1 Research in Astronomy and Astrophysics manuscript no. (L A TEX: tex; printed on June 7, 2021; 16:57) An Earth-mass Planet in a Time of Covid-19: KMT-2020-BLG-0414Lb Wei-Cheng Zang 1,34,35,37, Cheongho Han 2,34, Iona Kondo 3,36, Jennifer C. Yee 4,34,35, Chung-Uk Lee 5,6,34, Andrew Gould 7,8,34,35, Shude Mao 1,9,35,37, Leandro de Almeida 10,35, Yossi Shvartzvald 11,34,35, Xiang-Yu Zhang 1, Michael D. Albrow 12,34, Sun-Ju Chung 5,6,34, Kyu-Ha Hwang 5,34, Youn Kil Jung 5,34, Yoon-Hyun Ryu 5,34, In-Gu Shin 5,34, Sang-Mok Cha 5,13,34, Dong-Jin Kim 5,34, Hyoun-Woo Kim 5,34, Seung-Lee Kim 5,6,34, Dong-Joo Lee 5,34, Yongseok Lee 5,13,34, Byeong-Gon Park 5,6,34, Richard W. Pogge 8,34, John Drummond 14,35, Thiam-Guan Tan 15,35, José Dias do Nascimento Júnior 10,4,35, Dan Maoz 16,35, Matthew T. Penny 17,35,37, Wei Zhu 1,18,35,37, Ian A. Bond 19,36, Fumio Abe 20,36, Richard Barry 21,36, David P. Bennett 21,22,36, Aparna Bhattacharya 21,22,36, Martin Donachie 23,36, Hirosane Fujii 3,36, Akihiko Fukui 24,25,36, Yuki Hirao 3,36, Yoshitaka Itow 20,36, Rintaro Kirikawa 3,36, Naoki Koshimoto 26,27,36, Man Cheung Alex Li 23,36, Yutaka Matsubara 20,36, Yasushi Muraki 20,36, Shota Miyazaki 3,36, Greg Olmschenk 28,36, Clément Ranc 21,36, Nicholas J. Rattenbury 23,36, Yuki Satoh 3,36, Hikaru Shoji 3,36, Stela Ishitani Silva 29,28,36, Takahiro Sumi 3,36, Daisuke Suzuki 28,36, Yuzuru Tanaka 3,36, Paul J. Tristram 29,36, Tsubasa Yamawaki 3,36, Atsunori Yonehara 30,36, Andreea Petric 31,32,37, Todd Burdullis 31,37 and Pascal Fouqué 31,33,37 1 Department of Astronomy, Tsinghua University, Beijing , China; 2 Department of Physics, Chungbuk National University, Cheongju 28644, Republic of Korea; 3 Department of Earth and Space Science, Graduate School of Science, Osaka University, Toyonaka, Osaka , Japan 4 Center for Astrophysics Harvard & Smithsonian, 60 Garden St.,Cambridge, MA 02138, USA 5 Korea Astronomy and Space Science Institute, Daejon 34055, Republic of Korea 6 University of Science and Technology, Korea, (UST), 217 Gajeong-ro Yuseong-gu, Daejeon 34113, Republic of Korea 7 Max-Planck-Institute for Astronomy, Königstuhl 17, Heidelberg, Germany 8 Department of Astronomy, Ohio State University, 140 W. 18th Ave., Columbus, OH 43210, USA 9 National Astronomical Observatories, Chinese Academy of Sciences, Beijing , China 10 Universidade Federal do Rio Grande do Norte (UFRN), Departamento de F 1sica, , Natal, RN, Brazil 11 Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot 76100, Israel
2 2 12 University of Canterbury, Department of Physics and Astronomy, Private Bag 4800, Christchurch 8020, New Zealand 13 School of Space Research, Kyung Hee University, Yongin, Kyeonggi 17104, Republic of Korea 14 Possum Observatory, Patutahi, Gisbourne, New Zealand 15 Perth Exoplanet Survey Telescope, Perth, Australia 16 School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv , Israel 17 Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA USA 18 Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 St George Street, Toronto, ON M5S 3H8, Canada 19 Institute of Natural and Mathematical Sciences, Massey University, Auckland 0745, New Zealand 20 Institute for Space-Earth Environmental Research, Nagoya University, Nagoya , Japan 21 Code 667, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA 22 Department of Astronomy, University of Maryland, College Park, MD 20742, USA 23 Department of Physics, University of Auckland, Private Bag 92019, Auckland, New Zealand 24 Department of Earth and Planetary Science, Graduate School of Science, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo , Japan 25 Instituto de Astrofísica de Canarias, Vía Láctea s/n, E La Laguna, Tenerife, Spain 26 Department of Astronomy, Graduate School of Science, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo , Japan 27 National Astronomical Observatory of Japan, Osawa, Mitaka, Tokyo , Japan 28 Code 667, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA 29 Department of Physics, The Catholic University of America, Washington, DC 20064, USA 30 Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, Yoshinodai, Chuo, Sagamihara, Kanagawa, , Japan 31 University of Canterbury Mt. John Observatory, P.O. Box 56, Lake Tekapo 8770, New Zealand 32 Department of Physics, Faculty of Science, Kyoto Sangyo University, Kyoto, Japan 33 CFHT Corporation, Mamalahoa Hwy, Kamuela, Hawaii 96743, USA 34 Space Telescope Science Institute, Baltimore, MD Université de Toulouse, UPS-OMP, IRAP, Toulouse, France 36 The KMTNet Collaboration 37 The LCOGT & µfun Follow-up Teams 38 The MOA Collaboration 39 CFHT Microlensing Collaboration Received 2021 March 5 ; accepted 2021 June 4 Abstract We report the discovery of KMT-2020-BLG-0414Lb, with a planet-to-host mass ratio q 2 = = 3 4 q at 1σ, which is the lowest mass-ratio microlensing planet to date. Together with two other recent discoveries (4 q/q 6), it fills out the previous empty sector at the bottom of the triangular (log s, log q) diagram, where s is the
3 KMT-2020-BLG planet-host separation in units of the angular Einstein radius θ E. Hence, these discoveries call into question the existence, or at least the strength, of the break in the mass-ratio function that was previously suggested to account for the paucity of very low-q planets. Due to the extreme magnification of the event, A max 1450 for the underlying single-lens event, its light curve revealed a second companion with q and log s 3 1, i.e., a factor 10 closer to or farther from the host in projection. The measurements of the microlens parallax π E and the angular Einstein radius θ E allow estimates of the host, planet, and second companion masses, (M 1, M 2, M 3 ) (0.3M, 1.0M, 17M J ), the planet and second companion projected separations, (a,2, a,3 ) (1.5, 0.15 or 15) au, and system distance D L 1 kpc. The lens could account for most or all of the blended light (I 19.3) and so can be studied immediately with high-resolution photometric and spectroscopic observations that can further clarify the nature of the system. The planet was found as part of a new program of