© ESO 2021

1 Introduction

During the early phase of planetary microlensing experiments, for example the OGLE (Udalski et al. 1994), MACHO (Alcock et al. 1993), and EROS (Aubourg et al. 1993) surveys, for which the annual detection rate of lensing events was several dozen, individual events could be thoroughly inspected to check the existence of planet-induced anomalies in the light curves of the lensing events. With the advent of high-cadence surveys, that is, MOA (Sumi et al. 2013), OGLE-IV (Udalski et al. 2015), and KMTNet (Kim et al. 2016), the detection rate has soared to more than 3000 per year. With the greatly increased number of events, together with the large quantity of data for each event, planetary signals for some events may escape detection.

Two projects have been carried out since 2020 with the aim of finding missed planets in the survey data collected in and before the 2019 season. One project was conducted to find planet-induced anomalies via an automatized algorithm. Hwang et al. (in prep.) applied the automated AnomalyFinder software (Zang et al. 2021) to the 2018–2019 lensing light curves from the ~ 13 deg² of sky covered by the six KMTNet prime fields with cadences ≥2 h⁻¹. From this investigation, they reported six newly detected planets with mass ratios q < 2× 10⁻⁴, OGLE-2019-BLG-1053Lb, KMT-2019-BLG-0253Lb, OGLE-2018-BLG-0506Lb, OGLE-2018-BLG-0516Lb, OGLE-2019-BLG-1492Lb, and OGLE-2018-BLG-0977. The signals of these planets were not only very short but also weak, and thus they had not been noticed from visual inspections. More extensive searches for missing planetary signals via the application of the automated algorithm to all the lensing light curves detected by the KMTNet survey are underway (Jung et al., in prep.).

The other project was conducted by visually inspecting the previous survey data to find unnoticed planetary signals. This project led to the discoveries of ten planets. The first planet was KMT-2018-BLG-0748Lb (Han et al. 2020b), for which the planetary signal had been missed due to the faintness of the source combined with relatively large finite-sourceeffects. This discovery was followed by the detections of: KMT-2016-BLG-2364Lb, KMT-2016-BLG-2397Lb, OGLE-2017-BLG-0604Lb, and OGLE-2017-BLG-1375Lb, for all of which the lensing events involved faint source stars (Han et al. 2021a); KMT-2019-BLG-1339Lb, for which the planetary signal was partially covered (Han et al. 2020a); KMT-2018-BLG-1976Lb, OGLE-2019-BLG-0954Lb, and KMT-2018-BLG-1996Lb, for which the planetary signals were produced through a non-caustic-crossing channel and were thus weak (Han et al. 2021b); and KMT-2018-BLG-1025Lb, for which the planetary signal had been missed due to the low mass ratio of the planet (q ~ 0.8 × 10⁻⁴ or 1.6 ×10⁻⁴ for two degenerate solutions) together with the non-caustic-crossing nature of its planetary signal (Han et al. 2021c). Visually inspecting missing planets can provide various types of planetary signals that are prone to being missed and thus can help to develop a more complete algorithm for automatized planet detections.

In this paper we report three additional planets with similar planetary signatures and event characteristics found from the (visual) inspection of the 2017–2019 season data collected by the OGLE and KMTNet surveys. The common feature of these planetary events is that the planetary signals were produced by the crossings of faint source stars over the resonant caustics formed by giant planets located near the Einstein rings of the lens systems. Detecting such planetary signals requires a visual inspection of lensing light curves because the durations of the planet-induced anomalies comprise important portions of the event durations, making them difficult to be detected by an automatized system that is optimized to detect very short-term anomalies. An example of such a planetary event is found in the case of OGLE-2016-BLG-0596 (Mróz et al. 2017).

For the presentation of the work, we organize this paper as follows. In Sect. 2 we mention the observations of the lensing events and the acquired data. In Sect. 3 we detail the characteristics of the anomalies in the lensing light curves of the events and describe the detailed analyses conducted to explain the observed anomalies. In Sect. 4 we specify the source types of the events and constrain the angular Einstein radii. In Sect. 5 we determine the physical parameters of the planetary systems using the observables of the events. In Sect. 6 we discuss the role of high-resolution follow-up observations for faint-source planetary events with resonant caustic features in clarifying the nature of the lens system. A summary of the results and conclusion are presented in Sect. 7.

