Periodic Signals in Binary Microlensing Events
DDraft version October 26, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
PERIODIC SIGNALS IN BINARY MICROLENSING EVENTS
Xinyi Guo , , Ann Esin , Rosanne Di Stefano and Jeffrey Taylor Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA Department of Physics and Astronomy, Pomona College, 610 North College Avenue, Claremont, CA 91711, USA Department of Physics, Harvey Mudd College, 301 Platt Blvd., Claremont, CA 91711, USA
Draft version October 26, 2018
ABSTRACTGravitational microlensing events are powerful tools for the study of stellar populations. In partic-ular, they can be used to discover and study a variety of binary systems. A large number of binarylenses have already been found through microlensing surveys and a few of these systems show strongevidence of orbital motion on the timescale of the lensing event. We expect that more binary lenses ofthis kind will be detected in the future. For binaries whose orbital period is comparable to the eventduration, the orbital motion can cause the lensing signal to deviate drastically from that of a staticbinary lens. The most striking property of such light curves is the presence of quasi- periodic features,which are produced as the source traverses the same regions in the rotating lens plane. These repeatingfeatures contain information about the orbital period of the lens. If this period can be extracted, thenmuch can be learned about the lensing system even without performing time-consuming, detailed lightcurve modeling. However, the relative transverse motion between the source and the lens significantlycomplicates the problem of period extraction. To resolve this difficulty, we present a modification ofthe standard LombScargle periodogram analysis. We test our method for four representative binarylens systems and demonstrate its efficiency in correctly extracting binary orbital periods. INTRODUCTIONOver the past two decades gravitational microlensingsurveys have transformed from a bold theoretical idea(Paczynski 1986) into an important tool for studying thestellar population of our Galaxy. Following Paczy´nski’sproposal, four different monitoring programs, MACHO(Alcock et al. 1997), OGLE (Udalski et al. 1993), EROS(Aubourg et al. 1993) and MOA (Yock 1998), for yearshave collected data on the Galactic Center as well as theLarge Megellanic Cloud and the Small Magellanic Cloud,detecting many thousands of microlensing events. Twomicrolensing surveys that are still currently in operation,OGLE-IV and MOA, continue to add to this number at arate of roughly 2000 events per year. While the majorityof the detected events are consistent with being producedby a single-mass lens, a sizable fraction shows clear ev-idence of lens binarity (Alcock et al. 2000; Jaroszynskiet al. 2004; Jaroszy˜nski et al. 2010). These are partic-ularly interesting, because more complicated lightcurves(see, e.g. Mao & Paczynski 1991) produced by binarylenses allow us to learn considerably more about thesesystems (see, e.g. Mao & Di Stefano 1995; Di Stefano &Perna 1997).The signatures of lens binarity are most prominentwhen the projected orbital separation between the twobinary components, a , is of the order of the Einstein ra-dius: R E = (cid:115) GMc ( D s − D l ) D l D s , (1)where D s and D l are the distances to the source and thelens respectively, and M is the total mass of the lens.The timescale for the duration of the lensing event isdetermined by the Einstein radius crossing time t E = R E /v, (2) where v is the relative transverse velocity between thelens and the source as viewed by the observer. For solar-mass lenses in the Galactic disk t E is typically on theorder of months, while the orbital period of the lens-ing binary, T orb , is more likely to be measured in years.As a result, most binary lensing events can be modeledwhile completely neglecting the orbital motion of the twomasses (see, e.g. recent estimates by Penny et al. 2011b).Nevertheless, a handful of microlensing light curves doshow convincing evidence of binary phase change duringthe event (Dominik 1998; Albrow et al. 2000; An et al.2002; Jaroszynski et al. 2005; Hwang et al. 2010; Ryuet al. 2010; Park et al. 2013; Shvartzvald et al. 2014).Recently, there has been increased interest in the pos-sibility of finding lensing events for which the timescaleratio, defined as R = t E /T orb , (3)is of order the of or greater than unity (Penny et al.2011a; Di Stefano & Esin 2014; Nucita et al. 2014). Inaddition, new techniques for finding such systems usingastrometric microlensing are being developed (Sajadian2014).If the orbital period can be determined for such sys-tems, then it will provide a new avenue for extractingthe lens parameters. Specifically, the period relates thetotal mass to the orbital separation. If the mass ratioof the binary and their