high-cadence follow-up observations of high-magnification events. The detection of this planet, despite the considerable difficulties imposed by Covid-19 (two KMT sites and OGLE were shut down), illustrates the potential utility of this program. Key words: gravitational lensing: micro 1 INTRODUCTION It has long been known that intensive monitoring of high-magnification microlensing events is sensitive to planets of one-to-few Earth/Sun mass ratio, q = , planets. Dong et al. (2006) showed that the A max 3000 event OGLE-2004-BLG-343 would have had such sensitivity had it been observed over peak (see their Figure 9). Yee et al. (2009) showed such sensitivity for the actual data covering the peak of the A max 1600 event OGLE-2008-BLG-279 (see their Figure 7). In both cases (and, indeed, for high magnification events in general, Gaudi et al. 2002; Gould et al. 2010), the sensitivity diagrams have a triangular appearance that is symmetric in log s about the origin. That is, the contour limits meet at s = 1, where s is the planet-host separation in units of the Einstein radius θ E. Hence, the limiting sensitivity in q is via a so-called resonant caustic. For s 1, the caustic structure consists of a small quadrilateral caustic near the host and a larger quadrilateral caustic near the planet. For s 1, it consists of a similar small quadrilateral caustic near the host and two triangular caustics located on the opposite side of the planet. For s 1, these two sets of caustics merge into a relatively large 6-sided resonant caustic, which is what makes the detection feasible at the very limit of sensitivity 1. Neither of the above two events yielded any planet detection, but Udalski et al. (2018) did find a resonant-caustic planet in OGLE-2017-BLG-1434 (A max 23 for the underlying single-lens event). While its mass ratio was q (i.e., q 19 q ), Udalski et al. (2018) showed that a planet with exactly the same characteristics, but 30 times less massive, would have been detected (see their Figure 4). 1 In fact, Yee et al. (2021) showed that semi-resonant caustics, which have not quite merged but still have long magnification ridges extending from the central caustic, as well as exceptionally large planetary caustics, are just as sensitive as resonant caustics.
4 4 Zang et al. Nevertheless, in spite of the recognized theoretical possibility of such few-q planet detections, no planet with mass ratio below that of Uranus/Sun mass ratio q 15 q was actually detected prior to This failure gave rise to several different suggestions of a paucity of low mass-ratio planets. Suzuki et al. (2016) argued for a break in the mass-ratio function at q br 57 q based on a statistically welldefined sample of planets detected from MOA-discovered microlensing events. Jung et al. (2019) argued for a break, or possibly a pile-up at q 19 q based on the ensemble of planets with q < Udalski et al. (2018) used a new V/V max method to show that if planets with q < 10 4 were modeled as a power-law distribution in q, then the distribution was rising toward higher q, seemingly confirming the Suzuki et al. (2016) break. However, in 2018 and 2019, three planets were discovered with q below the previous record, and hence below the level of the conjectured break or pile-up : KMT-2018-BLG-0029Lb (q , Gould et al. 2020), KMT-2019-BLG-0842Lb (q , Jung et al. 2020), and OGLE-2019-BLG-0960Lb (q , Yee et al. 2021). All three were detected via resonant caustics at or near the peak of moderately high magnification events, with s 1 (0.000, 0.017, or 0.029). Moreover, Yee et al. (2021) showed that the recent discoveries of KMT-2018-BLG-0029Lb and OGLE-2019-BLG-0960Lb populate the previously-vacant lower region of the (log s, log q) diagram. See their Figure 11. It is therefore clear that the well-established sensitivity to q q depends primarily on relatively rare (i.e., high-magnification) microlensing events generated by relatively rare resonant lens configurations. And, therefore, it is possible that the previous paucity of planets near the detection limit was more a product of the difficulty of detection than the intrinsic rarity of the population. In this context, it is notable that over the last 10 years, microlensing planet searches have moved away from their previous focus on high-magnification events, which is one of the two rare elements just described for probing the low-q population. Prior to the inauguration of the wide-field-camera OGLE-IV survey (Udalski et al. 2015), substantial effort was expended, particularly by the Microlensing Follow Up Network (µfun), to find high-magnification events, and then to focus intensive observations over the peak (Gould et al. 2010). As a result, the projected spatial distribution of planetary events was drawn roughly uniformly from the OGLE-III and MOA-II surveys. See the blue circles in Figure 8 of Ryu et al. (2020a). That is, the surveys were able to detect events, more or less regardless of cadence, but were much less able to detect planets on their own, again regardless of cadence. Hence, planet-yielding events were unaffected by survey cadence. However, with the layered approach (higher cadence in more productive fields) made possible by the introduction of the larger-format OGLE-IV camera, planet searches came to rely more on survey data, so that planetary discoveries became more concentrated on high-cadence regions. See green and yellow points of the same diagram. This remained so with the advent of the still larger-format KMTNet (Kim et al. 2016) survey (magenta and black points). Beginning in 2016, KMTNet continuously monitors 97 deg 2 area of sky toward the Galactic bulge field from three 1.6m telescopes equipped with 4 deg 2 FOV cameras at CTIO in Chile, SAAO in South Africa, and SSO in Australia. In fact, KMTNet s three-observatory system is capable of detecting very low-q planets in its highest cadence, Γ = 4 hr 1 fields, as was proved by the case of KMT-2019-BLG- 2 Although OGLE-2017-BLG-0173 has a best-fit solution of q 8 q, its two degenerate solutions have q 21 q at χ 2 = 3.5 (Hwang et al. 2018).
