Optimal Survey Strategies and Predicted Planet Yields for the Korean Microlensing Telescope Network
Calen B. Henderson, B. Scott Gaudi, Cheongho Han, Jan Skowron, Matthew T. Penny, David Nataf, Andrew P. Gould
DD RAFT VERSION O CTOBER
9, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
OPTIMAL SURVEY STRATEGIES AND PREDICTED PLANET YIELDS FOR THE KOREAN MICROLENSINGTELESCOPE NETWORK C ALEN
B. H
ENDERSON , B. S COTT G AUDI , C HEONGHO H AN , J AN S KOWRON , M
ATTHEW
T. P
ENNY , D AVID N ATAF , AND A NDREW
P. G
OULD Draft version October 9, 2018
ABSTRACTThe Korean Microlensing Telescope Network (KMTNet) will consist of three 1.6m telescopes each with a4 deg field of view (FoV) and will be dedicated to monitoring the Galactic Bulge to detect exoplanets viagravitational microlensing. KMTNet’s combination of aperture size, FoV, cadence, and longitudinal coveragewill provide a unique opportunity to probe exoplanet demographics in an unbiased way. Here we presentsimulations that optimize the observing strategy for, and predict the planetary yields of, KMTNet. We findpreferences for four target fields located in the central Bulge and an exposure time of t exp = 120s, leading tothe detection of ∼ . ≤ M p / M ⊕ ≤ . ≤ a / AU ≤
16, respectively. Normalizingthese rates to the cool-planet mass function of Cassan et al. (2012), we predict KMTNet will be approximatelyuniformly sensitive to planets with mass 5 ≤ M p / M ⊕ ≤ ∼
20 planets per year per dexin mass across that range. For lower-mass planets with mass 0 . ≤ M p / M ⊕ <
5, we predict KMTNet willdetect ∼
10 planets per year. We also compute the yields KMTNet will obtain for free-floating planets (FFPs)and predict KMTNet will detect ∼ Subject headings: gravitational lensing: micro — planets and satellites: detection — planets and satellites:fundamental parameters INTRODUCTION
The past twenty years have witnessed a continual acceler-ation of the pace of the discovery of planets orbiting otherstars, resulting in an explosion in the number of known exo-planetary systems. To date, nearly ∼ . With these discov-eries, first using results from high-precision Doppler surveys(e.g., Cumming et al. 1999; Udry et al. 2003; Cumming et al.2008; Bonfils et al. 2013) and then using results from Kepler(e.g., Youdin 2011; Howard et al. 2012; Dong & Zhu 2013;Dressing & Charbonneau 2013; Morton & Swift 2013; Pe-tigura et al. 2013), we have been able to construct the first de-tailed determinations of the demographics of exoplanets overa broad range of planet masses and sizes based on large sam-ples of detections. These results have revolutionized our viewof exoplanetary systems, demonstrating a broad diversity ofarchitectures (e.g., Mayor & Queloz 1995; Butler et al. 1999;Lovis et al. 2006; Bakos et al. 2009; Charbonneau et al. 2009;Mayor et al. 2009; Lissauer et al. 2011; Orosz et al. 2012;Barclay et al. 2013) and revealing the ubiquity of small plan-ets with masses below a few times that of Earth (Howard et al.2012). [email protected] Department of Astronomy, The Ohio State University, 140 W. 18thAve., Columbus, OH 43210, USA Department of Physics, Chungbuk National University, Cheongju 361-763, Republic of Korea Warsaw University Observatory, Al. Ujazdowskie 4, 00-478Warszawa, Poland Research School of Astronomy and Astrophysics, The Australian Na-tional University, Canberra, ACT 2611, Australia See http://exoplanets.org and http://exoplanet.eu forcatalogs of known exoplanets with references
As exciting as these results are, they are nevertheless paint-ing an incomplete picture of the demographics of planetarysystems. In particular, the Doppler and transit methods arerestricted to relatively close orbits of less than a few AU, par-ticularly for low-mass planets. However, there are substan-tial reasons to believe that the physics of planet formation,and thus the population of exoplanets, may be substantiallydifferent in the outer regions of planetary systems that arenot currently being probed by these techniques. In particu-lar, in a bottom-up picture of planet formation the locationof the “snow line” in the protoplanetary disks plays a crucialrole. The snow line demarcates the distance from the hoststar at which it becomes cool enough for water to form as asolid in a vacuum. Beyond the snow line the surface densityof solid material is expected to increase by a factor of twoto three (Lissauer 1987). This reservoir of solids is crucialfor planet formation, facilitating the growth of more massiveprotoplanets and shorter formation time scales. In particular,under the core accretion model of giant planet formation itis thought that the majority of gas giants must form beyondthe snow lines in their protoplanetary disks (Ida & Lin 2005;Kennedy & Kenyon 2008). Furthermore, it is likely that theliquid water on habitable planets, thought to be a critical re-quirement for habitability, originated from outside the snowline and was subsequently delivered to such planets via dy-namical processes (Alexander et al. 2012; Jacquet & Robert2013). Thus, determining the demographics of planets be-yond the snow line is integral for understanding both the for-mation and habitability of planets (see Raymond et al. 2004and references therein).Aside from these more theoretically motivated arguments,it is of interest to survey the outer regions of planetary sys- a r X i v : . [ a s t r o - ph . E P ] J un Henderson, et al.tems on purely empirical grounds. For example, of the fourgiant planets in our solar system, current and near-future sur-veys using the Doppler or transit method will be sensitive toanalogs of only Jupiter. As a result, it is currently unknownhow common systems of giant planets like our own are in theGalaxy. More generally, protoplanetary disks are known toextend out to ∼ (cid:38) M Jup ) planets on rela-tively wide ( (cid:38)
10 AU) orbits, and then only in relatively young( (cid:46)
Gyr) planetary systems.Because microlensing is intrinsically sensitive to planetswith more distant orbits as well as very low-mass planets, itprovides a required complement to our present array of planetdetection methods, without which it is currently impossibleto obtain a complete picture of exoplanet demographics (seeGaudi 2012 for a review). The images created during a mi-crolensing event have an angular separation from the star thatis of order the angular Einstein ring, θ E ≡ ( κ M l π rel ) / , (1)where M l is the mass of the lens star, π rel is the relative lens-source parallax, given by π rel = AU( D − l − D − s ), D l and D s arethe distances to the lens and source, respectively, and κ ≡ G / ( c AU) = 8 .
144 mas / M (cid:12) . The presence of a planetarycompanion to this lens star can induce a perturbation of oneof these images, resulting in a deviation in the light curve fromthat which is expected from an isolated star (Mao & Paczynski1991; Gould & Loeb 1992). Since the angular distance of theplanet from the host star must place it near these images inorder to create a significant perturbation, and because theseimages are separated from the host star by ∼ θ E , microlensingis naturally most sensitive to planets with separations of orderthe physical Einstein ring radius at the lens, R E ≡ D l θ E . (2)By coincidence, these distances are of order the location of thesnow line for a wide range of host star masses and distances(Gould & Loeb 1992). Therefore, microlensing is an idealtechnique for probing exoplanet demographics at and beyondthe snow line.However, there are several practical challenges associatedwith conducting microlensing surveys for exoplanets. Theprimary events are rare (one per star per ∼ years) and,for the most part, unpredictable. Moreover, θ E is sufficientlysmall that the individual images are unable to be resolved( θ E (cid:46) mas for lens star masses and lens and source distancesthat are typical for microlensing events toward the GalacticBulge), forcing microlensing searches to rely solely on thetime evolution of the integral flux of the images. Tens of mil-lions of stars must thus be monitored on the time scales ofthe primary events (of order 25 days) simply to find severalhundred events per year. Furthermore, only a handful of theseprimary events contain planetary perturbations, and, with theimportant exception of high-magnification events, these im-age distortions are brief and unpredictable, so these primaryevents must be monitored at even higher cadence.Due to the relatively small detectors that were available atthe time when microlensing planet surveys were first initi- ated, they followed a two-tiered strategy that was first advo-cated by Gould & Loeb (1992). Survey telescopes with big-ger apertures and the largest available fields-of-view (FoVs)would monitor many tens of square degrees of high stellardensity, low extinction fields toward the Galactic Bulge withcadences of once or twice per night. These cadences were suf-ficient to detect and alert the primary events themselves, butinsufficient to accurately characterize planetary perturbationson these events. Networks of smaller telescopes with morereadily available narrow-angle detectors would then monitora subset of the most promising of these alerted events withthe cadence and wider longitudinal coverage necessary to ac-curately characterize these planetary perturbations.The first planet found by microlensing was published byBond et al. (2004), and since then a total of 29 planets or-biting 27 stars have been published , primarily using vari-ants of this strategy, including a Jupiter/Saturn analog (Gaudiet al. 2008; Bennett et al. 2010), a system with two Jovian-mass planets beyond the snow line (Han et al. 2013), andtwo super-Earths (Beaulieu et al. 2006; Bennett et al. 2008).The published microlensing planet detections have masses M p from 0 . (cid:46) M p / M Jup (cid:46) . a from0 . (cid:46) a / AU (cid:46) .
3. These detections have allowed for uniqueconstraints on the demographics of planets beyond the snowline (Gould et al. 2010; Sumi et al. 2011; Cassan et al. 2012)that are complementary to the constraints from other methods.Nevertheless, there are several problems that confront cur-rent microlensing surveys. The two-stage methodology intro-duces biases due to its reliance on human judgment for theselection of follow-up targets. Furthermore, the impact of mi-crolensing exoplanet surveys has been limited by a relativelylow number of detections. It is difficult to improve on theplanet yield using the current observational approach becauseits very design leads to the surveys missing the majority ofplanetary perturbations. Thus, while microlensing has pro-duced several interesting results, there is a strong need formore detections and for these to be obtained in an unbiasedand automated fashion.Recent technological developments have facilitated such atransition. Large format detectors, with FoVs of a few squaredegrees, on moderate aperture telescopes make it possibleto simultaneously image tens of millions of stars in a sin-gle pointing. With such a system, one can dispense with thetwo-tier strategy and instead enter into a “Next Generation”observationally, whereby larger-aperture telescopes monitor asignificant fraction of the Bulge with a small number of point-ings, thus achieving the cadence needed to detect the primarymicrolensing events as well as the planetary perturbations.Using these advances, exoplanetary microlensing has al-ready begun an observational evolution. There are currentlythree survey telescopes exclusively dedicated to monitor-ing the Galactic Bulge to detect exoplanetary microlensingevents. The Optical Gravitational Lens Experiment (OGLE-IV) telescope, located at Las Campanas Observatory in Chile,has a 1.3m aperture, a 1.4 deg FoV, and attains field-dependent observational cadences of 15–45 minutes (Udal-ski 2003). The Microlensing Observations in Astrophysics(MOA-II) telescope resides at Mt. John University Observa-tory in New Zealand and has a 1.8m aperture, a 2.18 deg FoV, and a field-dependent cadence of 15–45 minutes (Bondet al. 2001; Sumi et al. 2003). The Wise observatory near From http://exoplanet.eu as of 29/May/2014 ext Generation Microlensing Simulations 3
Table 1
KMTNet Telescope ParametersClear aperture Throughput f ratio[m] [%]1.6 66.2 a f/3.2 a For I -band and includes the effects of thetelescope optics, the central telescope ob-scuration, and the I -band filter throughput. Mitzpe Ramon, Israel (Gorbikov et al. 2010) has a 1m aper-ture, a 1 deg FoV, and a constant cadence of ∼
30 minutes(Shvartzvald & Maoz 2012). These three observatories workin concert to tile the Bulge, and they reduce data on dailytime scales and alert follow-up networks, including Micro-FUN (Gould et al. 2006), PLANET (Beaulieu et al. 2006),RoboNet (Tsapras et al. 2009), and MiNDSTEp (Dominiket al. 2010). Together, this current observational approachdetects ∼ FoV. With these characteristics, KMTNet will provide near-complete longitudinal coverage, and so nearly continuous ob-servations, of the Bulge for a significant portion of the ob-serving season and will obtain deeper photometry at a highercadence than the current network. KMTNet will thus signif-icantly increase the number of known planets at planet-stardistances near and beyond the snow line.Here we present the result of simulations that optimize theobserving strategy for KMTNet and predict the planet detec-tion rates that the full KMTNet will obtain. In §2 we detailthe characteristics and implementation of KMTNet. We ex-plain the details, ingredients, and methodology of our simula-tions in §3. We vary observational parameters in an attemptto converge on an optimal observing strategy in §4. In §5 weuse said observing parameters to compute fiducial planet de-tection rates, including rates for free-floating planets (FFPs).We then investigate the effects that varying different extrinsicparameters has on our fiducial detection rates in §6. Finally,we discuss our results in §7 and identify our assumptions andhow they affect our calculated detection rates. THE KOREAN MICROLENSING TELESCOPE NETWORK
KMTNet will use microlensing as a tool with which toprobe the demographics of exoplanets near and beyond thesnow line. The full KMTNet will consist of three survey tele-scopes that will be dedicated exclusively to monitoring theGalactic Bulge in the Cousins I -band for exoplanetary mi-crolensing events during the Galactic Bulge observing season,approximately early February through early November. Eachtelescope has a 1.6m aperture, a 4.0 deg FoV, and uses anequatorial mount. Tables 1 and 2 list the parameters for thetelescope and camera, respectively.The goal of the network will be to conduct a uniform surveythat has fewer selection biases and higher detection rates thanthe current surveys. Consequently, KMTNet will maintaina constant observational cadence across all target fields. Thefirst observatory will come online in August 2014 at Cerro
Figure 1.
Location and chronology of the KMTNet sites as viewed fromthe south pole. The corresponding wedges indicate the fraction of the nightduring which the Bulge will be visible (airmass < . ∼ xplanet . Tololo Inter-American Observatory (CTIO) near La Serena,Chile, the second in December 2014 at South Africa Astro-nomical Observatory (SAAO) at Sutherland, South Africa,and the third in February 2015 at Siding Spring Observatory(SSO) in Coonabarabran, Australia. Table 3 specifies the lo-cation of each observatory. Figure 1 shows the location ofeach of the three KMTNet observatories and also indicatesthe fraction of the night during which the Bulge will be visi-ble (airmass < . > ∼
35 days, from early June through early July. This representsan unprecedented step forward in the ability of a dedicatedmicrolensing survey to obtain complete longitudinal cover-age. Figure 2 also shows when the Bulge will be visible fromtwo observatories simultaneously. SIMULATION OVERVIEW
The primary goal of this paper is to simulate a large numberof microlensing light curves that resemble, as closely as pos-sible, those that KMTNet will obtain. From these simulatedmicrolensing events we can then determine the total numberof events (per year) that will result in planet detections, givenan assumed planet population. In order to accomplish thiswe must estimate the contribution of each of our simulatedevents to the total microlensing event rate, given realistic as-sumptions about the population of lenses and sources towardthe target fields in the Galactic Bulge.We estimate the event rate toward a given line of sight as Henderson, et al.
Table 2
KMTNet Camera ParametersFoV Plate scale Number of pixels Wavelength range Readout noise Full well depth t over QE[deg ] [ (cid:48)(cid:48) / pixel] [ (cid:48)(cid:48) / mm] [10 ] [nm] [electrons rms] [electrons] [s] [%]4 0.40 40 340 400-1000 5 80,000 a b c a For <
3% nonlinearity. b Encompasses readout time as well as telescope slew and settle time. c For the detector in Cousins I -band. Figure 2.