separation in units of the Ein-stein radius can also be determined, then we will havesome of the key elements needed to specify the binary.In cases when some additional information is available,e.g., because the lens is located nearby and/or becauseparallax or finite-source-size effects are detected, a fullbinary solution may be obtained. Unlike single-lens lightcurves which have a simple analytical form (Paczynski1986), binary lenses can produce very complex magnifi-cation patterns that cannot be expressed in a closed form. a r X i v : . [ a s t r o - ph . GA ] S e p Guo et al.Even for a stationary binary lens, the process of fitting anobserved light curve can be complex (Di Stefano & Mao1996). The addition of orbital motion complicates the fit-ting process even further. Fortunately, in the limit where R (cid:38) T orb to be determined via simple Lomb-Scargle (LS) pe-riodogram analysis (Di Stefano & Esin 2014). In this pa-per, however, we are primarily interested in the regimewhere R (cid:46)
1. For these longer-period binaries, the inter-play between the orbital motion and the projected mo-tion of the source causes the period determined via LSanalysis to differ significantly from T orb (Di Stefano &Esin 2014). Here, we describe a modified timing analysismethod which allows us to compensate for this relativemotion and extract the correct binary period of the lens. MICROLENSING BY A ROTATING BINARYLENSA microlensing event produced by a single lens can becompletely characterized by three parameters: R E , theEinstein radius of the system; b , the distance of clos-est approach between the lens and the projection of thesource onto the lens plane (measured in units of R E );and v , the transverse velocity of the source with respectto the lens. In general, we can express the separationbetween the lens and the source projection (measuredagain in units of R E ) as u ( t ) = (cid:112) b + [( t − t ) /t E ] , (4)where t is the time of closest approach. The resultinglight curve is described in terms of the amplification func-tion, defined as the ratio of the observed lensed and un-lensed fluxes, which takes the form (e.g. Paczynski 1986) A SL ( t ) = u + 2 u √ u + 4 . (5)When the lens is a binary, in addition to the totalmass of the system, M = M + M , we need to specifythe mass ratio q = M /M and the dimensionless semi-major axis of the binary, α = a/R E , to fully describethe microlensing event. For simplicity, in this paper, weconsider only face-on, circular binary orbits so that α isequal to the binary separation at all times. Since thelens is rotating, two more parameters are necessary: thevalue of the binary phase angle (shown as angle ϕ inFig. 1) at the moment of closest approach, ϕ , and thedirection of binary rotation with respect to the relativemotion vector. For the source velocity shown in Fig. 1,we define a binary to be prograde if it is rotating clockwiseand retrograde otherwise.Since general relativity is a nonlinear theory, the lightcurve produced by a binary lens is more complicated thana simple superposition of two amplification functions andcannot be described by an analytical formula. However,we can calculate the amplification value numerically as afunction of source position in the lens plane (Schneider& Weiss 1986; Witt 1990). The result depends only on α and q . Given b and v , we can calculate the trajectoryof the source projection in the plane of the binary lensand construct the resulting light curve.Fig. 2(a) shows a sample light curve produced by ourstandard binary lens system which consists of two 0 . M (cid:12) C o M M ϕ bxu M γθ Figure 1.
Geometry of the lensing system. The black dot showsthe projection of the source location onto the lens plane. Thetrajectory of the source in the lens plane is given by the dashedline. Note that we refer to the vertical line connecting the centerof mass of the binary lens and the point of closest approach of thesource as a CoM-b-axis.
Figure 2. (a) The black solid line shows a sample theoreticallight curve with b = 1 . χ fit single-lens light curve A SL ( t ). (b)The residual light curve, defined as δ ( t ) = A ( t ) − A SL ( t ), clearlyshows the effects of lens binarity. The oscillatory pattern containsinformation about the orbital motion of the lens. stars (i.e. M = 1 M (cid:12) , q = 1) in a circular orbit with a = 1AU. The lens-source system is set up (described indetail in Section 4) so that α = 0 .
25 and the timescaleratio is R = 0 .
31, representative of the R (cid:46) α = 0 .
25. However, subtracting the single-lens fit fromthe binary light curve reveals a clear oscillatory patternin the residuals, as shown in Fig. 2(b).To understand the relationship between this periodicpattern in the residuals and the orbital motion of thelens, it is instructive to examine the trajectory of theeriodic Signals in Binary Microlensing Events 3
Figure 3.