5 KMT-2020-BLG Lb. The substantially lower-q planet KMT-2018-BLG-0029Lb was discovered in KMT-only observations of a Γ = 1 hr 1 field. For the lowest-q planet OGLE-2019-BLG-0960Lb, although the planetary signal was first recognized by the µfun CT13 data and was most clearly delineated by the µfun Kumeu data, the detection would probably have been regarded as secure using the low-cadence survey observations (Γ KMT 0.4 hr 1, 3 sites; Γ OGLE = 0.17 hr 1, 1 site; Γ MOA = 0.6 hr 1, 1 site). An additional notable feature of these detections is that the magnification of the underlying 1L1S event at the time of the planetary anomaly was modest (A anom 37 for KMT-2018-BLG-0029Lb, A anom 22 for KMT-2019-BLG-0842Lb and A anom 45 for OGLE-2019-BLG-0960Lb), although A max 160 for the latter two cases. The source trajectory was oblique, i.e., close to parallel to the long axis of the resonant caustic, with α = 8.3 and α = 15.5 for KMT-2019-BLG-0842Lb and OGLE-2019-BLG-0960Lb, respectively. While such oblique trajectories are relatively rare, they can enhance detection efficiency by stretching out anomalies. Moreover, Yee et al. (2021) found that sensitivity to very low-q planets can be maximized by intensively monitoring events whenever they were magnified by a factor A > 10 (see also Abe et al. 2013). In order to probe the very low-q planets, KMTNet together with the LCOGT & µfun Follow-up Team developed a program for focusing on observations and analysis of A 20 events located in KMTNet Γ 1 hr 1 fields. However, with the advent of Covid-19, two of KMT s three observatories were shut down, leaving only KMT s Australia telescope as operational. Hence, the conditions became much more similar to those of the first decade of this century, when microlensing alerts came primarily from the OGLE-III survey, and planets were primarily discovered by follow-up observations of these single-site alerts, as well as some non-overlapping alerts from MOA. Of course, there were some differences. In particular, KMT has a much larger format camera than OGLE-III, and so it operates at substantially higher cadence. However, KMTA also has the worst conditions of KMT s three observatories, and so is inferior to OGLE-III in both weather interruptions and photometric precision. It was the specific orientation of this program, i.e., follow-up observations of A 20 events that led to the discovery of KMT-2020-BLG-0414Lb, the lowest mass-ratio microlensing planet discovered to date, q Due to the extreme magnification of the event, A max 1450 for the underlying single-lens event, and high-cadence observations by MOA over the peak, a second companion was also detected, with q OBSERVATIONS KMT-2020-BLG-0414 occurred at equatorial coordinates (α, δ) J2000 = (18:07:39.60, 28:29:06.8), corresponding to Galactic coordinates (l, b) = (2.82, 3.95). It was announced as a probable microlensing event by the KMTNet Alert-Finder system (Kim et al. 2018) on 1 June 2020, about 40 days before peak, when the event was manifested as an I 18.7 difference star. As mentioned in Section 1, by this date, KMTC and KMTS had been closed down for more than two months due to Covid-19. Hence, only KMTA data contributed to the alert and to subsequent monitoring of the event. The event lies in the KMNTet
6 6 Zang et al. BLG32 field, which has a cadence of Γ = 0.4 hr 1, with every tenth I-band observation complemented by one in the V band for the source color measurements 3. The event was independently identified by the Microlensing Observations in Astrophysics (MOA, Sumi et al. 2016) collaboration as MOA-2020-BLG-109 on 20 June 2020 (Bond et al. 2001). MOA observations are carried out with a 1.8m telescope at the Mt. John University Observatory in New Zealand, which is equipped with a 2.2 square degree camera. The nominal cadence for this field was Γ = 1.2 hr 1 by a MOA- Red filter (which is similar to the sum of the standard Cousins R- and I-band filters), and observations with the MOA V filter (Bessell V-band) were taken once every clear night. Earlier in the season, the MOA survey had also been closed for Covid-19 for almost 100 days, but it had re-opened 28 days before peak. Hence, the Covid-19 hiatus had very little effect on its observations of this event. Due to the poor seeing at KMTA and the rough real-time difference image analysis (DIA) photometry, the KMTNet Alert-Finder system found that a catalog star at (α, δ) J2000 = (18:07:39.60, 28:29:05.50) had the strongest signal and identified it as the preliminary source position of KMT-2020-BLG This catalog star is about 1.3 away from the true position of the source. As a result, the real-time on-line photometry was relatively noisy. Nevertheless, at UT 17:34 on 7 July 2020 (HJD = , HJD = HJD ), the LCOGT & µfun Follow-up Team found that this event had magnification A now > 10 based on the two KMTA points at HJD 9038 and could peak at a high magnification 2 3 days later 4. Thus, high-cadence follow-up observations were immediately scheduled by Las Cumbres Observatory global network (LCOGT) and Observatorio do Pico dos Dias (OPD) in Brazil (a µfun site). The LCOGT conducted observations from its 1.0m telescopes located at SAAO (LCOS), SSO (LCOA) and McDonald (LCOM), with the SDSS-i filter. Observations by OPD were taken from its 0.6m (OPD06) and 1.6m (OPD16) telescopes with