Systematic KMTNet Planetary Anomaly Search, Paper II: Six New $q<2\times 10^{-4}$ Mass-ratio Planets

There are several features of the light-curve analyses that are common to all six events, which we present in this preamble.

All six events exhibit short perturbations on otherwise standard Paczyński (1986) 1L1S light curves, which are characterized by three parameters (in addition to two flux parameters for each observatory). These are $(t_{0},u_{0},t_{\rm E})$, i.e., the time of lens-source closest approach, the impact parameter (in units of the Einstein radius, $\theta_{\rm E}$), and the Einstein timescale.

In each case, the anomaly can be localized fairly precisely by eye at $t_{\rm anom}$, which then yields the offset from the peak, $\tau_{\rm anom}$ and the offset from the host, $u_{\rm anom}$, both in units of $\theta_{\rm E}$, as well as the angle $\alpha$ of the source-lens motion relative to the binary axis (Gould & Loeb, 1992),

Note that this also implies that $u_{\rm anom}=|u_{0}/\sin\alpha|$.

We define the quantities $s^{\dagger}_{\pm}$ by

Note that $s^{\dagger}_{-}=1/s^{\dagger}_{+}$ and $u_{\rm anom}=s^{\dagger}_{+}-s^{\dagger}_{-}$. If the source crosses the minor-image (triangular) planetary caustics, then we expect $s\simeq s^{\dagger}_{-}$, and if it crosses the major image (quadrilateral) planetary caustic, then we expect $s\simeq s^{\dagger}_{+}$. Here, $s$ is the projected planet-host separation scaled to $\theta_{\rm E}$. However, only one of the six events analyzed here (OGLE-2018-BLG-0977) has such caustic-crossing features. For the rest, we may expect that the event may be subject to the inner/outer degeneracy identified by Gaudi & Gould (1997). That is, for minor-image perturbations, which are generally characterized by a “dip”, roughly the same anomaly can in principle be produced by the source passing “inside” the planetary caustics (closer to the central caustic) or “outside”. Then we may find two solutions, which obey,

For major-image perturbations, which are generally characterized by a “bump”, a similar formula applies for $s^{\dagger}_{+}$. However, in the present paper, all six planets give rise to minor-image perturbations (“dips”).

Gould & Loeb (1992) also show how to make by-eye estimates of $q$ for events in which the source crosses a planetary caustic. Here, we present an extension of their method to estimate $q$ for the case of non-caustic-crossing minor-image perturbations caused by low mass-ratio planets.

We first focus on caustic-crossing events. In this case, there is a de-magnification trough between the two flanking caustics, with separation (using notation from Han 2006),

Therefore, the duration of the dip in the light curve between the two caustics is $\Delta t_{\rm dip}=(\Delta u/\sin\alpha)t_{\rm E}$, implying

For the case of source trajectories that miss the caustic, but still experience a dip (by which the planet is detected), we apply the same formula, but with $s\rightarrow s^{\dagger}_{-}$, i.e., implicitly assuming that the width of the dip is approximately the same as the separation between the caustics. As shown by Figure 1 of Chung & Lee (2011), this approximation holds best for low mass-ratio lenses $q\lesssim 2\times 10^{-4}$, which is the main focus of interest here. We do not expect this mass-ratio estimate to be extremely accurate simply because the dip does not itself have precisely defined edges (except in the region of the trough that lies between the two triangular caustics). Moreover, $\Delta t_{\rm dip}$ enters quadratically into Equation (5). Thus, by contrast with Equations (1) and (2), which should be quite precise, we expect that Equation (5) should be accurate at the factor $\sim 2$ level.

Nevertheless, Equation (5) can be quite powerful. Note in particular that for events in which the anomaly occurs at magnifications of at least a few, $A_{\rm anom}\gtrsim 5$ and for which the blending is small compared to the magnified flux, we have $\sin\alpha=u_{0}/u_{\rm anom}\simeq A_{\rm anom}/A_{\rm max}\simeq 10^{-0.4\Delta I}$, where $\Delta I$ is the magnitude offset between the peak and the anomaly. For such cases $s^{\dagger}_{-}\rightarrow 1-0.5/A_{\rm anom}\simeq 1$, and so Equation (5) can be approximated by

