arXiv subjects

In-Gu Shin

Publications and source records attributed to In-Gu Shin.

At least 145 records · Page 8Linked to original sources

arXiv2019

OGLE-2016-BLG-1227L: A Wide-separation Planet from a Very Short-timescale Microlensing Event

We present the analysis of the microlensing event OGLE-2016-BLG-1227. The light curve of this short-duration event appears to be a single-lens event affected by severe finite-source effects. Analysis of the light curve based on single-lens single-source (1L1S) modeling yields very small values of the event timescale, $t_{\rm E}\sim 3.5$ days, and the angular Einstein radius, $θ_{\rm E}\sim 0.009$ mas, making the lens a candidate of a free-floating planet. Close inspection reveals that the 1L1S solution leaves small residuals with amplitude $ΔI\lesssim 0.03$ mag. We find that the residuals are explained by the existence of an additional widely-separated heavier lens component, indicating that the lens is a wide-separation planetary system rather than a free-floating planet. From Bayesian analysis, it is estimated that the planet has a mass of $M_{\rm p} = 0.79^{+1.30}_{-0.39} M_{\rm J}$ and it is orbiting a low-mass host star with a mass of $M_{\rm host}=0.10^{+0.17}_{-0.05} M_\odot$ located with a projected separation of $a_\perp=3.4^{+2.1}_{-1.0}$ au. The planetary system is located in the Galactic bulge with a line-of-sight separation from the source star of $D_{\rm LS}=1.21^{+0.96}_{-0.63}$ kpc. The event shows that there are a range of deviations in the signatures of host stars for apparently isolated planetary lensing events and that it is possible to identify a host even when a deviation is subtle.

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arXiv2019

OGLE-2015-BLG-1771Lb: A Microlens Planet Orbiting an Ultracool Dwarf?

We report the discovery and the analysis of the short (tE < 5 days) planetary microlensing event, OGLE-2015-BLG-1771. The event was discovered by the Optical Gravitational Lensing Experiment (OGLE), and the planetary anomaly (at I ~ 19) was captured by The Korea Microlensing Telescope Network (KMTNet). The event has three surviving planetary models that explain the observed light curves, with planet-host mass ratio q \~ 5.4 * 10^{-3}, 4.5 * 10^{-3} and 4.5 * 10^{-2}, respectively. The first model is the best-fit model, while the second model is disfavored by Δχ^2 ~ 3. The last model is strongly disfavored by Δχ^2 ~ 15 but not ruled out. A Bayesian analysis using a Galactic model indicates that the first two models are probably composed of a Saturn-mass planet orbiting a late M dwarf, while the third one could consist of a super-Jovian planet and a mid-mass brown dwarf. The source-lens relative proper motion is mu_rel ~ 9 mas/yr, so the source and lens could be resolved by current adaptive-optics (AO) instruments in 2021 if the lens is luminous.

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arXiv2019

Candidate Brown-dwarf Microlensing Events with Very Short Timescales and Small Angular Einstein Radii

Short-timescale microlensing events are likely to be produced by substellar brown dwarfs (BDs), but it is difficult to securely identify BD lenses based on only event timescales $t_{\rm E}$ because short-timescale events can also be produced by stellar lenses with high relative lens-source proper motions. In this paper, we report three strong candidate BD-lens events found from the search for lensing events not only with short timescales ($t_{\rm E} \lesssim 6~{\rm days}$) but also with very small angular Einstein radii ($θ_{\rm E}\lesssim 0.05~{\rm mas}$) among the events that have been found in the 2016--2019 observing seasons. These events include MOA-2017-BLG-147, MOA-2017-BLG-241, and MOA-2019-BLG-256, in which the first two events are produced by single lenses and the last event is produced by a binary lens. From the Bayesian analysis conducted with the combined $t_{\rm E}$ and $θ_{\rm E}$ constraint, it is estimated that the lens masses of the individual events are $0.051^{+0.100}_{-0.027}~M_\odot$, $0.044^{+0.090}_{-0.023}~M_\odot$, and $0.046^{+0.067}_{-0.023}~M_\odot/0.038^{+0.056}_{-0.019}~M_\odot$ and the probability of the lens mass smaller than the lower limit of stars is $\sim 80\%$ for all events. We point out that routine lens mass measurements of short time-scale lensing events require survey-mode space-based observations.

