OGLE-2015-BLG-1482L: the first isolated low-mass microlens in the Galactic bulge
S.-J. Chung, W. Zhu, A. Udalski, C.-U. Lee, Y.-H. Ryu, Y. K. Jung, I.-G. Shin, J. C. Yee, K.-H. Hwang, A. Gould, M. Albrow, S.-M. Cha, C. Han, D.-J. Kim, H.-W. Kim, S.-L. Kim, Y.-H. Kim, Y. Lee, B.-G. Park, R. W. Pogge, R. Poleski, P. Mróz, P. Pietrukowicz, J. Skowron, M.K. Szyma?ski, I. Soszy?ski, S. Koz?owski, K. Ulaczyk, M. Pawlak, C. Beichman, G. Bryden, S. Calchi Novati, S. Carey, M. Fausnaugh, B. S. Gaudi, Calen B. Henderson, Y. Shvartzvald, B. Wibking
DDraft version September 20, 2018
Preprint typeset using L A TEX style emulateapj v. 01/23/15
OGLE-2015-BLG-1482L: THE FIRST ISOLATED LOW-MASS MICROLENS IN THE GALACTIC BULGE
S.-J. Chung , , , W. Zhu , , , A. Udalski , , C.-U. Lee , , , Y.-H. Ryu , , Y. K. Jung , , I.-G. Shin , , J. C.Yee , , , K.-H. Hwang , , A. Gould , , , , andM. Albrow , S.-M. Cha , , C. Han , D.-J. Kim , H.-W. Kim , S.-L. Kim , , Y.-H. Kim , , Y. Lee , , B.-G. Park , ,R. W. Pogge (The KMTNet collaboration)R. Poleski , , P. Mr´oz , P. Pietrukowicz , J. Skowron , M.K. Szyma´nski , I. Soszy´nski , S. Koz(cid:32)lowski , K.Ulaczyk , , M. Pawlak (The OGLE collaboration)C. Beichman , G. Bryden , S. Calchi Novati , , S. Carey , M. Fausnaugh , B. S. Gaudi , Calen B.Henderson , , Y. Shvartzvald , , B. Wibking (The Spitzer team) Korea Astronomy and Space Science Institute, 776 Daedeokdae-ro, Yuseong-Gu, Daejeon 34055, Korea; [email protected] Korea University of Science and Technology, 217 Gajeong-ro, Yuseong-gu, Daejeon 34113, Korea Department of Astronomy, Ohio State University, 140 W. 18th Ave., Columbus, OH 43210, USA Warsaw University Observatory, AI. Ujazdowskie 4, 00-478 Warszawa, Poland Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138, USA Max-Planck-Institute for Astronomy, K¨onigstuhl 17, 69117 Heidelberg, Germany Department of Physics and Astronomy, University of Canterbury, Private Bag 4800 Christchurch, New Zealand School of Space Research, Kyung Hee University, Giheung-gu, Yongin, Gyeonggi-do, 17104, Korea Department of Physics, Chungbuk National University, Cheongju 361-763, Korea Department of Physics, University of Warwick, Gibbet Hill Road, Coventry, CV4 7AL, UK NASA Exoplanet Science Institute, MS 100-22, California Institute of Technology, Pasadena, CA 91125, USA Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA IPAC, Mail Code 100-22, Caltech, 1200 E. California Blvd., Pasadena, CA 91125 Dipartimento di Fisica “E. R. Caianiello”, Universit`a di Salerno, Via Giovanni Paolo II, 84084 Fisciano (SA), Italy Spitzer Science Center, MS 220-6, California Institute of Technology, Pasadena, CA, USA NASA Postdoctoral Program Fellow The KMTNet Collaboration The OGLE Collaboration and The
Spitzer
Team
Draft version September 20, 2018
ABSTRACTWe analyze the single microlensing event OGLE-2015-BLG-1482 simultaneously observed from twoground-based surveys and from
Spitzer . The
Spitzer data exhibit finite-source effects due to thepassage of the lens close to or directly over the surface of the source star as seen from
Spitzer . Suchfinite-source effects generally yield measurements of the angular Einstein radius, which when combinedwith the microlens parallax derived from a comparison between the ground-based and the
Spitzer lightcurves, yields the lens mass and lens-source relative parallax. From this analysis, we find that the lensof OGLE-2015-BLG-1482 is a very low-mass star with the mass 0 . ± . M (cid:12) or a brown dwarf withthe mass 55 ± M J , which are respectively located at D LS = 0 . ± .
19 kpc and D LS = 0 . ± .
08 kpc,where D LS is the distance between the lens and the source, and thus it is the first isolated low-massmicrolens that has been decisively located in the Galactic bulge. The degeneracy between the twosolutions is severe (∆ χ = 0 . Spitzer and this single data point gives rise to twosolutions for ρ , the angular size of the source in units of the angular Einstein ring radius. Because the ρ degeneracy can be resolved only by relatively high cadence observations around the peak, while the Spitzer cadence is typically ∼ − , we expect that events for which the finite-source effect is seenonly in the Spitzer data may frequently exhibit this ρ degeneracy. For OGLE-2015-BLG-1482, therelative proper motion of the lens and source for the low-mass star is µ rel = 9 . ± . − , whilefor the brown dwarf it is µ rel = 5 . ± . − . Hence, the degeneracy can be resolved within ∼
10 yrs from direct lens imaging by using next-generation instruments with high spatial resolution.
Keywords: brown dwarfs - gravitational lensing: micro - stars: fundamental parameters INTRODUCTION
Microlensing is sensitive to planets orbiting low-massstars and brown dwarfs (BDs) that are difficult to detectby other methods, such as the radial velocity and transitmethod. Although faint low-mass stars such as M dwarfscomprise ∼
70% of stars in the solar neighborhood and the Galaxy (Skowron et al. 2015), it is difficult to detectdistant M dwarfs due to their low luminosity. However,microlensing depends on the mass of the lens, not theluminosity, and thus it is not affected by the distanceand luminosity of the lens. Hence, microlensing is thebest method to probe faint M dwarfs in the Galaxy. Amajority of host stars of 52 extrasolar planets detected a r X i v : . [ a s t r o - ph . S R ] M a r Chung et al. by microlensing are M dwarfs, and they are distributedwithin a wide range of distances about 0 . − t E , which is the crossing time of theEinstein radius of the lens. With the observed t E , we canonly make a very rough estimate of the lens mass and socannot distinguish potential BDs from stars. To measurethe masses of isolated BDs in the isolated BDs events,two parameters are required: the angular Einstein radius θ E and microlens parallax π E . This is because (Gould1992, 2000) M L = θ E κπ E (1)and π E = π rel θ E ; π rel ≡ AU (cid:18) D L − D S (cid:19) , (2)where κ ≡ Gc AU ≈ .