2 Observations and data

The newly reported three planetary lensing events are KMT- 2017-BLG-2509, OGLE-2017-BLG-1099/KMT-2017-BLG-2336, and OGLE-2019-BLG-0299/KMT-2019-BLG-2735. The first event was detected solely by the KMTNet survey, and the latter two events were detected by both the OGLE and KMTNet surveys. Hereafter, we designate the events by the identification numbers of the surveys that first found the lensing events. In Table 1 we list the equatorial and galactic coordinates of the source stars, alert dates, and I-band baseline magnitudes, I_base, of the individual events. The notation “postseason” indicates that the event was detected from the postseason inspection of the data.

After one year of test observations in the 2015 season, the KMTNet group has been carrying out a lensing survey toward the Galacticbulge field since 2016 with the use of three identical 1.6 m telescopes that are distributed on the three continents of the Southern Hemisphere, at the Siding Spring Observatory in Australia, the Cerro Tololo Interamerican Observatory in Chile, and the South African Astronomical Observatory in South Africa. We designate the individual telescopes as KMTA, KMTC, and KMTS, respectively. The camera mounted on each telescope has a 4 deg² field of view. The OGLE team has been conducting a lensing survey since its commencement of the first phase experiment in 1992 (Udalski et al. 1994), and now the survey is in the fourth phase (OGLE-IV); it now uses an upgraded camera that yields a 1.4 deg² field of view(Udalski et al. 2015). The OGLE telescope with an aperture of 1.3 m is located at the Las Campanas Observatory in Chile. For both surveys, images of the events were obtained mainly in the I band, and a fraction of V -band images were acquired for the source color measurements. In Table 2 we list the data sets used in the analysis, the fields of the individual surveys, and the cadence of observations. We note that the KMTNet cadences of the events, ranging from 1.0–2.5 h, are substantially lower than the cadence of the prime fields, 15 min, and this partially contributed to the difficulty in identifying the planetary nature of the events.

The data used in the analyses were reduced using the photometry pipelines of the individual survey groups. These pipelines, developed by Albrow et al. (2009) for KMTNet and by Woźniak (2000) for OGLE, utilize the difference imaging technique (Tomaney & Crotts 1996; Alard & Lupton 1998), which is optimized for the photometry of stars in very dense star fields. A subset of the KMTC data was additionally processed using the pyDIA photometry code (Albrow 2017) to construct color-magnitude diagrams (CMDs) of stars around the source stars and to specify the source types (see Sect. 4 for details). For the data used in the analyses, we readjusted the error bars of the photometric data by $\sigma =k{({\sigma }_{\text{min}}^2+{\sigma }_0^2)}^{1/2}$ following the prescription outlined in Yee et al. (2012). Here, σ₀ denotes the error barfrom the pipelines, σ_min is a factor used to make the scatter of data consistent with error bars, and k is a scaling factor used to make the χ² per degree of freedom for each data set equal to unity. In Table 3 we list the error-bar readjustment factors and the number of data, N_data, of the individual data sets.

3 Analysis

In this section we present the detailed analyses of the individual lensing events. The light curves of all events exhibit anomaly features that include caustic crossings, and thus we model the light curves under a binary-lens and single-source (2L1S) interpretation.

The modeling was done by searching for the set of the lensing parameters that best explains the observed data. The first three of these parameters describe the source–lens approach, including (t₀, u₀, t_E), which represent the time of the closest source approach to a reference position of the lens, the separation (normalized to the angular Einstein radius, θ_E) between the source and the lens reference position at t₀, and the event timescale, respectively. For the reference position of the lens, we used the center of mass for a binary lens with a separation less than θ_E (close binary), and we used the effective position, defined by Di Stefano & Mao (1996) and An & Han (2002), for the lens with a separation greater than θ_E (wide binary). The next three parameters (s, q, α) describe the lens binarity, and they denote the projected separation (normalized to θ_E) and mass ratio between the lens components, M₁ and M₂, and the angle between the source motion and the binary axis (source trajectory angle), respectively. The last parameter is ρ, which is defined as the ratio of the angular source radius, θ_*, to θ_E, that is, ρ = θ_*∕θ_E (normalized source radius), and it is included to account for finite-source effects that affect the caustic-crossing parts of lensing light curves. We incorporated limb-darkening effects in the finite magnification computations by modeling the surface brightness variation of the source as S ∝ 1 − Γ(1−3cosϕ∕2), where Γ is the limb-darkening coefficient and ϕ represents the angle between two lines extending from the source center: one to the observer and the other to the source surface. As will be discussed in Sect. 4, the source stars of all events are of similar spectral types, early K-type main-sequence stars, and thus we adopted an I-band limb-darkening coefficient of Γ_I = 0.5 from Claret (2000), assuming that the effective temperature, surface gravity, and turbulence velocity are T_eff = 5000 K, log (g∕g_⊙) = 0.05, and v_turb = 2 km s⁻¹, respectively.