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arXiv2019

OGLE-2018-BLG-1700L: Microlensing Planet in Binary Stellar System

We report the discovery of a planet in a binary that was discovered from the analysis of the microlensing event OGLE-2018-BLG-1700. We identify the triple nature of the lens from the fact that the complex anomaly pattern can be decomposed into two parts produced by two binary-lens events, in which one binary pair has a very low mass ratio of $\sim 0.01$ between the lens components and the other pair has a mass ratio of $\sim 0.3$. We find two sets of degenerate solutions, in which one solution has a projected separation between the primary and its stellar companion less than the angular Einstein radius $\thetae$ (close solution), while the other solution has a separation greater than $\thetae$ (wide solution). From the Bayesian analysis with the constraints of the event time scale and angular Einstein radius together with the location of the source lying in the far disk behind the bulge, we find that the planet is a super-Jupiter with a mass of $4.4^{+3.0}_{-2.0}~M_{\rm J}$ and the stellar binary components are early and late M-type dwarfs with masses $0.42^{+0.29}_{-0.19}~M_\odot$ and $0.12^{+0.08}_{-0.05}~M_\odot$, respectively, and the planetary system is located at a distance of $D_{\rm L}=7.6^{+1.2}_{-0.9}~{\rm kpc}$. The planet is a circumstellar planet according to the wide solution, while it is a circumbinary planet according to the close solution. The projected primary-planet separation is $2.8^{+3.2}_{-2.5}~{\rm au}$ commonly for the close and wide solutions, but the primary-secondary binary separation of the close solution, $0.75^{+0.87}_{-0.66}~{\rm au}$, is widely different from the separation, $10.5^{+12.1}_{-9.2}~{\rm au}$, of the wide solution.

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arXiv2019

OGLE-2017-BLG-1186: first application of asteroseismology and Gaussian processes to microlensing

We present the analysis of the event OGLE-2017-BLG-1186 from the 2017 Spitzer microlensing campaign. This is a remarkable microlensing event because its source is photometrically bright and variable, which makes it possible to perform an asteroseismic analysis using ground-based data. We find that the source star is an oscillating red giant with average timescale of $\sim 9$ d. The asteroseismic analysis also provides us source properties including the source angular size ($\sim 27~μ{\rm as}$) and distance ($\sim 11.5$ kpc), which are essential for inferring the properties of the lens. When fitting the light curve, we test the feasibility of Gaussian Processes (GPs) in handling the correlated noise caused by the variable source. We find that the parameters from the GP model are generally more loosely constrained than those from the traditional $χ^2$ minimization method. We note that this event is the first microlensing system for which asteroseismology and GPs have been used to account for the variable source. With both finite-source effect and microlens parallax measured, we find that the lens is likely a $\sim 0.045~M_{\odot}$ brown dwarf at distance $\sim 9.0$ kpc, or a $\sim 0.073~M_{\odot}$ ultracool dwarf at distance $\sim 9.8$ kpc. Combining the estimated lens properties with a Bayesian analysis using a Galactic model, we find a $\sim 35$ per cent probability for the lens to be a bulge object and $\sim 65$ per cent to be a background disc object.

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arXiv2019

Spectroscopic Mass and Host-star Metallicity Measurements for Newly Discovered Microlensing Planet OGLE-2018-BLG-0740Lb

We report the discovery of the microlensing planet OGLE-2018-BLG-0740Lb. The planet is detected with a very strong signal of $Δχ^2\sim 4630$, but the interpretation of the signal suffers from two types of degeneracies. One type is caused by the previously known close/wide degeneracy, and the other is caused by an ambiguity between two solutions, in which one solution requires to incorporate finite-source effects, while the other solution is consistent with a point-source interpretation. Although difficult to be firmly resolved based on only the photometric data, the degeneracy is resolved in strong favor of the point-source solution with the additional external information obtained from astrometric and spectroscopic observations. The small astrometric offset between the source and baseline object supports that the blend is the lens and this interpretation is further secured by the consistency of the spectroscopic distance estimate of the blend with the lensing parameters of the point-source solution. The estimated mass of the host is $1.0\pm 0.1~M_\odot$ and the mass of the planet is $4.5\pm 0.6~M_{\rm J}$ (close solution) or $4.8\pm 0.6~M_{\rm J}$ (wide solution) and the lens is located at a distance of $3.2\pm 0.5$~kpc. The bright nature of the lens, with $I\sim 17.1$ ($V\sim 18.2$), combined with its dominance of the observed flux suggest that radial-velocity (RV) follow-up observations of the lens can be done using high-resolution spectrometers mounted on large telescopes, e.g., VLT/ESPRESSO, and this can potentially not only measure the period and eccentricity of the planet but also probe for close-in planets. We estimate that the expected RV amplitude would be $\sim 60\sin i ~{\rm m~s}^{-1}$.