14 mas M (cid:12) . Here M L is the lens mass, and D L and D S are the dis-tances to the lens and the source from the observer, re-spectively. However, it is usually quite difficult to mea-sure the two parameters θ E and π E .In general, θ E is obtained from the measurement ofthe normalized source radius ρ = θ (cid:63) /θ E , where θ (cid:63) isan angular radius of the source. The ρ measurement islimited to well-covered caustic-crossing events and high-magnification events in which the source passes close tothe lens, while θ (cid:63) is usually well measured through thecolor and brightness of the source. Because isolated BDevents are almost always quite short, π E can usually bemeasured only via so-called terrestrial parallax (Gould1997; Gould et al. 2009). Terrestrial parallax mea-surements are limited to well-covered high-magnification events. As a result, it is very hard to measure masses ofisolated BDs from the ground (Gould & Yee 2013). Thebest way to measure π E is a simultaneous observation ofan event from Earth and a satellite (Refsdal 1966; Gould1994b). Fortunately, since 2014, the Spitzer satellite hasbeen regularly observing microlensing events toward theGalactic bulge in order to measure the microlens paral-lax. The
Spitzer observations suggest a new opportunityto obtain the mass function of BDs from the simultane-ous observation from Earth and
Spitzer , although theyare not dedicated to BDs (Zhu et al. 2016).The simultaneous observation from the two observa-tories with sufficiently wide projected separation D ⊥ al-lows to measure the microlens parallax vector π E fromthe difference in the light curves as seen from the twoobservatories, π E = π E µ rel µ rel , (3)where µ rel is the lens-source relative proper motion and π E = AU D ⊥ (∆ τ, ∆ β ±± ) , (4)where∆ τ = t , sat − t , ⊕ t E ; ∆ β ±± = ± u , sat − ± u , ⊕ . (5)Here t is the time of the closest source approach to thelens (peak time of the event) and u is the separationbetween the lens and the source at time t (impact pa-rameter). The subscripts of “sat” and “ ⊕ ” indicate theparameters as measured from the satellite and Earth, re-spectively. Thus, ∆ τ and ∆ β represent the difference in t and u as measured from the two observatories. Par-allax measurements made by such comparisons betweenthe light curves are subject to a well-known four-fold de-generacy, which comes from four possible values of ∆ β in-cluding (+ u , sat , ± u , ⊕ ) and ( − u , sat , ± u , ⊕ ). However,there is only a two-fold degeneracy in the amplitude of π E because ∆ β −− = − ∆ β ++ and ∆ β − + = − ∆ β + − . Theonly exception to the four-fold degeneracy would be ifone of the two observatories has u consistent with zero,while the other has u inconsistent with zero. In thiscase, the four-fold degeneracy reduces to a two-fold de-generacy. For example, if u , sat = 0 (within errors), then∆ β + , + = ∆ β − , + → ∆ β , + (and similarly for ∆ β , − ).Then, since ∆ β , − = − ∆ β , + , there is no degeneracyin the mass (See e.g., Gould & Yee 2012). This case isvery important for point-lens mass measurements sincethe lens always passes very close to or over the source asseen by one observatory, so u (cid:39)
0, whether or not it isstrictly consistent with zero.Here we report the fifth isolated-star measurement de-rived from microlensing measurements of ρ and π E . Incontrast to the previous four measurements, this one hasa discrete degeneracy in ρ and therefore in mass. Wetrace the origin of this degeneracy to the fact that onlya single point is affected by finite-source effects, and weargue that it may occur frequently in future space-basedmicrolensing mass measurements, including BDs. Weshow how this degeneracy can be broken by future high-resolution imaging, regardless of whether the lens is darkor luminous. This paper is organized as follows. In Sec-tion 2, the observation of the event OGLE-2015-BLG-1482 is summarized, and we describe the analysis of thelight curve in Section 3. With the results of Section 3,we derive physical properties of the source and lens inSection 4 and then we discuss the results in Section 5.Finally, we conclude in Section 6. OBSERVATIONS
Ground-based observations
The gravitational microlensing event OGLE-2015-BLG-1482 was discovered by the Optical GravitationalLensing Experiment (OGLE) (Udalski 2003), and it wasalso observed by
Spitzer and Korea Microlensing Tele-scope Network (KMTNet, Kim et al. 2016). The mi-crolensed source star of the event is located at ( α , δ )= (17 h m s . , − ◦ (cid:48) . (cid:48)(cid:48)
3) in equatorial coordinatesand ( l , b ) = (358 . ◦ , − . ◦
92) in Galactic coordinates.OGLE observations were carried out using a 1.3 mWarsaw telescope with a field of view of 1.4 square de-grees at the Las Campanas Observatory in Chile. Theevent lies in the OGLE field BLG534 with a cadence ofabout 0 . − in I band. The Einstein timescale is quiteshort, t E ∼ −
12 minute cadence at the three sites and the expo-sure time was 60 s. The CTIO, SAAO, and SSO obser-vations were made in I -band filter, and for determiningthe color of the source star, the CTIO observations witha typical good seeing were also made in V -band filter.Thus, the light curve of the event was well covered bythe three KMTNet observation data sets. The KMT-Net data were reduced by the Difference Image Analysis(DIA) photometry pipeline (Alard & Lupton 1998; Al-brow et al. 2009). Spitzer observationsSpitzer observations in 2015 were carried out underan 832-hour program whose principal scientific goal wasto measure the Galactic distribution of planets (Gouldet al. 2014). The event selection and observational ca-dences were decided strictly by the protocols of Yee etal. (2015b), according to which events could be selectedeither “subjectively” or “objectively”. Events that meetspecified objective criteria must be observed accordingto a specified cadence. In this case all planets discov-ered, whether before or after
Spitzer observations aretriggered, (as well as all planet sensitivity) can be in-cluded in the analysis. Events that do not meet thesecriteria can still be chosen “subjectively”. In this case,planets (and planet sensitivity) can only be included inthe Galactic-distribution analysis based on data that be-come available after the decision. Like objective events,events selected subjectively must continue to be observed according to the specified cadence and stopping criteria(although those may be specified as different from thestandard, objective values at the time of selection).Because the current paper is not about planets orplanet sensitivities, the above considerations play no di-rect role. However, they play a crucial indirect role. Fig-ure 1 shows that despite the event’s very short timescale t E ∼ Spitzer slightly before it peaked from Earth, ob-servations began about 1 day prior to the peak. This isremarkable because, as discussed in detail by Udalski etal. (2015) (see their Figure 1), there is a delay betweenthe selection of a target and the start of the
Spitzer obser-vations. Targets can only be uploaded to the spacecraftonce per week, and it takes some time to prepare thetarget uploads. Therefore,
Spitzer observations begin aminimum of three days after the final decision is made toobserve the event with
Spitzer , and that decision is gen-erally based on data taken the night before, i.e. aboutfour days prior to the first
Spitzer observations. Hence,at the time that the decision was made to observe OGLE-2015-BLG-1482, the source was significantly outside theEinstein ring. It is notoriously difficult to predict the fu-ture course of such events. Therefore, such events cannotmeet objective criteria that far from the peak, but se-lecting them“subjectively” would require a commitmentto continue observing them for several more weeks of thecampaign, which risks wasting a large number of observa-tions if the event turns out to be very low-magnificationwith almost zero planet sensitivity (the most likely sce-nario). At the same time, if the event timescale is short,it could be over before the next opportunity to start ob-servations with
Spitzer ( 10 days later)Hence, Yee et al. (2015b) also specified the possibilityof so-called “secret alerts”. For these, an observationalsequence would be uploaded to
Spitzer for a given week,but no announcement would be made. If the event lookedpromising later (after upload), then it could be chosensubjectively. In this case,
Spitzer data taken after thepublic alert could be included in the parallax measure-ment (needed to enter the Galactic-distribution sample)but
Spitzer data taken before this date could not. Ifthe event was subsequently regarded as unpromising, itwould not be subjectively alerted, in which case the ob-servations could be halted the next week without violat-ing the Yee et al. (2015b) protocols.This was exactly the case for OGLE-2015-BLG-1482(see Figure 1). It was “secretly” alerted at the uploadfor observations to begin at HJD (cid:48) = HJD-2450000 =7206.73. It was only because of this secret alert that anobservation was made near peak, which became the basisfor the current paper. In fact, its subsequent rise was sofast (due to its short timescale) that it was subjectivelyalerted just prior to the near-peak
Spitzer observation.At the next week’s upload, it met the objective criteria.Note, however, that if we had waited for the event to be-come objective before triggering observations, we wouldnot have been able to make the mass measurement re-ported here, even though the planet sensitivity analysiswould have been almost identically the same (providedthat parallax could still be measured with the remaining
Spitzer observations). This is the first
Spitzer microlens-ing event for which a “secret alert” played a crucial role.