Both panels show the contour plot of the absolute residual magnification function, | δ ( x, y ) | = | A ( x, y ) − A SL ( x, y ) | , producedby the standard system (described in Section 4). The contour map is centered on the center of mass of the binary lens and the axes are inunits of R E . The two binary components are located on the x -axis at x = ± α/ ± . | δ ( x, y ) | that correspond to different contour colors. As the source moves past the lens, its trajectory in the lensplane spirals in and out, producing the oscillatory residual pattern seen in the lower panels. Note that the colors of the trajectories andthe residual curves are matched. Panel (a) shows the trajectory of the source with b = 1 . source in the lensing plane. In Fig. 3 we show thistrajectory superimposed onto the contour plot of theabsolute value of the residual magnification | δ ( x, y ) | = | A ( x, y ) − A SL ( x, y ) | . Here x and y are coordinates in thelens frame, scaled to R E . The residual contours exhibit anearly-symmetrical four-lobe shape. Note that the longerlobes (lined up with the y-axis) correspond to positive δ ( x, y ) values , while the shorter ones have δ ( x, y ) < α is time-independent and theamplification map is static. In the co-moving and co-rotating frame of the lens, the source appears to spiralin as it moves towards the point of closest approach andspiral out as it moves away from the lens. If the sourcewere stationary, one full rotation of the binary lens wouldresult in two full oscillations of the residual light curve.Thus, the periodicity of the residual signal is related tothe orbital period of the binary lens by T ∼ T orb / . Notethat the relation is not exact due to the modificationcaused by relative transverse motion between the sourceand the lens.To get a quantitative understanding of the interplaybetween source motion and binary rotation, we need toderive an equation for the time evolution of angle θ , asshown in Fig. 1. Assuming that the source moves withvelocity v in the lens plane, its distance to the CoM-b- axis, scaled to R E , takes the form x = t − t t E . (6)The angle from the CoM-b-axis to the source is thengiven by γ ( t ) = − arctan xb = − arctan t − t bt E (7)as shown in the geometry of the lensing system (Fig. 1).Notice that γ ( t ) > t < t and γ ( t ) < t > t . The phase angle from CoM-b-axis to the axis ofthe binary system is ϕ ( t ) = ϕ ∓ ω orb t, (8)where ω orb = πT orb is defined as the orbital frequencyof the binary lens and “ − ” sign is adopted for progrademotion while “+” sign corresponds to retrograde motion.Finally, the angle between the source position and theaxis of the binary is then θ ( t ) = ϕ ( t ) − γ ( t )= φ ∓ ω orb t + arctan t − t bt E . (9)The distance from the source to the center of mass inthe lens plane, u ( t ) is given by Eq. (4). Note that the Guo et al.position of the source in the co-rotating lens plane issimply x ( t ) = − u ( t ) cos θ ( t ) , y ( t ) = u ( t ) sin θ ( t ) . (10)A natural consequence of Eq. (9) is that the angularvelocity of the source in the lens plane, dθ/dt , is smallerfor a prograde system than that for a retrograde system.This effect manifests itself clearly in the apparent pe-riod of residual signals shown in Fig. 3, which comparesthe trajectories of the source for a prograde and retro-grade system with otherwise identical physical param-eters. The retrograde residual light curve shows muchfaster oscillations because it has a larger value of dθ/dt . TIMING ANALYSISWe now address the question of how to extract the in-formation about the orbital periodicity of the binary lensfrom the observed microlensing light curves. We start bygenerating a theoretical binary light curve, add noise tosimulate real survey data, subtract the dominant single-lens signal, perform spectral analysis on the the residuals,and examine the accuracy of the inferred orbital period.3.1.
Modeling the Observations
We can generate a theoretical light curve of arbitraryduration. To approximate realistic observing conditions,we take the photometric uncertainty to be 1% for mostof our calculations. (We discuss the results for high-precision photometry in Section 6.) At u = 3, Eq. (5)gives A SL ∼ .
01, and so the overall amplification levelis marginally distinguishable from random noise fluctu-ation. Based on this, we define the span of each lensingevent to be from 3 t E before the time of closest approachto 3 t E after the closest approach. For all of our lightcurves we assume a fairly conservative average samplinginterval of ∆ t = 2 days. With this average rate, we sam-ple the theoretical light curve at random times { t j } andadd random Gaussian noise with mean 0 and standarddeviation 0 .
01 to each data point.Finally, we fit Eq. (5) to our simulated light curve,both to obtain the residual light curve δ ( t ) as well asto determine our best-guess values for b, t , t E , whichare necessary for the subsequent timing analysis. Wefound that the fitting procedure works much better aftersmoothing out the sharp features, if any, produced bycaustic crossings.3.2. Modified Lomb-Scargle Analysis
Once we have the residual light curve we are readyto perform spectral analysis to extract the underlyingperiodicity of the binary. We base our period extrac-tion method on the classical LS periodogram analysis(Scargle 1982), a modification of the Fourier transformdesigned for unevenly sampled data that has the advan-tage of time-translation invariance.For time series { δ ( t j ) } with N data points, the LS In fact, the OGLE-IV program monitors a significant portionof the Galactic Bulge with ∆ t < http://ogle.astrouw.edu.pl/sky/ogle4-BLG/ ) and the upcoming Korean MicrolensingTelescope Network will achieve even higher cadence with its widerlongitude coverage (Henderson et al. 2014).
Figure 4.