the I filter. At UT 05:58 on 10 July 2020 (HJD = ), the LCOGT & µfun Follow-up Team identified that this event was currently undergoing an anomaly and would peak at a very high magnification soon, based on the real-time LCOGT and MOA data. Noting that SSO was predicted to be rainy that night, the Team issued an alert to the MOA collaboration. MOA responded to the alert and densely observed this event 60 times over the peak. Due to the very high brightness, the MOA observer decreased the exposure time from 60s to 5.2s over the peak. We carefully inspected these 60 MOA images and excluded 19 data points from the analysis due to saturation or bad seeing. At UT 13:53 on 10 July 2020 (HJD = ), the LCOGT & µfun Follow-up Team also issued an alert to all µfun observers. As a result, the 0.3m Perth Exoplanet Survey Telescope (PEST) in Australia and the 0.4m Possum Observatory (Possum) at New Zealand responded to the alert and took intensive observations without a filter. Finally, the event was also observed by the 3.6m Canada-France-Hawaii Telescope (CFHT) with the SDSS-i filter. For the light curve analysis, the KMTNet, MOA, CFHT and LCOGT data were reduced using custom implementations of the DIA technique (Tomaney & Crotts 1996; Alard & Lupton 1998): pysis (Albrow 3 In fact, this V -band to I-band ratio applies only to the normal cycle of KMT observations. During the latter part of the season (including the peak of KMT-2020-BLG-0414), these normal-cycle observations were supplemented by an end-of-night sequence of Eastern fields, which was purely in the I-band. For the low-cadence field BLG32, these end-of-night observations accounted for 30% of the total near the peak of the event. 4 The LCOGT & µfun Follow-up Team recognized the KMTA photometric centroid shift by its noisy curve and thus started follow-up observations for security, although A now did not meet the A 20 threshold. In fact, the actual magnification at that time was about 35.
7 KMT-2020-BLG Table 1: Data used in the analysis with corresponding data reduction method and rescaling factors Collaboration Site Filter Coverage (HJD ) N data Reduction Method (k, e min) for 2L1S (k, e min) for 3L1S KMTNet SSO I pysis 1 (1.23, 0.010) (1.11, 0.020) KMTNet SSO V pydia 2 MOA Red Bond et al. (2001) (1.74, 0.006) (1.35, 0.020) MOA V Bond et al. (2001) LCOGT SSO i ISIS 2 (1.08, 0.002) (0.35, 0.010) LCOGT SAAO i ISIS (1.31, 0.001) (1.05, 0.005) LCOGT McDonald i ISIS (1.03, 0.003) (1.10, 0.005) CFHT i ISIS (3.25, 0.000) (1.10, 0.020) µfun OPD06 I DoPHOT 3 (1.15, 0.002) (1.33, 0.000) µfun OPD16 I DoPHOT µfun PEST unfiltered DoPHOT (0.82, 0.016) (0.71, 0.020) µfun Possum unfiltered DoPHOT (0.81, 0.027) ( ) HJD = HJD MOA V -band data are only used to determine the source color. KMT SSO V -band data are not used due to poor seeing. OPD16 data are not used for the analysis due to bleeding from a bright star. For the MOA data, the 41 points on the peak ( < HJD < ) are exlucded in the 2L1S analysis. 1 Albrow et al. (2009) 2 MichaelDAlbrow/pyDIA: Initial Release on Github, doi: /zenodo Alard & Lupton (1998); Alard (2000); Zang et al. (2018) 4 Schechter et al. (1993) et al. 2009) for the KMTNet data, Bond et al. (2001) for the MOA data and ISIS (Alard & Lupton 1998; Alard 2000; Zang et al. 2018) for the CFHT and LCOGT data. The OPD, PEST and Possum data were reduced using DoPHOT (Schechter et al. 1993). On the OPD16 images, the target was affected by a bleed trail from a saturated star, resulting in some systematics. We therefore do not include OPD16 data in the analysis. The I-band magnitude of the KMTA light curve has been calibrated to the standard I-band magnitude. For the source color measurements, we use the MOA V -band data, while KMTA V -band data are not used due to poor seeing. The errors from photometric measurements for each data set i were renormalized using the formula σ i = k i σi 2 + e2 i,min, where σ i and σ i are original errors from the photometry pipelines and renormalized error bars in magnitudes, and k i and e i,min are rescaling factors. We obtained the rescaling factors using the procedure of Yee et al. (2012), which enables χ 2 /dof for each data set to become unity. We derived the rescaling factors using the binary lens (2L1S) and triple lens (3L1S) models, respectively, in order to understand how the event would have been interpreted in the absence of MOA data on the peak. The data used in the analysis, together with corresponding data reduction method and rescaling factors are summarized in Table L1S ANALYSIS Figure 1 shows the observed light curve of KMT-2020-BLG Although the light curve can be regarded as single peak in the sense that it monotonically rises and then falls, its significantly asymmetric shape cannot be fitted by a single-lens single-source (1L1S) model. A 1L1S model is usually described by three Paczyński (1986) parameters (t 0, u 0, t E ), i.e., the time of closest lens-source approach, the impact