Spitzer observations were made in 3 . µ m channel on Chung et al. the IRAC camera from HJD (cid:48) = HJD - 2450000 = 7206.73to 7221.04. The data were reduced using specialized soft-ware developed specifically for this program (Calchi No-vati et al. 2015). Even though the
Spitzer data are rel-atively sparse, there is one point near the peak, whichproves to be essential to determine the normalized sourceradius ρ . LIGHT CURVE ANALYSIS
Event OGLE-2015-BLG-1482 was densely, and almostcontinuously covered by ground-based data, but showedno significant anomalies (See Figure 1). This has twovery important implications. First, it means that theground-based light curve can be analyzed as a point lens.Second, it implies that it is very likely (but not absolutelyguaranteed) that the
Spitzer light curve can likewise beanalyzed as a point lens. The reason that the latter con-clusion is not absolutely secure is that the
Spitzer andground-based light curves are separated in the Einsteinring by ∆ β ∼ .
15. Thus, even though we can be quitecertain that the ground-based source trajectory did notgo through (or even near) any caustics of significant size,it is still possible that the source as seen from
Spitzer didpass through a significant caustic, but that this causticwas just too small to affect the ground-based light curve.Nevertheless, since the closest
Spitzer point to peakhas impact parameter u spitzer ∼ .
06 and it is quiterare for events to show caustic anomalies at such sep-arations when there are no anomalies seen in denselysampled data u > .
15, we proceed under the assump-tion that the event can be analyzed as a point lens fromboth Earth and
Spitzer . Thus, we conduct the singlelens modeling of the observed light curve by minimizing χ over parameter space. For the χ minimization, weuse the Markov Chain Monte Carlo (MCMC) method.Thanks to the simultaneous observation from the Earthand satellite, we are able to measure the microlens par-allax π E = ( π , N + π , E ) / , which are the north andeast components of the parallax vector π E , respectively.The Spitzer light curve has a point near the peak of thelight curve, and thus we can also measure the normal-ized source radius ρ . Hence, we put three single lensingparameters of t , u , and t E , the parallax parameters of π E , N and π E , E , and the normalized source radius ρ as freeparameters in the modeling. In addition, there are twoflux parameters for each of the 5 observatories ( Spitzer ,OGLE, KMT CTIO, KMT SAAO, KMT SSO). One rep-resents the source flux f s,i as seen from the i th observa-tory, while the other, f b,i is the blended flux within theaperture that does not participate in the event. That is,the five observed fluxes F i ( t j ) at epochs t j are simulta-neously modeled by F i ( t j ) = f s,i A i ( t j ; t , u , t E , ρ, π E ) + f b,i , (6)where A i ( t ) is the magnification as a function of timeat the i th observatory. In principle, these magnificationsmay differ because the observatories are at different loca-tions. However, in this event the separations of the obser-vatories on Earth are so small compared to the projectedsize of the Einstein ring that we ignore them and considerall Earth-based observations as being made from Earthscenter. That is, we ignore so-called “terrestrial paral-lax”. At the same time, the distance between the Earth and Spitzer remains highly significant, so A Spitzer ( t ) isdifferent from A Earth ( t ). As is customary (e.g., Dong etal. 2007; Udalski et al. 2015; Yee et al. 2015a), we de-termine the parameters in the “geocentric” frame at thepeak of the event as observed from Earth (Gould 2004),and likewise adopt the sign conventions shown in Figure4 of Gould (2004).In addition, we conduct the modeling for the point-source/point-lens, because only a single point of Spitzer contributes to the finite-source effect. We find that the∆ χ between the best-fit models of the point- and finite-sources is ∆ χ = 31 .
47. Hence, OGLE-2015-BLG-1482strongly favors the finite-source model.
Limb Darkening
As we will show, the lens either transits or passes veryclose to the source as seen by
Spitzer , which inducesfinite-source effects near the peak of the
Spitzer lightcurve. To account for this, we adopt a limb-darkenedbrightness profile for the source star of the form S λ ( θ ) = ¯ S λ (cid:20) − Γ (cid:18) −
32 cos θ (cid:19)(cid:21) , (7)where ¯ S λ ≡ F S,λ / ( πθ (cid:63) ) is the mean surface brightness ofthe source, F S,λ is the total flux at wavelength λ , Γ is thelimb darkening coefficient, and θ is the angle between thenormal to the surface of the source star and the line ofsight (An et al. 2002). Based on the estimated color andmagnitude of the source, which is discussed in Section 4,assuming an effective temperature T eff = 4500 K, solarmetallicity, surface gravity log g = 0 .
0, and microturbu-lent velocity v t = 2 km/s, we adopt Γ . µ m = 0 .
178 fromClaret & Bloemen (2011). (2 ×
2) = 4 highly degenerate solutions
As discussed in Section 1, space-based parallax mea-surements for point lenses generically give rise to foursolutions, which can be highly degenerate. However, incases for which one of two observations has u (cid:39)
0, whilethe other has u (cid:54) = 0, the four solutions reduce to two so-lutions. Since for event OGLE-2015-BLG-1482, Spitzer has u , sat (cid:39)
0, we expect the event to have two degen-erate solutions, u , ⊕ > u , ⊕ <
0. However, whatwe see in Table 1 is not two degenerate solutions butfour. For each of the two expected degenerate solutions[(+ , , ( − , ρ ( ρ (cid:39) .
06 and ρ (cid:39) . Spitzer data sets. The best-fit solution is (+ ,
0) solution for ρ (cid:39) .