The fraction of time that the Bulge is observable before (left) and after (right) accounting for weather, for different combinations of observatories forthe order in which the KMTNet sites will come online. We take the center of the Bulge to be ( α , δ ) = (18 h , − ◦ ), which corresponds to ( l , b ) ≈ (1 . ◦ , − . ◦ ).In the left panel we assume it is visible if it has an airmass < . < >
50% of each day/night for about four and a half months, from the middle of April through the end of August.
Table 3
KMTNet Site ParametersSite Longitude Latitude Altitude[ddd:mm:ss.ss] [dd:mm:ss.ss] [m]CTIO 70:42:06 -29:00:01.2 2400SAAO 339:11:21.5 -32:22:46 1798SSO 210:56:19.70 -31:16:24.10 1149 follows. Following Peale (1998), we consider a slab of thick-ness d D l located a distance D l from the observer. The numberof potential lenses d N l in this slab with mass within d M l of M l and within a solid angle d Ω isd N l = d n l ( M l , D l )d M l d M l d V , (3)where d V = d Ω d D l D l (4)is the volume element at a distance D l and n l is the volume number density of compact objects with mass within d M l of M l at a distance D l .We define a microlensing event to occur if a source at dis-tance D s passes within an angular separation of u , max θ E of agiven lens, where u , max is the impact parameter in units of θ E .Conventionally, the microlensing optical depth and event rateare defined for u , max = 1, which corresponds to a minimummagnification of (cid:39) θ E u , max µ rel , where µ rel is the geocentric relative lens-sourceproper motion.Thus, the microlensing event rate per solid angle d Ω , fromlenses located within d D l of D l and with mass within d M l of M l , for sources at a distance D s , and with relative lens-sourceext Generation Microlensing Simulations 5proper motion within d µ rel of µ rel isd Γ d µ rel d M l d Ω d D l d N s = d n l ( M l , D l )d M l D l u , max θ E µ rel . (5)Using equation (2) and the relation between µ rel and the rel-ative transverse velocity, v rel , between a given source and anintervening lens, v rel = µ rel D l , (6)we can rewrite this asd Γ d v rel d M l d Ω d D l d N s = d n l ( M l , D l )d M l u , max R E v rel . (7)We next consider a distribution of source magnitudes anddistances. We adopt a luminosity function (LF) Φ ∗ , whichgives the number of sources with absolute magnitude withind M I , s of M I , s per unit solid angle d Ω . As discussed further in§3.1.1, we employ the LF of Holtzman et al. (1998), whichis an empirical determination of the number of stars per abso-lute magnitude per solid angle toward Baade’s Window (BW).We call this LF Φ ∗ , BW . To obtain the LF Φ ∗ toward an arbi-trary line-of-sight (l.o.s.), we use our Galactic density mod-els, which are primarily based on the models of Han & Gould(1995a,b, 2003) and discussed in §3.1.2, to compute ξ , theratio of the total integrated mass density along the l.o.s. to-ward the given ( l , b ) to that toward BW, and assume that theLF scales by that ratio, Φ ∗ = ξ Φ ∗ , BW . (8)We assume the areal LF Φ ∗ applies at all D s , but weight thesource distances by the fraction of sources f s at each distance,given a volume density of sources ρ s ( D s ) as a function of D s .The volume element increases as D s d D s , making the fractionof sources within d D s of D s f s = ρ s ( D s ) D s d D s (cid:82) ∞ ρ s ( D s ) D s d D s . (9)Thus, each bin of the LF represents a population of sourceswith fixed luminosity, and within each bin we allow for thesources to be distributed across the full range of distances be-ing considered. The differential number density of sources ata distance D s with a given luminosity is thend N s = Φ ∗ f s d M I , s . (10)Combining this with equation (7), the differential event ratefor a population of lenses and sources isd Γ d v rel d M l d Ω d D l d D s d M I , s =d n l ( M l , D l )d M l u , max R E v rel ξ Φ ∗ , BW ρ s ( D s ) D s (cid:82) ∞ ρ s ( D s ) D s d D s . (11)The total event rate is then given by Γ = (cid:90) d Γ = (cid:90) d v rel (cid:90) d M l (cid:90) d Ω (cid:90) d D l (cid:90) d D s (cid:90) d M I , s d Γ d v rel d M l d Ω d D l d D s d M I , s . (12) However, we are interested in simulating individual eventsand thus are interested in the differential contribution of eachevent to the total event rate.To estimate the microlensing event rates, we perform aMonte Carlo (MC) simulation of a large number of microlens-ing events. In general, there are two possible approaches tocreating an ensemble of microlensing events that accountsfor the various contributions to the differential event rate inequation (11). One approach would be to draw the microlens-ing event parameters according to their contributions to theevent rate. The second is to draw parameters from, e.g., uni-form distributions, and then weight each event by equation(11). We adopt a hybrid approach: we draw some variablesfrom our assumed input distribution functions while others aredrawn uniformly and weighted accordingly.An outline of our MC simulation is as follows. We assumea population of planetary companions with fixed mass M p andon a circular orbit with semimajor axis a . Then, we begin bystepping through each absolute magnitude bin j of the LF. Foreach bin j , we simulate a large number of MC trials N MC , j .For the i -th MC trial we independently draw D l , i and D s , i uni-formly, giving d D l d D s = ∆ D l ∆ D s N MC , j , (13)where ∆ D l and ∆ D s represent the full range of D l and D s be-ing considered, respectively. We draw lens and source veloc-ities from the distributions described in §3.1.2 and then com-pute v rel , i from these velocities and D l , i and D s , i .We assume that the mass function of lenses is independentof location in the Galaxy, and thus separate the volume num-ber density of lenses into two components,d n l ( M l , D l )d M l d M l = d N d M l n l . (14)Here d N / d M l is the (normalized) mass function of lenses, i.e.,the fraction of lenses with mass within d M l of M l , and n l is thenumber density at a distance D l , i . Our models actually spec-ify the mass volume density ρ l , and therefore we substitute n l = ρ l / M l and draw a value of M l , i from M l (d N / d M l ), whered N / d M l is the Gould (2000) mass function as described in§3.1.3. From M l , i , D l , i , and D s , i we compute R E , i accord-ing to equation (2). We then evaluate ρ l at the value of D l , i .Each event is randomly assigned a pair of Galactic coordi-nates ( l i , b i ) within the FoV of the detector. With D l , i and ( l i , b i )we use our Galactic models to compute ρ l , i , the mass densityof lenses at D l , i in the direction of ( l i , b i ).We similarly use our density models to compute ρ s , i as wellas the total mass of sources, M s , tot , i , across ∆ D s toward ( l i , b i ), M s , tot , i ≡ (cid:90) ∆ D s ρ s ( D s ) D s d D s . (15)Combining equations (13) and (15) with ρ s , i at D s , i allows usto specify equation (9) via f s , i = ∆ D s ρ s , i D s , i N MC , j M s , tot , i . (16)Finally, the contribution to the event rate from the i -th MCtrial is then ∆Γ i = 2 ∆ D l ∆ D s N MC , j M s , tot , i u , max R E , i v rel , i ρ l , i ( D l , i ) ρ s , i ( D s , i ) D s , i M l , i . (17) Henderson, et al.Fundamentally, equation (17) gives the weight of a microlens-ing event that is taken to be representative of all possibleevents with physical characteristics within the same infinites-imal range of parameter values. This must be multiplied by ξ i Φ ∗ , BW , j to account for the number of sources toward that( l i , b i ) with the same fixed luminosity. The total planet detec-tion rate is given by summing across all MC trials, all LF bins,and finally all target fields N fld , yielding Γ tot = u , max Ω FoV N fld (cid:88) k N LF bins (cid:88) j Φ ∗ , BW , jN MC (cid:88) i ξ i ∆Γ i H ( ∆ χ > ∆ χ , th ) · H ( ∆ χ > ∆ χ , th ) , (18)where Ω FoV represents KMTNet’s FoV. We have also includedtwo Heaviside step functions H ( x ). The first requires that theimprovement in χ for a microlensing fit relative to a constantfit, ∆ χ , is larger than some minimum threshold ∆ χ , th inorder to detect the primary event. The second requires thatthe improvement in χ for a binary-lens fit relative to a single-lens fit ∆ χ is larger than some threshold ∆ χ , th in order tosubsequently detect the planetary signature. In practice, if weare interested solely in the overall microlensing event rate, wedo not include the second step function. We also include afew additional cuts on our events that we discuss below, butdo not specify explicitly here.Our MC simulation includes many different ingredients, in-cluding • using Galactic models to generate populations of sourceand lens stars with physical properties such as massdensities, distances, velocities, masses, and apparentmagnitudes that match empirical constraints, • populating each lens system with a planetary compan-ion and assigning microlensing parameters in orderto compute the magnification of the given binary mi-crolensing event as a function of time, accounting forthe effects of a source of finite size when appropriate, • using realistic observing conditions to create “ob-served” light curves for these binary microlensingevents by determining the photon rate normalization,including all contributing sources of noise and back-ground such as the Moon, the dark sky, the lens, andunassociated blend stars, and modeling the effects ofvisibility, gaps due to weather, and seeing at each site,and • implementing a detection algorithm for each light curveto determine first whether the primary microlensingevent is detected and if so whether the signal ofthe planetary perturbation is subsequently robustly de-tected.We thus divide the discussion of our simulation into thesefour primary components. The first generates a populationof lens and source stars drawn from a Galactic model thatmatches empirical constraints. In the second we compute pa-rameters for binary microlensing events. For each of thesemicrolensing events we calculate the magnification as a func-tion of time and then simulate realistic observing conditionsand effects, including all relevant sources of uncertainty in Figure 3.
The differential and cumulative luminosity function of Holtzmanet al. (1998) toward BW. The apparent magnitude assumes a distance of 8.2kpc (Nataf et al. 2013) and an extinction of A I = 1. Here we see the bumpdue to red clump giants at M I , s ≈
0, which leads into the subgiant branch andultimately the main sequence, at M I , s ≈
3. For M I , s (cid:38) I s (cid:38)
19) the sourcepopulation is dominated by main sequence stars. the flux measurements, and compute the observed light curve.Finally, we subject these simulated light curves to a series ofdetection criteria. We describe the details of each of thesecomponents in the subsequent subsections.
Galactic Model
The first step in our MC simulation is to generate microlens-ing events with parameter distributions that are consistentwith those expected based on empirical constraints on Galac-tic structure. As described in §3, we do this by drawing andweighting individual event parameters by their contribution tothe total microlensing event rate. This requires the followingingredients: a source LF, density distribution models for theGalactic Disk and Bulge, models for the kinematics of Diskand Bulge stars, and a mass function of lenses. Furthermore,in order to predict the lens and source fluxes and the flux ofblended light, we must adopt an extinction map as well as amass-luminosity relation for the lenses. Finally, we must in-clude a radius-luminosity relation to obtain the physical andangular size of the sources.
Luminosity Function
We use the LF of Holtzman et al. (1998), who use
Hub-ble Space Telescope data to obtain Φ ∗ , BW , j ( M I , s ), the num-ber density of stars for different bins, j , of absolute I -bandmagnitude, M I , s , toward BW near the Galactic Bulge. Fig-ure 3 shows both the cumulative and differential LF. Forthe j -th bin we generate N MC , j microlensing events, where N MC , j = C Φ ∗ , BW , j ( M I , s ) M − / p . We scale N MC , j with Φ ∗ , BW , j toensure that the number of simulated events is proportional tothe number of sources with a given absolute magnitude M I , s and thus proportional to the event rate, thereby producing afixed fractional accuracy in the event rate per bin of M I , s ofext Generation Microlensing Simulations 7the LF. The scaling with M − / p arises from the fact that theduration of a planetary perturbation is given by ∆ t p ≈ q / t E , (19)where q is the mass ratio of the lens system, given by q = M p M l . (20)We consequently expect the observational coverage and thusthe planet detection rate to roughly scale as M ν p , where ν ≈ /
2. In reality, as we will discuss further in §5, the planetdetection rate scaling is closer to ν ≈ /
4. Regardless, forsimplicity we scale our number of sampled events accordingto our naive expectation that ν = 1 / Bulge and Disk Models
We base our Galactic Bulge model on that of Han & Gould(1995a), derived from the “boxy” Gaussian triaxial G2 modelof Dwek et al. (1995), which has the functional form (seeequations (3) and (4) of Dwek et al. 1995) ρ B = ρ , B exp( − . r s ) , (21)where r s = (cid:34)(cid:18) x (cid:48) x (cid:19) + (cid:18) y (cid:48) y (cid:19) (cid:35) + (cid:18) z (cid:48) z (cid:19) / . (22)Here the origin is at the Galactic Center (GC) and the threeaxes x (cid:48) , y (cid:48) , and z (cid:48) point along the three axes of the triaxialBulge. The values for the scale lengths of the three differentaxes, x , y , and z , as well as the normalization for the Bulgestellar mass density, ρ , B , are given in Table 4. We adopt thevalues of the scale lengths from the 2.2 µ m fit of Dwek et al.(1995) (see their Table 1) but renormalize them to a Galacto-centric distance R GC of 8.2 kpc (Nataf et al. 2013). We takethe position angle of the major axis of the triaxial Bulge tobe 25 ◦ (Nataf et al. 2013) and normalize the stellar mass den-sity of the Bulge, which includes main sequence stars (MSSs),brown dwarfs (BDs), and remnants—white dwarfs (WDs),neutron stars (NSs), and black holes (BHs)—such that we ob-tain a column density of stars, BDs, and remnants in the Bulgetoward BW equal to the value of 2086 M (cid:12) pc − obtained byHan & Gould (2003).For our Galactic Disk model we follow the prescription ofHan & Gould (1995b) and adopt the Bahcall (1986) model,which has the form ρ D = ρ , D exp (cid:20) − (cid:18) R − R GC R + zz , D (cid:19)(cid:21) , (23)where R = ( x + y ) / , R is the radial scale length of the disk, z , D is the vertical scale height of the disk, and ρ , D is the massdensity in the Solar neighborhood, all of which are specifiedin Table 4. The values for R , R GC , z , D , and ρ , D come fromHan & Gould (1995a,b). This coordinate system has its originat the GC, and the x -axis points toward Earth, the y -axis to-ward increasing Galactic longitude, and the z -axis toward theNorth Galactic Pole.We simulate microlensing events for two populations oflens systems and assume the source is in the Bulge for both.In the first case we assume the lens to also be located in theBulge and refer to these events as Bulge-Bulge (BB) events. The second case consists of lens systems in the Disk and arecalled Disk-Bulge (DB) events. In each case, we randomlydraw the Galactic coordinates ( l , b ) of the event from withinthe FoV. We randomly draw the distance from the observer tothe lens, D l , from the range D l , min ≤ D l ≤ D l , max , (24)where D l , min and D l , max are different for BB and DB events.Their values are specified in Table 4. We assume that allsources are in the Bulge and so, for both BB and DB events,draw the distance from the observer to the source, D s , from thesame range as D l for BB events. Events for which D l ≥ D s are discarded. We compute the total mass of sources acrossthe full range of D s toward the given ( l , b ), M s , tot , according toequation (15). Using the appropriate models for BB and DBevents, we then calculate the mass density of sources, ρ s ( D s ),and lenses, ρ l ( D l ), along the l.o.s. to and at the distance of thesource and lens, respectively.With D s and D l in hand we calculate v rel for each populationof lenses, BB and DB. We assume a Gaussian velocity distri-bution for both the y - and z -direction of motion with mean anddispersion adopted from Han & Gould (1995b) and listed inTable 5. Finally, we add the two components in quadrature,obtaining the relative velocity of the lens-source system in theplane of the sky, v rel . Lens Mass
We draw M l from the mass function of Gould (2000).Specifically, we adopt a power-law mass function of the fol-lowing form d N d M l ∝ (cid:18) M l M brk (cid:19) (cid:15) , M brk = 0 . (cid:12) , (25)where (cid:15) = − . . < M l M (cid:12) < M brk ) , (26a) (cid:15) = − . M brk < M l M (cid:12) (cid:46) . . (26b)As in Gould (2000), we assume that all MSSs in the range 1 < M l / M (cid:12) < < M l / M (cid:12) < < M l / M (cid:12) <
100 havebecome BHs, and adopt the same distributions for each classof remnants, which are shown in Figure 4. We include allclasses of objects—BDs, MSSs, and stellar remnants—in thecalculation of the total microlensing event rate. However, weonly consider MSSs as planet hosts and thus exclude objectswith mass outside of the range 0 . < M l / M (cid:12) < M I , l , the absolutemagnitude of the lens, from M l using a 1 Gyr isochrone ofBaraffe et al. (1998, 2002).We have hitherto described the models and correspondingparameters we use to determine ρ s , ρ l , D l , D s , M s , tot , v rel ,and M l . These are the physical characteristics of the lens andsource necessary to compute the weight of an individual mi-crolensing event via equation (17), which describes the rate atwhich events with parameters in the same infinitesimal rangeoccur. Extinction Map
Henderson, et al.