We generally search for seven-parameter “static” (i.e., without lens orbital motion or microlens parallax effects) 2L1S solutions on an $(s,q)$ grid, characterized by $(t_{0},u_{0},t_{\rm E},s,q,\alpha,\rho)$, where $\rho$ is the angular source radius scaled to $\theta_{\rm E}$, i.e., $\rho=\theta_{*}/\theta_{\rm E}$. In the grid search, $(s,q)$ are held fixed at the grid values while $(t_{0},u_{0},t_{\rm E},\rho)$ are allowed to vary in a Markov chain Monte Carlo (MCMC). We use the advanced contour integration code VBBinaryLensing (Bozza, 2010; Bozza et al., 2018) to calculate the magnification of the 2L1S model, and we identify the best-fit solution via the Markov chain Monte Carlo (MCMC) method emcee (Foreman-Mackey et al., 2013). The three Paczyński parameters $(t_{0},u_{0},t_{\rm E})$ are seeded at their 1L1S values, and $\rho$ is seeded using the method of Gaudi et al. (2002) at a value near $\rho=\theta_{*,\rm est}/\mu_{\rm typ}t_{\rm E}$, where $\mu_{\rm typ}\equiv 6\,{\rm mas}\,{\rm yr}^{-1}$ is a typical value of the lens-source relative proper motions, $\mu_{\rm rel}$, and $\theta_{*,\rm est}$ is estimated based on the value of $I_{s}$ from the 1L1S fit and the KMT-website extinction. For example, $\theta_{*,\rm est}=0.6\,\mu{\rm as}$ for $I_{s}-A_{I}=18.65$ (Sun-like star) and $\theta_{*,\rm est}=6\,\mu{\rm as}$ for $I_{s}-A_{I}=14.5$ (clump-like star). The final parameter, $\alpha$ is seeded at a grid of values around the unit circle, and it is either held fixed or allowed to vary with the chain, depending on circumstances.

After finding one or more local minima on the $(s,q)$ plane, each local is further refined by allowing all seven parameters to vary in an MCMC.

If it is suspected that microlens parallax effects can be detected or constrained, then four additional parameters are initially added, ${\mbox{\boldmath$\pi$}}_{\rm E}=(\pi_{{\rm E},N},\pi_{{\rm E},E})$ and ${\mbox{\boldmath$\gamma$}}=((ds/dt)/s,d\alpha/dt)$. Here, ${\mbox{\boldmath$\pi$}}_{\rm E}$ parameterizes the effects of Earth’s orbital motion on the light curve (Gould, 1992, 2000, 2004),

where $M$ is the lens mass and $(\pi_{\rm rel},{\mbox{\boldmath$\mu$}}_{\rm rel})$ are the lens-source relative (parallax, proper motion), while $(ds/dt,d\alpha/dt)$ are the first derivatives on lens orbital motion. The four parameters must be introduced together, at least initially, because the orbital motion of the lens can mimic the effects on the light-curve of orbital motion of Earth (Batista et al., 2011; Skowron et al., 2011). Lens orbital motion can be poorly constrained by the light curve, so that we generally impose a constraint $\beta<0.8$ where $\beta$ is the absolute value of the ratio of transverse kinetic to potential energy (An et al., 2002; Dong et al., 2009),

and where $\pi_{S}$ is the source parallax. Strictly speaking $\beta$ is only physically constrained to be $\beta<1$, but combinations of orbits and viewing angles that would generate $\beta>0.8$ are extremely rare.

When (as usual) the parallax vector ${\mbox{\boldmath$\pi$}}_{\rm E}$ is evaluated in the geocentric frame, it often has elongated, nearly straight contours whose minor axis is aligned with the projected position of the Sun at $t_{0}$, i.e., the (negative of the) direction of the Sun’s instantaneous apparent direction of acceleration on the plane of the sky. This is because $\pi_{{\rm E},\parallel}$, the component of ${\mbox{\boldmath$\pi$}}_{\rm E}$ that is aligned to this acceleration, induces an asymmetry in the light curve, which leads to strong constraints, whereas $\pi_{{\rm E},\perp}$ induces more subtle, symmetric distortions (Gould et al., 1994; Smith et al., 2003; Gould, 2004). Because the right-handed coordinate system $(\pi_{{\rm E},\parallel},\pi_{{\rm E},\perp})$ is rotated with respect to the (North, East) equatorial coordinates, the tightness of the error ellipse is often not fully reflected when $(\pi_{{\rm E},N},\pi_{{\rm E},E})$ and their errors are tabulated in papers. In this paper, we therefore also tabulate $(\pi_{{\rm E},\parallel},\pi_{{\rm E},\perp})$ and their errors in addition to those of $(\pi_{{\rm E},N},\pi_{{\rm E},E})$. For this purpose, we define $\psi$ as the angle of the minor axis of the parallax ellipse (North through East) rather than the angle of the projected position of the Sun. Of course, there are two such directions, separated by $180^{\circ}$. We choose the one closer to the Sun’s apparent acceleration on the sky. In all four events for which the parallax is measurable in the current work, $\psi$ is the same as the Sun’s acceleration within one or two degrees. See, e.g., Figure 3 of Park et al. (2004) for an explicit example of the coordinate system.

Even when these so-called “one-dimensional (1-D) parallax” measurements fail to constrain the magnitude $|{\mbox{\boldmath$\pi$}}_{\rm E}|=\pi_{\rm E}$, which is what directly enters the mass and distance estimates, they can interact with the Bayesian priors from a Galactic model to constrain the final estimates of these quantities substantially better than ${\mbox{\boldmath$\pi$}}_{\rm E}$ or the Galactic priors do separately.

More generally, ${\mbox{\boldmath$\pi$}}_{\rm E}$ measurements can provide valuable information even when they are consistent with zero. As discussed in the Appendix to Han et al. (2016), no “evidence” of parallax is required to introduce the two ${\mbox{\boldmath$\pi$}}_{\rm E}$ parameters because it is known a priori that $\pi_{\rm E}$ is strictly positive. Hence, measurements that constrain it to be small provide additional information, even if they are formally consistent with zero.