06, which means u , ⊕ > u , sat (cid:39)
0. The biggest ∆ χ between thefour solutions is ∆ χ (cid:39) . Spitzer are in rectilinear motion and 2) they have zero relativeprojected velocity (Gould 1995). For events that arevery short compared to a year (like this one), the ap-proximation of rectilinear motion is excellent. And whileEarth and
Spitzer had relative projected motion of or-der v ⊕ ∼
30 km s − , this must be compared to the lens-source projected velocity ˜ v ,˜ v ≡ AU π E t E (cid:39) − . (8)Hence, these two solutions are almost perfectly degener-ate. On the other hand, the ρ degeneracy was completelyunexpected. It is also very severe. The origins of the ρ degeneracy are discussed in Section 5. To illustrate the ρ degeneracy, the light curve of the best-fit model (+ , ρ (cid:39) .
09 is also presented in Figure 1. In Table 1, wepresent the parameters of all the four solutions. PHYSICAL PROPERTIES
Source properties
The color and magnitude of the source are estimatedfrom the observed ( V − I ) source color and best-fit mod-eling of the light curve, but they are affected by ex-tinction and reddening due to the interstellar dust alongthe line of sight. The dereddened color and magnitudeof the source can be determined by comparing to thecolor and magnitude of the red clump giant (RC) un-der the assumption that the source and RC experiencethe same amount of reddening and extinction (Yoo etal. 2004). Figure 2 shows the instrumental KMT CTIOcolor-magnitude diagram (CMD) of stars in the observedfield. The color and magnitude of the RC are obtainedfrom the position of the RC on the CMD, which cor-respond to [( V − I ) , I ] RC = [1 . , . V − I ) RC , = 1.06 (Bensby et al. 2011) and I RC , = 14.50 (Natafet al. 2013). The instrumental source color obtainedfrom a regression is ( V − I ) s = 1 .
74 and the magni-tude of the source obtained from the best-fit model is I s = 17 .
37. The measured offset between the source andthe RC is [∆( V − I ) , ∆ I ] = [0 . , . I kmt − I ogle = 0 . V − I ) , I ] s , = [1 . , . V − K ) source color by using the color-color relationof Bessell & Brett (1988) is ( V − K ) s , = 2 .
61. Thenadopting ( V − K ) s , to the the color-surface brightnessrelation of Kervella et al. (2004), we determine the sourceangular radius θ (cid:63) = 5 . ± . µ as. The estimated colorand magnitude of the source suggest that the source is aK type giant. The error in θ (cid:63) includes the uncertainty inthe source flux, the uncertainty in the conversion fromthe observed ( V − I ) color to the surface brightness, andthe uncertainty of centroiding the RC. The uncertaintyin the source flux is about 1% and the uncertainty of themicrolensing color is 0 .
02 mag, which contributes 1 . θ (cid:63) measurement. The scatter of the source an-gular radius relation in ( V − K ) s , is 5% (Kervella &Fouqu´e 2008), and centroiding the RC contributes 4% tothe radius uncertainty (Shin et al. 2016).As mentioned above, since the degeneracy between twodifferent ρ solutions is very severe as ∆ χ (cid:46) .
3, weshould consider both ρ solutions. The two ρ values yieldtwo different Einstein radii, θ E = θ (cid:63) /ρ = (cid:40) . ± .
022 mas for ρ (cid:39) . . ± .
006 mas for ρ (cid:39) . . (9) Because of the two different Einstein radii, all the phys-ical parameters related to the lens take on two discretevalues. The relative proper motions of the lens andsource are, µ rel = θ E /t E = (cid:40) . ± .
88 mas yr − for ρ (cid:39) . . ± .
48 mas yr − for ρ (cid:39) . . (10) Lens properties
The mass and distance of the lens can be obtained fromthe measured Einstein radius θ E and microlens parallax π E . As discussed in the introduction, the four-fold de-generacy in π E usually leads to a two-fold degeneracy inits amplitude π E . However, in the case of events that aremuch higher magnification (much lower u ) as seen fromone observatory than the other, the two-fold degeneracycollapses as well. This is because, under these conditions, | ∆ β ±± | (cid:39) | ∆ β ±∓ | . The present case is consistent withthe lens passing exactly over the center of the source asseen by Spitzer (to our ability to measure it). Then,according to Equation (1), we measure the lens mass, M = θ E κπ E = (cid:40) . ± . M (cid:12) for ρ (cid:39) . . ± . M (cid:12) for ρ (cid:39) . . The lens-source relative parallax for the two cases is π rel = θ E π E = (cid:40) . ± .
003 mas for ρ (cid:39) . . ± .
001 mas for ρ (cid:39) . . (11)These values of π rel are very small compared to the sourceparallax π s ∼ .
12 mas. This implies that the distancebetween the lens and the source is determined much moreprecisely than the distance to the lens or the source sep-arately. That is, D LS ≡ D S − D L = π rel AU D S D L (12) (cid:39) (cid:40) . ± .
19 kpc for ρ (cid:39) . . ± .
08 kpc for ρ (cid:39) . . Since the source is almost certainly a bulge clump star(from its position on the CMD), and the lens is (cid:46) DISCUSSION
Future Resolution of the ρ Degeneracy UsingAdaptive Optics
Event OGLE-2015-BLG-1482 has a very severe two-fold degeneracy in ρ , in which the ∆ χ between the twosolutions ( ρ (cid:39) .
06 and ρ (cid:39) .
09) is ∆ χ ∼ .
3. For thesolutions with u , ⊕ > u , ⊕ <
0, the microlens par-allax vectors π E are different from one another, but theyhave almost the same amplitude π E . Therefore, the twosolutions yield almost the same physical parameters ofthe lens. However, each of the two solutions also has twodegenerate ρ solutions: ρ (cid:39) .
06 and ρ (cid:39) .
09. Each ρ solution yields different physical parameters of the lens, Chung et al. in particular the lens mass. For ρ (cid:39) .
06, the lens isa very low-mass star, while for ρ (cid:39) .
09 it is a browndwarf. The degeneracy of the lens mass due to the two ρ can be resolved from direct lens imaging by using instru-ments with high spatial resolution (Han & Chang 2003;Henderson et al. 2014), such as the VisAO camera of the6.5m Magellan telescope with the resolution ∼ . (cid:48)(cid:48)
04 inthe J band (Close et al. 2013) and the GMTIFS of the24.5 m Giant Magellan Telescope (GMT) with resolution ∼ . (cid:48)(cid:48)
01 in the NIR (McGregor et al. 2012). In general, di-rect imaging requires 1) that the lens be luminous, and 2)that it be sufficiently far from the source to be separatelyresolved. In the present case, (1) clearly fails for the BDsolution. Hence, the way that high-resolution imagingwould resolve the degeneracy is to look for the luminous(but faint) M dwarf predicted by the other solution atits predicted orientation (either almost due north or duesouth of the source – since | π E , N | (cid:29) | π E , E | ) and with itspredicted separation ( t AO − × (9 mas yr − ). If theM dwarf fails to appear at one of these two expected po-sitions, the BD solution is correct. Since the source is aclump giant, and hence roughly 10 times brighter thanthe M dwarf, it is likely that the two cannot be separatelyresolved until they are separated by at least 2.5 FWHM.This requires to wait until 2015 + 2 . × (40 /
9) = 2026 forMagellan or 2015 + 2 . × (10 /
9) = 2018 for GMT.