8 8 Zang et al. parameter scaled to θ E, and the Einstein crossing time, t E = θ E µ rel ; θ E = κm L π rel ; κ 4G c 2 au 8.144mas, (1) M where M L is the mass of the lens and (π rel, µ rel ) are the lens-source relative (parallax, proper motion). In the present case, we also consider finite-source effects (Gould 1994; Witt & Mao 1994; Nemiroff & Wickramasinghe 1994), which occur when the source passes close to singular structures in the magnification pattern. This requires a fourth parameter ρ = θ /θ E, where θ is the angular radius of the source. As we will show in Section 4, the full light curve cannot be explained by a 2L1S model, and in fact requires 3L1S. However, if we exclude the MOA data on the peak ( < HJD < , the only data set to cover the peak), then the remaining data are quite well fit by a 2L1S model. We therefore begin by analyzing this restricted (non-moa-peak) data set. We are motivated by two considerations. First, and most importantly, 3L1S models often factor into two 2L1S models (Han 2005; Gaudi et al. 2008; Gould et al. 2014; Han et al. 2013, 2017, 2019a). In particular, the 3L1S caustic is often very nearly the superposition of the two 2L1S caustics. In such cases, one can often exclude the data from the neighborhood of the anomaly from the second 2L1S model in order to accurately determine the parameters of the first 2L1S model. In these cases, the first 2L1S model provides a powerful basis for finding the full 3L1S model, by one of several techniques. This proves to be the case for KMT-2020-BLG Second, it is of independent scientific interest to understand how the event would have been interpreted and check the so-called higherorder effects in the absence of MOA data on the peak. For example, MOA could have been weathered out on the night of the peak (as was KMTA). The analysis would have led to a report of a single low-massratio planet and a lens that is much brighter than the blended light. The comparison of this reconstructed report with the full model can inform our understanding of other 2L1S events with incomplete light-curve coverage. Therefore, for the remainder of this section, we will exclude the MOA data on the peak. These data will then be incorporated in Section Static Binary Lens Model The 2L1S model requires three additional parameters (s, q, α), which are respectively the separation of the two lens bodies scaled to θ E, the mass ratio between these bodies, and the angle of the source trajectory relative to the binary axis. For modeling, we use the advanced contour integration code (Bozza 2010; Bozza et al. 2018), VBBinaryLensing 5. We initially carry out a sparse grid searches for the parameters (log s, log q, α). The grid consists of 21 values equally spaced between 1.0 log s 1.0, 20 values equally spaced between 0 α < 360, and 61 values equally spaced between 6.0 log q 0.0. For each set of (log s, log q, α), we find the minimum χ 2 by Markov chain Monte Carlo (MCMC) χ 2 minimization using the emcee ensemble sampler (Foreman-Mackey et al. 2013), with fixed log q, log s and free t 0, u 0, t E, ρ, α. We identify one local minimum at (log s, log q) (0.0, 5.1), similar to the case of Yee et al. (2021). We thus conduct a similar dense grid search as Yee et al. (2021) that consists of 51 values equally spaced between 0.02 log s 0.03 and 41 values equally spaced between 6.0 log q 4.0. Often, such a grid search yields two local minima at s > 1 and s < 1 (e.g., Jung 5
9 KMT-2020-BLG I-Mag KMTA MOA OPD06 PEST Possum LCOA LCOS LCOM CFHT Residuals I-Mag KMTA MOA OPD06 PEST Possum LCOA LCOS LCOM CFHT 1L1S 2L1S 3L1S Residuals L1S 2L1S 3L1S I-Mag Residuals HJD Fig. 1: Light curve of KMT-2020-BLG-0414 with lensing models. The circles with different colors are the observed data points for different data sets. The black solid line is the best-fit 3L1S model using all the data, the cyan solid line is the best-fit 2L1S model excluding the MOA data on the peak, and the black dashed line is the 1L1S model derived using the same (t 0, u 0, t E, ρ) as the best-fit 3L1S model. The middle and bottom panels show a close-up of the main perturbations from the q 10 5 planet and the third body (q 0.05), respectively. et al. 2020; Yee et al. 2021), which must then be individually further explored and compared. However, in the present case, there is only one local minimum at s < 1, while the s > 1 model is disfavored by χ 2 > 900. We refine this minimum by allowing all parameters to vary. The parameters with their 68%
10 10 Zang et al. uncertainty range from the MCMC are shown in Table 2, and the fit and residuals are shown in Figure 1. The very low mass ratio q 10 5 indicates that the companion is a very-low-mass planet. Table 2: Parameters for 2L1S Model Parameters Static (non-parallax) Parallax Parallax + Orbital Motion u 0 > 0 u 0 < 0 u 0 > 0 u 0 < 0 χ 2 /dof 948.6/ / / / /900 t 0 (HJD ) u 0(10 3 ) t E (days) s q (10 5 ) α (rad) ρ (10 4 ) π E,N π E,E ds/dt(yr 1 ) dα/dt(yr 1 ) I S HJD = HJD Uncertainties are given in the second line for each parameter. t 0 represents the time of closest approach of the source to the lens mass center. u 0 is the closest distance of the source to the lens mass center. 3.2 Microlens Parallax Model Even without detailed analysis, the results for the static model, that are listed in Table 2, imply a large (and so potentially measurable) microlens parallax (Gould 1992, 2000), π E π rel θ E µ rel µ rel. (2) According to θ and the blended light in Section 5, θ E = θ /ρ 1.68 mas and the lens light I L 18.9 at 3σ level. The two limits correspond roughly to an M 0.5M at a distance of 1.2 kpc. Thus, we can expect 6 π E = θ E κm L (3) 6 One exception to this limit would be if the lens were a massive remnant.