Periodogram of the light curve shown in Fig. 2. Panel(a) shows the results produced by the modified LS analysis. Thehighest-power peak corresponds to a period of 362 days, which isvery close to the actual orbital period of the binary shown witha dashed line. Panel (b) displays the results of the standard LSperiodogram analysis (here the power spectrum for negative periodvalues is simply a mirror image of the positive part). The best-fitperiod of 266 days is clearly very different from either T orb or T orb / periodogram as a function of frequency ω is defined as P ( ω ) = 12 (cid:16)(cid:80) Nj =1 δ ( t j ) C j (cid:17) (cid:80) Nj =1 C j + (cid:16)(cid:80) Nj =1 δ ( t j ) S j (cid:17) (cid:80) Nj =1 S j (11)where C j and S j are given by C j = cos( ω ( t j − τ )) , (12) S j = sin( ω ( t j − τ )) , (13)and τ is defined as τ = 12 ω tan − N (cid:88) j =1 sin 2 ωt jN (cid:88) j =1 cos 2 ωt j (14)Unsurprisingly, the unmodified LS periodogram anal-ysis does not generally yield the correct binary period,since the frequency of the observed oscillatory signal, ω ,is affected by the relative source-lens motion (see Fig.3) and is therefore not equal to the the binary orbitalfrequency, ω orb . Figure 4(b) shows the LS power spec-trum for a microlensing light curve with b = 0 . T orb = 372 . T orb / θ ,defined by Eq. (9), which takes into account both bi-nary rotation and source motion. Thus, we propose tomodify the oscillating coefficients C j and S j , essentiallyreplacing ωt j with θ ( t j ): C j = cos[2 { ω orb ( t j − τ ) − γ ( t j ) } ] , (15) S j = sin[2 { ω orb ( t j − τ ) − γ ( t j ) } ] . (16)We dropped the ϕ term in Eq. (9) since it simply pro-duces a time translation of the signal without affectingthe results of the LS analysis. The extra factor of 2originates from the fact that the binary magnificationpattern repeats itself twice over one binary revolution,and so the period directly observed in the data would beroughly half of the actual binary period, as we alreadypointed out above. Note that the definition of τ remainsthe same apart from replacing ω with ω orb in Eq. (14).Finally, the approximate values of b , t and t E necessaryfor evaluating γ were obtained when fitting Eq. (5) tothe original light curve.Unlike the standard LS periodogram analysis for whichonly positive ω values have meaning, ω orb can have ei-ther sign. Following the sign convention in Eq. (9), wheninterpreting the results of the modified periodogram, ω orb < ω orb > T = 2 π/ω orb =362 days, within 3% of the actual binary period. In addi-tion, a positive period value correctly identifies the lensas a retrograde binary.3.3. False Alarm Probability
The final issue we need to address is how to determinethe minimum power level, P cut , such that any peak witha power above P cut is unlikely to be produced by chance.Scargle (1982) derived a formula relating the false alarmprobability p f and P cut : P cut = − ln[1 − (1 − p f ) /N ω ] , (17)where N ω is the number of independent frequenciessearched. For p f (cid:28)
1, we can approximate Eq. (17)as P cut ≈ ln( N ω /p f ) = ln( N ω ) + ln(1 /p f ) . (18)For a given value of p f , P cut depends only on N ω . Thereis no clear consensus in the literature on how to calculate N ω for a given set of data, since the precise answer de-pends on the sampling method (Gilliland & Fisher 1985;Baliunas et al. 1985; Horne & Baliunas 1986). However,it is clear that N ω must be on the order of the numberof data points, N . We can thus rewrite Eq. (18) as P cut − ln( N ) = ln( N ω /N ) + ln(1 /p f ) , (19)and determine the term ln N ω /N empirically, using thesame sampling scheme as we do for our light curves.In order to do this, we simulated 5000 time series, ran-domly sampling pure Gaussian noise with 0 mean and σ = 1. Each time series consisted of N data points,with N randomly chosen between 10 and 1000 using a -2 0 2 4 6 8P-ln(N)0100200300400500 N u m be r o f T i m e S e r i e s Figure 5.