Another important application of 1-D parallax measurements is that they can yield a very precise mass estimate when the source and lens are separately resolved many years after the event (Ghosh et al., 2004; Gould, 2014). That is, such imaging generally yields a very precise measurement of ${\mbox{\boldmath$\mu$}}_{\rm rel}$, which has the same direction as ${\mbox{\boldmath$\pi$}}_{\rm E}$. Thus, if $\pi_{{\rm E},\parallel}$ is well constrained (and assuming that the ${\mbox{\boldmath$\pi$}}_{\rm E}$ error ellipse is not by chance closely aligned to ${\mbox{\boldmath$\mu$}}_{\rm rel}$), both $\pi_{\rm E}$ and $\theta_{\rm E}=\mu_{\rm rel}t_{\rm E}$ can be precisely determined from such imaging. See Dong et al. (2009) and Bennett et al. (2020) for a case in which the original 1-D parallax measurement was turned into lens mass and distance measurements by successively better determinations of ${\mbox{\boldmath$\mu$}}_{\rm rel}$.

In the original automated search for anomalies (Zang et al., 2021b), the deviation in KMT-019-BLG-0253 from the 1L1S model (Figure 1) was entirely due to KMTC data forming a 0.23 day, roughly linear rising track that lay below the model, centered on HJD${}^{\prime}\equiv$ HJD - 2450000 = 8588.81, i.e., $\Delta t_{\rm anom}=-1.8$ days before the peak. The method aggressively removes bad-, or even questionable-seeing and high-background points, which very frequently produce such “features”. Nevertheless, because the algorithm examines ${\cal O}(10^{2})$ nights of data from three observatories on ${\cal O}(10^{3})$ events, there are numerous such single-observatory deviations even after this aggressive automated culling of the data. Hence, they must be treated with caution. However, after the TLC re-reductions, the previously excluded KMTS points on the same night could be re-included, and these confirmed the dip, whose duration can be estimated $\Delta t_{\rm dip}=0.6\,$days. Moreover, there are five OGLE points on this night, which generally track the KMTC deviation, including three at the beginning of the night when the dip is pronounced. Finally, dense MOA coverage confirms the subtle “ridge” feature immediately following the dip. Thus, there is no question that the deviation is of astrophysical origin. See Figure 1.

Figure 3 shows a clear $\Delta I\sim 0.06$ mag dip in the KMTC data at $\Delta t_{\rm anom}\sim 0.54\,$days after the peak, with a duration $\Delta t_{\rm dip}=0.4\,$days. The depression defined by these 26 data points, which are taken in good seeing ($1.2^{\prime\prime}$–$1.6^{\prime\prime}$) and low background is confirmed by the two contemporaneous OGLE points. Hence, the anomaly is secure.

Again, there is clear dip in the light curve, which has $\Delta t_{\rm anom}=2.2\,$days and duration $\Delta t_{\rm dip}=0.5\,$days. It is most clearly defined by KMTC data, but these are supported by contemporaneous OGLE data. Moreover, the beginning of the dip is traced by KMTS data, while the end of the dip is supported by both MOA and KMTA data. Hence, the anomaly is secure.

Again, there is clear dip in the light curve, which has $\Delta t_{\rm anom}=+1.6\,$days and duration $\Delta t_{\rm dip}=0.8\,$days. It is most clearly defined by KMTC data, but these are supported by contemporaneous OGLE data. Moreover, there are “ridges” around the dip, which are supported by all observatories.

The KMTS data show a strong and sudden dip ending in a flat minimum, which is characteristic of the source passing into the trough between the two triangular minor-image caustics, with the entrance close to one of these caustics. One therefore expects the source to pass over the other caustic, an expectation that is confirmed by the rapid drop of the KMTA data during the first three points of the night, which span 69 minutes. Thus, KMTS and KMTA give independent, and completely consistent evidence for a minor-image caustic crossing.

The center of the dip occurs $\Delta t_{\rm anom}=+4.0\,$days after peak. Because the source crosses the two triangular caustics, the $\Delta t_{\rm dip}=0.5\,$day depression is a much more clearly defined structure than in the four previous cases.

In contrast to the previous five events, the amplitude of the anomaly (a dip at HJD${}^{\prime}\simeq 8643.7$, i.e., $\Delta t_{\rm anom}=+5.4\,$days) is not much larger than the individual KMT error bars. See Figure 7. However, this event lies in a small $(0.4\,{\rm deg}^{2})$ zone that is covered by four fields and so is observed at $\Gamma=8\,{\rm hr}^{-1}$. All four fields from two observatories (KMTC and KMTS) participate in the same overall trend during the night of the dip. Moreover the dip is confirmed by eight OGLE points. Hence, it is secure.