Table 4
Density Model ParametersBulge Disk x y z ρ , B D l , min D l , max R GC R z , D ρ , D D l , min D l , max [pc] [pc] [pc] [M (cid:12) pc − ] [pc] [pc] [pc] [pc] [pc] [M (cid:12) pc − ] [pc] [pc]1580 620 430 1.25 4200 12200 8200 3500 325 0.06 0 12200 Table 5
Velocity Distribution ParametersLocation µ v y , rel σ y , rel µ v z , rel σ z , rel Bulge − − η ) 82 . · (1 + η ) 0 66 . · (1 + η )Disk 200 η + (82 . η ) + (66 . η ) Note . — All values are in km s − and η ≡ D l D s . Figure 4.
Our input event rate distribution as a function of the lens mass M l ,which goes as M / l d N d M l , adapted from Gould (2000). We draw M l from themass range 0 . < M l / M (cid:12) ≤
10. We exclude BDs and remnants as planetaryhosts but include them in the total microlensing event rate.
The dust map we employ combines two different methodsof using red clump giants (RCGs) to determine the Galac-tic extinction in the I -band, A I . The first is the Bulge RCG-derived map of Nataf et al. (2013) that uses optical and near-IR (NIR) photometry to derive A I for the inner Milky Way.However, this map is incomplete in the region | b / deg | (cid:46) A I . We thus complement it with an extinctionmap that uses mid-IR and NIR data (Majewski et al. 2011;Nidever et al. 2012) to determine A K for | l / deg | ≤ A K to A I we sample the overlap region of these two maps,4595 points at the resolution of the optical map, and find abest-fit slope of A I / A K = 4 .
78 (with an error in the mean of ± Figure 5.
Our extinction map derived from red clump giants, which coversa significant fraction of the inner Galactic Bulge. Extinction data come fromthe I -band map of Nataf et al. (2013) and the mid-IR and NIR map of Majew-ski et al. (2011); Nidever et al. (2012). Note the deleteriously high extinctionfor | b | (cid:46)
2. We have overlaid the OGLE-IV target fields, which are grey-scaled according to cadence, with white representing occasional observationsand black representing 10–30 observations per night. in Figure 5.It should be noted that our resulting extinction map doesnot contain any information about the distribution of the dustalong the l.o.s., forcing us to estimate A I specifically at D l and D s for every ( l , b ) that we might sample. The optical map ofNataf et al. (2013) explicitly and exclusively targets RCGs inthe Bulge, and we have utilized the RCG stellar populationof the RJCE map. While the median distance to the RCGsample of the IR map lies ∼ A I at a given distance D by adopting a model forthe dust distribution and normalizing to the total extinctionat the GC. Though for the bulk of microlensing events thesource will lie behind all of the extinction and the lens willbe faint, we include this treatment to account for those eventswith sources on the near side of the Bulge and/or unusuallybright lenses.To do so, we assume that our extinction map provides A I atthe distance of the Galactic Center, D GC = 8 . D < D GC is given by the totalcolumn density of dust to D . We assume the dust is distributedext Generation Microlensing Simulations 9exponentially with vertical distance from the plane and inte-grate the dust volume density along the l.o.s. to D in order toobtain the column density. Then, in order to estimate A I alonga given l.o.s. at D , we take the extinction to be distributed as A I = − . e τ ( D ) · κ ] , (27)where τ is the optical depth to dust at D , which we take asproportional to the dust column density, and κ is a normal-ization factor. For a given ( l , b ) we first integrate along thel.o.s. to D GC to obtain τ at the distance of the Bulge. We usea dust distribution model that is distributed vertically with anexponential profile but that is constant with radius at a givenheight above the plane. We assume a vertical scale height of125pc (Marshall et al. 2006). Given that we have assumed ourdust map to give A I at D GC , we interpolate across the map todetermine A I for the given ( l , b ) and compute κ , the value ofwhich is required to reproduce A I toward the given l.o.s. forthe optical depth τ ( D GC ). For a source at D s , we subsequentlyintegrate our dust model along the l.o.s. once more, this timeto D s , to obtain τ at D s , and finally apply κ to get A I at D s .Given D s , M I , s , and A I , s , we calculate the apparent magnitudeof the source, I s . We similarly compute the apparent magni-tude of the lens, I l . Microlensing Parameters
After the physical parameters of each lensing event and itsevent rate contribution have been determined, we assign themicrolensing parameters, which we ultimately use to com-pute the magnification of the source as a function of time.First we calculate the basic single lens parameters. Then weadd a planetary companion to the lens star and determine thestatic binary lens parameters. For all cases we do not considerhigher-order dynamical effects such as parallax, xallarap, orlens orbital motion, in our simulations.
Primary Event
We refer to the magnification structure that arises from amicrolensing event that is due to a single lensing mass as theprimary event. There are four parameters that specify such asingle-lens primary event and allow for the derivation of themagnification as a function of time. They are t , the timeof closest approach of the source to the lens, u , the angulardistance of the closest approach of the source to the lens, nor-malized by θ E , the Einstein crossing time t E , and ρ , the angularsize of the source star normalized to θ E . In our simulations wecompute the annual planet detection rate, so we compute timein the reference frame of a generic year and randomly draw t from the range 0 . ≤ t days ≤ . . (28)We draw u randomly from the range0 . ≤ u ≤ u , max , (29)adopting a maximum impact parameter of u , max = 3. The Ein-stein crossing time t E is calculated as t E ≡ θ E µ rel . (30)The set of these three parameters, t , u , and t E , is sufficientfor microlensing events in which the source is point-like. For the case of a single-lens event, the lens-source separation as afunction of time, u ( t ), is given by u ( t ) = u + (cid:18) t − t t E (cid:19) . (31)The magnification for a point-source is then (Paczynski1986) A [ u ( t )] = u + u √ u + . (32)If the source passes sufficiently near the lens mass such thatthere is a significant second derivative of the magnificationacross its surface and the size of the source can be resolved,the additional parameter ρ must be specified. We use the 10Gyr isochrone of Girardi et al. (2000), assuming solar metal-licity, to obtain a relation between M I , s and R ∗ . As mentionedin Gaudi (2000), reasonable variations in age and metallic-ity do not have appreciable effects on the conversion between M I , s and R ∗ . We use R ∗ to determine the physical size of thesource star, given its absolute magnitude from the LF. Theangular size of the source star normalized to θ E is then ρ = θ ∗ θ E , (33)where θ ∗ = R ∗ D s . (34) Binary Lens
The next step is to populate the lens system with a planetand compute the three additional parameters that determinea static binary lens. These are q , the mass ratio of the lenssystem, s , the instantaneous projected separation of the lenscomponents in units of θ E at the time of the event, and α , theangle of the source trajectory with respect to the binary axis atthe time of the event. The binary axis points from the primary,the lens star, to the secondary, the planet. The mass ratio ofthe lens system is given by q = M p M l . (35)We assume a circular orbit for the planetary companion andcompute s as s = aR E (cid:112) − cos ζ, (36)where ζ is the angle between the semimajor axis a and theplane of the sky. For randomly oriented orbits, cos ζ is uni-formly distributed. We therefore draw cos ζ from a uniformrandom deviate in the range [0 − α ,which specifies the direction of the lens-source relative mo-tion, is measured counter-clockwise from the binary lens axis.We draw α randomly from the range0 . ≤ α ≤ π. (37) Magnification Calculation
We then calculate the magnification of the source due to thestatic binary lens system as a function of time. We first checkwhether it is appropriate to make use of either of two approx-imations that use a series of point-source calculations to ap-proximate a source of finite size. In each case it is necessaryto solve a complex fifth-order polynomial, whose coefficients0 Henderson, et al.are given by Witt & Mao (1995), in order to obtain the magni-fication of a point-like source due to a binary lens (PSBL). Toexpedite this procedure we employ the root-solving methodof Skowron & Gould (2012), which is of order a few timesfaster than the root-solving subroutine ZROOTS containedwithin
Numerical Recipes . This process allows us to circum-vent using a computationally expensive algorithm to calculatethe full finite-source binary-lens (FSBL) magnification for thevast majority of data points without loss of precision, in turnboosting the number of light curves we can simulate per unittime and improving our derived primary event and planet de-tection rate statistics.We employ a tiered magnification algorithm that balancescomputational efficiency with robustness in order to effica-ciously model the large number and wide variety of binarylens systems our simulations generate. At a given time t we first compute the PSBL magnification A psbl . We also es-timate the finite-source magnification using the quadrupoleapproximation (Pejcha & Heyrovský 2009), which uses fivepoint-source magnification calculations—one at the centerof the source and four at equally spaced points along theperimeter of the source—to approximate the magnificationof a source of extended size. If the fractional difference be-tween the point-source and quadrupole approximations forthe magnification due to a binary lens is below the tolerance δ A ≡ | A quad − A psbl A quad | ≤ A tol , where A tol = 10 − is our arbitrary butconservative choice, we adopt the quadrupole magnificationfor that data point. Otherwise, we compute the magnificationusing the hexadecapole approximation (Pejcha & Heyrovský2009; Gould 2008), which approximates an extended sourceusing thirteen point-source magnification calculations—eightequally spaced along the perimeter of the source, four equallyspaced along the perimeter defined by ρ/
2, and one at thecenter of the source. If the fractional difference between thequadrupole and hexadecapole magnifications is below A tol weuse the hexadecapole magnification for that point.If both the quadrupole and subsequently the hexadecapoleapproximations fail the fractional tolerance criterion, the mag-nification at that time t requires the use of a full FSBL mag-nification computation. We utilize an inverse ray-shooting al-gorithm that “shoots” rays from the images of the magnifiedsource on the image plane and computes the FSBL magnifica-tion by using the binary lens equation to determine how manyof these rays can be traced back to the interior of the unmag-nified source star on the source plane. In order to further ex-pedite our inverse ray-shooting algorithm, we use the hexade-capole approximation at each time to help determine the ap-propriate geometry of our coordinate system. If A hex satisfies A hex ≤ A thresh , we presume the resulting FSBL magnificationof the source will be sufficiently low that it is optimal to cre-ate a grid in a rectangular coordinate system. If A hex > A thresh ,the FSBL magnification is taken to be sufficiently high that agrid in a polar coordinate system centered on the primary lensmass is more appropriate, as the majority of the magnifica-tion arises from two images that form extended arcs centeredon the more massive lens component at the distance ∼ θ E . Inthis high-magnification regime we utilize a variable axis ra-tio that decreases the grid resolution in the angular directionrelative to that in the radial direction to more accurately cap-ture the image morphology and increase computational effi-ciency without loss of precision, following Bennett (2010).We adopt a threshold of A thresh = 100 based on the fact thatBennett (2010) find that the precision of a polar-based algo- rithm increases with the axis ratio for magnifications higherthan this. We set the resolution of each grid such that eventhe largest numerical errors due to finite sampling from eitherinverse ray-shooting algorithm are, fractionally, ≤ − , morethan an order-of-magnitude below the fractional photometricprecision expected from KMTNet. Finite Source Effects in the Single Lens Model
As described in §3.4.2, in order to determine whether agiven planetary perturbation is detectable, we fit our simulatedbinary-lens lightcurve to a single lens model. Before doingso, however, we must first determine whether or not we needto consider finite-source effects in the comparison model. Inmost cases, a point-source single-lens (PSSL) model providesa sufficiently good approximation. However, if the sourcepasses very near to or over the primary lens, and the centralcaustic due to the planet is sufficiently small, the resultinglight curve will closely resemble that due to a single lens withfinite-source effects (FSSL) and have no significant deviationsfrom the planet (at least during the peak of the event). Thus, ifone were to fit such a light curve to a PSSL model, one wouldfind large deviations, resulting in a spurious planet detection.For each event, we first determine whether or not u ≥ ρ .If so, then we assume that finite-source effects for a single lensare completely negligible and therefore adopt a PSSL modelas the best-fit comparison, leaving t , u , and t E as free param-eters and computing the magnification according to equation(32). In particular, we do not include ρ as a free parameterin this case. It is straightforward to demonstrate that, for asingle lens, the fractional deviation in magnification betweena point-like source and a finite source at the closest point ofapproach u = u is (cid:46) × − for u ≥ ρ , assuming ρ (cid:28) u < ρ , we determine whether or not the fractionaldifference between the magnification for a point-like source,computed according to equation (32), and that for a finitesource, computed numerically via the elliptic integrals givenin Witt & Mao (1994), is greater than our tolerance A tol forat least one data point. If so, and if the primary event passesthe initial detection criteria, we use a FSSL model to fit to thelightcurve, including ρ as a free parameter. If not, then we as-sume finite-source effects are negligible, again adopt a PSSLmodel, and do not include ρ as a free parameter. Light Curve Creation
With all of the physical and microlensing parameters inhand, the next task is to generate the light curve for eachevent. This requires turning the magnification of the sourceas a function of time into a measured flux by determining theflux contributions of the lens, the source, and all sources ofblended and background light, accounting for the effects ofthe Moon and weather, and accurately modeling the flux mea-surement uncertainties.
Observational Parameters
For each observatory we divide a generic year into the totalnumber of possible data points, assuming a constant cadence(the choice of which is discussed in §4). For each target fieldwe determine whether that field is observable for each of thetotal possible data points from each observatory. The criteriaare that the field center be at or above an airmass of 2.0 andthat the Sun is at least 12 degrees below the horizon (nauticaltwilight).ext Generation Microlensing Simulations 11
Figure 6.
The fraction of clear nights for each of the three KMTNet ob-servatory sites as a function of time. These data were taken from Peale(1997) and we assume the weather at La Silla is a good approximation forthe weather at the KMTNet observatory at CTIO.