For a fraction of events with long timescales comprising an important portion of a year, lensing light curves may deviate from the form expected from a rectilinear lens-source motion. One cause for this deviation is the motion of an observer induced by the orbital motion of Earth (microlens-parallax effect; Gould 1992), and the other is the orbital motion of the binary lens (lens-orbital effects; Dominik 1998). It is expected that the signature of the microlens parallax, π_E, will be small for OGLE-2017-BLG-1099 and OGLE-2019-BLG-0299 due to their short timescales, which are ~ 19 days and ~ 30 days, respectively,but the signature might not be small for KMT-2017-BLG-2509 due to its relatively long timescale of ~ 67 days. We checked these higher-order effects and found that it is difficult to detect the higher-order signatures for any of the events, mainly because the photometric quality is not high enough to detect the subtle deviations induced by these higher-order effects. A microlens parallax can provide an important constraint on the physical lens parameters. As will be discussed in Sect. 5, the uncertainties of the physical lens parameters are large for all events due to the absence of the π_E constraint.

The 2L1S modeling was carried out in two steps. In the first step, the binary lensing parameters s and q were searched for using a grid approach with different starting values of α evenly distributed in the 0 – 2π range with 21 grids, while the five non-grid lensing parameters, that is, (t₀, u₀, t_E, α, ρ), were found using a downhill approach based on the Markov chain Monte Carlo (MCMC) algorithm. The ranges of the grid parameters are − 1.0 ≤ logs < 1.0 and − 4.0 ≤ logq < 1.0, and they were divided into 70 grids. We then identified local solutions that appear in the Δχ² map on the s–q parameter plane. In the second step, we refined the individual local solutions found in the first step by letting all parameters vary. Then, the global solution was found by comparing χ² values of the local solutions if there were more than one. Figure 1 shows the Δχ² maps on the log s– logq parameter plane obtained from the first-step modeling of the individual events. It shows that there exists a single local solution for all events.

Fig. 1 Δχ2 maps on the logs– logq parameter plane obtained from the first-round modeling of the individual events. Color coding is set to represent points with Δχ2 ≤ n(12) (red), Δχ2 ≤ n(22) (yellow), Δχ2 ≤ n(32) (green), Δχ2 ≤ n(42) (cyan), and Δχ2 ≤ n(52) (blue), where n = 5, 3, and 9 for KMT-2017-BLG-2509, OGLE-2017-BLG-1099, and OGLE-2019-BLG-0299, respectively.

3.1 KMT-2017-BLG-2509

Figure 2 shows the lensing light curve of KMT-2017-BLG-2509. It is characterized by two distinctive caustic features that appear at t₁ ~ 7858 and t₂ ~ 7870 in HJD′≡HJD − 2 450 000. For the first caustic feature, both the rising side (covered by four KMTA data points) and the falling side (covered by a single KMTC point) were captured, while only the rising side was covered (by five KMTC points) for the second feature (see the zoomed-in view of the anomaly region shown in Fig. 3). The region between the two caustic features exhibits negative deviations from a single-lens single-source (1L1S) light curve. Regarding the light curve, it should be noted that the error bars of the data are substantial due to the faintness of the source and that the anomaly region occupies a large portion of the magnified region of the light curve. This made it difficult for the anomalies to be readily noticed as a planetary (rather than a stellar-binary) signal¹.

The 2L1S modeling yields binary parameters of (s, q) ~ (0.93, 4.4 × 10⁻³), indicating that the companion to the primary lens is a planetary-mass object lying near the Einstein ring of the primary. We find a unique solution without any degeneracy. For a lensing event produced by a binary lens with a small mass ratio, there often exists a pair of close (s < 1.0) and wide (s > 1.0) solutions arising from the similarity between the central caustics induced by the companions with s and s⁻¹: the close–wide degeneracy (Griest & Safizadeh 1998; Dominik 1999). It is found that KMT-2017-BLG-2509 is not subject to this degeneracy because the light-curve morphology (two pairs of caustic crossings that flank a de-magnified region) is strictly characteristic of an s < 1 geometry (see Fig. 4). The normalized source radius, ρ ~ 1.9 × 10⁻³, is measured by analyzing the caustic-crossing parts of the light curve.