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arXiv2019

OGLE-2018-BLG-1011L\lowercase{b,c}: Microlensing Planetary System with Two Giant Planets Orbiting a Low-mass Star

We report a multiplanetary system found from the analysis of microlensing event OGLE-2018-BLG-1011, for which the light curve exhibits a double-bump anomaly around the peak. We find that the anomaly cannot be fully explained by the binary-lens or binary-source interpretations and its description requires the introduction of an additional lens component. The 3L1S (3 lens components and a single source) modeling yields three sets of solutions, in which one set of solutions indicates that the lens is a planetary system in a binary, while the other two sets imply that the lens is a multiplanetary system. By investigating the fits of the individual models to the detailed light curve structure, we find that the multiple-planet solution with planet-to-host mass ratios $\sim 9.5\times 10^{-3}$ and $\sim 15\times 10^{-3}$ are favored over the other solutions. From the Bayesian analysis, we find that the lens is composed of two planets with masses $1.8^{+3..4}_{-1.1}~M_{\rm J}$ and $2.8^{+5.1}_{-1.7}~M_{\rm J}$ around a host with a mass $0.18^{+0.33}_{-0.10}~M_\odot$ and located at a distance $7.1^{+1.1}_{-1.5}~{\rm kpc}$. The estimated distance indicates that the lens is the farthest system among the known multiplanetary systems. The projected planet-host separations are $a_{\perp,2}=1.8^{+2.1}_{-1.5}~{\rm au}$ ($0.8^{+0.9}_{-0.6}~{\rm au}$) and $a_{\perp,3}=0.8^{+0.9}_{-0.6}~{\rm au}$, where the values of $a_{\perp,2}$ in and out the parenthesis are the separations corresponding to the two degenerate solutions, indicating that both planets are located beyond the snow line of the host, as with the other four multiplanetary systems previously found by microlensing.

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arXiv2019

OGLE-2015-BLG-1670Lb: A Cold Neptune beyond the Snow Line in the Provisional WFIRST Microlensing Survey Field

We present the analysis of the microlensing event OGLE-2015-BLG-1670, detected in a high-extinction field, very close to the Galactic plane. Due to the dust extinction along the line of sight, this event was too faint to be detected before it reached the peak of magnification. The microlensing light-curve models indicate a high-magnification event with a maximum of $A_\mathrm{max}\gtrsim200$, very sensitive to planetary deviations. An anomaly in the light curve has been densely observed by the microlensing surveys MOA, KMTNet, and OGLE. From the light-curve modeling, we find a planetary anomaly characterized by a planet-to-host mass ratio, $q=\left(1.00^{+0.18}_{-0.16}\right)\times 10^{-4}$, at the peak recently identified in the mass-ratio function of microlensing planets. Thus, this event is interesting to include in future statistical studies about planet demography. We have explored the possible degeneracies and find two competing planetary models resulting from the $s\leftrightarrow1/s$ degeneracy. However, because the projected separation is very close to $s=1$, the physical implications for the planet for the two solutions are quite similar, except for the value of $s$. By combining the light-curve parameters with a Galactic model, we have estimated the planet mass $M_2=17.9^{+9.6}_{-8.8}\,\mathrm{M}_\oplus$ and the lens distance $D_\mathrm{L}=6.7^{+1.0}_{-1.3}\,\mathrm{kpc}$, corresponding to a Neptune-mass planet close to the Galactic bulge. Such events with a low absolute latitude ($|b|\approx 1.1\,\mathrm{deg}$) are subject to both high extinction and more uncertain source distances, two factors that may affect the mass measurements in the provisional Wide Field Infrared Survey Telescope fields. More events are needed to investigate the potential trade-off between the higher lensing rate and the difficulty in measuring masses in these low-latitude fields.