Distribution of P max − ln( N ) values for 5000 Gaussianrandom noise time series. The probability of detecting a spuriouspeak with a power P > N ) is less than 0.36%. uniform distribution in log space. We then applied thestandard LS analysis to each time series by searchingthrough frequencies from 2 π to N π , and examined thedistribution of P max − ln( N ) values, where P max is thepower corresponding to the highest peak in each time se-ries. We found that P max − ln( N ) tended to increase aswe increased N ω but above N ω = 16 N the distributionsremained virtually unchanged. This worst case scenariois shown in Fig. 5. Based on this simulation, if we choose P cut = 6 + ln( N ) as our significance criterion, then only18 out of 5000 time series register P max above this thresh-old. This corresponds to the false alarm probability of18 / . p f = 0 . N ω = N . Weadopted P cut = ln( N ) + 6 as our detection threshold, i.e.only the peaks with power exceeding P cut were treatedas real detections. THE STANDARD SYSTEMHaving established our method for timing analysis, wefirst test it for what we call our standard binary lensingsystem. It consists of two 0 . M (cid:12) stars separated by adistance of 2 a = 2 AU. We would expect this type ofbinary system to be fairly common since 0 . M (cid:12) starslie near the peak of the IMF. We take the source to benear the Galactic Center at a distance of 8 kpc and setthe lens at 4 kpc. For the relative transverse velocity, weadopt the value v = 60 km/s. For these parameters, theratio of the binary separation to the Einstein radius is α = 0 .
25 and the timescale ratio is R = 0 .
31. These arefairly conservative choices. On the one hand, α is lowenough that the signatures of binarity will not be veryprominent, as demonstrated in Figs. 2 and 3 (see alsothe discussion in Di Stefano & Esin 2014). In addition, arelatively low value of the timescale ratio puts this binaryfirmly in the regime where both the orbital motion andthe transverse motion of the source will play importantroles in the formation of the light curve, making it anideal testing system for our timing analysis.Our goal is to investigate the success of our proposedperiod extraction method for this system as a functionof the impact parameter b . We want to determine howweak must be the overall amplification of a microlensingevent must be for the periodicity in the light curve to nolonger be detected. To this effect, we vary the value of b from 0 .
05 to 4 .
0. Since low b means stronger events andpotentially more identified light curves in the existinglensing archives, we adopt a finer mesh for b < . Figure 6.
The efficiency of our timing analysis for the standardsystem. The dashed black line shows the period detection rate, i.e.the fraction of the simulated light curves for which of a significantperiodic signal is detected. The solid red line shows the correctperiod detection rate, i.e. the probability of extracting a significantorbital period that is within 10% of the true period of 372.5 days. coarser mesh for larger values of b .For each value of b we generated 2000 light curvesequally split between prograde and retrograde rotationdirections. For each light curve, the initial phase angle ϕ was randomly chosen from the interval [0 , π ]. Eachlight curve was then analyzed using our method to deter-mine (1) whether the power spectrum contains at leastone peak with P > P cut , as defined in the previous sec-tion; and (2) whether the period corresponding to thepeak with the highest power falls within 10% of the trueorbital period of the lensing binary.4.1.
Period Detection Rates
The results are summarized in Fig. 6. As expected,the overall period detection rate (shown as a dotted blackline) falls off with increasing b . This makes perfect sensebecause as the lensing event becomes weaker, the bina-rity signal becomes buried in the noise. Nevertheless,it is significant that most light curves are identified asperiodic past b = 3.Of greater interest to us is the correct period detectionrate (shown as a solid red line), i.e. the probability ofextracting a significant orbital period that is reasonablyclose to the true period of the binary. For the purposesof this paper, we defined a detected period to be “rea-sonably close” if it lies within 10% of the true binaryperiod of the lensing system (this also includes havingthe correct sign, i.e. distinguishing between progradeand retrograde rotation). This rate is falling faster withincreasing b than the overall detection rate, but even at b = 3, we are still correctly identifying periods for 30%of all the light curves. Figure 7 shows the distributionof detected periods for light curves with b = 1 .
0. It isclear that for most of these events we can use our methodto determine the binary period to an accuracy of a fewpercent.While the behavior of the detection rates for b > . b < .
0. Firstly, for b < . , < b < . ∼
50% drop in the correctdetection rate, while the overall detection rate remainsnear 100%. We investigate these in the remainder of thissection.
Figure 7.
Distribution of periods detected for b = 1 . High-frequency Signal in Low- b Events
To diagnose the precipitous drop in the detection ratesfor b < .
2, we examine the trajectory and the corre-sponding residual light curve curves of an event with b = 0 . Figure 8.
Source trajectory and the corresponding residual lightcurve for a prograde standard system with b = 0 . The problem arises because the overall shape of theresidual curve (lower panel of Fig. 8) shows a singlestrong high-frequency oscillation near the peak of themicrolensing event. By relating the residual curve to thetrajectory (upper panel of Fig. 8), we can see that itis due to the crossing of the red region between the twobinary companions. Because of its relatively large ampli-tude, this high-frequency feature dominates our timingeriodic Signals in Binary Microlensing Events 7
Figure 9.