11 KMT-2020-BLG Moreover, given the long Einstein timescale, t E 103 days, the projected velocity on the observer plane, ṽ au/(π E t E ) < 42 km s 1 is close to the changes of Earth s velocity over the course of the event, taking account of which could impact other parameters as well. Therefore, it is essential to include microlens-parallax effects in the fit. We fit the annual microlensparallax effect by introducing two additional parameters π E,N and π E,E, the North and East components of π E in equatorial coordinates (Gould 2004). We also fit u 0 > 0 and u 0 < 0 models to consider the ecliptic degeneracy (Jiang et al. 2004; Poindexter et al. 2005). Table 2 shows the results of fitting the light curve with the microlens parallax effect. Because the annual parallax effect can be degenerate with the effects of lens orbital motion (Batista et al. 2011; Skowron et al. 2011), we also introduce two linearized parameters (ds/dt, dα/dt), the instantaneous changes in the separation and orientation of the two lens components defined at t 0. We restrict the MCMC trials to β < 0.8, where β is the ratio of projected kinetic to potential energy (Dong et al. 2009) β KE PE = κm yr 2 ( ) 3 ( π E 8π 2 γ 2 s ds/dt ; γ, dα ), (4) θ E π E + π S /θ E s dt where we adopt the source parallax π S = mas based on the mean distance to clump giant stars in this direction (Nataf et al. 2013). See Table 2 for the results. We find that the addition of lens orbital motion effect provides improvements of χ 2 = 2.4 and 5.4 for the u 0 > 0 and u 0 < 0 solution, respectively, and π E is basically the same compared to the parallax model. Although the angular Einstein radius θ E estimated from the parallax modeling is smaller than the value from the static model, the resulting parallax for 2L1S is still strongly inconsistent with the constraint of the blended light at about 3σ. We will further discuss the implication of the 2L1S results in Section Binary-Source (1L2S) Model In some cases, planetary (2L1S) light curves can be imitated by binary-source (1L2S) events (Gaudi 1998). We do not expect that this will be case for KMT-2020-BLG-0414 because the planetary anomaly is mainly characterized by sharp changes in slope, rather than a smooth short-lived bump. Nevertheless, as a matter of due diligence, we search for such models including both microlens-parallax and microlens-xarallap effects (Griest & Hu 1992; Han & Gould 1997; Poindexter et al. 2005). We find that while the introduction of a second source yields a huge improvement with respect to the 1L1S model with χ 2 = χ 2 (1L1S) χ 2 (1L2S) > 10000, the 1L2S model still does not compete with the 2L1S model with χ 2 = χ 2 (1L2S) χ 2 (2L1S) > L1S ANALYSIS While the 2L1S models described in Section 3 fit the non-moa-peak data very well, they completely fail to explain the features of the MOA data in the peak region. Moreover, a 2L1S grid search that includes all the data fails to return any model that even approximately traces the data over the peak. See Figure 1. We therefore conduct a 3L1S grid search. Relative to static 2L1S models, 3L1S models have three additional parameters, (s 3, q 3, ψ). These are, respectively, the normalized separation of the third body from the primary, the mass ratio of the third body to the primary, and the angle of the second from the third body, as seen from the primary. Note that, to avoid confusion, we rename (s, q) (s 2, q 2 ).
12 12 Zang et al L1S Static Models We begin by conducting a grid search for static 3L1S solutions that is analogous to the one carried out previously for 2L1S solutions, but is substantially more computationally intensive. In a grid search (whether 2L1S or 3L1S), the lens geometry is held fixed at each grid point. However, for 2L1S, the geometry is specified by just two parameters, (s 2, q 2 ), whereas for 3L1S, five geometric parameters are required, (s 2, q 2, s 3, q 3, ψ). To reduce the grid of geometries from five to three dimensions, we consider a ( ) grid in (s 3, q 3, ψ). We hold (s 2, q 2 ) fixed at the best-fit 2L1S model. We seed the remaining five parameters, (t 0, u 0, t E, ρ, α), at the best-fit 2L1S model, and we then allow these to vary. We initially consider only close models for the third body, i.e., s 3 < 1. For the grid-search phase, which relies on fixed geometries, we apply the map-making technique of Dong et al. (2006) to evaluate the magnifications. This grid search yields only one local region of candidate solutions. We then seed an additional MCMC with the best grid point from this region and allow all 10 parameters to vary. Because the geometry now varies with each step in the MCMC, we use the adaptive-image inverseray-shooting technique to evaluate the magnifications. The resulting parameters are shown in Table 3, and the model light curve is compared to the data in Figure 1. The corresponding caustic structure in the upper panel of Figure 2 shows that KMT-2020-BLG-0414 is a classic case of caustic factorization. The caustic is nearly the superposition of two well-known caustic types: a large resonant caustic associated with the planet, and a smaller, nearly Chang-Refsdal (Chang & Refsdal 1979), caustic associated with the third body. As is often the case, the two caustic structures interact and become intertwined where they overlap. Because the caustic factors and q 3 1, it is straightforward to guess the alternate wide (s 3 > 1) solution according to the prescription of Griest & Safizadeh (1998): s 3 s 1 3. We seed this guess into an MCMC to yield the alternate wide solution, whose parameters are given in Table 3 and whose geometry is shown in the lower panel of Figure 2. The most striking difference between the 3L1S close and wide solutions is that s 2,close = ± , while s 2,wide = ± , which appears to be a 100σ difference. We address this issue in Appendix Section A L1S Parallax-only Models As discussed in Section 3.2, the microlens-parallax parameters π E (Equation 2) can be degenerate with the orbital-motion parameters γ (Equation 4). Hence, both should be considered together. However, as in that section, we proceed step-by-step, in part due to the increasing computational load as more parameters are introduced, and therefore the importance of understanding which are really necessary. When only π E is added to the 3L1S static model, there are 12 parameters. As was the case for 2L1S, adding parallax to the fit results in doubling the number of solutions, i.e., there is a ±u 0 pair of solutions for each of the close and wide solutions found in Section 4.1. Hence there are four solutions altogether. The resulting parameters and χ 2 values are given in Table 3. It is found that including parallax significantly improves the fit by χ 2 > 130. The wide solutions are disfavored by χ 2 > 17. Between the two close solutions, the u 0 > 0 solution is significantly favored by χ We note that the magnitude of π E for 3L1S is substantially larger than the value estimated from the 2L1S modeling regardless of the lensorbital effect. Figure 3 shows the cumulative distribution of χ 2 = χ 2 (static) χ 2 (parallax) for the