The model curve of the 2L1S solution is drawn over the data points in Figs. 2 and 3. In Table 4 we list the full lensing parameters of the model together with the flux parameters of the source, f_s, and blend, f_b, in which the flux is approximately scaled to that of I = 18, that is, f = 10^{−0.4(I−18)}. Figure 4 shows the configuration of the lens system corresponding to the solution. The planet is located close to the Einsteinring, and this results in a single, six-sided resonant caustic. The caustic appears as the merging of the single central caustic and the two sets of planetary caustics. The source crossed the tip of the lower planetary caustic at t₁ and then crossed the slim bridge connecting the upper planetary caustic and the central caustic at t₂. The negative deviation region between t₁ and t₂ occurred when the source passed through the negative deviation region formed between the two planetary caustics.

Fig. 2 Light curve of KMT-2017-BLG-2509. The dotted and solid curves plotted over the data points represent the 1L1S and 2L1S models, respectively. The residual of the 2L1S model is shown in the lower panel. The colors of the telescopes in the legend are set to match those of the data points.

Fig. 3 Zoomed-in view of the major anomaly region of KMT-2017-BLG-2509. Notations are the same as those in Fig. 2.

Fig. 4 Lens system configuration of KMT-2017-BLG-2509. The inset shows the whole view of the lens system, and the main panel shows the enlarged view around the caustic. The red closed figure is the caustic, the line with an arrow represents the source motion, the two filled blue dots indicate the positions of the lens components, and the dashed circle is the Einstein ring. The gray curves around the caustic represent equi-magnification contours. The empty magenta dots on the source trajectory represent the source positions at t1 and t2, marked in Fig. 2. The size of the dot is not scaled to the source size.

Fig. 5 Light curve of OGLE-2017-BLG-1099. Notations are the same as those in Fig. 2.

3.2 OGLE-2017-BLG-1099

The lensing light curve of OGLE-2017-BLG-1099 is shown in Fig. 5. The online data of the event posted on the alert web pages of the individual survey groups² did not show an obvious anomaly feature in the light curve. Nevertheless, the event was selected for a detailed analysis because it reached a very high magnification, A_max ~ 140, during which the chance of detecting a planet-induced anomaly was high (Griest & Safizadeh 1998). The optimal light curve obtained by re-reducing the data revealed that the light curve was anomalous. The anomaly is characterized first by the asymmetry of the light curve and second by the caustic-involved feature at t₂ ~ 7917.4. The zoomed-in view of the regionaround t₂ presented inthe right panel of Fig. 6 shows that the five KMTS data points exhibit the characteristic pattern of magnification variation that occurs when a source exits a caustic (see Fig. 1 of Gould & Andronov 1999).

The 2L1S modeling yields a unique solution with the binary parameters of (s, q) ~ (1.1, 6.4 × 10⁻³), indicating that the companion is a planetary-mass object with a separation similar to θ_E. We note that the analysis of the event was done in the 2017 season and that its planetary nature was realized by one of the coauthors (Y.-H. Ryu), but the result was not shared with the other coauthors. As a result, the analysis presented in this work was carried out independently, reaching a result that is consistent with the previous one. The full lensing parameters are listed in Table 4, and the model curve is presented in Figs. 5 and 6. Due to the proximity of the binary separation to unity, thebinary lens pair forms a single resonant caustic. According to the model, the source entered the caustic at t₁ ~ 7911 and exited the caustic at t₂. Due to the weakness of the caustic combined with the low photometric precision, it was difficult to notice the anomaly feature that occurred at around the caustic entrance in the preliminary modeling using the online data, but the re-reduced data showed that the caustic was covered by four KMTA data points, although the uncertainties of the data points around the caustic were still large (see the zoomed-in view around t₁ shown in the left panel of Fig. 6). The coverage of both the caustic entrance and exit yields a normalized source radius of ρ ~ 1.5 × 10⁻³.

Figure 7 shows the configuration of the lens system. As in the case of KMT-2017-BLG-2509, the caustic appears as the merging of the planetary and central caustics. The source moved approximately in parallel with the binary axis (α ~ 3.7°). It entered the caustic by passing the lower right side of the planetary caustic, generating a weak caustic spike at t₁. Then, the source exited the caustic by passing the left side of the central caustic and then passed the region close to the back-end cusp of the central caustic, and this produced the caustic feature at t₂ (see the zoomed-in view of the central region presented in the left inset).

Fig. 6 Zoomed-in view of the major anomaly region of OGLE-2017-BLG-1099.