KMT-2019-BLG-0253 has significant blended light, which raises the question of whether or not this blend could be due to the lens. The CMD position of the baseline-object $[(V-I),I]_{\rm base}=(3.30,18.84)\pm(0.18,0.06)$ was determined by transforming the KMTC02 pyDIA field-star photometry at the position of the event to the OGLE-III system²²2Note that the CMD position of the baseline object is below the threshold of detection (roughly $V\sim 21.5$) on the OGLE-III CMD, but still has only moderate color error in the KMT field-star photometry. This is because the $V$-band photometry is carried out by different methods. For OGLE-III, sources on the $V$ and $I$ templates are found independently and then matched based on astrometry. For KMT sources are identified only in $I$ band. Then the $V$-band measurement is derived from the $V$-band flux at this $I$-band location, averaged over all available images.. We also measured the SDSS $i$ flux at this position from seven images taken with the Canada-France-Hawaii Telescope (CFHT) in $0.5^{\prime\prime}$ seeing, transformed to the OGLE-III system. This confirmed the KMTC02 $I_{\rm base}$ measurement within errors. Subtracting the source flux from the baseline flux yields the blended flux $[(V-I),I]_{b}=(3.67,19.42)\pm(0.46,0.10)$. The baseline object and blended light are shown in Figure 8 as green and magenta points, respectively.

We transform the source position (as determined from KMTC02 difference images) to the CFHT $0.5^{\prime\prime}$ images and measure its offset from the baseline object, finding,

This offset is formally inconsistent with zero at more than $4\,\sigma$, which would nominally rule out the lens as the origin of the blended light. In fact, as we show below, there are physical effects that can account for this offset, even if the blended light is due to the lens.

However, first we note that this small offset can be compared with the overall surface density of field stars that are brighter than the blend, $I_{b}=19.42$, which we find from the OGLE CMD to be $N=0.012\,{\rm arcsec}^{-2}$. Taking Equation (18) at face value and noting that the blend:source flux ratio is 1.75, the separation of the source and blend is 50 mas. Thus, the probability of such a chance superposition is $p=\pi N((11/7)\Delta\theta)^{2}\sim 1\times 10^{-4}$, which is to say, virtually ruled out. Hence, the blended light is due either to the lens itself, a companion to the lens, or a companion to the source.

One possibility for the offset in Equation (18), just mentioned, is that it is due to a companion to the source or the lens. For the source (at $D_{S}\sim 8\,{\rm kpc}$), the projected separation would correspond to $\sim 250\,{\rm au}$. For the lens, it would be proportionately closer, e.g., $\sim 150\,{\rm au}$ at $D_{L}\sim 5\,{\rm kpc}$. Such $\log(P/{\rm day})\sim 6$ stellar companions are among the most common. See Figure 7 of Duquennoy & Mayor (1991). On the other hand, if the blend were a companion to the source, it would be a subgiant, i.e., of almost identical mass to the source. From Table 7 of Duquennoy & Mayor (1991), this would be quite rare. It is more plausible that the blend is a companion to the lens, simply because there are no independent constraints on its mass ratio or distance.

However, another possibility is that an ambient star (contributing some, but not all, of the blended light) is corrupting the measurement of ${\mbox{\boldmath$\theta$}}_{\rm base}$. Suppose, for example, that 10% of the baseline flux were due to an ambient star at $\sim 300\,{\rm mas}$ from the source (and lens). This would be too faint and too close to be separately resolved, even in 500 mas seeing. Therefore, the fitting program would find the flux centroid displaced by $0.1\times 300\,{\rm mas}=30\,{\rm mas}$, which is what is observed. We can estimate the surface density of stars that are $2.5\pm 0.5$ mag fainter than the baseline object (i.e., $M_{I}=4.5\pm 0.5$) using the Holtzman et al. (1998) luminosity function, based on Hubble Space Telescope (HST) observations of Baade’s Window. We must first multiply by a factor 3 based on the relative surface density of clump stars in the KMT-2019-BLG-0253 field compared to Baade’s Window (Nataf et al. 2013, D.Nataf 2019, private communication). We then find a surface density $N=1.0\,{\rm arcsec}^{-2}$. Hence, the probability that one such star falls within $0.3^{\prime\prime}$ of the lens is about 25%.

We conclude that the blend is most likely either the lens itself or a companion to the lens. It could be a companion to the source with much lower probability. And it is very unlikely to be an ambient star.

We note that the pyDIA reductions of the KMTC data are only approximately calibrated. However, this has no practical importance because the filters are near standard and the only quantity entering the calculations that follow is the offset between the source and the clump. Moreover, this offset is small. See Figure 8. For reference, we note that in the other other four events, the KMT$\rightarrow$OGLE-III correction was found to be $[(V-I),I]_{\rm s,kmt}-[(V-I),I]_{\rm s,ogle-iii}=$ [0.06, 0.22], [0.17, 0.12], [0.14, 0.01], and [0.03 ,0.22], for KMT-2019-BLG-0253, OGLE-2018-BLG-0516, OGLE-2019-BLG-1492, and OGLE-2018-BLG-0977, respectively.