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arXiv2019

OGLE-2018-BLG-0532Lb: Cold Neptune With Possible Jovian Sibling

We report the discovery of the planet OGLE-2018-BLG-0532Lb, with very obvious signatures in the light curve that lead to an estimate of the planet-host mass ratio $q=M_{\rm planet}/M_{\rm host}\simeq 1\times10^{-4}$. Although there are no obvious systematic residuals to this double-lens/single-source (2L1S) fit, we find that $χ^2$ can be significantly improved by adding either a third lens (3L1S, $Δχ^2=81$) or second source (2L2S, $Δχ^2=65$) to the lens-source geometry. After thorough investigation, we conclude that we cannot decisively distinguish between these two scenarios and therefore focus on the robustly-detected planet. However, given the possible presence of a second planet, we investigate to what degree and with what probability such additional planets may affect seemingly single-planet light curves. Our best estimates for the properties of the lens star and the secure planet are: a host mass $M\sim 0.25\,M_\odot$, system distance $D_L\sim 1\,$kpc and planet mass $m_{p,1}= 8\,M_\oplus$ with projected separation $a_{1,\perp}=1.4\,$au. However, there is a relatively bright $I=18.6$ (and also relatively blue) star projected within $<50\,$mas of the lens, and if future high-resolution images show that this is coincident with the lens, then it is possible that it is the lens, in which case, the lens would be both more massive and more distant than the best-estimated values above.

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arXiv2019

$Spitzer$ Parallax of OGLE-2018-BLG-0596: A Low-mass-ratio Planet around an M-dwarf

We report the discovery of a $Spitzer$ microlensing planet OGLE-2018-BLG-0596Lb, with preferred planet-host mass ratio $q \sim 2\times10^{-4}$. The planetary signal, which is characterized by a short $(\sim 1~{\rm day})$ "bump" on the rising side of the lensing light curve, was densely covered by ground-based surveys. We find that the signal can be explained by a bright source that fully envelops the planetary caustic, i.e., a "Hollywood" geometry. Combined with the source proper motion measured from $Gaia$, the $Spitzer$ satellite parallax measurement makes it possible to precisely constrain the lens physical parameters. The preferred solution, in which the planet perturbs the minor image due to lensing by the host, yields a Uranus-mass planet with a mass of $M_{\rm p} = 13.9\pm1.6~M_{\oplus}$ orbiting a mid M-dwarf with a mass of $M_{\rm h} = 0.23\pm0.03~M_{\odot}$. There is also a second possible solution that is substantially disfavored but cannot be ruled out, for which the planet perturbs the major image. The latter solution yields $M_{\rm p} = 1.2\pm0.2~M_{\oplus}$ and $M_{\rm h} = 0.15\pm0.02~M_{\odot}$. By combining the microlensing and $Gaia$ data together with a Galactic model, we find in either case that the lens lies on the near side of the Galactic bulge at a distance $D_{\rm L} \sim 6\pm1~{\rm kpc}$. Future adaptive optics observations may decisively resolve the major image/minor image degeneracy.

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arXiv2019

KMT-2018-BLG-1990Lb: A Nearby Jovian Planet From A Low-Cadence Microlensing Field

We report the discovery and characterization of KMT-2018-BLG-1990Lb, a Jovian planet $(m_p=0.57_{-0.25}^{+0.79}\,M_J)$ orbiting a late M dwarf $(M=0.14_{-0.06}^{+0.20}\,M_\odot)$, at a distance $(D_L=1.23_{-0.43}^{+1.06}\,\kpc)$, and projected at $2.6\pm 0.6$ times the snow line distance, i.e., $a_{\rm snow}\equiv 2.7\,\au (M/M_\odot)$, This is the second Jovian planet discovered by KMTNet in its low cadence ($0.4\,{\rm hr}^{-1}$) fields, demonstrating that this population will be well characterized based on survey-only microlensing data.