Fig. 7 Lens system configuration of OGLE-2017-BLG-1099. Notations are the same as those in Fig. 4, except that an additional inset (left) is presented to show a zoomed-in view of the central region. The source positions corresponding to t1 and t2 designated in Fig. 5 are marked by the empty magenta dots. The size of the dot in the main panel is arbitrary, but the dot in the left inset is scaled to the source size.

3.3 OGLE-2019-BLG-0299

Figure 8 shows the light curve of OGLE-2019-BLG-0299. The anomalous nature of the light curve was known when the event was proceeding, and planetary-lensing models found with the use of the OGLE data were circulated to the microlensing community by C. Han and V. Bozza before the end of the event. The analysis in this work was done with the addition of the KMTNet data obtained from the optimized photometry of the event. The enlarged view of the light curve around the anomaly region is shown in Fig. 9. The anomaly exhibits a very complex pattern that is characterized by six peaks or bumps, at t₁ ~ 8556.2, t₂ ~ 8556.7, t₃ ~ 8559.4, t₄ ~ 8560.5, t₅ ~ 8562.5, and t₆ ~ 8563.5.

A 2L1S modeling yields a unique solution with the binary parameters of (s, q) ~ (0.99, 10.0 × 10⁻³). The binary separation is very close to unity, as in the cases of the two previous events. The mass ratio is about ten times the Jupiter/Sun ratio of the Solar System, but the mass of the companion is still in the planet-mass regime, considering that the measured event timescale, t_E ~ 30 days, is not much longer than those of typical lensing events produced by low-mass stars. The full lensing parameters are presented in Table 4. Althoughall the major anomaly features were delineated, none of the caustics was resolved densely enough for the secure measurement of ρ, and only the upper limit, ρ_max ~ 3.5 × 10⁻³, is constrained.

Figure 10 shows the lens system configuration of OGLE-2019-BLG-0299. According to the model, the source crossed the binary axis with a source trajectory angle of α ~ 73° and passed through the six-fold resonant caustic four times, at t₂, t₃, t₅, and t₆, where (t₂, t₃) and (t₅, t₆) are the two time pairs of caustic entrance and exit. These caustic crossings produced the spikes at the corresponding times, and the regions of the light curve between the individual caustic-crossing pairs exhibited characteristic U-shape trough patterns. The bump at t₁ was produced by the source approach close to the lower cusp of the caustic before the first caustic crossing at t₂. The other bump, at t₄, was produced when the source passed through the outer-caustic region near the left-side on-axis cusp of the caustic. The source positions corresponding to the six epochs of the anomaly are marked by empty magenta dots (not to scale) on the source trajectory.

Fig. 8 Lensing light curve of OGLE-2019-BLG-0299. Notations are the same as those in Fig. 2.

Fig. 9 Enlarged view around the anomaly region of the OGLE-2019-BLG-0299 light curve. The positions marked by arrows and labeled as t1, t2, t3, t4, t5, and t6 represent the epochs of major anomaly features.

Fig. 10 Lens system configuration of OGLE-2019-BLG-0299. The empty magenta points represent the source positions corresponding to the six epochs of t1, t2, t3, t4, t5, and t6 that are marked by arrows in Fig. 9. The size of the dots is not scaled.

4 Source stars and Einstein radii

In this section we estimate the angular Einstein radii of the events. The Einstein radius was determined by θ_E = θ_*∕ρ. The ρ value was measured orconstrained from the analysis of the caustic-crossing parts, which are affected by finite-source effects. With the measured ρ, it was then required to estimate the angular source radius, θ_*. The value ofθ_* was deduced from the color and magnitude of the source.

In general, the source color is estimated by measuring the source magnitudes in two passbands, the I and V bands in our case, which in turn is done by regressing the data of the individual passbands with the variation of the lensing magnification. The I-band magnitudes were securely measured by this method for all events. However, measuring the V -band magnitudes is difficult with this method due to the large uncertainties of the data, making it difficult to securely estimate the source colors. We, therefore, estimated the source color using the Bennett et al. (2008) method, which utilizes the CMD constructed from the Hubble Space Telescope (HST) observations (Holtzman et al. 1998). In the first step of this method, the HST CMD is aligned with the CMD obtained from ground-based observations using the well-defined centroid of the red giant clump (RGC). In the second step, the source position in the CMD is interpolatedfrom the branch of main-sequence or giant stars on the HST CMD using the well-measured I-band magnitude difference between the RGC centroid and the source. In the final step, the source color and its uncertainty are estimated as the mean and standard deviation of stars located on the branch.