Origin of the ρ Degeneracy
The degeneracy in ρ was completely unexpected. In-deed we discovered it accidentally because ρ had onevalue in one of two degenerate parallax solutions andthe other value in another one. Originally, this led usto think that it was somehow connected to the paral-lax degeneracy. However, by seeding both solutions withboth values of ρ we discovered that it was completelyindependent of the parallax degeneracy.In retrospect, the reason for this degeneracy is “obvi-ous”. There is only a single point that is strongly im-pacted by the finite size of the source. The value of u atthis time is well predicted by the rest of the light curve,in particular because Spitzer data begin before peak (seeSection 2), u = (cid:113) τ + u w here τ = ( t − t , sat ) t E . (13)Hence, the magnification (for point-lens/point-source ge-ometry in a high magnification event) is also known A ps (cid:39) /u . Moreover, both f s and f b for Spitzer are alsowell measured, so that the measured flux at the near-peak point F directly yields an empirical magnification A obs = ( F − f b ) /f s (i.e. the magnification in the pres-ence of finite-source effects). Following Gould (1994a),the ratio of A obs and A ps can therefore be derived di-rectly from the light curve B ( z ) ≡ A obs A ps (cid:39) A obs u. ( z ≡ u/ρ ) (14)As shown by Figure 1 of Gould (1994a), B ( z ) reaches Close et al. (2014) have obtained a diffraction limited FWHMin ground-based 6m R band images, which gives hope for opticalAO. However, it is premature to claim that this technique can beapplied to faint stars in the Galactic bulge a peak at z (cid:39) .
91, with B = 1 . Therefore, if oneinverts a measurement of B ( z ) to infer a value of z , thereare respectively one, two and zero solutions for B obs < < B obs < .
34, and B obs > . Spitzer , i.e., the finite-source effect is only seen by
Spitzer , only the trajectory of
Spitzer is considered. Fig-ure 3 (adapted from Gould 1994a) shows the finite-source effect function B ( z ) as a function of z . For thisevent, B ( z ) = A obs u = 19 . × .
06 = 1 .
15 at the nearestpoint to the peak, which is indicated by the horizontaldotted line in the figure. As shown in Figure 3, the func-tion B ( z ) = 1 .
15 is satisfied at two different values of z = 0 .
64 and z = 1 .
12, which implies (as outlined above)that there are two ρ values. The two z values yield twonormalized source radii of ρ = 0 .
094 (for z = 0 .
64) and ρ = 0 .
054 (for z = 1 . ρ values arealmost the same as those obtained numerically from thebest-fit solutions. Because high-magnification events canbe alerted in real time, the high-magnification events ob-served from Earth are often well covered around the peakby intensive follow-up observations, and thus ρ is almostalways well measured if there are significant finite-sourceeffects (i.e., B (cid:54) = 1 for some points). This means that the ρ degeneracy will often be resolved in high-magnificationevents observed from the ground. On the other hand,since the observation cadence of Spitzer is much lowerthan those of ground-based observations, the ρ degen-eracy can occur frequently in high-magnification eventsobserved by Spitzer . Note that, in contrast to Figure 1 ofGould (1994a), our Figure 3 shows B ( z ) with and with-out the effects of limb darkening. The effect is hardlydistinguishable by eye, in particular because limb dark-ening at 3 . µ m is very weak. Nevertheless this effectshould be included.If finite-source effects are reliably detected from a sin-gle measurement near peak, how often will ρ be ambigu-ous, and if it is ambiguous, how often will the value fall inthe upper versus lower allowed ranges? We might judgethere to be a“reliable detection” of finite-source effectsfrom a single point if | B − | > X , where X might betaken as 5%. For high-magnification events including thelimb darkening effect, we can Taylor expand B for z > B ( z ) = 1 + 18 (cid:18) − Γ5 (cid:19) z + 364 (cid:18) − (cid:19) z + . . . , (15)where Γ is the limb darkening coefficient, as mentionedin Section 3. Truncating at the second term, we have B ( z ) (cid:39) − Γ / / (8 z ). For Spitzer Γ / (cid:28)
1, so wecan ignore it. Then B ( z ) = 1+1 / (8 z ). Thus, B − X ,i.e., B = 1 + X , implies z = (1 / (8 X )) / → . X =5%). To next order, z = (4 / √ X − − / =1 .
685 which is very close to the numerical value, 1.7.Hence, we have three ranges of recognizable finite-sourceeffects. The ranges are presented in Table 3. Table 3shows that 0 . / (0 .
51 + 0 .
34 + 0 .
79) = 31% of the finite-source effects will be unambiguous. And of the timesthey are ambiguous 0 . / (0 .
34 + 0 .
79) = 30% will have While Figure 1 from Gould (1994a) shows the correct quali-tative behavior, it has a quantitative error in that the peak is at1.25, rather than 1.34 (the correct value) the higher value of ρ .Figure 4 shows the χ distribution of u , sat versus ρ from the MCMC chains of the four degenerate solutionsin Table 1. The figure shows that the distribution iscentered on u , sat = 0 . u , sat = 0 .
0, although there is scatter.Therefore, it is correct to label u , sat as “0”. The fig-ure also shows that the nearest point to the peak of Spitzer light curve favors u , sat = 0, but can accom-modate other values of u , sat , up to about 0.03 at < σ .In this case, the bigger u , sat makes B ( z ) bigger because B ( z ) = uA obs , and so allowing values of z between thetwo best-fit values. At the nearest point to the peak, τ = | ( t − t ,sat ) | /t E = | (7207 . − . | / .
26 = 0 . B ( u , sat = 0 . B ( u , sat = 0) = u ( u , sat = 0 . u ( u , sat = 0)= (cid:114) . + 0 . . = 1 .
12 (16)Since B ( u , sat = 0) = 1 .
15 from Figure 3, B ( u , sat =0 .
03) = 1 . × .
12 = 1 .
29, and it is the maximum valueallowed B , and thus the maximum allowed u , sat . Theallowed maximum B ( z ) = 1 .
29 yields z = 0 .
79 and z =0 .
98 and hence two ρ values, ρ = 0 .
085 and ρ = 0 . ρ (cid:39) .
09 solutions have the lower limit of ρ =0 . ρ (cid:39) .
06 solutions have the upper limit of ρ = 0 . ρ degeneracy of OGLE-2015-BLG-0763 OGLE-2015-BLG-0763 is the only other event with asingle lens mass measurement based on finite-source ef-fects observed by
Spitzer (Zhu et al. 2016). As withOGLE-2016-BLG-1482, the
Spitzer light curve showsonly one point that is strongly affected by finite-sourceeffects (i.e., B (cid:54) = 1). Zhu et al. (2016) report ρ =0 . t E = 33 days and their solution implies t , sat (cid:39) .
60 and u , sat = 0 . t − .
60 = ( − . , . , .
72) days, have respectively, u = (0 . , . , . z ≡ u/ρ = 0 .
73. Inspection of Figure 3 showsthat this implies B ( z ) = 1 .
25, which (since
B > z = 1 .
01 andtherefore with ρ = 0 . z = (1 . , . , .
15) (adopted) and z = (1 . , . , .
57) (other). These yield values of B (from Figure 3) of B ( z ) = (1 . , . , .
14) (adopted)and B ( z ) = (1 . , . , .
