is estimated from the baseline flux, $D_{s}$ is estimated from the mean distance of bulge sources in the direction of the event, and $A_{H}$ is measured in the standard way from field-star photometry. Clearly, then, this estimate of $L_{H}$ can only be in error due to some combination of errors in $f_{s}$, $D_{s}$, and $A_{H}$.

Before assessing these three error sources, we note that because $V_{H,\rm lum}=L_{H}/4\pi R_{s}^{3}$ is approximately invariant over the relevant range, $0.9\,M_{\odot}>M_{s}>0.2\,M_{\odot}$, we have $R_{s}\propto L_{H}^{1/3}$ and so, $S_{H}\propto L_{H}/R_{s}^{2}\propto L_{H}^{1/3}$. Hence, $\theta_{\rm E}^{2}\propto 10^{0.4\,A_{H}}\Delta f/S_{H}\propto 10^{0.4\,A_{H}}L_{H}^{-1/3}$, i.e., $\theta_{\rm E}\propto 10^{0.2\,A_{H}}L_{H}^{-1/6}$. Assuming for the moment (as is almost always the case) that $A_{H}$ is well measured, this implies that errors in the luminosity estimate propagate to the $\theta_{\rm E}$ measurement only as the sixth-root.

Now, let us consider the three sources of error in $L_{H}$. First, if there is no parallax estimate for the source, then the rms error in the source distance $D_{s}$ (due to the depth of the bulge) is about 10%, leading to a 20% error in $L_{H}$ and therefore a 3% error in $\theta_{\rm E}$, which is negligible in the current context.

Second, if the estimated $A_{H}$ is higher than the true one by $\Delta H$, then $L_{H}$ will have been overestimated by a factor $10^{0.4\,\Delta H}$, and therefore $\theta_{\rm E}$ will be overestimated by a factor $10^{0.2\,\Delta A_{H}}\times(10^{0.4\,\Delta A_{H}})^{-1/6}=10^{\Delta A_{H}/7.5}\sim 1+0.12\Delta A_{H}$. Given that typical errors in $A_{H}$ are $\sigma(A_{H})\lesssim 0.1$, this factor is also negligible.