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arXiv2019

KMT-2018-BLG-1292: A Super-Jovian Microlens Planet in the Galactic Plane

We report the discovery of KMT-2018-BLG-1292Lb, a super-Jovian $M_{\rm planet} = 4.5\pm 1.3\,M_J$ planet orbiting an F or G dwarf $M_{\rm host} = 1.5\pm 0.4\,M_\odot$, which lies physically within ${\cal O}(10\,\pc)$ of the Galactic plane. The source star is a heavily extincted $A_I\sim 5.2$ luminous giant that has the lowest Galactic latitude, $b=-0.28^\circ$, of any planetary microlensing event. The relatively blue blended light is almost certainly either the host or its binary companion, with the first explanation being substantially more likely. This blend dominates the light at $I$ band and completely dominates at $R$ and $V$ bands. Hence, the lens system can be probed by follow-up observations immediately, i.e., long before the lens system and the source separate due to their relative proper motion. The system is well characterized despite the low cadence $Γ=0.15$--$0.20\,{\rm hr^{-1}}$ of observations and short viewing windows near the end of the bulge season. This suggests that optical microlensing planet searches can be extended to the Galactic plane at relatively modest cost.

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arXiv2019

Spitzer Microlensing parallax reveals two isolated stars in the Galactic bulge

We report the mass and distance measurements of two single-lens events from the 2017 Spitzer microlensing campaign. The ground-based observations yield the detection of finite-source effects, and the microlens parallaxes are derived from the joint analysis of ground-based observations and Spitzer observations. We find that the lens of OGLE-2017-BLG-1254 is a $0.60 \pm 0.03 M_{\odot}$ star with $D_{\rm LS} = 0.53 \pm 0.11~\text{kpc}$, where $D_{\rm LS}$ is the distance between the lens and the source. The second event, OGLE-2017-BLG-1161, is subject to the known satellite parallax degeneracy, and thus is either a $0.51^{+0.12}_{-0.10} M_{\odot}$ star with $D_{\rm LS} = 0.40 \pm 0.12~\text{kpc}$ or a $0.38^{+0.13}_{-0.12} M_{\odot}$ star with $D_{\rm LS} = 0.53 \pm 0.19~\text{kpc}$. Both of the lenses are therefore isolated stars in the Galactic bulge. By comparing the mass and distance distributions of the eight published Spitzer finite-source events with the expectations from a Galactic model, we find that the Spitzer sample is in agreement with the probability of finite-source effects occurrence in single lens events.

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arXiv2019

OGLE-2018-BLG-0022: First Prediction of an Astrometric Microlensing Signal from a Photometric Microlensing Event

In this work, we present the analysis of the binary microlensing event OGLE-2018-BLG-0022 that is detected toward the Galactic bulge field. The dense and continuous coverage with the high-quality photometry data from ground-based observations combined with the space-based {\it Spitzer} observations of this long time-scale event enables us to uniquely determine the masses $M_1=0.40 \pm 0.05~M_\odot$ and $M_2=0.13\pm 0.01~M_\odot$ of the individual lens components. Because the lens-source relative parallax and the vector lens-source relative proper motion are unambiguously determined, we can likewise unambiguously predict the astrometric offset between the light centroid of the magnified images (as observed by the {\it Gaia} satellite) and the true position of the source. This prediction can be tested when the individual-epoch {\it Gaia} astrometric measurements are released.

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arXiv2019

Spitzer Microlensing of MOA-2016-BLG-231L : A Counter-Rotating Brown Dwarf Binary in the Galactic Disk

We analyze the binary microlensing event MOA-2016-BLG-231, which was observed from the ground and from Spitzer. The lens is composed of very low-mass brown dwarfs (BDs) with $M_1 = 21^{+12}_{-5} \ M_J$ and $M_2 = 9^{+5}_{-2}\ M_J$, and it is located in the Galactic disk $D_{\rm L} = 2.85^{+0.88}_{-0.50}\ {\rm kpc}$. This is the fifth binary brown dwarf discovered by microlensing, and the BD binary is moving counter to the orbital motion of disk stars. Constraints on the lens physical properties come from late time, non-caustic-crossing features of the Spitzer light curve. Thus, MOA-2016-BLG-231 shows how Spitzer plays a crucial role in resolving the nature of BDs in binary BD events with short timescale ($\lesssim 10$ days).