We model the length of weather patterns as a Poisson pro-cess and compute the cumulative distribution function (CDF)of their duration as e − λ (cid:80) ki =0 λ i / i !. We adopt a mean weatherpattern coherence length of λ = 4 days and compute the CDFto k = 20. Beginning with the first data point for each ob-servatory, we randomly draw from the CDF to obtain the du-ration of a given weather pattern at the given observing site.Then, using the fraction of clear and cloudy nights for eachsite, taken from Peale (1997) and shown in Figure 6, we ran-domly draw to determine whether it will be cloudy or clearfor the weather pattern. Here we have assumed that the frac-tion of clear nights at La Silla is a good approximation ofthe same fraction for CTIO. If cloudy, we skip all data pointsthat occur during the pattern and repeat this process beginningwith the next data point after the weather pattern. Otherwise,the weather for those data points is considered clear. Figure2 shows the observability of the Bulge for the chronology ofthe KMTNet sites, which convolves the visibility of the Bulgewith the distribution of clear nights at each site that is shownin Figure 6.For each observable data point we draw the seeing froma Gaussian with site-dependent minimum, mean, and sigmaseeing values, listed in Table 6, and modify the seeing as(airmass) . (Woolf 1982), where airmass = sec( z ) and z is thezenith angle. We calculate the total background sky bright-ness in I -magnitudes per square arcsecond, including the con-tributions from the mean dark sky at zenith, which we assumeis µ sky = 19 . / (cid:3) (cid:48)(cid:48) for each site, and the phase and dis-tance of the Moon to the field center, according to the pre-scription of Krisciunas & Schaefer (1991). Flux Determination
The final photometric reduction pipeline that KMTNet willimplement is based on difference image analysis (DIA) pho-tometry (Alard & Lupton 1998; Alard 2000). Aperture pho-
Table 6
Fiducial Site-dependentSeeing DistributionParametersSite min. µ σ
CTIO 0.8 1.4 0.26SAAO 0.9 1.6 0.30SSO 1.3 2.0 0.40
Note . — All values are inarcseconds. tometry breaks down in crowded stellar fields, and even theapproach of point-spread function (PSF) photometry becomesquite difficult in the regime of extreme blending that is typi-cal of the Galactic Bulge. To circumvent this issue, DIA con-structs a reference template frame by combining the subsetof images with the best seeing and then measures the PSFsolely on this reference image. For each observation, it thendetermines a convolution kernel that transforms the PSF ofthe reference image into that of the given frame, “matching”the two, and subtracts the given image from the convolvedreference image. Each resulting difference image thus yieldsthe difference in flux between the given observation and thetemplate image, which causes only photometrically variableobjects to have a non-zero difference flux. Performing PSFphotometry on the reference image then provides a flux zero-point (see §3.1 of Hartman et al. 2004 for a more completediscussion of DIA).In the case of a microlensing event, the object flux that ismeasured on the reference image includes the light from thesource, the lens, and any other interloping stars that are unas-sociated with the event but still unresolved and thus blendedwith the source. For typical Bulge fields with high stellardensity, there will also be a quasi-smooth background fluxproduced by faint unresolved stars scattered across the en-tire frame, even for a reference image with excellent seeing.DIA treats this “sea” of unresolved stars in the same way itdoes the Moon and the dark sky, fitting and subtracting thesesmooth backgrounds from the reference image prior to themeasurement of the object flux. These sources of backgroundflux—the Moon, the dark sky, and the “sea” of faint unre-solved stars—will thus not contribute to the flux measured onthe reference image. They will, however, still contribute tothe measured flux uncertainty for each individual frame in away that varies with the seeing of the image. Often there areinterloping stars blended with the event that are brighter thanthe limiting magnitude that defines this “sea” of faint unre-solved stars, so their brightness is not (entirely) subtracted offwith the smooth stellar background and so does not vary withseeing from image to image.We approximate these populations of unassociated and un-resolved stars as a dichotomy between faint stars that con-tribute solely to the noise of an individual flux measurementand bright stars that contribute to the object flux as well asthe noise. Under this dichotomy we assume each microlens-ing event to be blended with, on average, one bright interlop-ing star as well as a smooth surface brightness of faint stars.The flux from the interloping star will contribute to the objectflux measured on the reference frame as well as its uncer-tainty, while the flux from the smooth stellar background willonly contribute to the flux measurement uncertainty in a man-ner that depends on the seeing. To determine their respectiveflux contributions, we estimate the apparent brightness above2 Henderson, et al.which there is, on average, one unassociated and unresolvedstar per seeing disc. The first step in this process is to simulatethe construction of a reference image.As shown in Table 6, CTIO will have the best seeing ofthe three KMTNet sites, so we assume that the template willbe comprised entirely of images taken at CTIO. From 12 and10 random light curves from 2011 and 2012 OGLE-IV data,respectively, we find that the 1st percentile value of seeingis 0 . (cid:48)(cid:48) , which we take as the seeing of the reference im-age, σ ref . We make the approximation that all non-lens stellarblend flux, resolved or unresolved, is due to stars at the dis-tance of the Bulge and interpolate across our dust map to get A I at the location of the event assuming a distance of 8.2kpc.We then modify our LF to give the areal density of stars to-ward a given l.o.s. as a function of apparent magnitude. Todo so, first we apply the 8.2 kpc distance to the Bulge andthe computed extinction. Then we use our Bulge model, de-scribed in §3.1.2, to calculate the ratio of the stellar surfacedensity for a given l.o.s. to that of BW, ξ ∗ , the region for whichHoltzman et al. (1998) determined the LF.Next we use a Moffat function to model the star’s light pro-file (Moffat 1969). A Gaussian falls off more steeply at largerradii and so is insufficient for capturing the full extension ofthe wings of a realistic PSF. We find that adopting a Moffatprofile is crucial for regions of such high stellar density in theBulge. The intensity I as a function of radius r from the centerof the profile is I ( r ) = I (cid:20) + (cid:16) r α (cid:17) (cid:21) − β , (38)where I is the intensity at the central peak, α is the width pa-rameter, related to the full width at half maximum (FWHM),i.e., the seeing, of the light profile via α = FW HM · √ / β − , (39)and β is the atmospheric scattering parameter. From examin-ing bright and isolated stars across a series of OGLE-III Diskreference images, we find β = 3 . Ω eff = 5 πα π · FW HM · (2 / −
1) (40)for β = 3 .
0. This corresponds to an increase in Ω eff by a factorof ∼ FW HM .Using this and taking
FW HM = σ ref to be the diameter of aseeing disc, we start with the brightest bin of our LF and sumacross bins of decreasing brightness to obtain the cumulativenumber of stars that fall in a seeing disc, according to N ∗ , disc = (cid:88) j =1 Φ ∗ , BW , j Ω eff ξ ∗ d m . (41)The apparent magnitude at which N ∗ , disc = 1 defines the cut-off magnitude I cut . We take there to be one interloping starbrighter than I cut that is unresolved and unassociated butblended with the event. We draw from the cumulative dis-tribution N ∗ , disc to obtain I int , the brightness of the interlopingblend star. Stars below I cut contribute to the blend flux witha smooth surface brightness equal to the cumulative flux be-low I cut divided by Ω eff . It should be noted that this approach Figure 7.
The distributions of the different sources of blend flux across thefinal set of target fields for all three observatories. The top panel shows thedistribution of the apparent magnitude of the bright interloping blend star, I int . We see that, under our assumptions about and treatment of unassociatedbut blended stars, there is a floor for the brightness of objects that KMTNetwill detect toward the Bulge at I obj ≈
20. The middle panel panel gives thecontributions to the total surface brightness from the Moon, the dark sky atzenith, faint unresolved stars, and the additional smooth blend we include tomatch the OGLE-III photometric uncertainties. These will contribute to thenoise of each data point, but not to the flux measurement itself. The contri-bution from the Moon and the sky overwhelms that due to unresolved stars,although the additional smooth blend is the dominant contributing source ofnoise from the smooth backgrounds. In the bottom panel we sum the pho-ton rate from all sources of background—the lens, the interloping blend star,the Moon and sky, the faint sea of unresolved stars, and also the additionalsmooth blend—and compare it to the photon rate of the source. In doing sowe see that KMTNet will be background-dominated. will underestimate the number of blended interloping stars insome cases but will overestimate the total blend flux. Figure7 shows our resulting distributions of I int and the backgroundsurface brightness contributions from the Moon, the dark skyat zenith, and faint unresolved stars for our final set of targetfields, discussed in §4.For each data point, the total object flux is calculated as thecombination of the flux of the source, F s , the lens, F l , and theinterloping blend star, F int , F obj ( t ) = F s A ( t ) + F l + F int , (42)where A is the magnification at time t . The photometric un-certainty is the combination of the Poisson photon uncertaintyof the number of all object photons collected in an exposure, N obj = ˙ γ obj t exp , (43)where ˙ γ obj is the combined photon rate of the magnifiedsource, the lens, and the interloping blend star and t exp is theexposure time, and the total smooth background N back = ˙ ω back Ω eff t exp , (44)where ˙ ω back is the total photon rate per steradian of the Moon,the dark sky, and the smooth stellar background. Here weuse the seeing of the individual data point as modified by theairmass, described in §3.3.1, as the FW HM for equation (40).ext Generation Microlensing Simulations 13
Figure 8.
The result of matching the expected KMTNet photon rate nor-malization to OGLE-III photometric data. The green line represents the 5thpercentile of the RMS seen in OGLE-III light curves and the blue points rep-resent our simulated KMTNet photometric data assuming the same photonrate normalization (2.11 ph s − at I = 22 . t exp = 120s) as OGLE-III. We find that in order to match theirphotometric uncertainties we have to including a fractional systematic errorfloor of σ sys = 0 .
004 mag, scale the flux measurement Poisson uncertaintiesup by a factor of 1.3, and include an additional smooth background blendcomponent of µ I , sm = 18 . (cid:3) (cid:48)(cid:48) . The OGLE-III data shown here are forchip 5 from OGLE-III field 190, centered on (l,b)=(1.2136, -3.8694), nearBW. The fractional Poisson photometric uncertainty σ Poi is then σ Poi = (cid:112) N obj + N back N obj . (45)We also include a fractional systematic uncertainty, σ sys , toaccount for the limit of precision with which it is possibleto measure even the brightest stars due to uncertainties fromscintillation, flat-fielding, the determination of the PSF on thereference image, and other factors. The final fractional uncer-tainty σ obs on a given flux measurement is given by σ obs = (cid:113) σ + σ . (46)To calibrate the expected KMTNet flux in a realistic fash-ion we use OGLE-III photometric data. The 1.3m aperture ofOGLE-III obtains a photon rate of 2 . ˙ γ for I = 22. We findthat this photon rate normalization alone does not account forthe photometric uncertainties seen in OGLE-III photometry.We find that we are able to match the reported OGLE uncer-tainties only by introducing an additional smooth blend com-ponent of µ I , sm = 18 . / (cid:3) (cid:48)(cid:48) , scaling the resulting Poissonuncertainty up by a factor of 1.3, and including a systematicerror floor of σ sys = 0 .
004 magnitudes. Figure 8 shows an ex-ample using data from chip 5 of OGLE-III field 190, centeredon ( l , b ) = (1 . , − . µ I , sm in our fiducial simulations to account for realis- tic observational hurdles. We also use σ sys = 0 .
004 mag as apessimistic assumption of what the precision limit of KMT-Net might be and scale the Poisson photometric uncertain-ties by the same factor of 1.3 as a conservative estimate. Werun another set of simulations, discussed in §6.3, in which weremove this extra source of background, lower the value of σ sys , and make other optimistic assumptions in an attempt tobracket our expectations of the detection yields that KMTNetwill obtain.We base the final photon rate of KMTNet on OGLE-IV,which obtains 3 . ˙ γ for I = 22, higher than for OGLE-III dueto newer-generation CCD chips and an improved photometrypipeline. Scaling this by the ratio of the KMTNet to OGLEapertures, (1 . / . , yields a final photon rate normalizationof ˙ γ = 4 .