Comparison of the detection rates with (panel [a]) andwithout (panel [b]) removing the high magnification ( A ≥
5) por-tion of the light curves. The dotted black and solid red curves aredefined as in Fig. 6. analysis and no significant period is detected in the datasince it is never repeated .A simple way of dealing with this issue is to simplyremove the high-frequency part of the light curve so thatthe global low-frequency signal is given more weight inthe timing analysis. To determine which part of the lightcurve to remove, we estimated the average radius of thered region to be ∼ . R E , and removed the correspondingportion of the light curve, i.e. the region with the overallmagnification higher than A SL (0 . ≈
5, where A SL isgiven by Eq. (5). The detection rates with and with-out this modification to the timing analysis are shown inFig. 9. Cutting out high magnification points appears tobe highly effective; it restores the period detection rateto near 100% and raises the correct detection rate to atworst 50% (Fig. 9(a)). Our investigation of other sys-tems (Section 5) showed that his high-frequency signalcontamination during close-approach events appears tobe an universal problem. Thus, any results we show fromthis point on include this step in the timing analysis.4.3. Prograde–Retrograde Period Confusion
To investigate the drop in the correct detection ratein the region 0 . < b < .
0, we examined the statisticsfor prograde and retrograde systems separately. Fig. 10demonstrates that this effect is mainly due to the pro-grade systems.Next we examined the statistically significant thoughincorrect periods detected for the prograde light curves.It turns out that for many of these events the highestpeak in the periodograms corresponds to a longer ret-rograde period ∼ +500 days instead of the correct pro-grade period of − . b , t E and t introducesan extra bump near the peak of the light curve. Thisextra feature causes the prograde residual light curves tobe mistaken for a retrograde one because the introduc-tion of the extra peak in the prograde case will mimicthe effect of higher frequency oscillations characteristicof retrograde light curves. In Fig. 12 we plot the proba-bility that one of the two highest-power peaks lies within10% of the correct period (dashed blue line), in addi-tion to the two detection rates we have been discussingso far. This new detection rate shows that indeed mostof the missing periods for low- b light curves appear inthe timing analysis as second-highest-power peaks. In- Figure 10.
Detection efficiency for the standard system for lightcurves with prograde (panel (a)) and retrograde (panel (b)) rota-tion.
Figure 11.
Periodogram of a prograde standard system with b =0 .
4. The highest peak corresponds to a longer retrograde periodof +491 days while the true prograde period of − . Figure 12.
Detection efficiency in standard system. The blueline represents the probability of either one of the highest peakand second highest peak lie in within 10% of the correct period ofthe system. terestingly, the detection rate of periods for higher b lightcurves also significantly improves if we are willing to set-tle for two possible answers for the binary period.It turns out that this confusion between prograde andretrograde rotation is specific to our standard systemrather than universal to binary lensing systems. In mostof the other example systems that we discuss in Section5, this issue hardly arises, except for one that is mostsimilar to the standard binary, i.e. the probability of thehighest peak being the correct period (red curve in Fig-ure 13) is the same as one of the highest two peaks beingthe correct period (blue curve in Figure 13) for systemsother than the standard and unequal. Guo et al. Figure 13.
Period detection rates for four systems summarizedin Table 1. OTHER EXAMPLE SYSTEMSWe now examine the efficiency of our timing analysistechnique for four other example lensing systems. Forall of them we take D l and D s to be the same as forour standard binary. The rest of their physical parame-ters are summarized in Table 1 and the resulting perioddetection rates are plotted in Fig. 13. Note that panel(a) shows the results for our standard system for ease ofcomparison. 5.1. Unequal Mass Binary
This binary system is essentially equivalent to the stan-dard binary lens system we discussed in the previous sec-tion except the masses of the two companions are M (cid:12) and M (cid:12) , respectively. The effect of having q (cid:54) = 1 is thechange in the shape of the residual amplification map;as illustrated in Fig. 14, the spatial asymmetry of massdistribution about the y -axis leads to the tilting of the“petals” in the residual pattern. As the source followsits spiraling trajectory in the lens plane, this asymmetrywill certainly affect the spacing of the peaks, as illus-trated in the light curves shown in the lower panels inFig. 14, and can complicate the process of extractingthe orbital period. However, it is encouraging that thedetection efficiency for this system, shown in Fig. 13(b),is only marginally different than for the standard equal-mass binary (Fig. 13[a]). In fact, the overall shapes ofthe detection rates for the two cases are essentially iden-tical when b <
1. At larger values of b , for which theasymmetry is more pronounced, the unequal-mass sys-tem shows slightly lower detection rates, but the dropdoes not exceed 5-10%.5.2. Tight Binary
This system differs from the standard case only in itssmaller binary separation. With α = 0 .
15, the charac-teristic binarity features in the residual light curves havevery low amplitudes (see e.g. Di Stefano & Esin 2014).On the other hand, smaller binary separation also meanssmaller orbital period. As a result, the timescale ratio ismore than twice as large for this system as for our stan-dard case, so the periodicity is easier to detect even inshorter duration light curves.