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arXiv2019

OGLE-2016-BLG-0156: Microlensing Event With Pronounced Microlens-Parallax Effects Yielding Precise Lens Mass Measurement

We analyze the gravitational binary-lensing event OGLE-2016-BLG-0156, for which the lensing light curve displays pronounced deviations induced by microlens-parallax effects. The light curve exhibits 3 distinctive widely-separated peaks and we find that the multiple-peak feature provides a very tight constraint on the microlens-parallax effect, enabling us to precisely measure the microlens parallax $π_{\rm E}$. All the peaks are densely and continuously covered from high-cadence survey observations using globally located telescopes and the analysis of the peaks leads to the precise measurement of the angular Einstein radius $θ_{\rm E}$. From the combination of the measured $π_{\rm E}$ and $θ_{\rm E}$, we determine the physical parameters of the lens. It is found that the lens is a binary composed of two M dwarfs with masses $M_1=0.18\pm 0.01\ M_\odot$ and $M_2=0.16\pm 0.01\ M_\odot$ located at a distance $D_{\rm L}= 1.35\pm 0.09\ {\rm kpc}$. According to the estimated lens mass and distance, the flux from the lens comprises an important fraction, $\sim 25\%$, of the blended flux. The bright nature of the lens combined with the high relative lens-source motion, $μ=6.94\pm 0.50\ {\rm mas}\ {\rm yr}^{-1}$, suggests that the lens can be directly observed from future high-resolution follow-up observations.

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arXiv2019

$Spitzer$ Microlensing Parallax for OGLE-2017-BLG-0896 Reveals a Counter-Rotating Low-Mass Brown Dwarf

The kinematics of isolated brown dwarfs in the Galaxy, beyond the solar neighborhood, is virtually unknown. Microlensing has the potential to probe this hidden population, as it can measure both the mass and five of the six phase-space coordinates (all except the radial velocity) even of a dark isolated lens. However, the measurements of both the microlens parallax and finite-source effects are needed in order to recover the full information. Here, we combine $Spitzer$ satellite parallax measurement with the ground-based light curve, which exhibits strong finite-source effects, of event OGLE-2017-BLG-0896. We find two degenerate solutions for the lens (due to the known satellite-parallax degeneracy), which are consistent with each other except for their proper motion. The lens is an isolated brown dwarf with a mass of either $18\pm1M_J$ or $20\pm1M_J$. This is the lowest isolated-object mass measurement to date, only $\sim$45\% more massive than the theoretical deuterium-fusion boundary at solar metallicity, which is the common definition of a free-floating planet. The brown dwarf is located at either $3.9\pm0.1$ kpc or $4.1\pm0.1$ kpc toward the Galactic bulge, but with proper motion in the opposite direction of disk stars, with one solution suggesting it is moving within the Galactic plane. While it is possibly a halo brown dwarf, it might also represent a different, unknown population.

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arXiv2018

KMT-2017-BLG-0165Lb: A Super-Neptune mass planet Orbiting a Sun-like Host Star

We report the discovery of a low mass-ratio planet $(q = 1.3\times10^{-4})$, i.e., 2.5 times higher than the Neptune/Sun ratio. The planetary system was discovered from the analysis of the KMT-2017-BLG-0165 microlensing event, which has an obvious short-term deviation from the underlying light curve produced by the host of the planet. Although the fit improvement with the microlens parallax effect is relatively low, one component of the parallax vector is strongly constrained from the light curve, making it possible to narrow down the uncertainties of the lens physical properties. A Bayesian analysis yields that the planet has a super-Neptune mass $(M_{2}=34_{-12}^{+15}~M_{\oplus})$ orbiting a Sun-like star $(M_{1}=0.76_{-0.27}^{+0.34}~M_{\odot})$ located at $4.5~{\rm kpc}$. The blended light is consistent with these host properties. The projected planet-host separation is $a_{\bot}={3.45_{-0.95}^{+0.98}}~{\rm AU}$, implying that the planet is located outside the snowline of the host, i.e., $a_{sl}\sim2.1~{\rm AU}$. KMT-2017-BLG-0165Lb is the sixteenth microlensing planet with mass ratio $q<3\times10^{-4}$. Using the fifteen of these planets with unambiguous mass-ratio measurements, we apply a likelihood analysis to investigate the form of the mass-ratio function in this regime. If we adopt a broken power law for the form of this function, then the break is at $q_{\rm br}\simeq0.55\times10^{-4}$, which is much lower than previously estimated. Moreover, the change of the power law slope, $ζ>3.3$ is quite severe. Alternatively, the distribution is also suggestive of a "pile-up" of planets at Neptune-like mass ratios, below which there is a dramatic drop in frequency.

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