06) (other). That is, for OGLE-2015-BLG-0763, the two nearest points to the peak willboth be about 0.08 mag brighter in the adopted solu-tion than in the other solution. Since the
Spitzer pho-tometric errors are small compared to these inferred dif-ferences (Figure 2 of Zhu et al. 2016), we expect that,in the case of OGLE-2015-BLG-0763 (and in contrastto OGLE-2015-BLG-1482), the near-peak points resolvethe degeneracy between the two solutions.Armed with the above understanding, which wasderived without any detailed modeling, we reanalyzeOGLE-2015-BLG-0763 and find only an upper limit of0.01 for the second ρ . However, as discussed in Zhu et al. (2016), solutions of the second ρ result in inconsistencywith observations, and thus they are not physically cor-rect. As a result, there is no ρ degeneracy for the eventOGLE-2015-BLG-0763. As mentioned before, this is be-cause of the near-peak points. This implies that althoughfor events in which the finite-source effect is seen only inthe Spitzer the ρ degeneracy can occur frequently due tolow observation cadence of the Spitzer , it can be resolvedby a few data points near the peak.
Error analysis in ρ measurement The error in the ρ measurement of the event OGLE-2015-BLG-1482 is 19 .
8% for ρ (cid:39) .
06 and 6 .
6% for ρ (cid:39) .
09. These errors are quite big relative to mea-surements in high-magnification events from the ground.We therefore study the source of these errors in ρ bothto determine why they are so different and to make surethat we are properly incorporating all sources of error inthis measurement.As outlined above, the train of information is basicallycaptured by ρ = u/z ( B ) where z ( B ) is the inverse of B ( z ) and both u and B can be regarded approximately as“measured” quantities. It is instructive to further expandthis expression ρ = uz ( A obs u ) . (17)In this form, it is clear that the contribution from anerror in determining u tends to be suppressed if z (cid:48) ≡ dz/dB > z < .
91, so ρ (cid:39) .
09 in our case),and it tends to be enhanced if z (cid:48) <
0. Hence, thisfeature of Equation (17) goes in the direction of ex-plaining the larger error in the ρ (cid:39) .
06 case. Sec-ond, if we for the moment ignore the error in u , thenEquation (17) implies σ (ln ρ ) = | z (cid:48) /z | σ ( A obs ). For thetwo cases, ρ = (0 . , . z = (0 . , . z (cid:48) = (0 . , − .
18) and so | z (cid:48) /z | = (1 . , . ρ (larger z ) solution. This is intuitively clear from Fig-ure 3: the shallow slope of B ( z ) toward large z makes itdifficult to estimate z from a measurement of B . Hence,the fact that the fractional error in ρ is much larger forthe large z (small ρ ) solution is well understood.Ignoring blending, we can write A obs = F/f s . Theerror in F (i.e., the flux measurement at the high point)is uncorrelated with any other error. Since in our case, u ,spitzer (cid:39)
0, we can write u = ( t − t ) /t E , and so B = A obs u = | t − t | Ff s t E . (18)Since t is known extremely well, and t is known essen-tially perfectly, there would appear to be essentially noerror in | t − t | . The denominator is a near-invariant inhigh-magnification events (Yee et al. 2012). That is, theerrors in this product are generally much smaller than theerrors in either one separately. This means that the errorin B (and so z ( B )) is dominated by the flux measurementerror of the single point that is affected by finite-sourceeffects.Nevertheless, it is important to recognize that Yee etal. (2012) derived their conclusion regarding the invari-ance of f s t E under conditions that the error in f s is com-pletely dominated by the model, and not by the fluxmeasurement errors. Indeed, as a rather technical, but Chung et al. very relevant point, it is customary practice to ignorethe role of flux measurement errors in the determina-tion of f s . That is, f s and f b are normally not includedas chain variables when modeling microlensing events.Instead, the magnification is determined at each pointalong the light curve from microlens parameters that arein the chain, and then the two flux parameters (fromeach observatory) are determined from a linear fit. Thisis a perfectly valid approach for the overwhelming major-ity of microlensing events because the errors arising fromthis fit (which are returned but usually not reported fromthe linear fit routine) are normally tiny compared to theerror in f s due to the model. Moreover, there are usuallymany observatories contributing to the light curve, andif all the flux parameters were incorporated in the chain,it would increase the convergence time exponentially.However, in the present case t E is essentially deter-mined from ground-based data, which are both numerousand very high precision, while f s is determined from just16 Spitzer points (i.e., all the points save the one nearpeak). If the usual (linear fit) procedure were applied, itwould seriously underestimate the error in f s and so over-estimate its degree anticorrelation with t E . We thereforeinclude ( f s , f b ) spitzer as chain parameters and remodelthis event. The result of the remodeling is presented inTable 2. By comparing to runs in which we treat theseflux parameters in the usual way, we find that includ-ing these parameters in the chain contributes about 41%to the ρ error compared to all other sources of ρ errorcombined. That is, in the end, this does not dramati-cally increase the final error, since (1 + 0 . ) / = 1 . f s , f b ) spitzer in aformally proper way since this contribution could easilybe the dominant one in other cases. Impact of the ρ Degeneracy
The ρ degeneracy was not realized until now for severalreasons. First of all, although single lens finite-sourceevents have been routinely detected from ground-basedobservations, they are not scientifically very interestingwithout the measurement of π E . However, π E mea-surements of single lens events based on ground-baseddata alone are intrinsically rare and technically difficult(Gould & Yee 2013). Second, prior to the establish-ment of second-generation microlensing surveys, observa-tions of high-magnification microlensing events were usu-ally conducted under the survey+followup mode, whichwas first suggested by Gould & Loeb (1992). High-magnification events with their nearly 100% sensitivityto planets (Griest & Safizadeh 1998) were therefore of-ten followed up with intensive ( ∼ ρ degeneracy, ifit exists.The ρ degeneracy is nevertheless important for thescience of second-generation ground-based and futurespace-based microlensing surveys. The majority ofevents found by these surveys will not be followed upat all, and thus the ρ degeneracy can appear because thetypical source radius crossing time, t (cid:63) , is comparable to the observing cadences that these surveys adopt. Here t (cid:63) ≡ θ (cid:63) µ rel = 45 min (cid:18) θ (cid:63) . µas (cid:19) (cid:18) µ rel − (cid:19) − , (19)where 0 . µ as is the angular source size of a Sun-like starin the Bulge, and 7 mas yr − is the typical value forlens-source relative proper motion of disk lenses. Forsecond-generation microlensing surveys like OGLE-IVand KMTNet, although a few fields are observed onceevery <