We find a best estimate for the blended light $I_{b}=21.08$ on the OGLE-IV system and $I_{b}=21.11$ on the KMTC41 pyDIA system. Even ignoring the uncertainties in this estimate due to the mottled background light, which can be considerable (e.g., Gould et al. 2020), this measurement would play very little role when treated as an upper limit on lens light. For example, even at $M=1\,M_{\odot}$ (the $1\,\sigma$ upper limit derived in Section 5), and at $D_{L}=5.6\,{\rm kpc}$, the lens magnitude would be $I_{L}\sim 22.2$. Therefore, we do not pursue a detailed investigation of this blend.

The KMTC01 pyDIA analysis formally yields a negative blended flux that, after conversion to the OGLE-III system, corresponds to an $I_{b}=21.1$ “anti-star”. This level of negative blending would be easily explained by errors in the PSF-fitting photometry of faint stars in the baseline images, even without taking account of the unresolved mottled background. Hence, it is of no concern in itself, and can be regarded as consistent with zero. The field has extinction $A_{I}=1.87$. Hence, a lens with $M=0.84\,M_{\odot}$ (the $1\,\sigma$ upper limit derived in Section 5) at $D_{L}\sim 7.0\,{\rm kpc}$ would have brightness $I_{L}=20.8$. This is comparable to the error in the measurement of the blended flux (as demonstrated by its negative value). Hence, no useful limits on the lens flux can be derived.

For four of the six events, the position of the source on the CMD corresponds to what is expected for a reasonably common class of bulge star. These are turnoff stars (KMT-2019-BLG-0253 and OGLE-2018-BLG-0516), clump giant (OGLE-2018-BLG-0506), and mid-G dwarf (OGLE-2018-BLG-0977). However, the source position for OGLE-2019-BLG-1492 is somewhat puzzling and deserves further investigation.

The source has a similar color to the Sun, but is about 1.45 mag fainter than the Sun would be if it were at the mean distance of the clump. This could, in principle, be resolved by it being 1.45 mag dimmer than the Sun or 1.45 mag more distant than the clump, or some combination. For example, for the first, it could be that the source is a relatively unevolved metal-poor star. Another potential explanation is that the source color has not been correctly measured. For example, if the source were 0.2 mag redder, it would be expected to be 1 mag or more dimmer than the Sun. However, the color is determined from regression of about a dozen well-magnified $V$-band points that lie along the regression line. Hence this explanation appears unlikely. Regardless of the exact explanation of the discrepancy, it has little practical effect. That is, even if $\theta_{*}$ were double the value that we have estimated in Table 9, the constraints provided by the resulting limits on $\theta_{\rm E}$ and $\mu_{\rm rel}$ would still be too weak to matter.

The blended light has $I_{b}=19.47$ on the calibrated OGLE-III scale. However, the baseline $V$-band photometry is not precise enough to reliably determine $(V-I)_{b}$. In principle, this blended light might be due primarily to the lens. However, we find from CFHT baseline images in $0.45^{\prime\prime}$ seeing that the blend is displaced from the event by $0.33^{\prime\prime}$. Therefore, it cannot be the lens, and so must be either a companion to the lens or to the source, or an ambient star that is unrelated to the event. We find that surface density of stars that are brighter than the blend $(I<I_{b})$ is $0.33\,{\rm arcsec}^{-2}$. Thus, the probability of an ambient star within a circle of radius equal to the separation is 11%. This is the most likely explanation for the blended light.

In Table 9, we only list a $3\,\sigma$ upper limit on $\rho$ because the light curve is formally consistent with $\rho=0$ at this level. However, it should be noted that at the $1\,\sigma$ and $2\,\sigma$ levels, $\rho$ is constrained by $\rho=1.9^{+0.3}_{-0.6}\times 10^{-3}$ and $\rho=1.9^{+0.6}_{-1.6}\times 10^{-3}$, respectively. These tighter limits on $\rho$ result primarily from the source crossing a caustic. When, evaluating the physical parameters in Section 5, we will make full use of the $\chi^{2}$ fit values as a function of $\rho$. The blended light is a relatively bright star on the foreground main sequence $[(V-I),I]_{b,OGLE-III}=(17.39,1.98)$, which raises the possibility that this is the lens. However, we find from examining the CFHT baseline image that this blended light is a resolved star with a separation of $\sim 0.5^{\prime\prime}$. Thus, it is definitely not the lens, nor can it be a companion to the source. This leaves two possibilities: companion to the lens or ambient star. The surface density of foreground main-sequence stars brighter than the blend is $N=0.0085\,{\rm arcsec}^{-2}$. Hence, the probability of such an ambient star within $0.5^{\prime\prime}$ is $p=1/150$. This is relatively low, but the prior probability that a given star has a substantially more massive companion at separation $a_{\perp}\gtrsim$1000–2000 au (for distance 2–4 kpc) is also of order 1% or less. Thus, no information about the lens system can be inferred from the presence of this bright blend.