System
M/M (cid:12) q T orb (day) v (km/s) α R Standard 1 1 372.54 60 0.25 0.3135Unequal 1 0.5 372.54 60 0.25 0.3135Tight 1 1 172.76 60 0.15 0.676Wide 1 1 634.73 60 0.36 0.184
Table 1
Summary of relevant parameters for the four example systems.
Looking at the detection rates shown in Fig. 13(c), itis clear that our results are dominated by the decreasein α , the parameter which determines the amplitude ofthe features in the residual light curves. At best, we candetect periodicity in ∼
60% of all light curves and theprobability of detection drops nearly to zero for b > . R , whena period is detected the chances of it being correct arevery high. The only exceptions are light curves with b < .
1, where the correct detection rate drops to about halfof the total rate, just like it does for the standard system.We show in Section 6 that the detection rates for thissystem are very sensitive to the photometric precision ofthe data; with smaller noise the detectability can becomevery high (see Fig. 15(c)).5.3.
Wide Binary
We now consider the effect of having a larger binaryseparation while keeping all the other parameters thesame. With α = 0 .
36, this lensing system will producelight curves characterized by much more pronounced de-viations from the single-lens form. However, larger a im-plies longer T orb and therefore smaller R , making perioddetection more difficult.The results for the detection rates are shown in Fig.13(d). In contrast to the small-separation system dis-cussed in the section above, the overall detection ratefor the wide binary light curves shows only a moderatedecrease compared to the standard case. However, thecorrect detection rate drops significantly in the regimewhere b (cid:38)
1, reaching a maximum of 50% near b = 1 . b > b (cid:38) R = 0 .
184 this binary straddles the boundary of de-tectability since its orbital period is slightly longer thanthe length of the microlensing event. In prograde bina-ries the effective period of the residuals is larger than T orb , so detection becomes virtually impossible. For ret-rograde binaries, the effective period is smaller than T orb and can therefore be still detected. The correct detec-tion rate in the regime b < LOW NOISE DETECTION RATESWhile monitoring observations from the ground areunlikely to have photometric errors smaller than ∼ eriodic Signals in Binary Microlensing Events 9 Figure 14.
Magnification map for the unequal-mass binary system. Since the two masses are not the same, the magnification map is notsymmetrical with respect to the y-axis. Also plotted are the source trajectories with b = 1 . Figure 15.
Period detection rates for our four systems with0.001% photometric uncertainties. The dashed black line showsthe period detection rate and the solid red line shows the correctperiod detection rate as defined in Fig. 6.
Kepler , TESS , and
WFIRST , can do roughly 1000 times better (Koch et al.2010; Ricker et al. 2014). It is highly likely that mul-tiple microlensing events will occur in the field of thosespace-based telescopes over the lifetime of the mission,so it is worthwhile to consider how our period detectionmethod would fare for data with photometric precisionon the order of 0 . t E , insteadof 6 t E we adopted for higher noise levels.Fig. 15 shows our new detection rates for the foursystems. It is clear that the main effect of increasingphotometric precision is a significant increase in detec-tion rates, especially at high b . This result makes perfectsense: at large b , the overall light curve amplificationas well as deviations from the point-lens form becomesmall enough as to be non-detectable from the ground,but perfectly discernible from space.Note that the overall rate of period detection now re-mains flat at essentially 100% all the way beyond b = 4.Examining retrograde and prograde light curves sepa-rately, we again find that most of the gap between theoverall and correct detection rates is attributable to pro-grade rotation.The systems affected the most by the change in noiselevels are the tight and wide binaries. The results for theformer are particularly striking; where before we coulddetect at most 60% of the periods, we now can detectthe correct period for any light curve. The main rea-son for this improvement is again entirely due to the factthat very small amplitude signal can now be clearly de-tected. The value of α for this system places the binaryfeatures in the microlensing light curves at the edge ofdetectability with 1% errorbars, but this is not the casewith improved precision.For the wide binary, the periods for prograde lightcurves were largely not detectable with higher noise lev-els since the length of the event was simply too long0 Guo et al. Figure 16.