20 min, the majority of fields are observed at > ρ degeneracy appears.Fortunately, however, the result of event OGLE-2015-BLG-0763 showed that a few additional data points (be-fore/after crossing the source) around the peak play acrucial role in resolving the ρ degeneracy. When we ob-serve typical microlensing events with a cadence of 1 hr,we can obtain 2 more data points right before and af-ter crossing the source, except one source-crossing datapoint. In this case, the ρ degeneracy will be resolved asin the event OGLE-2015-BLG-0763. This implies that 1hr is the upper limit of the observing cadence to resolvethe ρ degeneracy in typical single lens events to be ob-served from the second-generation ground-based surveys,whereas for events with high µ rel , such as events causedby a fast moving lens object or a high-velocity sourcestar, it is not enough to resolve the ρ degeneracy.Since about half of KMTNet fields have (cid:54) (cid:62) . ρ degeneracy will be resolved in the majority of singlehigh-magnification events to be observed by KMTNet.Although π E is still intrinsically hard to measure evenwith second-generation surveys, the fraction of eventswith finite-source effects can be used as an indicator ofthe properties of the lens population, which is especiallyimportant for validating the short-timescale events suchas the population of free-floating planets (FFPs) (Sumiet al. 2011). The Wide-Field InfraRed Survey Telescope ( WFIRST )is likely to have six microlensing campaigns, each with72 days observing a ∼ microlensing field at 15min cadence (Spergel et al. 2015). WFIRST microlens-ing is expected to detect thousands of bound planetsand hundreds of FFPs. At first sight, the 15 min ca-dence that
WFIRST microlensing is currently adoptingsuggests that it will not be affected by the ρ degener-acy. However, since WFIRST will go much fainter thanground-based surveys, most of the sources for
WFIRST events will be M dwarfs, which are a half or even a quar-ter the size of Sun. Then the typical t (cid:63) for WFIRST events is ∼
15 min, which is the same as the adoptedcadence. Hence, as mentioned above, although the ρ de-generacy can be usually resolved by obtaining more than2 data points around the peak, it will be severe for asignificant fraction of events with high µ rel . What makesthis ρ degeneracy more important for WFIRST is that π E can be measured relatively easily once ground-basedobservations are taken simultaneously (Yee 2013; Zhu& Gould 2016). Therefore, the degeneracy in ρ will di-rectly lead to a degeneracy in the mass determination ofisolated objects, including FFPs, BDs, and stellar-massblack holes. Potential for the second body
As discussed in Section 3, it is possible that the de-viation of the highly magnified
Spitzer point might becaused by a caustic structure rather than being entirelydue to finite-source effects. It is easy to show qualita-tively that this could affect the exact nature of the sys-tem, but is unlikely to significantly change the conclusionthat the lens is a low-mass object in the bulge. First, ifthere were a caustic perturbation, there would be a sec-ond body in the lens system. However, we do not seeany evidence for the second body in the ground-basedlight curve, and therefore the dominant lensing effect stillcomes from a single star. Second, consider the effect onthe inferred physical properties of the lens (e.g., massand distance). If there were a caustic structure, then itis likely to be small since it does not affected the ground-based data. In that case ρ would be smaller and therefore θ E would be larger. At the same time, t E is clearly de-termined from the dense, ground-based observations, soif θ E is larger, µ rel must also be larger. However, µ rel is already 9 mas yr − for the smaller ρ solution. Largervalues of µ rel are increasingly improbable and will eventu-ally become unphysical. Hence, OGLE-2015-BLG-1482is likely an event caused by the single lens star.However, since a binary lens system could simulta-neously reproduce the single lens-like light curve fromground-based observations and the poorly sampled lightcurve from Spitzer , we conduct binary lens modeling.As a result, we find that the best-fit binary lens solu-tion is the BD binary lens system composed of a pri-mary star M L , = 0 . ± . M (cid:12) and a secondary star M L , = 0 . ± . M (cid:12) with their projected separation19 AU, which correspond to lensing parameters of themass ratio between the binary components q = 0 .
78 andthe projected separation in units of θ E of the lens system s = 24. The estimated distance to the BD binary is 7.5kpc, and thus it is also located in the Galactic bulge. Al-though χ of the binary lens model is smaller than thatof the single lens model by 35, it is a very wide binarysystem with large ρ = 0 . ρ (cid:39) . Spitzer is explained by finite-source effects on the tiny caustic ofthe very wide BD binary that “replaces” the point caus-tic of the point lens in the single lens solution. But the χ improvement for the binary solution comes entirely fromground-based data, while the χ of the Spitzer becomesslightly worse than that of the single lens model (see Fig-ure 1). Thus, the
Spitzer high point is not caused by thebinary, as we originally sought to test. The χ improve-ment could in principle be due to a distant companion.However, low-level systematics can also easily produce∆ χ = 35 improvements in microlensing light curves,which could then mistakenly be attributed to planets,binaries, etc. For this reason, Gaudi et al. (2002) andAlbrow et al. (2001) already set a threshold at ∆ χ (cid:62) Spitzer is actually explainedby the caustic of a binary, we also conduct binary lensmodeling in which ρ ∼ .
0. From this, we find that thereis no valid binary lens solution with small ρ . This isbecause although we find two solutions with better χ relative to the single lens model, the best fit lens-sourcerelative proper motions are µ rel = 177 mas yr − and µ rel = 583 mas yr − for the ρ = 0 . ρ = 0 . ρ = 0 . M L , =1 . ± . M (cid:12) and a planet M L , = 1 . ± . M Jupiter with their projected separation 9.3 AU, while for theother solution (for ρ = 0 . M L , = 5 . ± . M (cid:12) anda planet M L , = 5 . ± . M Jupiter with the separa-tion 14.7 AU, and these binaries are respectively locatedat 3.7 kpc and 1.6 kpc. The very large proper motion isdue to large θ E , while t E is clearly determined from denseground-based observations, as mentioned in the previousparagraph. Moreover, the χ of the Spitzer data for thetwo binary lens models becomes worse. CONCLUSION
We analyzed the single lens event OGLE-2015-BLG-1482 simultaneously observed from two ground-basedsurveys and from
Spitzer . The
Spitzer data exhibit thefinite-source effect due to the passage of the lens directlyover the surface of the source star as seen from
Spitzer .Thanks to the finite-source effect and the simultaneousobservation from Earth and
Spitzer , we were able to mea-sure the mass of the lens. From this analysis, we foundthat the lens of OGLE-2015-BLG-1482 is a very low-massstar with the mass 0 . ± . M (cid:12) or a brown dwarfwith the mass 55 ± M J , which are respectively locatedat D LS = 0 . ± .
19 kpc and D LS = 0 . ± .