91 ph s − · − . I − . . (47)For both our fiducial and more optimistic simulation resultswe do not include any scatter in the photometric measure-ments and instead take them to be exactly equal to what isexpected for the binary-lens model. If we were to includescatter, finding the best-fit FSBL model for each light curvewould be prohibitively time-consuming. Furthermore, addingsuch noise to the measured fluxes would cause different sub-sets of light curves to be scattered into and out of our sam-ple of planet detections for each realization generated by thesimulations. We are interested in a more precise global es-timate of the planet detection rate that KMTNet will obtain,one that is not dependent on such fluctuations, and so do notinclude any Gaussian (or otherwise) random noise in our sim-ulated photometry. This is equivalent to assuming noise inthe photometry that is uncorrelated and scattered symmetri-cally about the binary lens model and averaging over a largenumber of realizations of our simulations. In such a case, foreach realization of the KMTNet detection rates, some detec-tions would be scattered into, and others out of, the sampledue to the inclusion of symmetric photometric noise. On av-erage, these competing effects would cancel out. Thus, bynot including any noise in our photometry we are both expe-diting the fitting process—we know the best-fit FSBL modelexactly—and implicitly assuming our predictions for the de-tection rates to be an average over a large number of realiza-tions of our simulations that include uncorrelated symmetricscatter, Gaussian or otherwise, in the photometry. Detection Algorithm
A microlensing event will ultimately result in a planet de-tection only if the initial increase in brightness due to theprimary event is detected and if the light curve subsequentlydisplays sufficiently significant deviations from a single-lensmicrolensing event. There are several criteria that must besatisfied for this to be true. The most important are that themicrolensing event itself must be detected and the perturba-tion due to the planetary companion must distinguish itselffrom a best-fit single-lens event, both according to predeter-mined ∆ χ thresholds. If a given microlensing event passesthese cuts, it is considered robustly detected and its event rateis added to the total event rate of detected planets. Detection of the Primary Event
Before a microlensing event can be probed for planetarysignatures, the initial increase in brightness due to the primarymicrolensing event must be reliably detected. We establish4 Henderson, et al.three criteria to determine this. We first compute the error-weighted mean flux of the light curve and subsequently thedifference in χ between this constant model and the best-fitbinary-lens model, ∆ χ . Since we do not include any scatterin our photometry, as discussed in §3.3.2, the best-fit binary-lens model is simply the light curve itself, and this curve has ∆ χ ≡
0. Thus, ∆ χ is just the ∆ χ of the constant (error-weighted mean) model. For the microlensing event to be ini-tially detected, ∆ χ must satisfy ∆ χ ≥ ∆ χ , th , (48)where we choose a threshold of ∆ χ , th = 500. It should benoted that there will be a subset of events that will pass thisthreshold solely due to the planetary perturbation. For certainlow-magnification events the primary event will not be suf-ficiently distinct from a constant model, given our choice of ∆ χ , th . However, a significant planetary signature can itselfincrease ∆ χ to cause the microlensing event to be initiallydetected, despite having a weakly magnified primary event.FFPs represent the extreme limit of this case, when the signaldue to the planet is the sole source of magnification over thecourse of the event. Our second criterion is that there must bemore than 100 points in the light curve, which we establishas a crude proxy for the precise determination of the lensingparameters. Thirdly, t must fall within the time coverage ofthe light curve, which also improves upon the precision of theparameters measured from the light curve. This final criterionis not automatically satisfied due to the limited visibility ofthe Bulge at the beginning and end of each year. If the eventsatisfies all of these criteria, its event rate is added to the to-tal microlensing event rate, which is given by equation (18)without the second Heaviside step function. This is separatefrom our final planet detection rate, as it considers solely theidentification of primary microlensing events, independent ofwhether their planetary signature is ultimately detected. Detection of the Planetary Perturbation
The next step is to search the primary event for the signatureof a planetary companion. First we use the input values for t , u , and t E from the event and compute a comparison PSSLlight curve. We then compute the ∆ χ of the FSBL light curvefrom this initial PSSL model, ∆ χ . We discard events with ∆ χ ≤ ∆ χ , th , (49)where we choose a threshold of ∆ χ , th = 100. We make thiscut because we do not consider any events below ∆ χ , th tobe robustly detected, and finding a best-fit model will onlydecrease from the value of ∆ χ that was found using thevalues from the binary lens as input for the parameters t , u ,and t E . Moreover, finding a best-fit light curve is generally thesecond-most computationally expensive operation, after theFSBL magnification calculation, and extraneous fitting shouldbe avoided. Otherwise, if ∆ χ > ∆ χ , th , we determinea best-fit PSSL or FSSL model, depending on the algorithmdescribed in §3.2.4.In the case of a PSSL model, we must find the PSSL lightcurve whose observed flux best matches that of the FSBL lightcurve. The model flux is given by F ( t ) = F s · A ( t ) + F b (50) where F s is the base flux of the un-magnified source, A ( t )is the magnification, and F b is the total blend flux. For asingle-lens microlensing event, F ( t ) is uniquely determinedby the five parameters t , u , t E , F s , and F b . The observedflux is a linear function of F s and F b but is non-linear in t , u , and t E , which together specify A ( t ). We use a downhill-simplex method (Press et al. 1992) to explore the parameterspace for combinations of t , u , and t E . At each point ofthe simplex, which represents a unique set of ( t , u , t E ), weuse matrix inversion to solve for F s and F b at each observa-tory (Gould 2003). We determine the PSSL light curve thatyields the lowest ∆ χ between itself and the simulated lightcurve and take that to be the best-fit PSSL model. The pro-cedure is identical when finding the FSSL model except that ρ is included as a fourth non-linear free parameter during thedownhill-simplex process. If the ∆ χ between this best-fitmodel and the FSBL light curve is greater than ∆ χ , th , wecount the planetary event as being detected and its event rateis added to the total event rate of detected planets, accordingto equation (18). We adopt ∆ χ , th = 160 as our canonicalvalue (Bennett et al. 2003) and note that while this value isappropriate for detections that arise from perturbations dueto the source passing over a planetary caustic, it is likely toolow for detections resulting from high-magnification events(Gould et al. 2010; Yee et al. 2013). However, we aver thisto be a valid threshold since the vast majority of planet de-tections will result from events with a peak magnification of A (cid:46) M p ≈ ⊕ and a ≈ I s ≈ u , ρ , and q indicates that finite-source effects are important, andin this case the best-fit is an FSSL model. OBSERVATIONAL PARAMETER OPTIMIZATION
We use the simulations as described above to first determinethe optimal observing strategy for KMTNet. The parametersover which we optimize are the locations of the target fields,the number of target fields, N fld , and the exposure time, t exp .KMTNet is designed to conduct a uniform survey to exploreexoplanet demographics in the regimes of parameter space towhich microlensing is most sensitive. Given its homogeneousapproach, we perform our optimizations assuming each fieldto have the same value of t exp , which we ultimately optimize.First we determine the importance of field placement, thenwe investigate trade-offs between different numbers of fieldsas well as observational cadence.ext Generation Microlensing Simulations 15 Figure 9.
An example light curve for an analog of Jupiter. The top panelshows the physical, lensing, and observational parameters for this event. Wehave included Gaussian scatter in the photometry solely for the purposes ofvisualizing the quality of data that KMTNet will actually obtain. This typeof perturbation, with a smaller amplitude and a longer time scale, is commonfor larger planet masses. Even a perturbation with such a low amplitude isrobustly distinguished from the best-fit single-lens model due to the combi-nation of KMTNet’s photometric precision and cadence, allowing for preciseand dense coverage of such deviations.
Figure 10.
An example light curve for an analog of Earth. The top panelshows the physical, lensing, and observational parameters for this event. Wehave included Gaussian scatter in the photometry solely for the purposes ofvisualizing the quality of data that KMTNet will actually obtain. This typeof perturbation, with a larger amplitude and a shorter time scale, is commonfor smaller planet masses. Even a planet for an event with such a faint source( I s ≈
21) and short perturbation ( ∆ t p ≈ Figure 11.
An example light curve for an analog of Mars. The top panelshows the physical, lensing, and observational parameters for this event. Wehave included Gaussian scatter in the photometry solely for the purposes ofvisualizing the quality of data that KMTNet will actually obtain. Here thebest-fit single-lens model is for a source of finite size. Even such a low-massplanet can be robustly detected given the photometric precision and cadenceof KMTNet.
Field Locations
There is an arbitrary number of possible tilings for a givennumber of target fields. Fortunately, as we will show, the pre-cise locations of the field centers do not significantly affectthe planet detection rate, provided that we restrict attention tofields with high event rate and low extinction. For definite-ness, we consider five tilings with a maximum of 13 fieldseach. In choosing the field centers we avoid regions where A I (cid:38) . ∼ ◦ . Determining the maximum planet yield across a largegrid of planet mass and separation pairings for each of the fivetilings would be prohibitively time-consuming. Therefore, weinstead use the total primary microlensing event rate down to I s = 22 as a proxy for the planet detection rate. For each fieldin each of these tilings we run 10 MC trials, drawing sourcesand lenses from within the field and computing their parame-ters and contributions to the event rate as described in §3. Thetotal event rate for all sources brighter than I s = 22 per field isthus the product of the total event rate for all events, the solidangle of the FoV, and the number density of source stars downto I s = 22 for the ( l , b ) of each event, Γ tot , I s ≤ = Ω FoV N MC (cid:88) i =1 ∆Γ i ξ i Φ ∗ , BW , I s ≤ , i , (51)6 Henderson, et al. Figure 12.
The cumulative microlensing event rate for sources with I s ≤ N fld for our five different tilings of 13 fieldseach. While there is a slight preference for different tilings at certain fixednumbers of fields, the differences between all tilings for a given value of N fld are nearly all within three sigma of the Poisson error on the rates. Given this,we consider the event rates across all tilings to be essentially equivalent. where, in ∆Γ i (see equation (17)), we have taken u , max = 1.We then rank all fields within a given tiling by this eventrate. Figure 12 shows the cumulative event rate for each ofthe five different tilings. While there is a slight preference fordifferent tilings at certain fixed numbers of fields, the differ-ences between all tilings for a given value of N fld are nearlyall within three sigma of the Poisson error on the rates. Giventhis, we consider that the event rates across all tilings are es-sentially equivalent. In other words, from our model and setof assumptions there does not appear to be an optimal tiling oftarget fields, provided they are chosen to lie within regions ofsufficiently low extinction and sufficiently high stellar density.This is a consequence of KMTNet’s FoV, which is generallylarger than the features in the morphology of the optical depthand stellar density in the regions of the Bulge we are consid-ering. To converge on an optimal tiling we turn to the resultsof Sumi et al. (2013), who use two years of MOA-II data tomeasure the microlensing event rate and optical depth towardthe Bulge. They identify and model a peak in the event rateper square degree per year that is located at ( l , b ) ≈ (1 . , − . M p / M ⊕ ) = 0 . , . , and 2 . a /AU) = 0.15, 0.40,and 0.65 for each mass. In linear units this corresponds toplanet masses of M p = 1, 10, and 100M ⊕ and planet-host starseparations of a ≈ Table 7
Fiducial Tiling of KMTNet FieldsField Rank l b α δ [deg] [deg] [deg] [deg]1 1.0500 -2.3000 269.28090 -29.208372 -0.7846 -3.1263 269.04706 -31.208373 3.6969 -2.9066 271.36107 -27.208374 2.0361 -4.0408 271.57224 -29.208375 -2.6225 -3.9495 268.80282 -33.208376 -2.0000 3.8000 261.52480 -28.572507 -4.9212 -3.9919 267.44505 -35.208378 4.4500 3.3500 265.81065 -23.393839 5.7235 -4.1018 273.62035 -26.0083710 0.2021 -4.8673 271.38545 -31.2083711 4.0587 -5.2401 273.86816 -28.0083712 -1.6355 -5.6909 271.19321 -33.2083713 -6.6162 -5.0639 267.52308 -37.20837
Note . — The fields are ranked by primary event rate. turn, we create and fit the light curves and implement the ∆ χ cuts discussed in §3.4. We then rank the fields according totheir primary event rate as well as their planet detection rate.The rankings using the two different metrics are nearly identi-cal, and for those field rankings that are not, the differences inthe rates between the two different fields with the same rankare within one sigma of the Poisson uncertainty in the rates.We ultimately select the primary event rate as our indicatorfor field ranking and order the fields accordingly. These finalrankings and the corresponding coordinates are given in Table7. Figure 13 shows the field locations of this tiling, overlaidon our extinction map, with field rankings labeled. Number of Fields
Now that a single tiling of target fields has been selected, wedetermine the optimal number of fields, N fld , within this tiling.We run our full simulation over the same 3x3 grid of M p and a as in §4.1, again creating and fitting the light curves and im-plementing the ∆ χ cuts. We vary the total number of fieldsfrom 1–13 and add fields according to the ranks determinedin §4.1, as shown in Table 7. We include the overhead timefor KMTNet of t over = 30s (Table 2). In addition, we assumea fixed exposure time of t exp = 120s. As a result, the cadence t cad with which the light curves in a given field are sampledincreases with the number of fields N fld as t cad = 150s N fld .Figure 14 shows the cumulative primary event rate andplanet detection rate as a function of the number of targetfields, the latter for three different planet masses. The cu-mulative primary event rate increases monotonically with N fld as a power-law N ev ∝ N α fld with α ≈ .
45 for N fld (cid:46)
9, atwhich point the slope becomes more gradual, α ≈ . N fld depends on the planet mass.For M p = 100M ⊕ , the planet detection rate increases almostmonotonically with additional target fields up until N fld ≈ M p = 10M ⊕ , the detection rate has approximately leveled offby N fld ≈
5. For M p = 1M ⊕ , the planet detection rate peaks at N fld ≈ N fld > N fld for a fixed valueof t exp , as shown in Figure 14. In other words, the total num-ext Generation Microlensing Simulations 17 Figure 13.
Our fiducial tiling of 13 KMTNet fields, ranked by their primaryevent rate, overlaid on our extinction map.
Figure 14.
The upper left panel shows the cumulative primary event rate asa function of the number of fields N fld for our fiducial tiling. Here the blackline shows the broken power-law fit, with a slope of α ≈ .
45 for 1 ≤ N fld ≤ α ≈ .
069 for 9 ≤ N fld ≤
13. The remaining three panels show the planetdetection rate for three different planet masses, summed across semimajoraxis a , as a function of N fld . Here we have assumed a constant exposuretime of 120s and an overhead of 30s, and thus the cadence varies with thenumber of fields as 150s N fld . The cumulative primary event rate increaseswith N fld as a broken power-law with the shift in slope occurring at N fld ≈ N fld shows a dependence on planet mass, reaching maxima at lower numbersof target fields for planets of lower mass. ber of primary microlensing events monitored and detectedincreases monotonically as the number of fields increases, atleast up to N fld = 13, and therefore there is a larger number ofprimary events in which to detect perturbations from plane-tary companions. On the other hand, each of these primarylight curves will be more poorly sampled, resulting in a de-crease in the signal-to-noise ratio (i.e., ∆ χ ) of the planetaryperturbations. According to equation (19), the planetary Ein-stein ring crossing time ∆ t p for an Earth-mass planet orbitinga 0.3M (cid:12) host star ( q ∼ − ) for an event with a time scaleof t E = 20d is ∼
90 minutes, and the durations of the plane-tary perturbations are a factor of a few times larger than this.Therefore, given the scaling of t cad with N fld , this would leadto many dozens of observations on a typical perturbation for asingle target field, but fewer than 10 observations for 13 targetfields.To investigate these two contributions more quantitatively,we first examine the dependence of the primary event rateon N fld . Revisiting the broken power-law behavior of Figure14, we find that the cumulative primary event rate goes as N ev = k F N α fld , with a change in α at N fld ≈
9. For 1 ≤ N fld ≤ k F ≈ α ≈ .
45. Over the range 9 ≤ N fld ≤
13 theslope flattens to α ≈ . k F ≈ α is always positive, indicating that in-cluding additional target fields acts to increase the microlens-ing event rate monotonically up to at least N fld ≈
13. How-ever, the break at N fld ≈ N fld , α < ∆ χ distribution, werun a set of higher-fidelity simulations across the same gridof M p and a for each number of fields. We increase N MC andlower the primary event detection threshold to ∆ χ , th = 10in order to increase the total number of simulated events anddecrease the statistical noise. Figure 15 shows the cumulativemicrolensing event rate as a function of ∆ χ for four valuesof N fld for each value of M p , summed across a . We see thatabove ∆ χ , th the ∆ χ distribution is well-characterized by apower-law, such that the fraction of events with ∆ χ above aminimum value ∆ χ scales as f det ∝ (cid:0) ∆ χ (cid:1) − β with β > ∆ χ , cut , with the distribution drop-ping off precipitously above this value due to finite-source ef-fects. This cutoff occurs at lower values of ∆ χ for lowerplanet masses because as q decreases, the presence of finite-source effects increasingly hampers our ability to detect theplanetary signals.Even conservative choices of ∆ χ , cut encapsulate ≥ t exp , t cad ∝ N fld , which in turn implies that the number of data pointsper unit of time, and thus ∆ χ , is inversely proportional to thenumber of fields ∆ χ ∝ N − . As a result, for our approximatepower law form for the cumulative distribution of ∆ χ , thefraction of events with ∆ χ > ∆ χ scales as f det ∝ N − β fld .Finally, we can combine these two results to estimate the8 Henderson, et al. Figure 15.
Primary event rate as a function of the final ∆ χ for each event,summed across semimajor axis a , for four different numbers of fields usinga value of ∆ χ , th = 10, denoted by the solid black line. Above ∆ χ , th andup until some ∆ χ cutoff, the event rate obeys a power law whose slopeincreases in magnitude with decreasing planet mass. We find that the slopeof this power law increases with N fld and is steeper for smaller planet masses,indicating the increasing importance of finite-source effects. total planet detection rate, N det = N ev f det ∝ N α − β fld . (52)Both α and β are positive for all values of ∆ χ , N fld , and M p we examined. Thus, when α > β , including additional tar-get fields will increase the planet detection rate, because theloss in the number of detections due to the poorer samplingof the planetary perturbations is more than compensated forby the increase in the number of primary events. On the otherhand, if α < β , then adding more fields will result in a net de-crease in the number of detected planets since the additionalfields have an optical depth that is sufficiently low that theincrease in the number of primary events they contribute isoverwhelmed by the decrease in planet detections due to thepoorer sampling. If α (cid:39) β , the total number of planet detec-tions is essentially independent of the number of fields. Pic-torially, we can understand the result of these two competingeffects by examining the morphology of Figure 15. As N fld in-creases, the overall ∆ χ distribution gets shifted up to higherevent rates due to the increase in the number of detected pri-mary events and to the left to lower values of ∆ χ due to theincrease in t cad and thus decrease in the number of data pointsand resulting ∆ χ per light curve.The value of β gradually increases as N fld increases from N fld = 1 to N fld = 13, steepening from β ≈ . β ≈ . M p = 100M ⊕ over the range of N fld , β ≈ . β ≈ . M p = 10M ⊕ , and β ≈ .