There was significant difficulty measuring the source color of KMT-2019-BLG-0953. As with the other four events that lie in the OGLE-III catalog (Szymański et al., 2011) footprint, our approach is to first measure the source color in the KMT system by regression and then to transform this color to the OGLE-III system by comparison of field stars. The results for all events are shown as blue points in Figure 8. However, for KMT-2019-BLG-0953, we applied this procedure five times, using the independent data sets from KMTC02, KMTC42, KMTC03, KMTC43, and KMTS02. Strikingly, these five determinations are comprised of two groups: (KMTC02, KMTC43) were closely grouped near $(V-I)_{s,\rm OGLE-III}\simeq 1.94$, while (KMTC03, KMTC42, KMTS02) were closely grouped near $(V-I)_{s,\rm OGLE-III}\simeq 2.05$, The errors in each case were near 0.04. One approach would simply be to take the average of these five measurements, which would yield a mean and standard error of the mean of $2.006\pm 0.018$. This joint fit has $\chi^{2}=9.1$ for four degrees of freedom, which has a Gaussian probability of $p=(1+\chi^{2}/2)\exp(-\chi^{2}/2)=6\%$. Under normal circumstances, this value would not be low enough to raise serious concerns.

However, there are two further issues that must be taken into account. First, using the clump centroid from Table 9, this would lead to a dereddened source color $(V-I)_{s,0}=0.51\pm 0.02$. Such blue stars are extremely rare in the bulge. Given that the source is 2.3 magnitudes below the clump, it would have to be either a blue straggler or a relatively young star. Alternatively it could be an F dwarf in the disk, relatively close to the Galactic bulge. This would also be unexpected.

The second issue is that the color measurement may be impacted by the presence of a bright neighbor at $1.0^{\prime\prime}$, which is about 2 mag brighter than the source. DIA works by first subtracting the reference image convolved with a kernel and then multiplying the resulting difference image by the point spread function (PSF). If the subtraction were perfect, the only impact of the neighbor would be to add photon noise, which is already taken into account in the measurement process. However, the PSF varies over the field, particularly near the edge of the field (recall that this event lies in the overlap of two neighboring fields), so that the PSF (and resulting kernel) are inevitably less than perfect. Then, residuals from an imperfectly subtracted, bright, overlapping neighbor can impact the photometry. Thus, the difficulties induced by this neighbor might plausibly account for the relatively high $\chi^{2}/$dof of the joint fit, in which case the high scatter would be due to systematics, so that the errors could not be treated as Gaussian.

We resolve these challenges as follows. First, we derive the color from the three redder measurements and ignore the two bluer measurements as plausibly due to systematics. Second, we adopt the standard deviations of these measurements, rather then the standard error of the mean, in order to reflect the non-Gaussian errors. Third, with these assumptions, the source is very plausibly a bulge star, and so we assign it bulge kinematics in the Bayesian analysis of Section 5.

However, we also briefly discuss the implications for both present and future work, if the source is later proved by followup observations to be bluer than we have assumed.

First, such a bluer source would yield a smaller $\theta_{*}$. In particular, it would be $\sim 10\%$ smaller if the source were 0.11 mag bluer. Under normal circumstances, this would affect the Bayesian estimates by reducing both $\theta_{\rm E}$ and $\mu_{\rm rel}$ by 10%. However, in the present case, we have only weak lower limits on these quantities (see Table 9), so there is essentially no impact. Furthermore, if the lens and source are separately resolved by future adaptive optics (AO) observations, then $\mu_{\rm rel}$ (and so $\theta_{\rm E}=\mu_{\rm rel}t_{\rm E}$) will be directly determined from these observations, so that the current $\theta_{*}$ estimate will become moot.

The main impact would be that a blue source would almost certainly imply that the source is in the disk, which then implies that the lens (which must lie in front of the source) is also in the disk. Thus, adopting a blue color would affect the Bayesian estimates. The error bars of these estimates are already quite large, and we regard the probability of a blue source to be sufficiently small to ignore this possibility. Hence, the main role of this section will be to alert workers carrying out future followup observations to the difficulties of the color measurement, so that they can properly take these into account in their analyses.

Our approach yielded a total of 11 planets with $q<2\times 10^{-4}$ in the KMT prime fields during 2018-2019, including OGLE-2019-BLG-1053Lb, which was reported by Zang et al. (2021b), the six reported here, and four previously discovered planets³³3Not including KMT-2018-BLG-1025Lb (Han et al., 2021a), which, as mentioned above, is excluded from the sample because it has two degenerate solutions that are well-separated in $q$. , OGLE-2018-BLG-0532Lb (Ryu et al., 2020), OGLE-2018-BLG-0596Lb (Jung et al., 2019b), OGLE-2018-BLG-1185Lb (Kondo et al., 2021), and KMT-2019-BLG-0842Lb (Jung et al., 2020b).

There are four known $q<2\times 10^{-4}$ planets from 2018-2019 that were not found in this search: KMT-2018-BLG-0029Lb (Gould et al., 2020), OGLE-2019-BLG-0960Lb (Yee et al., 2021), KMT-2018-BLG-1988Lb (Han et al., 2021d), and OGLE-2018-BLG-0677Lb (Herrera-Martin et al., 2020). The first three of these do not lie in prime fields, while the last failed our $\Delta\chi^{2}_{0}>75$ cut in the initial automated anomaly search. Note that KMT-2018-BLG-1988Lb has a very large (factor 2) $1\,\sigma$ error in $q$, so that it will almost certainly not be included in mass-ratio studies.