08 kpc,and thus it is the first isolated low-mass object locatedin the Galactic bulge. The degeneracy between the twosolutions is very severe (∆ χ = 0 . Spitzer and this data point has the finite-source effectfunction B ( z ) = A obs u >
1, where z = u/ρ . We showedthat whenever B ( z ) >
1, there are two solutions for z and hence for ρ = u/z . Because the ρ degeneracy canbe resolved only by relatively high cadence observationsaround the peak, while the Spitzer cadence is typically ∼ − , we expect that events where the finite-sourceeffect is seen only in the Spitzer data may frequentlyexhibit the ρ degeneracy.In the case of OGLE-2015-BLG-1482, the lens-sourcerelative proper motion for the low-mass star is µ rel =9 . ± . − , while for the brown dwarf it is5 . ± . − . Hence, the severe degeneracy canbe resolved within ∼
10 yr from direct lens imaging byusing next-generation instruments with high spatial res-0
Chung et al. olution.Work by S.-J. Chung was supported by the KASI (Ko-rea Astronomy and Space Science Institute) grant 2017-1-830-03. Work by W.Z. and A.G. was supported byJPL grant 1500811. Work by C.H. was supported bythe Creative Research Initiative Program (2009-0081561)of National Research Foundation of Korea. This re-search has made use of the KMTNet system oper-ated by KASI and the data were obtained at threehost sites of CTIO in Chile, SAAO in South Africa,and SSO in Australia. The OGLE has received fund-ing from the National Science Centre, Poland, grantMAESTRO 2014/14/A/ST9/00121 to A.U. The OGLETeam thanks Professors M. Kubiak, G. Pietrzy`nski, and(cid:32)L.Wyrzykowski for their contribution to the collection ofthe OGLE photometric data over the past years.REFERENCES
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FINITE-SOURCE EFFECTS
In the high-magnification limit, the magnification of a point source is A ps (cid:39) /u . Considering the limb darkeningeffect in high-magnification events, the ratio between the magnifications with and without the finite-source effect isexpressed as B ( z ) = u (cid:82) ρ dr r (cid:82) π dθA ps ( | u + r ˆ n ( θ ) | ) (cid:104) − Γ(1 − . (cid:112) − ( r/ρ ) ) (cid:105) πρ , (A1)where u is the normalized separation between the lens and the center of the source, ˆ n ( θ ) = (cos θ, sin θ ), and r and θ are the position vector and the position angle of a point on the source surface with the respect to the source center,respectively (see Gould 1994a, 2008). Changing variables to x = r/ρ , B ( z ) = z (cid:82) dx x (cid:82) π dθ | z + x ˆ n ( θ ) | − (cid:2) − Γ(1 − . √ − x ) (cid:3) π = (cid:82) dx x (cid:2) − Γ(1 − . √ − x ) (cid:3) (cid:82) π dθ (cid:2) x/z ) cos θ + ( x/z ) (cid:3) − / π . (A2)1Here we change x/z = Q , and Taylor expand the first factor in the integrand,(1 + 2 Q cos θ + Q ) − / = 1 −
12 (2 Q cos θ + Q ) + 38 (2 Q cos θ + Q ) − Q cos θ + Q ) + 105384 (2 Q cos θ + Q ) + . . . Then, keeping terms only to Q , (cid:90) π dθ (1 + 2 Q cos θ + Q ) − / = (cid:90) π dθ (cid:20) − Q cos θ + (cid:18) −
12 + 32 cos θ (cid:19) Q + 12 cos θ (cid:0) − θ (cid:1) Q + (cid:18) −
154 cos θ + 358 cos θ (cid:19) Q (cid:21) = 2 π (cid:20) (cid:18) − (cid:19) Q + (cid:18) −
158 + 10564 (cid:19) Q (cid:21) = 2 π (cid:18) Q + 964 Q (cid:19) . With y ≡ x , Equation (A2) becomes B ( z ) = (cid:90) dy (cid:20) − Γ(1 − . − y ) / ) (cid:18) yz + 964 y z (cid:19)(cid:21) = (cid:90) dy (cid:18) yz + 964 y z (cid:19) − Γ (cid:90) dy (cid:20)(cid:16) − . − y ) / (cid:17) (cid:18) yz + 964 y z (cid:19)(cid:21) . Noting that (cid:82) x a (1 − x ) b = a ! b ! / ( a + b + 1)!, we get B ( z ) = 1 + 18 1 z + 364 1 z − Γ (cid:20) z + 364 1 z − . (cid:18)
23 + 115 1 z + 34 ×
35 1 z (cid:19)(cid:21) = 1 + 18 1 z + 364 1 z − Γ (cid:18)
140 1 z + 3364 ×
35 1 z (cid:19) . Then, we finally get B ( z ) = 1 + 18 (cid:18) − Γ5 (cid:19) z + 364 (cid:18) − (cid:19) z . (A3)2 Chung et al.
Figure 1.
Light curves of the best-fit single lens model for OGLE-2015-BLG-1482. The light curves of the best-fit binary lens model arealso shown in the figure, and they are drawn by a dark grey dotted line. The finite-source effect is constrained by only one single
Spitzer data point, which leads to two models with different ρ values. The grey vertical lines represent the times when secret, subjective, andobjective alerts were issued. -2 0 2 42220181614 Figure 2.
Color-magnitude diagram (CMD) of stars in the observed field. The field stars are taken from KMTNet CTIO data. We notethat there exists an offset between the instrumental magnitudes of OGLE and KMTNet as I kmt − I ogle = 0 .
045 mag. The red and bluecircles mark the centroid of the red clump giant and the microlensed source star, respectively. Chung et al.
Figure 3.
Ratio between the actual magnification including finite-source effects to the magnification of a point source, B ( z ), as afunction of z ≡ u/ρ , i.e., the ratio of the lens-source projected separation to the source radius. In contrast to Gould (1994a) from whichthis figure is adapted, we show the magnification both with (solid) and without (dashed) limb darkening. The horizontal dotted lineindicates B ( z ) = 1 . -0.04 -0.02 0 0.02 0.040.020.040.060.080.1 Figure 4. χ distribution of u , sat versus ρ from the MCMC chains of four degenerate solutions in Table 1. Chung et al. T a b l e B e s t - fi t p a r a m e t e r s . F i t p a r a m e t e r s S o l u t i o n s χ / d o f t ( H J D (cid:48) ) u t E ( d a y s ) ρ ( − ) π E , N π E , E f s , o g l e f b , o g l e ( + , ) . . ± . . ± . . ± . . ± . − . ± . . ± . . ± . − . ± . . . ± . . ± . . ± . . ± . − . ± . . ± . . ± . − . ± . ( − , ) . . ± . − . ± . . ± . . ± . . ± . . ± . . ± . − . ± . . . ± . − . ± . . ± . . ± . . ± . . ± . . ± . − . ± . N o t e . — ( + , ) i nd i c a t e s u , ⊕ > nd u , s a t (cid:39) . H J D (cid:48) i s H J D − . T a b l e B e s t - fi t p a r a m e t e r s f o rr e m o d e li n g . F i t p a r a m e t e r s S o l u t i o n s χ / d o f t ( H J D (cid:48) ) u t E ( d a y s ) ρ ( − ) π E , N π E , E f s , o g l e f b , o g l e ( + , ) . . ± . . ± . . ± . . ± . − . ± . . ± . . ± . − . ± . . . ± . . ± . . ± . . ± . − . ± . . ± . . ± . − . ± . ( − , ) . . ± . − . ± . . ± . . ± . . ± . . ± . . ± . − . ± . . . ± . − . ± . . ± . . ± . . ± . . ± . . ± . − . ± . N o t e . — T h i s i s t h e r e s u l t o f r e m o d e li n g i n c l ud e d ( f s , f b ) s p i t z e r a s c h a i np a r a m e t e r s Chung et al.
Table 3
Ranges of recognizable finite-source effects for a single data point.Range (B) 0 < B < .
95 1 . < B < .
34 1 . > B a > . < z < .
51 0 . < z < .
91 0 . < z < . ρ solution single two (higher ρ ) two (lower ρ ) a The range 1 . > B > .
05 represents the decreasing range of B ( z ) curve (i.e., from B = 1 .
34 (peak) to B = 1 ..