56 to β ≈ .
75 for M p = 1M ⊕ . Thisindicates that the drop-off of planet detection rate with ∆ χ becomes more severe as N fld increases and as M p decreases.Recalling that α ≈ .
45 for 1 ≤ N fld ≤ α ≈ .
069 for 9 ≤ N fld ≤
13, we see that α < β for for the entire range of N fld for M p = 1M ⊕ and that α < β by N fld ≈ M p = 10M ⊕ , causingthe planet detection rate to level off. For M p = 100M ⊕ , thebalance is not reached over our range of N fld . The expectationsbased on these approximate analytic scalings are roughly inaccord with the detailed results shown in Figure 14.Thus we conclude that there is no unique value for the op-timum number of fields, because the choice that maximizesthe planet detection rate depends on the planet mass. Further-more, there are additional considerations. In all cases, eventhough the planet detection rate may remain roughly constantor increase slightly as one increases the number of fields be-yond 2–4, it is clear that these perturbations will be less wellsampled, and therefore less well characterized. Therefore, wechoose four target fields as our fiducial optimal value becausealthough the detection rates for massive planets may increaseslightly for a larger number of fields, the increase is modest,the sampling of the perturbations is worse, and these detec-tions will be less well characterized. Furthermore, we aremore interested in low-mass planets, whose detection ratesand characterizations will be improved for fewer fields. Exposure Time
In the absence of overheads (e.g., detector readout time andtelescope slew and settle time), pixel saturation, and system-atic precision limits, conservation of information dictates thatthe exposure time should not affect the total microlensing rateor the planet detection rate, provided that t exp is sufficientlyshorter than the duration of typical magnification features.However, in reality, the value of t exp can significantly affectboth the number and type of microlensing events detected,given the existence of overheads, saturation, precision lim-its, and planetary signatures of varying durations. The choiceof t exp is, then, a balance between number of data points perlight curve and the resulting uncertainties in the flux measure-ment, both of which contribute to the ∆ χ of the event andultimately the uncertainties in the derived parameters. Wenow investigate the effect of varying t exp for a fixed number offields. The combined slew, settle, and readout time for KMT-Net will be t over = 30s, so we investigate the effect that choicesof t exp within a factor of several of t over will have on the eventand detection rates.We run our simulation across the same 3x3 grid of M p and a for a range of cadences. In §4.2 we found that a smallernumber of fields is preferable for planets with M p (cid:46) ⊕ .Higher-mass planets have only a slight increase in detectionrates with more target fields, and that boost would be miti-gated to some extent by larger parameter uncertainties. Wethus explore cadences of 300, 600, 900, 1200, and 1500s for1–6 fields.Figure 16 shows how the primary event rate and planet de-tection rate for each planet mass M p , summed across semi-major axis a , changes as a function of cadence and exposuretime for different numbers of fields. For M p = 10 and 100M ⊕ for N fld (cid:38)
4, the event rate increases with exposure time upto t exp ≈ t exp (cid:38) N fld (cid:46)
3, showa steady decline of event rate with t exp . For M p = 1M ⊕ , thesame structure is present for N fld (cid:38)
4, but the decline with t exp for N fld (cid:46) t exp decreases for N fld (cid:38) t exp approaches t over , the total time spenton a given observation becomes dominated by overhead time,so fewer photons are collected, driving the measured flux un-ext Generation Microlensing Simulations 19 Figure 16.
The upper two panels show the primary event rate as a functionof the cadence t cad (left) and the exposure time t exp (right) for five valuesof t cad . The lower three rows each represent the planet detection rate for adifferent planet mass. Each line represents a different value of N fld and iscolor-coded according to the legend at the top. For all planet masses andcadences there is a preference for 100 (cid:46) t exp / s (cid:46) N fld continue to gradually decay for higher values of t exp . certainties up and consequently lowering the ∆ χ . The dropin event rate as t exp / t over increases is slightly more subtle.Though a larger t exp decreases the number of data points perlight curve, information should be conserved, as the extantdata points would have reduced noise. However, this gain inprecision is bounded by both the saturation depth of an indi-vidual pixel, which renders individual data points with suf-ficiently high flux unusable, as well as the systematic errorfloor, which establishes the lower precision limit of the pho-tometry.We note that since our simulation does not strongly prefer aspecific choice of t exp , any value in the range ∼ × t over , correspond-ing to a cadence of 10 minutes for 4 fields) as the fiducial ex-posure time to minimize noise and maximize number of datapoints. FIDUCIAL SIMULATION RUN
Having converged on the field locations, N fld , and the ex-posure time t exp , we run a set of simulations across a larger9x17 grid of planet mass and planet-star separation, witheach variable again equally separated in log-space. This gridspans the range − . ≤ log( M p / M ⊕ ) ≤ .
00 and − . ≤ log( a / AU) ≤ .
20, with 0.50 and 0.10 dex spacing, respec-tively. In linear units, to which we will refer for the re-mainder of the text, this corresponds to planet masses of0 . ≤ M p / M ⊕ ≤ . (cid:46) a / AU (cid:46) .
8. For the sake of brevity we will limit eachquoted linear value of M p and a for all grid points to a maxi-mum of three significant figures. This grid of M p and a con- stitutes our fiducial results, which we take to be a conserva-tive estimate of the KMTNet event and planet detection rates.For this final run we implement the methodology describedin Penny (2013) to expedite our simulations. Their CausticRegion Of INfluence (CROIN) parametrization of binary mi-crolensing events identifies an area centered on the planetarycaustics outside of which there will be no (detectable) devi-ation due to the presence of a planetary companion. We usetheir CROIN parametrization to avoid modeling events whosesource does not pass through the CROIN but we include high-magnification events, which have u (cid:28)
1. We also run a moreoptimistic simulation on the same grid, discussed in §6.3, andanticipate that these two will bracket the microlensing eventrates and planet detection rates that KMTNet will obtain.After running our full grid of simulations, we find a hand-ful of cases for which our magnification computation algo-rithm, described in §3.2.3, fails, yielding data points with in-correct FSBL magnification according to the known underly-ing FSBL model. A systematic search reveals that this affects (cid:46)
10 points for 28 total detections across the grid points for thethree lowest masses, M p = 0 . , . , and 1M ⊕ . The sum ofthe detection rate for the affected events, from equation (18),is ∼ M p = 0 . ⊕ , ∼ M p = 0 . ⊕ , and ∼ M p = 1M ⊕ , which is only ∼ ∼ ∼
2% ofthe total planet detection rate for the respective planet masses.Furthermore, after recomputing the magnification and refit-ting the events we find that in all cases the fractional changein the value of ∆ χ is (cid:46) (cid:46) ∆ χ thatare below our threshold of 160. Planet Detection Rates
Figure 17 shows the annual planet detection rates as a func-tion of semimajor axis for all planet masses. Here we assumethat each lens star hosts exactly one planet, of the specifiedmass, at the specified separation. The detection rate for each[ M p , a ] grid point is calculated via equation (18) and is listedin Table 8. We compute the uncertainty in the detection rate asthe Poisson fluctuation of each individual event, weighted byits rate. Thus, the total detection rate for a given combinationof M p and a is given by equation (18) and its correspondinguncertainty by σ Γ tot = (cid:118)(cid:117)(cid:117)(cid:116) N det (cid:88) i =1 Γ i . (53)The peak in sensitivity occurs at a ≈ . M p (cid:38) M Jup , whileplanets with M p (cid:46) M ⊕ would constitute 4% of the detectionrate.We then normalize our planet detection rates assumingan underlying distribution of one planet per dex per starin log( M p ) and log( a ). The detection rate as a function oflog( M p ) is approximately a power-law with a slope of ∼ M p (cid:46) M ⊕ , likely due to finite-source effects. We also rescale our planet detection rates ac-cording to Cassan et al. (2012), who combine the three planetdetections found in PLANET data from 2002–2007 with pre-vious estimates of the slope of the mass-ratio function (Sumiet al. 2010) and its normalization (Gould et al. 2010) to derive0 Henderson, et al. Table 8
Fiducial Planet Detection Rates a /AU M p / M ⊕ M p / M ⊕ M p / M ⊕ M p / M ⊕ M p / M ⊕ M p / M ⊕ M p / M ⊕ M p / M ⊕ M p / M ⊕ Note . — Here we assume one planet per star at each grid point.
Figure 17.
Planet detection rate as a function of semimajor axis a for dif-ferent planet masses M p for the full grid of our fiducial simulations. Herewe have assumed that each lens star hosts exactly one planet, of the specifiedmass, at the specified separation. The detection rate increases with planetmass and peaks at a semimajor axis of a ≈ . a . a cool-planet mass function: f [log ( a ) , log ( M p )] ≡ d N det dlog ( a ) dlog ( M p )= 10 − . ± . (cid:18) M p ⊕ (cid:19) − . ± . . (54)We saturate the cool-planet mass function in equation (54) at5M ⊕ , corresponding to ∼ , as a conserva-tive approximation that is congruent with the fact that Cassanet al. (2012) have no measurements below that mass. Fig-ure 18 shows our planet detection rates as a function of M p Figure 18.
Planet detection rate as a function of planet mass M p , summedacross semimajor axis a , for the full grid of our fiducial simulations. The solidblack line represents an assumed planet frequency of one planet per dex perstar. The dashed black line represents our predictions for the detection ratesafter normalizing to the cool-planet mass function of Cassan et al. (2012),where we have saturated it at M p = 5M ⊕ (corresponding to ∼ ). We see that the steep dependence that the detection rates have as afunction of mass assuming one planet per dex is nearly exactly canceled outby the increasing frequency of planets with decreasing planet mass accordingto the cool-planet mass function for M p (cid:38) ⊕ . We find that KMTNet willbe approximately uniformly sensitive to planets with mass in the range 5 ≤ M p / M ⊕ ≤ ∼
20 planets per year per dex in mass acrossthat range. For lower-mass planets with mass in the range 0 . ≤ M p / M ⊕ < ∼
10 planets per year, the rate beingdominated by planets with mass near the upper end of this range. assuming both an underlying planet population of one planetper dex per star as well as our modified version of the cool-planet mass function. Interestingly, the steep dependence thatthe detection rates have as a function of mass assuming oneplanet per dex is nearly exactly canceled out by the increas-ext Generation Microlensing Simulations 21 Table 9
Normalized Planet Detection Rates M p / M ⊕ N det a N det b ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± a Here we assume an underlying planet pop-ulation of one planet per dex per star and therates are per year. b Here we apply the cool-planet mass func-tion of Cassan et al. (2012), where we havesaturated it at M p = 5M ⊕ (corresponding to ∼ ), and the rates are peryear. ing frequency of planets with decreasing planet mass that isfound by Cassan et al. (2012). This causes the detection ratesto be relatively independent of mass for M p (cid:38) ⊕ . Adopt-ing equation (54) and saturating it at M p = 5M ⊕ , we find thatKMTNet will be approximately uniformly sensitive to planetswith mass in the range 5 ≤ M p / M ⊕ ≤ ∼
20 planets per year per dex in mass across that range. Forlower-mass planets with mass in the range 0 . ≤ M p / M ⊕ < ∼
10 planets per year, therate being dominated by planets with mass near the upper endof this range. The detection rates as a function of M p , assum-ing both one planet per dex per star and also applying ourmodified version of the cool-planet mass function, are givenin Table 9. Parameter Distributions
Figure 19 shows the microlensing event rate as a functionof ∆ χ for all planet masses. For all planet masses there isa pileup in the rates just below ∆ χ , th , as expected. Above ∆ χ , th the distribution follows a power-law across several dexof ∆ χ before falling off steeply due to the prominence offinite-source effects.Figure 20 shows the planet detection rate as a functionof D s and D l for all planet masses. The mean value of D s for all planet detections is 8.7kpc, somewhat larger than ouradopted value of the Galactocentric distance, D GC = 8 . D l varies as a functionof planet mass. Lower-mass planets are found around prefer-entially closer lenses, with the average lens distance decreas-ing from D l ≈ . M p = 1000M ⊕ to D l ≈ . M p = 0 . ⊕ . For a fixed source distance D s and lens mass M l ,smaller values of D l increase the size of the Einstein ring θ E ,from equation (1). The larger the size of θ E , the less impor-tant finite-source effects will be, accentuating the deviationsdue to the presence of the planetary companion that are typ-ically more subtle for lower-mass planets (i.e., lens systemswill smaller mass ratios, q ). Consequently, as planet mass de-creases, or more appropriately, as q decreases, detections willoccur for closer lens systems. The percentage of Bulge ver-sus Disk lenses correspondingly also depends on planet mass.The fractions of Bulge and Disk lenses are 52% and 48%, re-spectively, for M p = 1000M ⊕ . But as M p decreases, the lenspopulation becomes increasingly dominated by Disk lenses, Figure 19.
Microlensing event rate as a function of ∆ χ for different planetmasses for our fiducial simulations, assuming one planet per star per dex .The solid black line shows our initial single-lens fitting threshold ∆ χ , th =100, above which the rates drops off sharply. The dashed black line denotesour final best-fit detection threshold ∆ χ , th = 160. with the percentage reaching 68% for M p = 0 . ⊕ .Figure 21 shows the planet detection rate as a function of I l , I s , and ∆ I ≡ I l − I s for all planet masses. The morphol-ogy of the source brightness distribution roughly follows thatof the LF, shown in Figure 3. Only 6% of the detection ratearises from sources with I s <
17, which we take as a crude cut-off for giant stars. The majority of the detections will comefrom fainter sources, with 64% having I s >
20. The bulk ofthe detection rate for lower-mass planets comes from brightersources, for which the photometric precision is higher, on av-erage, making it easier to robustly detect lower amplitude per-turbations. However, the vast majority of detections will suf-fer from severe blending, with 76% of detections coming fromevents with a source that is fainter than the combination of thelens plus the interloping blend star. Furthermore, a substan-tial fraction of the lenses will be faint, with 3% of detectionshavine I l ≥
20, making it more difficult to follow-up the lenssystems and obtain direct flux measurements.Figure 22 shows the planet detection rate as a function of q and s for all planet masses. For a given planet mass, thespread in q is slightly larger than one dex, corresponding tothe spread in the primary masses we consider. We find thatKMTNet will have moderate sensitivity down to mass ratiosas low as ∼ − . For planets with mass M p (cid:38) ⊕ , the dis-tribution of s roughly follows a broken power-law with achange in slope where the instantaneous projected separationis approximately unity. For s < s , scaling roughly as s . On the otherhand for s >
1, the detection rate scaling is less steep, go-ing roughly as s − . The asymmetry in this distribution arisesfrom the scaling of the size of the planetary caustic(s) with s .As shown in Han (2006), for s (cid:28) s and for s (cid:29) Figure 20.