Figure 11 shows the source locations (black filled dots with error bars) with respect to the RGC centroids (red filled dot) in the combined CMD. In Table 5 we list the positions of the source, (V − I, I), and the RGC centroid, ${(V-I,I)}_{\text{RGC}}$, measured on the instrumental CMD. The reddening and extinction corrected (de-reddened) color and magnitude of the source, ${(V-I,I)}_0$, were then determined using the offsets from the RGC centroid, Δ(V − I, I), together with the known de-reddened values of RGC, ${(V-I,I)}_{\text{RGC},0}$ (Bensby et al. 2013; Nataf et al. 2013), by the relation $${(V-I,I)}_0={(V-I,I)}_{\text{RGC},0}+\Delta (V-I,I).$$(1)

The values of ${(V-I,I)}_0$, Δ(V − I, I), ${(V-I,I)}_{\text{RGC},0}$ for the individual events are listed in Table 5. We note that the de-reddened I-band magnitudes of the RGC centroids, that is, I_RGC,0, vary depending on the event because we consider the varying distance depending on the source location using Table 1 of Nataf et al. (2013). The measured colors and magnitudes, which are in the ranges of $0.9\lesssim {(V-I)}_0\lesssim 1.1$ and 19.5 ≲ I₀ ≲ 20.2, respectively, indicate that the source stars of the events are of similar spectral types, early K-type main-sequence stars.

The measured (V − I) color was then converted into (V − K) color using the color-color relation of Bessell & Brett (1988), and then the source radius was deduced from the (V − K)–θ_* relation of Kervella et al. (2004). With the measured source radius, the Einstein radius and the relative lens-source proper motion were then estimated using the relations θ_E = θ_*∕ρ and μ = θ_E∕t_E, respectively. We list the estimated values of θ_*, θ_E, and μ in Table 5. We note that the lower limits θ_E and μ are presented for OGLE-2019-BLG-0299 because only the upper limit of ρ is constrained for the event.

Fig. 11 Source positions (filled black dot with error bars) in the instrumental CMD with respect to the centroids of the RGC (filled red dots) for the individual lensing events. In each panel, the ground-based and HST CMDs are marked by gray and brown dots, respectively.

5 Physical lens parameters

We estimated the physical lens parameters of the lens mass, M, and distance, D_L, by conducting a Bayesian analysis. The lensing observables that can be used to constrain M and D_L include t_E, θ_E, and π_E, where the first two observables are related to M and D_L by $$t_{\text{E}}=\frac{{\theta }_{\text{E}}}{\mu };{\theta }_{\text{E}}={(\kappa M{\pi }_{\text{rel}})}^{1/2};\text{}{\pi }_{\text{rel}}=\text{AU}\left(\frac{1}{D_{\text{L}}}-\frac{1}{D_{\text{S}}}\right),$$(2)

and the additional measurement of π_E would allow one to uniquely determine the lens parameters by $$M=\frac{{\theta }_{\text{E}}}{\kappa {\pi }_{\text{E}}};D_{\text{L}}=\frac{\text{AU}}{{\pi }_{\text{E}}{\theta }_{\text{E}}+{\pi }_{\text{S}}}.$$(3)

Here, κ = 4G∕(c²AU), π_S = AU∕D_S, and D_S denotes the distance to the source (Gould 2000). The available observables vary depending on the events: t_E and θ_E for KMT-2017-BLG-2509 and OGLE-2017-BLG-1099, and t_E and the lower limit of θ_E for OGLE-2019-BLG-0299. The value of π_E is not measured for any of the events.

In the first step of the Bayesian analysis, we produced artificial lensing events by conducting a Monte Carlo simulation. In the simulation, we used a prior Galactic model, which describes the physical and dynamical distributions and the mass function of Galactic objects. We adopted the Jung et al. (2021) Galactic model, in which the physical distribution of disk and bulge objects are described by the Robin et al. (2003) and Han & Gould (2003) models, respectively, the dynamical distributions of the disk and bulge objects are depicted by the Jung et al. (2021) and Han & Gould (1995) models, respectively, and the Jung et al. (2018) mass function model is commonly used for both populations. In the second step, we chose events with t_E and θ_E values consistent with the measured observables and constructed the posterior distributions of M and D_L for these events. Then, the representative values of the lens parameters were determined as the median values of the posterior distributions, and the lower and upper limits were determined as 16 and 84% of the distributions, respectively.