Finally, although it is not directly germane to the present study, we note that our approach failed to recover (or fully recover) three known planets with $q>2\times 10^{-4}$. The first is KMT-2019-BLG-1715LABb (Han et al., 2021c) which is a 3L2S event. The automated AnomalyFinder identified the anomaly generated by the $q_{2}=0.25$ binary companion, which dominated (and so suppressed recognition of) the planetary companion $q_{3}=4\times 10^{-3}$. The AnomalyFinder would have to be modified to take account of the possibility of planet-binary systems to have detected this planet. The second case is that of KMT-2019-BLG-1953Lb (Han et al., 2020c), which is a $q\sim 2\times 10^{-3}$ “buried planet” (Dong et al., 2009), i.e., its signature is submerged in the finite-source effects near the peak of a high-magnification event, $A_{\rm max}\sim 1000$. It was identified in our search as a finite-source-point-lens (FSPL) anomaly. In principle, all such events should be systematically searched for planets because of their overall strong sensitivity, particularly to Jovian mass-ratio planets. However, this stage was not pursued in the present context because of the weak sensitivity of such events to very low-$q$ planets, which is the focus of the present effort.

The third case is OGLE-2018-BLG-1011Lb,c (Han et al., 2019). This is a two-planet system for which the AnomalyFinder reports a single anomaly that contains both planets. However, in contrast to the case of the binary+planet system KMT-2019-BLG-1715LABb that was mentioned above, this is not really a shortcoming. That is, the AnomalyFinder identifies the principal anomaly, but does not itself classify the anomaly as “planetary” or “binary”. Rather, the operator must make the decision that an anomaly is “potentially planetary”. All such events must be thoroughly investigated before they can be published, and such detailed investigations, which are carried out using TLC reductions, are far more sensitive to multiplicity of planets than any potential machine search based on pipeline reductions. By contrast, the detection of a binary system will not, of itself, trigger such a detailed investigation. Because OGLE-2018-BLG-1011 was a well-known planetary system at the time that the AnomalyFinder was run on 2018 data, no investigation was needed. However, it would have been triggered if the planets were not already known.

For completeness, we note that all three of these planetary events, i.e., OGLE-2018-BLG-1011, KMT-2019-BLG-1715, and KMT-2019-BLG-1953, were designated as “anomalous” by the AnomalyFinder. However, because there were multiple anomalies in the three cases (either a third body or finite source effects that dominated over the planet), the AnomalyFinder did not identify all of the planetary anomalies.

The cumulative mass-ratio distribution of these 11 planets is presented in Figure 9, which shows that they span the range $-5<\log q<-3.7$. We have not yet measured the efficiency of our selection, but the automatic selection should be less sensitive to lower-$q$ planets.

In particular, the sensitivity $\xi(q)$ of any wide-angle ground-based survey that depends primarily on a $\chi^{2}$ threshold to detect planets will be approximately a power law, which gradually steepens toward lower $q$. First, in the regime where the size of the sources (dominated by upper-main-sequence and turnoff stars) is smaller than the caustic size, the cross section for the source to interact with the caustics, or with the magnification structures that extend from them, scales as either $q^{1/2}$ or $q^{1/3}$ for resonant and planetary caustics, respectively. Second, as $q$ falls, the duration of the anomaly declines according to the same power laws, which for fixed source brightness $I_{s}$, proportionately reduces $\chi^{2}$. Hence, for otherwise similar events, the $\chi^{2}$ threshold restricts lower-$q$ planets to the brighter end of the luminosity function. Thus, if the luminosity function were a strict power law (as it roughly is for $0.5<M_{I}<3$, Figure 5 of Holtzman et al. 1998), then the efficiency would also be a power law, $\xi(q)\propto q^{\gamma}$. In fact, the luminosity becomes much shallower for $M_{I}\gtrsim 3$, which makes $\xi(q)$ become gradually shallower toward higher $q$. Finally, for low mass ratios ($\log q\lesssim-4$) where turnoff sources become comparable to or larger than the caustic size, two new effects take hold. The first is that the cross section of interaction is fixed by the source size, rather than declining with $q$ as the caustic size. By itself, this would make the slope shallower toward lower $q$. However, the second effect is that the flux deviation due to the caustic is washed out by the extended source, which at fixed $I_{s}$ reduces $\chi^{2}$ and so restricts detections to the brighter end of the luminosity function, i.e., more so than for the case analyzed above of small sources. The net effect is to maintain a steepening $\xi(q)$ in this regime. A corollary of this logic is that if one overestimates (or underestimates) the source sizes in the efficiency calculation, then one will derive a $\xi(q)$ that is too steep (or too shallow) in the $\log q\lesssim-4$ regime. However Figure 7 of Suzuki et al. (2016) shows that this is a relatively modest effect.

Therefore, while no rigorous conclusions can yet be drawn, a declining sensitivity function suggests that the low mass-ratio planets in this homogeneously selected sample represents a substantial underlying population. This raises questions about the previously inferred paucity of planets at low $q$ (Suzuki et al., 2016; Jung et al., 2019a).