Planet detection rate as a function of source distance D s andlens distance D l for different planet masses for the full grid of our fiducialsimulations, assuming one planet per dex per star. The mean value of D s forall planet detections is 8.7kpc, somewhat larger than our adopted value of theGalactocentric distance, D GC = 8 . D l varies as a function of planet mass. Lower-mass planets are found aroundpreferentially closer lenses, with the average lens distance decreasing from D l ≈ . M p = 1000M ⊕ to D l ≈ . M p = 0 . ⊕ . For a fixedsource distance D s and lens mass M l , smaller values of D l increase the sizeof the Einstein ring θ E , from equation (1). The larger the size of θ E , the lessimportant finite-source effects will be, accentuating the deviations due to thepresence of the planetary companion that are typically more subtle for lower-mass planets (i.e., lens systems will smaller mass ratios, q ). Consequently, asplanet mass decreases, or more appropriately, as q decreases, detections willoccur for preferentially closer lens systems. The percentage of Bulge versusDisk lenses correspondingly also depends on planet mass. The fractions ofBulge and Disk lenses are 52% and 48%, respectively, for M p = 1000M ⊕ .But as M p decreases, the lens population becomes increasingly dominated byDisk lenses, with the percentage reaching 68% for M p = 0 . ⊕ . caustic shrinks as s − .For low-mass planet with mass M p (cid:46) ⊕ , the behaviorwith s is qualitatively different from that for more massiveplanets. In particular, for s <
1, the detection rate for low-mass planets is strongly suppressed. This is a consequenceof finite-source effects. As shown in Gould & Gaucherel(1997), when the source is larger than and fully encloses thetwo triangular-shaped planetary caustics for s <
1, the frac-tional difference in the magnification from the single lens iszero, up to fourth order in the source size in units of the plan-etary Einstein ring radius, ρ p ≡ ρ q − / . On the other hand,when the source size is larger than and fully encloses the sin-gle, diamond-shaped planetary caustic that exists for s > ρ − p . There-fore, in the presence of strong finite source effects, planetaryperturbations for s < s > ρ p (cid:29)
1, the detectionrate for s > ∝ ρ rather than ∝ q / . This is likely the cause of the more gradualfall-off of the detection rate for low-mass planets for s > Figure 21.
Planet detection rate as a function of apparent source magnitude I s , apparent lens magnitude I l , and their difference ∆ I ≡ I l − I s for differ-ent planet masses for the full grid of our fiducial simulations, assuming oneplanet per dex per star. The morphology of the source brightness distributionroughly follows that of the LF, shown in Figure 3. Only 6% of the detectionrate arises from sources with I s ≤
17, which we take as a crude cut-off for gi-ant stars. The majority of the detections will come from fainter sources, with64% having I s ≥
20. The bulk of the detection rate for lower-mass planetscomes from brighter sources, for which the photometric precision is higher,on average, making it easier to robustly detect lower amplitude perturbations.However, the vast majority of detections will suffer from severe blending,with 76% of detections coming from events with a source that is fainter thanthe combination of the lens plus the interloping blend star. Furthermore, asubstantial fraction of the lenses will be faint, with 3% of detections havine I l ≥
20, making it more difficult to follow-up the lens systems and obtaindirect flux measurements. u for all planet masses. Using u as a proxy for peak mag-nification and taking A max ≈ u − , we find that 79% of eventswill have A max (cid:46)
10 and 95% of events will have A max (cid:46) Free-floating Planets
We also explore the detection rates KMTNet will obtain forFFPs. We run a set of simulations across the same mass rangeas for bound planets, 0 . ≤ M p / M ⊕ ≤ ∆ χ of eachFFP event from its flux-weighted mean brightness be greaterthan 500: ∆ χ ≥ . (55)We choose a higher detection threshold for FFPs than forbound planets because a much larger sample of light curvesmust be searched over a longer time baseline for perturba-tions from FFPs than bound planets. An example light curvefor an Earth-mass FFP is shown in Figure 24. As with thelight curves for bound planets, the scatter in the photometryis Gaussian and is solely for the purposes of visualizing theext Generation Microlensing Simulations 23 Figure 22.
Planet detection rate as a function of mass ratio q and instan-taneous projected separation s for different planet masses for the full gridof our fiducial simulations, assuming one planet per dex per star. The dis-tribution of s roughly follows a broken power-law with a change in slopewhere the instantaneous projected separation is approximately unity. Thesescalings, s for s < s − for s >
1, arise from the dependence of thesize of the planetary caustics on s , as shown in Han (2006). This behaviorchanges for low-mass planets with mass M p (cid:46) ⊕ . While finite-source ef-fects strongly suppress the detection rates for s <
1, they act to potentiallyenhance the perturbation for s >
1, enhancing the resulting detection rate(Bennett & Rhie 1996; Gould & Gaucherel 1997).
Table 10
Detection Rates for FFPs M p / M ⊕ N det ± ± ± ± ± ± ± ± ± Note . — Here we assumean underlying planet popula-tion of one planet per star inthe Galaxy and the rates areper year. quality of data that KMTNet will actually obtain. The highcadence of KMTNet produces densely sampled light curveseven for time scales as short as t E ≈ . M p for allplanet masses, assuming an underlying planet frequency ofone such planet per star in the Galaxy. We find that the de-tection rate as a function of planet mass to have a slope andnormalization similar to that for bound planets. The detectionrates are given in Table 10, and we predict that, assuming oneFFP per star in the Galaxy, KMTNet will detect ∼
100 withthe mass of Jupiter per year and ∼ Figure 23.
Planet detection rate as a function of impact parameter u fordifferent planet masses for the full grid of our fiducial simulations, assumingone planet per dex per star. The left-most solid black line marks a maximummagnification of A max ≈
100 while the right-most solid black line denotes amaximum magnification of A max ≈
10. We find that 79% of events will have A max (cid:46)
10 and 95% of events will have A max (cid:46) Figure 24.
An example light curve for an Earth-mass FFP. The top panelshows the physical, lensing, and observational parameters for this event. Wehave included Gaussian scatter in the photometry solely for the purposes ofvisualizing the quality of data that KMTNet will actually obtain. This demon-strates the ability of KMTNet to find FFPs down to the mass of Earth evenfor events with low peak magnifications and short time scales.
Figure 25.
Planet detection rate for FFPs as a function of planet mass M p .Here we have assumed an underlying FFP frequency of one such planet perstar in the Galaxy. The distribution roughly follows the same power-law asfor bound planets, with a steeper slope for M p (cid:46) ⊕ due to the prominenceof finite-source effects. EXTRINSIC PARAMETER VARIATION
While we have converged on an optimal set of observationalparameters, discussed in §4, there are other factors that willalso impact KMTNet’s detection rates. Here we examine thedependence of the detection rates on the number of observa-tories, to understand the gain in detection rates as the threeKMTNet telescopes successively come online, and the sys-tematic error floor, ultimately set by the quality of the dataand the photometry pipeline that KMTNet will employ. Fi-nally, we run a full grid of simulations with a more optimisticset of assumptions regarding seeing distributions, the system-atic error floor, the photon rate normalization, and the photo-metric precision achieved. We believe that this last set of sim-ulations in conjunction with our fiducial results will bracketthe detection rates that KMTNet will obtain.
Observatory Chronology
Figure 26 shows the planet detection rates as a function ofthe number of observatories, following the order in which theKMTNet telescopes will come online. The rates have beensummed across all semimajor axes. The rate of increase ofthe number of detections shows a dependence on planet mass.The fraction of planets that are detected by the first obser-vatory and by the first and second observatories together de-creases with the planet mass. For M p = 1000M ⊕ , 66% ofthe planet detections come from CTIO alone, and 86% comefrom CTIO in conjunction with SAAO. For M p = 1M ⊕ , only43% of the detections come from CTIO alone, and the frac-tion of detections that arise from CTIO and SAAO workingin concert has dropped to 73%. For high-mass planets with M p (cid:38) ⊕ , the planetary perturbation typically lasts sub-stantially longer than a day (see, e.g., Figure 9), and thus canbe detected from a single site, given sufficient photometricprecision and good weather. In this regime, particularly given Figure 26.
Planet detection rate as a function of the number of observato-ries for four different planet masses. The rates on the left-hand side of eachplot represent an assumed planet frequency of one planet per dex per star.The rates on the right-hand side of each plot assume the cool-planet massfunction of Cassan et al. (2012), where we have saturated it at M p = 5M ⊕ (corresponding to ∼ ). The red dashed line denotes thedetection rates for our fiducial simulations, which includes all three observa-tories. The fraction of planets that are detected by the first observatory andby the first and second observatories together decreases with the planet mass.For M p = 1000M ⊕ , 66% of the planet detections come from CTIO alone,and 86% come from CTIO in conjunction with SAAO. For M p = 1M ⊕ , only43% of the detections come from CTIO alone, and the fraction of detec-tions that arise from CTIO and SAAO working in concert has dropped to73%. For high-mass planets with M p (cid:38) ⊕ , the planetary perturbationtypically lasts substantially longer than a day (see, e.g., Figure 9), and thuscan be detected from a single site, given sufficient photometric precision andgood weather. In this regime, particularly given the cadence and photometricprecision of KMTNet, adding additional observatories will only marginallyincrease the total number of detected planets. On the other hand, planetswith mass M p (cid:46) ⊕ create perturbations that typically last a day or less(see Figures 10 and 11), and thus for such low-mass planets it is more prob-able for perturbations to be detected and observed by a single observatory,approaching the limit that each observatory contributes and equal fraction ofthe overall planet detection rate. the cadence and photometric precision of KMTNet, addingadditional observatories will only marginally increase the to-tal number of detected planets. On the other hand, planetswith mass M p (cid:46) ⊕ create perturbations that typically lasta day or less (see Figures 10 and 11), and thus for such low-mass planets it is more probable for perturbations to be de-tected and observed by a single observatory, approaching thelimit that each observatory contributes and equal fraction ofthe overall planet detection rate. We conclude that, since thedetection rates for the three observatories become more nearlyindependent as the planet mass decreases, having the full net-work is critical for maximizing the low-mass planet yield. Systematic Error Floor
Figure 27 shows the planet detection rate as a function ofthe systematic error floor, summed across semimajor axis. Westep through the range 0 . ≤ σ sys / mag ≤ .
02 in steps of0.001 mag. The value adopted for our fiducial simulations is0.004 mag, denoted by the dashed red line. Here we see thatext Generation Microlensing Simulations 25
Figure 27.
Planet detection rate as a function of the systematic error floorfor four different planet masses. The rates on the left-hand side of each plotrepresent an assumed planet frequency of one planet per dex per star. Therates on the right-hand side of each plot assume the cool-planet mass func-tion of Cassan et al. (2012), where we have saturated it at M p = 5M ⊕ (corre-sponding to ∼ ). The red dashed line denotes the detectionrates for our fiducial simulations, for which we use a systematic error floor of0.004 mag. Here we see two trends. First, the slope of the decrease in detec-tion rate as a function of the photometric precision limit increases as planetmass decreases. Secondly, the morphology of that dependence becomes moreasymptotic and less linear as planet mass decreases. the decrease in detections as a function of σ sys depends onmass. Over the range of systematic error floors we consider,we find that the trend is relatively weak and consistent withlinear for planets with mass M p ≥ ⊕ . On the other hand,for lower mass planets the trend is much stronger and appearsto exhibit an asymptotic behavior toward larger error floors.In particular, the fraction of detections that remain even forthe highest value (0.02 mag) of the systematic floor is 86%for M p = 1000M ⊕ , but drops to 51% for M p = 1M ⊕ .This behavior is analogous to that seen in the distribution of ∆ χ values for the detected planets shown in Figure 19, wherethe low-mass planets produce perturbations with a steeper dis-tribution of ∆ χ than high-mass planets. Both trends arelikely a consequence of the underlying spectrum of planetaryperturbations that is being probed in the simulations. In par-ticular, the bulk of the detections from high-mass planets arisefrom perturbations that have lower amplitudes and longer timescales. Such perturbations have very broad but shallow sig-nals, and thus the detection rate is less sensitive to the pre-cise value of systematic error floor or threshold ∆ χ . On theother hand, the detectable signals from smaller-mass planetsare more often due to short-duration but high-amplitude per-turbations. The spectrum of such perturbations is narrow andsteep, and thus the detection rate is more sensitive to the sys-tematic error floor or detection threshold. Optimistic Simulation Run
We also run a set of simulations across the same 9x17 gridof M p and a and with the same spacing, this time using a set Table 11
Optimistic Site-dependentSeeing DistributionParametersSite min. µ σ
CTIO 0.5 0.80 0.16SAAO 0.6 0.92 0.20SSO 0.6 1.2 0.40
Note . — All values are inarcseconds. of more optimistic input assumptions to bracket the planet de-tection rates that KMTNet will obtain. For these simulationswe • decrease the assumed value of the systematic errorfloor, from σ sys = 0 .
004 mag to σ sys = 0 .
002 mag, • remove the extra component of the smooth backgroundof µ I = 18 . (cid:3) (cid:48)(cid:48) , discussed in §3.3.2, that we hadpreviously included to match OGLE’s photometric un-certainties, which were higher than expected due to ad-ditional noise of unknown origin, • adopt seeing distributions that approximate the nativeseeing of each site, shown in Table 11, and • implement a photon rate normalization, derived below,that yields a photon rate of 9 . ˙ γ for I = 22.Here we derive the simplified theoretical photon rate nor-malization that we employ for the optimistic simulations. Wecompute the rate of detected photons for a star with an appar-ent magnitude of I = 22 as ˙ γ I =22 = f I =22 · E λ eff , I · ∆ λ I · A KMTNet · T E · QE · AE , (56)where f I =22 is the flux per unit wavelength of a 22nd mag-nitude star, E λ eff , I is the energy of a photon with a wave-length equal to the effective wavelength λ eff of the Cousins I -band filter, ∆ λ I is the width of the Cousins I -band filter, and A KMTNet is the effective collecting area of KMTNet. The lastthree terms encapsulate assumptions made about the photonloss rate from the top of the atmosphere to the production of aphotoelectron. Here
T E is the telescope efficiency, QE is the I -band quantum efficiency of a given pixel on the KMTNetCCD, and AE is the atmospheric extinction.Using these and the flux received from a 0-magnitudestar through the Cousins I -band filter, f I =0 = 112 . · − erg s − cm − Å − (Bessell et al. 1998), we obtain the fluxfor a star with I = 22: f I =22 = 1 . · − erg s − cm − Å − .From Bessell (2005) we obtain λ eff , I = 7980Å, making E λ eff , I =2 . · − erg γ − , and ∆ λ I = 1540Å. The effective diam-eter of the clear aperture of each KMTNet telescope is 1.6m,given in Table 1, yielding A KMTNet = 20106 . . Finally,we adopt T E = 0 . QE = 0 .
7, listedin Table 2. While in reality AE depends on the airmass of agiven observation, we make the simplifying assumption thatit is constant and use a slightly pessimistic value of AE = 0 . ˙ γ = 9 .
25 ph / s · − . I − . , (57)which is a factor of ∼ Figure 28.