In Figs. 12 and 13 we present the Bayesian posterior distributions of M₁ and D_L, respectively.In Table 6 we list the estimated masses of the host (M_host ≡ M₁) and planet (M_planet ≡ M₂), the distance, and the projected planet-host separation (a_⊥). The projected separation was calculated by a_⊥ = sD_Lθ_E. It is estimated that the host masses are in the range of 0.45 ≲ M_host∕M_⊙≲ 0.59, which corresponds to early M to late K dwarfs, and thus the host stars are less massive than the Sun. On the other hand, the planet masses, which are in the range of 2.1 ≲ M_planet∕M_J ≲ 6.2, are heavier than the mass of the heaviest planet of the Solar System, that is, Jupiter. Considering that the snow line distance is a_sl ~ 2.7(M∕M_⊙) AU and that a_⊥ is the separation in projection, the planets in all systems lie beyond the snow lines of the hosts. Therefore, the discovered planetary systems, together with many other microlensing planetary systems, support the idea that massive gas-giant planets are commonplace around low-mass stars (see the distribution of exoplanets with respect to a_⊥ ∕a_sl presented in Fig. 6 of Gaudi 2012). The disk and bulge contributions are, respectively, 28 and 72% for KMT-2017-BLG-2509, 33 and 67% for OGLE-2017-BLG-1099, and 48 and 52% for OGLE-2019-BLG-0299. The relatively low disk contributionfor KMT-2017-BLG-2509 arises due to the constraint of the low relative lens-source proper motion, μ ~ 1.3 mas yr⁻¹, because low proper motion is difficult to produce for disk lenses.

Fig. 12 Bayesian posteriors of the primary lens mass, M1, for KMT-2017-BLG-2509 (top panel), OGLE-2017-BLG-1099 (middle panel), and OGLE-2019-BLG-0299 (bottom panel). In each panel, the blue and red curves represent the contributions by the disk and bulge lens populations, respectively, and the black curve represents the sum of the two lens populations. The solid vertical line represents the median, and the two dotted lines indicate the 1σ range of the distribution.

Fig. 13 Bayesian posteriors of the lens distance, DL. Notations are the same as those in Fig. 12.

6 High-resolution follow-up observation

In the general case of a lensing event, high-resolution observations using space-borne telescopes or ground-based adaptive optics instruments allow one to measure the lens flux and the lens-source separation. The lens flux allows one to estimate the lens mass, M, and the lens-source separation allows one to estimate the relative lens-source proper motion, μ, which in turn constrains the Einstein radius by θ_E = μt_E.

For faint-source planetary lensing events with resonant caustic features, high-resolution follow-up observations are especially important for clarifying the planetary lens systems. The size of a resonant planetary caustic scales to the planet/host mass ratio as Δζ_c ∝ q^1∕3θ_E, and thus the duration of the planetary anomaly is related to the event timescale by Δt = Δζ_c∕μ ∝ q^1∕3t_E. For a given anomaly duration, then, the event timescale and the mass ratio are related by $q\propto t_{\text{E}}^{-3}$. For faint-source events, the very faint sources can make it difficult to precisely determine t_E, and this leads to a large uncertainty of q because Δq ∝ 3Δt_E. Late-time observations in two passbands can yield the source flux and color and, thus, the I-band source-flux estimate. The well-estimated source flux can further constrain t_E, and this leads toa tight constraint on the planet mass ratio. High-resolution image data could have resulted in best constraints if they had been acquired at the time of the event to provide a comparison. Unfortunately, no such images were taken because the planetary nature of the events was not known at the times of the event discoveries. However, post-event imaging can still help to constrain the physical parameters of the lens systems.

7 Summary and conclusion

We have reported three microlensing planets, KMT-2017-BLG-2509Lb, OGLE-2017-BLG-1099Lb, and OGLE-2019-BLG-0299Lb, that were found from the reinvestigation of the microlensing data collected by the KMTNet and OGLE surveys during the 2017–2019 seasons. For all of these lensing events, the planetary signals deviated from the typical form of short-term anomalies because they were produced by the crossings of the source stars over the resonant caustics formed by the giant planets located around the Einstein rings of host stars. The faintness of the source stars and the relatively low observational cadences also contributed to the difficulty in finding the planetary signals. Due to the resonant nature of the caustics, the lensing solutions were uniquely determined without any degeneracy. The estimated masses of the planet hosts are in the range of 0.45 ≲ M∕M_⊙≲ 0.59, which corresponds to early M to late K dwarfs, and thus the host stars are less massive than the Sun. On the other hand, the planets, with masses in the range of 2.1 ≲ M_planet∕M_J ≲ 6.2, are heavier than Jupiter. The planets in all systems lie beyond the snow lines of the hosts, and thus the discovered planetary systems support the conclusion that massive gas-giant planets are commonplace around low-mass stars.