We note that it is already known that the low-$q$ 2018-2019 planets KMT-2018-BLG-0029Lb ($\log q=-4.74$) and OGLE-2019-BLG-0960Lb ($\log q=-4.90$) will be recovered when we apply our approach to the non-prime KMT fields. See Figure 9. While we cannot include these in the homogeneously selected distribution because we do not yet know what new planets will be discovered in the non-prime fields, these detections prove that such planets exist. So, it will not be the case that only larger mass-ratio planets are found in these fields. Therefore, it would require many such discoveries with $\log q\gtrsim-4.3$ to maintain a strong case for a mass-ratio function that declines with declining $q$ in the $q<2\times 10^{-4}$ regime. We briefly discuss the reasons that we do not consider this to be likely in Section 6.3.

An interesting difference between the recovered versus discovered low-$q$ planets is that the former all have source trajectories that are closer to the planet-host axis than the latter. The two groups have angular offsets ($6^{\circ}$, $8^{\circ}$, $15^{\circ}$, $27^{\circ}$), and ($31^{\circ}$, $36^{\circ}$, $45^{\circ}$, $50^{\circ}$, $58^{\circ}$, $59^{\circ}$, $74^{\circ}$), respectively. In particular, the two $q<10^{-4}$ recovered planets have trajectories very close to the planet-host axis ($6^{\circ}$ and $8^{\circ}$). Although we have excluded KMT-2018-BLG-1025, we note that if the favored $q<10^{-4}$ solution is indeed correct, then it also has a source trajectory close to the binary axis ($5^{\circ}$). This probably means that it is much easier to spot by eye the extended anomalies due to such oblique encounters, compared to the short dips or bumps in the events containing the newly discovered planets. This higher sensitivity of oblique trajectories was already noted by Jung et al. (2020b), Yee et al. (2021), and Zang et al. (2021a).

It is overall less likely that the seven newly discovered (as opposed to recovered) planets would have been discovered if exactly the same event had occurred in lower cadence fields. All seven planets were newly discovered because a short-lived, low-amplitude perturbation was densely covered in the prime fields. Their short duration and low amplitude severely diminished⁴⁴4We say “diminished” rather than “prevented” because OGLE-2018-BLG-0677 (Herrera-Martin et al., 2020) was recognized by eye despite the fact that it was below the threshold of our machine search. the chance that they would be noticed in by-eye searches, while their dense coverage enabled relatively high $\Delta\chi^{2}_{0}$ during the small, short deviation. The same event would likely still escape by-eye detection in a lower-cadence field, while the $\Delta\chi^{2}_{0}$ would be lower by the same factor as the cadence. Four of the seven discovered, planets had nominal cadences of $\Gamma=4\,{\rm hr}^{-1}$, two (OGLE-2018-BLG-0516 and OGLE-2018-BLG-0977) had $\Gamma=2\,{\rm hr}^{-1}$, and one (KMT-2019-BLG-0953 had $\Gamma=8\,{\rm hr}^{-1}$. If we simply scale $\Delta\chi^{2}_{0}$ by $\Gamma$, then for KMTNet’s seven $\Gamma=1\,{\rm hr}^{-1}$ fields, the seven $q<2\times 10^{4}$ discoveries listed in Table 12 would have $\Delta\chi^{2}_{0}=(22,56,41,98,65,35,75)$. For these lower-cadence fields, the detection threshold will be set to $\Delta\chi^{2}_{0}=50$ (rather than 75) because the problem of contamination by low-level systematics is substantially reduced. Hence, four of the seven planetary events would have been investigated, and plausibly published. On the other hand, for KMTNet’s 11 $\Gamma=0.4\,{\rm hr}^{-1}$ fields, the corresponding numbers are $\Delta\chi^{2}_{0}=(9,22,16,39,26,14,30)$, which are clearly hopeless. The same applies, ipso facto, to KMTNet’s three $\Gamma=0.2\,{\rm hr}^{-1}$ fields.

Regarding higher mass-ratio planets, $q>2\times 10^{-4}$, these are generally much easier to identify by eye. Our preliminary review of the higher mass-ratio sample from the high-cadence fields confirms this assessment. The ratio of recovered-to-discovered planets that unambiguously have mass ratios $q<2<10^{-4}$ is 4:7. For comparison there are 16 recovered planets with $2\times 10^{-4}<q<0.03$ in Table 11. If the recovered-to-discovered ratio were the same as for low-$q$ planets, then we would expect 28 high-$q$ discoveries. While we have not yet completed our investigation, we are confident that the actual number of high-$q$ discoveries will be well below this number.

Thus, we expect that when our method is applied to lower-cadence fields, it will generally recover the by-eye detected events, but the fraction of new discoveries will likely be smaller.

The KMTNet AnomalyFinder was applied to 2018-2019 prime-field data and returned 11 low mass-ratio ($q<2\times 10^{-4}$) planets, of which four were previously discovered by eye, one was previously reported in Paper I (Zang et al., 2021b), and six are reported here for the first time. This 7:4 ratio suggests that many low mass-ratio planets are missed in by-eye searches. By contrast there was only one by-eye discovery that was missed by AnomalyFinder, and this because it was below the $\chi^{2}$ threshold of the algorithm.