13 KMT-2020-BLG Fig. 2: Geometries of the 3L1S Close (upper panel) and Wide (lower panel) models. In each panel, the red dashed line represents the caustic structure, the black solid line is the trajectory of the source, and the arrow indicates the direction of the source motion. The 3L1S caustic is nearly the superposition of a large 6-sided resonant caustic associated with the q 10 5 planet and a small quadrilateral caustic associated with the third body (q 0.05). four solutions. Overall, χ 2 grows steadily over time, giving credence to the parallax measurement. An important feature of this diagram is that the contribution to χ 2 during the short time interval starting from one day before the peak and ending two days after the peak is about 40% of the total χ 2. By contrast, one normally expects the parallax signal to be dominated by the wings of the light curve. This time interval is essentially the duration of contact with the planetary caustic, in particular as the source rides the caustic for two days after the peak. This gives a plausible explanation for the sensitivity of π E to the near-peak region of the light curve. Hence, it is essential to include orbital motion L1S Parallax and Planet Orbital Motion We now include both π E and γ for the planet in the 3L1S fit, for a total of 14 chain parameters. We show the parameters of this fit in the Table 4. It is found that including planet orbital motion significantly changes π E in both magnitude and direction, and π E of the u 0 < 0 solution is 1.8 times greater than that of the u 0 > 0 solution. The close u 0 < 0 solution has the best fit to the observed data, while other solutions are only disfavored by χ 2 < 7. Thus, we cannot exclude any solution from the light-curve analysis. The ratio β of projected kinetic to potential energy is well measured, and all of the solutions have β < 0.1 at 3σ. This relatively low value of the ratio suggests that the planet and the host may be aligned along the line of sight. 4.4 Possible Orbital Motion of the Third Body The third body (with a brown-dwarf-like mass ratio q ) must also undergo orbital motion. In the wide solutions, the period would be of order 100 years, implying that orbital motion of the third body would
14 14 Zang et al. Table 3: Parameters for 3L1S Static and Parallax Model Parameters Static (non-parallax) Parallax Close Wide Close u 0 > 0 Close u 0 < 0 Wide u 0 > 0 Wide u 0 < 0 χ 2 /dof / / / / / /940 t 0 (HJD ) u 0(10 3 ) t E (days) s x q 2(10 5 ) α (rad) s q ψ (rad) ρ(10 4 ) π E,N π E,E I S HJD = HJD Uncertainties are given in the second line for each parameter. t 0 represents the time of closest approach of the source to the lens mass center. u 0 is the closest distance of the source to the lens mass center. x 2 is a derived quantity and is not fitted independently. See Appendix Section A for the definition of x 2. not affect the lensing light curve. For the close solutions, the period would be of order 50 days, and thus its orbital motion could affect the light curve. Nevertheless, we do not attempt to model orbital motion of the third body for several reasons. First, the duration of its pronounced perturbation ( 0.3 days over the peak) is 10 times shorter than for the duration of the planetary signal, and its impact on the light curve is quadratic in the duration. Second, if there were clear prospects of a scientifically important result, such work would be warranted, but there are no such prospects (see Section 5.4). Finally, the results of 3L1S parallax + planet orbital motion required about two weeks of computations with 400 processors. The already prodigious use of computer time (which scales (n/2)! where n is the number of chain parameters) would increase by a factor eight. We therefore decline to pursue this aspect of the problem.
15 KMT-2020-BLG I-Mag Close u 0 > 0 Close u 0 < 0 Wide u 0 > 0 Wide u 0 < Cumulative HJD Fig. 3: Cumulative distribution of χ 2 = χ 2 static χ2 parallax between the four 3L1S parallax solutions and the two 3L1S static solutions. Overall, χ 2 grows steadily over time and does so steeply from one day before peak until two days after peak, when the source touches the planetary caustic. The upper panel shows the best-fit 3L1S parallax model. 5 PHYSICAL PARAMETERS Normally, if the angular Einstein radius θ E and the microlens parallax π E are well measured, one can simply determine the lens total mass M L and the lens distance D L by (Gould 1992, 2000) M L = θ E AU ; D L =. (5) κπ E π E θ E + π S However, in the present case, the lens system is very close (D L 1 kpc), and the large symmetric errors of π E can lead to an asymmetric distribution in inferred lens distance. Hence, we conduct a Bayesian analysis to estimate the lens physical parameters in Section 5.3. Before doing so, we estimate θ by a color-magnitude diagram (CMD) analysis (Yoo et al. 2004) in Section 5.1, in order to estimate the angular Einstein radius by θ E = θ /ρ. We also study the blended light in Section 5.2 to obtain constraints on the
16 16 Zang et al. Table 4: Parameters for 3L1S Model with Parallax and Planet Orbital Motion Parameters Close u 0 > 0 Close u 0 < 0 Wide u 0 > 0 Wide u 0 < 0 χ 2 /dof 939.7/ / / /938 t 0 (HJD ) u 0(10 3 ) t E (days) s x q 2(10 5 ) α (rad) s q ψ (rad) ρ(10 4 ) π E,N π E,E ds/dt(yr 1 ) dα/dt(yr 1 ) β I S HJD = HJD Uncertainties are given in the second line for each parameter. t 0 represents the time of closest approach of the source to the lens mass center. u 0 is the closest distance of the source to the lens mass center. β and x 2 are derived quantities and are not fitted independently. See Equation (4) and Appendix Section A for the definitions of β and x 2, respectively. lens light. Finally in Section 5.4, we illustrate how future high-resolution photometric and spectroscopic observations would clarify the nature of the system.