Acknowledgements

Work by C.H. was supported by the grants of National Research Foundation of Korea (2019R1A2C2085965 and 2020R1A4A2002885). This work was conducted during the research year of Chungbuk National University in 2021. This research has made use of the KMTNet system operated by the Korea Astronomy and Space Science Institute (KASI) and the data were obtained at three host sites of CTIO in Chile, SAAO in South Africa, and SSO in Australia. The OGLE project has received funding from the National Science Centre, Poland, grant MAESTRO 2014/14/A/ST9/00121 to AU.

References

All Tables

All Figures

	Fig. 1 Δχ2 maps on the logs– logq parameter plane obtained from the first-round modeling of the individual events. Color coding is set to represent points with Δχ2 ≤ n(12) (red), Δχ2 ≤ n(22) (yellow), Δχ2 ≤ n(32) (green), Δχ2 ≤ n(42) (cyan), and Δχ2 ≤ n(52) (blue), where n = 5, 3, and 9 for KMT-2017-BLG-2509, OGLE-2017-BLG-1099, and OGLE-2019-BLG-0299, respectively.
In the text

	Fig. 2 Light curve of KMT-2017-BLG-2509. The dotted and solid curves plotted over the data points represent the 1L1S and 2L1S models, respectively. The residual of the 2L1S model is shown in the lower panel. The colors of the telescopes in the legend are set to match those of the data points.
In the text

	Fig. 3 Zoomed-in view of the major anomaly region of KMT-2017-BLG-2509. Notations are the same as those in Fig. 2.
In the text

Fig. 4 Lens system configuration of KMT-2017-BLG-2509. The inset shows the whole view of the lens system, and the main panel shows the enlarged view around the caustic. The red closed figure is the caustic, the line with an arrow represents the source motion, the two filled blue dots indicate the positions of the lens components, and the dashed circle is the Einstein ring. The gray curves around the caustic represent equi-magnification contours. The empty magenta dots on the source trajectory represent the source positions at t1 and t2, marked in Fig. 2. The size of the dot is not scaled to the source size.

In the text

	Fig. 5 Light curve of OGLE-2017-BLG-1099. Notations are the same as those in Fig. 2.
In the text

	Fig. 6 Zoomed-in view of the major anomaly region of OGLE-2017-BLG-1099.
In the text

Fig. 7 Lens system configuration of OGLE-2017-BLG-1099. Notations are the same as those in Fig. 4, except that an additional inset (left) is presented to show a zoomed-in view of the central region. The source positions corresponding to t1 and t2 designated in Fig. 5 are marked by the empty magenta dots. The size of the dot in the main panel is arbitrary, but the dot in the left inset is scaled to the source size.

In the text

	Fig. 8 Lensing light curve of OGLE-2019-BLG-0299. Notations are the same as those in Fig. 2.
In the text

	Fig. 9 Enlarged view around the anomaly region of the OGLE-2019-BLG-0299 light curve. The positions marked by arrows and labeled as t1, t2, t3, t4, t5, and t6 represent the epochs of major anomaly features.
In the text

	Fig. 10 Lens system configuration of OGLE-2019-BLG-0299. The empty magenta points represent the source positions corresponding to the six epochs of t1, t2, t3, t4, t5, and t6 that are marked by arrows in Fig. 9. The size of the dots is not scaled.
In the text

	Fig. 11 Source positions (filled black dot with error bars) in the instrumental CMD with respect to the centroids of the RGC (filled red dots) for the individual lensing events. In each panel, the ground-based and HST CMDs are marked by gray and brown dots, respectively.
In the text

Fig. 12 Bayesian posteriors of the primary lens mass, M1, for KMT-2017-BLG-2509 (top panel), OGLE-2017-BLG-1099 (middle panel), and OGLE-2019-BLG-0299 (bottom panel). In each panel, the blue and red curves represent the contributions by the disk and bulge lens populations, respectively, and the black curve represents the sum of the two lens populations. The solid vertical line represents the median, and the two dotted lines indicate the 1σ range of the distribution.

In the text

	Fig. 13 Bayesian posteriors of the lens distance, DL. Notations are the same as those in Fig. 12.
In the text