Planet detection rate as a function of planet mass M p , summedacross semimajor axis a , for the full grid of our optimistic simulations. Thesolid black line represents an assumed planet frequency of one planet perdex per star. The dashed black line represents our predictions for the detec-tion rates after normalizing to the cool-planet mass function of Cassan et al.(2012), where we have saturated it at M p = 5M ⊕ (corresponding to ∼ ). The grey points and lines represent the detection rates for ourfiducial simulations, also shown in Figure 18. As with our fiducial detec-tion rates, we see that the steep dependence that the optimistic detection rateshave as a function of mass assuming one planet per dex is nearly exactlycanceled out by the increasing frequency of planets with decreasing planetmass according to the cool-planet mass function for M p ≥ ⊕ . For thehighest-mass planets with M p = 1000M ⊕ , the detection rate is roughly twotimes larger for the optimistic simulations than for our fiducial simulations.This boost factor increases monotonically as planet mass decreases, result-ing in a gain in detection rate by a factor of ∼
10 for the lowest-mass planetswith M p = 0 . ⊕ . This trend is likely a result of the evolution of the de-pendence of the detection rate on the photometric precision limit shown inFigure 26, given that the optimistic simulations have a systematic error floorthat is half of the value assumed for the fiducial simulations, coupled with theimproved photometric precision arising from the removal of the additionalsmooth background component. Figure 28 shows the planet detection rates as a function ofplanet mass for our optimistic simulations, summed acrosssemimajor axis. For M p = 1000M ⊕ our optimistic assump-tions result in an increase in planet detection rates by a fac-tor of ∼ ∼ M p = 1M ⊕ and ∼
10 for M p = 0 . ⊕ . These results show that even modest improve-ments in technology over that used by the extant generationof observational microlensing that boost the photon rate, de-crease the systematics, and limit the background noise leadto an significant increase in KMTNet’s planet detection rates,particularly for low-mass planets. DISCUSSION
If our fiducial and optimistic simulations reasonablybracket the expected KMTNet detection rates, we find thatKMTNet will substantially increase the annual detection ratesof exoplanets via gravitational microlensing. Adopting thecool-planet mass function of Cassan et al. (2012) and levelingit off at M p = 5M ⊕ (corresponding to ∼ ), wefind that the slope of the mass function almost exactly cancels Figure 29.
Planet mass M p as a function of semimajor axis a for knownexoplanets as of 29/May/2014. The data for planets discovered via transits orRV come from http://exoplanets.org while the data for planets discovered viamicrolensing, imaging, or timing come from http://exoplanet.eu. The thicksolid green line marks the KMTNet detection contour of 10 planets per yearwhile the thick dashed green line represents the KMTNet detection contourfor one planet per year, both assuming that each lens star hosts exactly oneplanet, of the specified mass, at the specified separation. The thick horizontalgreen dashed line denotes the mass at which KMTNet will detect one FFPper year, assuming one such planet per star in the Galaxy. KMTNet willsignificantly augment the number of known exoplanets near and beyond thesnow lines of their host stars, facilitating synoptic studies of exoplanet demo-graphics across an unprecedented range in both planet mass and planet-starseparation. the power-law slope of detections as a function of mass thatwe find with our simulations. For M p (cid:38) ⊕ , the regime inwhich we believe our rates to be robust and less sensitive tosmall number statistics, the detection rate is roughly flat, in-dicating that KMTNet will detect approximately ∼
20 planetsper year per dex in mass across this range. For lower-massplanets with mass 0 . ≤ M p / M ⊕ <
5, we predict KMTNetwill detect ∼
10 planets per year.Figure 29 shows all known exoplanets , color-coded by dis-covery technique. We overlay contours for our fiducial KMT-Net detection rates of one and 10 planets per year for boundplanets, assuming that each lens star hosts exactly one planet,of the specified mass, at the specified separation, and showthe planet mass at which the detection rate for FFPs is ap-proximately unity, assuming one such planet per star in theGalaxy. Even for our more conservative set of assumptionswe find that KMTNet’s annual detection rate will significantlyaugment the sample of known planets at planet-star distancesnear and beyond the snow line, and will do so even for planetsless-massive than Earth. This explosion in the microlensingdetection rate will complement the Kepler transiting planetpopulation, specifically at semimajor axes of a few AU all theway down to Earth-mass planets, allowing for studies of thedemographics of exoplanets across a range of over four dexin mass and three dex in separation. Furthermore, we predict Data are taken from http://exoplanets.org and http://exoplanet.eu as of 29/May/2014 ext Generation Microlensing Simulations 27that KMTNet will have the ability to detect FFPs with massbelow that of the Earth, providing a statistically large samplewith which to improve our understanding of their prevalencewithin the Galaxy.This material is based upon work supported by the Na-tional Science Foundation (NSF) Graduate Research Fellow-ship Program under Grant No. DGE-0822215 and an inter-national travel allowance through the Graduate Research Op-portunities Worldwide. Any opinions, findings, and conclu-sions or recommendations expressed in this material are thoseof the authors and do not necessarily reflect the views ofthe NSF. Work by B. S. G. was partially supported by NSFCAREER grant AST-1056524. C. Han was supported bythe Creative Research Initiative Program (2009-0081561) ofthe National Research Foundation of Korea. J. S. acknowl-edges support of the Space Exploration Research Fund ofThe Ohio State University, and also from the OGLE project’sfunding received from the European Research Council underthe European Community’s Seventh Framework Programme(FP7/2007–2013)/ERC grant agreement No. 246678. Workby B.S.G. and A.G. was partially supported by NSF grantAST 1103471. A. OPTICAL DEPTH COMPARISON
In this appendix we compare the optical depth predictedby our Galactic model to that measured in recent microlens-ing studies. Figure 30 shows the optical depth τ as a func-tion of Galactic latitude as measured from RCG stars fromthe MACHO (Popowski et al. 2005), OGLE-II (Sumi et al.2006), EROS (Hamadache et al. 2006), and MOA-II (Sumiet al. 2013) surveys, as well as a simple linear fit to thesecombined data. To compare to these results, we also show theoptical depth produced by our model in the range − . ≤ b / deg ≤ − .
75 with spacing of 0.25 degrees, averaging τ across − . ≤ l / deg ≤ .
0, with 0.5 degree spacing, at eachvalue of b . Our gradient of τ with b is slightly less steepthan that found by a simple linear fit to the RCG microlens-ing survey data. For the inner survey fields of b (cid:38) −
3, whichconstitute the majority of the event rate, our model under-estimates the optical depth as inferred from RCGs by up to ∼ b ∼ − .
5, these optical depths are ∼ l , b ) ≈ (1 . , − .
5) (see theirFigures 3 and 12). While we acknowledge that these resultswould affect our detection rates, we assert that it would nothave an impact with regard to the lack of a clear preferencefor the tiling of the observational fields, because the shifts be-tween the fields are not substantial and the tilings generallycover similar regions of the inner Bulge. On the other hand, itis possible that the steep gradient in the event rate implied bythese results would have an appreciable effect on our chosenoptimal number of fields. However, we argue that, given thesubstantial uncertainties in the measured optical depths andevent rates, the prudent strategy is to initially monitor the fourfields we have advocated here. The results of this initial sur-vey can then be used to more accurately determine the eventrate in these fields, and this information can then be used tofurther optimize a second phase of the KMTNet survey. In-
Figure 30.
Optical depth as a function of absolute Galactic latitude. Allpoints except the grey ones represent RCG-derived measurements of τ . Thedashed black line is a best-fit to the combined MACHO, OGLE, EROS, andMOA data. The solid black line is the optical depth of our Galactic model,in steps of 0.25 degrees in b , averaged over − . ≤ l / deg ≤ .
0, with 0.5degree spacing. The discrepancy between our model and the best-fit wouldcorrespondingly affect our event rates. The grey points show the optical depthderived using all sources in the MOA-II catalog, and increases more steeplytoward higher Galactic latitude. deed, it may be useful initially to monitor a dozen or so addi-tional “outrigger” fields with a much lower cadence, in orderto map the event rate over a much larger area of the bulge. B. ONLINE LIGHT CURVE ATLAS
In this appendix we describe a large-scale atlas of lightcurves we have created to facilitate visualization of and im-prove intuition about the planetary systems that we predictKMTNet will detect. For each grid point combination ofplanet mass M p and planet-star separation a we have gener-ated dozens of light curves from a random selection of thedetections. Figure 31 shows an example for a detection with M p = 0 . ⊕ and a = 2 . I s , I l , I int , and an estimate of the peak magnification8 Henderson, et al. Figure 31.
An example light curve from our online atlas a . The middlepanel contains an overview of the magnification of the event as a function oftime, with the data color-coded to the respective observatories from whichthey were taken. We have included Gaussian scatter in the photometry solelyfor the purposes of visualizing the quality of data that KMTNet will actuallyobtain. The second panel from the bottom highlights the planetary perturba-tion in greater detail and also includes the curves for three different models.The solid black line represents the FSBL model, about which the data arescattered. The dashed dark grey line shows the PSBL model, indicating theeffect of a finite source. The dotted light grey line represents the best-fitsingle-lens (SL) model, either PSSL or FSSL as determined by our simu-lation in §3.2.4. The lowest panel shows the residuals and models over thesame temporal range as the second panel. The fourth panel contains two plotsthat show an overview (left) and a zoomed-in version (right) of the geometryof the event. Here the blue circles represent the primary lensing mass (left)and planetary companion (right). The absolute sizes of the lensing masses arearbitrary but the radius of the primary is fixed while the radius of the planetscales as log( q ). The origin is located at either the center of mass (if s < s >
1) of the lens system. The orange curves represent thecaustic(s) while the black line and arrow together specify the source trajec-tory. In the right plot of the fourth panel the purple open circle is the source,scaled appropriately, centered on the time of the maximum deviation of theobserved light curve. The top panel shows the physical, lensing, and obser-vational parameters for the event. We have generated dozens of such figuresfor a random selection of planet detections for each grid point combinationof planet mass M p and semimajor axis a . a To view the light curve atlas in its entirety, please visit from the bottom panel it is possible to estimate the max-imum brightness of the event. It is also possible to esti-mate by-eye from where the signal of the planet arises, bothin time and in the source plane with respect to the caus-tics, and to estimate the ∆ χ . To view the light curve at-las in its entirety, please visit . REFERENCESAlard, C. 2000, A&AS, 144, 363Alard, C., & Lupton, R. H. 1998, ApJ, 503, 325Alexander, C. M. O. ., Bowden, R., Fogel, M. L., Howard, K. T., Herd,C. D. K., & Nittler, L. R. 2012, Science, 337, 721Bahcall, J. N. 1986, ARA&A, 24, 577Bakos, G. Á., et al. 2009, ApJ, 707, 446 Baraffe, I., Chabrier, G., Allard, F., & Hauschildt, P. H. 1998, A&A, 337,403—. 2002, A&A, 382, 563Barclay, T., et al. 2013, Nature, 494, 452Beaulieu, J.-P., et al. 2006, Nature, 439, 437Bennett, D. P. 2010, ApJ, 716, 1408Bennett, D. P., & Rhie, S. H. 1996, ApJ, 472, 660Bennett, D. P., et al. 2003, in Society of Photo-Optical InstrumentationEngineers (SPIE) Conference Series, Vol. 4854, Society of Photo-OpticalInstrumentation Engineers (SPIE) Conference Series, ed. J. C. Blades &O. H. W. Siegmund, 141–157Bennett, D. P., et al. 2008, ApJ, 684, 663—. 2010, ApJ, 713, 837Bessell, M. S. 2005, ARA&A, 43, 293Bessell, M. S., Castelli, F., & Plez, B. 1998, A&A, 333, 231Bond, I. A., et al. 2001, MNRAS, 327, 868—. 2004, ApJ, 606, L155Bonfils, X., et al. 2013, A&A, 549, A109Butler, R. P., Marcy, G. W., Fischer, D. A., Brown, T. M., Contos, A. R.,Korzennik, S. G., Nisenson, P., & Noyes, R. W. 1999, ApJ, 526, 916Cassan, A., et al. 2012, Nature, 481, 167Charbonneau, D., et al. 2009, Nature, 462, 891Cumming, A., Butler, R. P., Marcy, G. W., Vogt, S. S., Wright, J. T., &Fischer, D. A. 2008, PASP, 120, 531Cumming, A., Marcy, G. W., & Butler, R. P. 1999, ApJ, 526, 890Dominik, M., et al. 2010, Astronomische Nachrichten, 331, 671Dong, S., & Zhu, Z. 2013, ApJ, 778, 53Dressing, C. D., & Charbonneau, D. 2013, ApJ, 767, 95Dwek, E., et al. 1995, ApJ, 445, 716Gaudi, B. S. 2000, ApJ, 539, L59—. 2012, ARA&A, 50, 411Gaudi, B. S., et al. 2008, Science, 319, 927Girardi, L., Bressan, A., Bertelli, G., & Chiosi, C. 2000, A&AS, 141, 371Gorbikov, E., Brosch, N., & Afonso, C. 2010, Ap&SS, 326, 203Gould, A. 2000, ApJ, 535, 928—. 2003, ArXiv Astrophysics e-prints—. 2008, ApJ, 681, 1593Gould, A., & Gaucherel, C. 1997, ApJ, 477, 580Gould, A., & Loeb, A. 1992, ApJ, 396, 104Gould, A., et al. 2006, ApJ, 644, L37—. 2010, ApJ, 720, 1073Hamadache, C., et al. 2006, A&A, 454, 185Han, C. 2006, ApJ, 638, 1080Han, C., & Gould, A. 1995a, ApJ, 449, 521—. 1995b, ApJ, 447, 53—. 2003, ApJ, 592, 172Han, C., et al. 2013, ApJ, 762, L28Hartman, J. D., Bakos, G., Stanek, K. Z., & Noyes, R. W. 2004, AJ, 128,1761Holtzman, J. A., Watson, A. M., Baum, W. A., Grillmair, C. J., Groth, E. J.,Light, R. M., Lynds, R., & O’Neil, Jr., E. J. 1998, AJ, 115, 1946Howard, A. W., et al. 2012, ApJS, 201, 15Ida, S., & Lin, D. N. C. 2005, ApJ, 626, 1045Jacquet, E., & Robert, F. 2013, Icarus, 223, 722Kalas, P., et al. 2008, Science, 322, 1345Kennedy, G. M., & Kenyon, S. J. 2008, ApJ, 673, 502King, I. R. 1983, PASP, 95, 163Krisciunas, K., & Schaefer, B. E. 1991, PASP, 103, 1033Lagrange, A.-M., et al. 2010, Science, 329, 57Lissauer, J. J. 1987, Icarus, 69, 249Lissauer, J. J., et al. 2011, Nature, 470, 53Lovis, C., et al. 2006, Nature, 441, 305Majewski, S. R., Zasowski, G., & Nidever, D. L. 2011, ApJ, 739, 25Mao, S., & Paczynski, B. 1991, ApJ, 374, L37Marois, C., Macintosh, B., Barman, T., Zuckerman, B., Song, I., Patience, J.,Lafrenière, D., & Doyon, R. 2008, Science, 322, 1348Marshall, D. J., Robin, A. C., Reylé, C., Schultheis, M., & Picaud, S. 2006,A&A, 453, 635Mayor, M., & Queloz, D. 1995, Nature, 378, 355Mayor, M., et al. 2009, A&A, 493, 639Moffat, A. F. J. 1969, A&A, 3, 455Morton, T. D., & Swift, J. J. 2013, ArXiv e-printsNataf, D. M., et al. 2013, ApJ, 769, 88Nidever, D. L., Zasowski, G., & Majewski, S. R. 2012, ApJS, 201, 35Orosz, J. A., et al. 2012, Science, 337, 1511Paczynski, B. 1986, ApJ, 304, 1Peale, S. J. 1997, Icarus, 127, 269 ext Generation Microlensing Simulations 29ext Generation Microlensing Simulations 29