17 KMT-2020-BLG Color-Magnitude Diagram (CMD) Figure 4 shows the CMD of stars from the OGLE-III catalog (Szymański et al. 2011) located within a square region with one side length of 240 centered at the location of KMT-2020-BLG-0414, together with the source position (blue) and the centroid of the red giant clump (red). We measure the centroid of the red giant clump as (V I, I) cl = (2.04 ± 0.01, ± 0.02). For the intrinsic centroid of the red giant clump, we adopt (V I, I) cl,0 = (1.06, 14.35) (Bensby et al. 2013; Nataf et al. 2013). This implies that A I = 1.20 and E(V I) = 0.98 toward this direction. For the source color, which is independent of any model, we get (V I) S = 1.82 ± 0.01 by regression of MOA V versus R flux as the source magnification changes and a calibration to the OGLE-III scale using the field-star photometry from the same reductions. Because the source apparent brightness slightly depends on the model, for simplicity, we explicitly derive results for I S = and then present a scaling relation for different source magnitudes. From this procedure, we obtain the intrinsic color and brightness of the source as (V I, I) S,0 = (0.84 ± 0.03, ± 0.03), suggesting that the source is a mid-g type dwarf (Bessell & Brett 1988). Using the color/surface-brightness relation of Adams et al. (2018), we obtain θ = ± µas. (6) where the 5% error is given by Table 3 of Adams et al. (2018). Then, for any particular model with source magnitude I S, one can infer θ = (IS 19.12). 5.2 The Blended Light For KMT-2020-BLG-0414, the baseline object was detected by the OGLE-III survey, (V, I) base = (20.70± 0.08, 18.46±0.03). We further consider the statistical errors due to the mottled background from unresolved stars (Park et al. 2004). We follow the approach of Ryu et al. (2020b) using the GalSim package (Rowe et al. 2015) with the readout noise of Udalski et al. (2015). We find σ I = 0.08 mag and σ V = mag, and thus the baseline object has (V, I) base = (20.70 ± 0.14, ± 0.09), yielding the blended light of (V I, I) B = ( , ± 0.20). This value is consistent with the lens properties that are predicted by the microlensing light-curve and CMD analyses. For example, using the median θ E and π E values of the close u 0 < 0 solutions, the host mass M 1 = 0.25 M and it would have rough intrinsic brightness and color of M I 9.8, (V I) Assuming an extinction curve with a scale height of 120 pc, it would have (A I, E(V I)) L = (0.38, 0.31) at the lens distance D L = 0.74 kpc. These corresponds to (V I, I) L (3.3, 19.5), which is shown as the cyan point ( naive lens ) in Figure 4 and is quite consistent with the blend. We also check the astrometric alignment between the source and the baseline object from KMTA imaging. We find that the baseline object lies (0.18, 0.01 ) west and south of the source. Because the source position is derived from difference image analysis on highly magnified images, the uncertainty in the source position ( 0.01 ) is negligible relative to the error in the baseline position. We estimate the error of baseline position by the fractional astrometric error being equal to the fractional photometric error (Jung et al. 2020), σ ast = 0.39σ I FWHM = Hence, the baseline object is astrometrically consistent with the source (and thus lens) at 2σ level. Thus, it is plausible that most or all of the blended light is due to the lens.
18 18 Zang et al source red giant clump naive lens blend baseline IOGLE (V I) OGLE Fig. 4: Color-magnitude diagram (CMD) for field stars in a 240 square centered on KMT-2020-BLG-0414 using the OGLE-III star catalog (Szymański et al. 2011). The red asterisk, blue dot, magenta dot and green dot represent the positions of the centroid of the red giant clump, the microlens source, the baseline object and the blended light, respectively. The cyan dot shows the position of a naive lens host with M 1 = 0.25M and D L = 0.74 kpc, estimated using the median θ E and π E values of the close u 0 < 0 solution. The alignment between the source and the baseline object can be immediately checked (i.e., 2021 bulge season) by the Hubble Space Telescope (HST) or by ground-based adaptive optics (AO) mounted on large ground-based telescopes (e.g., Keck, Subaru). Even if the alignment was demonstrated, the blended light could in principle come from a stellar companion to either the source or the lens. However, if the alignment is 50 mas, an additional stellar companion to the lens would have generated significant deviations on the peak, so the confirmation that the blend is well aligned to the lens could probably rule out the lens-companion scenario. Because the source and the blended light have significantly different colors, the possibility of the source companion can be checked by a measurement of the astrometric offset in different bands (Bennett et al. 2006). It is a priori unlikely that the blended light is primarily due to ambient stars that are unassociated with the event because of the low surface density of stars relative to the 180 mas offset. If