Ratio of the detected to actual orbital periods of thebinary lens as a function of R for LS (black triangles) and modifiedLS (red squares) periodogram methods. The results for the formerwere divided by a factor of 2 to compensate for the symmetry ofthe amplification pattern. Top and bottom panels show the resultsfor retrograde and prograde light curves, respectively. compared with the orbital period of the lens. Improvedsensitivity allows us to follow the event for a much longertime, effectively decreasing the minimum value of R forwhich the periods can be reliably detected. While the minimum value of R for which our timinganalysis can be applied is set by the photometric preci-sion of the data, we can place a firmer limit on the uppervalue of R for which our method is important. Clearly,when R (cid:29)
1, the lens-source relative motion becomesirrelevant and standard LS analysis should yield the cor-rect period (up to a factor of 2) for such a binary lens.Tosee when the standard LS results begin to deviate signif-icantly from the actual orbital period, we analyze, usingboth our modified LS and the standard LS methods, aseries of light curves all with b = 1, produced by lensingbinaries with R spanning the range between 0 . a , whichwas adjusted to vary T orb and therefore R . Fig. 16compares the ratio of detected orbital period to the trueorbital period of the binary T /T orb using our modified LSmethod (red curve) and the standard LS method (blackcurve) as a function of R .We see that our proposed timing analysis method isvery successful in extracting the correct orbital periodsfor the entire range of timescale ratios above the min-imum set by the length of the light curves , while the We point out that the drop of the correct detection rate in the b < t E and was thus not searched for in the analysis of high-noise lightcurves. Note that for high values of R , α becomes very small, but the standard LS periodogram algorithm clearly begins to fail(i.e. the detected period differs from the actual value bymore than 10%) when R (cid:46) . DISCUSSION AND CONCLUSIONSWhen the Einstein crossing time is comparable to theorbital period of a binary lens, microlensing light curvesdisplay quasi-periodic features which can be used to de-termine T orb . In the limit when R (cid:38)
1, the standardLS periodogram analysis can be used to find the period.However, in this paper we focus on the regime with R (cid:46) In this paper, we pro-posed a modification to the standard LS timing analysismethod designed to compensate for the source motionand extract the correct orbital period. We tested ournew timing analysis method on simulated light curvesat two different noise levels for four different binary lenssystems and calculated period detection rates for a widerange of impact parameter values averaged over randominitial phase angles. The results are very encouraging. Aslong as the orbital period is detectable (i.e. not longerthan the length of the simulated light curve) our pro-posed method finds and identifies it correctly in a largefraction of cases.We find that the period detection rates are determinedprimarily by the photometric precision and three interde-pendent parameters, α , R and b . Perhaps unsurprisingly,the photometric precision is the most important. It de-termines the minimum value of α for which binarity ofthe lens is detectable (Di Stefano & Esin 2014) and setsthe maximum value of b for which a microlensing eventis distinguishable from the noise. It also essentially setsthe maximum practical length for a light curve which inturn determines the maximum detectable orbital periodand therefore places a lower limit on R .We consider two levels of photometric uncertainty: 1%,characteristic of ground-based data, and 0.001%, relevantfor dedicated space missions such as Kepler . In the firstcase, the reasonable event duration is 6 t E (beyond this,the wings disappear into the noise) and, correspondingly,the detection rate drops off for R (cid:46) /
6, as illustrated inFig. 16 and well as by our results for the wide binary lensin Section 5.3. High-precision photometry allows us to periods are still detectable with the low-noise data. This would notbe the case for 1% photometry; however, other lensing parameterscan be adjusted to ensure that the value of α remains within adetectable range (see Di Stefano & Esin 2014). So our conclusionsfor R are still applicable. It remains to be explored how sensitively the data-removalmethod depends on the configuration of the source-lens system,e.g. the impact parameter and the initial phase angle, and thenoise level and sampling frequency under realistic observationalconditions. eriodic Signals in Binary Microlensing Events 11significantly expand the parameter space for which theorbital periods can be reliably detected with our method.We note that in our study, we model the light curvesby treating the lensed star as a point source. This iswell motivated by the fact that, for all the systems weconsidered in this paper, the angular size of a typicalsource star with radius ∼ R (cid:12) is only ∼ .
06% of theEinstein radius of the binary lens, negligible comparedto any of the feature size in the magnification map. Un-der certain circumstances, e.g. when the source is a giantstar located very close to the lens, finite-source effect canbecome important, reducing the amplitude and broaden-ing the features in the residual curve. However, Pennyet al. (2011a) showed that even in such extreme cases,the finite-source effect only affects a small portion of thebinary features in the light curve. More importantly,since the finite-source effect does not significantly alterthe timing of the periodic features, the result of the tim-ing analysis shall not be affected, as long as the softenedfeatures are still detectable above the noise level.As a proof-of-concept for our modified LS analysis, theresults we discuss in this paper are based on the analy-sis of light curves produced by binary lenses in face-oncircular orbits. We are now in the process of extendingour calculations to include elliptical and inclined orbits.In these more complex situations, the projected binaryseparation, and with it the shape of the lensing magnifi-cation pattern, will vary with time during the microlens-ing event. This effect will certainly affect the spacing ofthe features in the residual light curves, possibly affect-ing the extraction of the orbital period. Our preliminaryresults indicate that the orbital period should still be de-tectable in a large fraction of light curves, but the fullreport will be the subject of a follow-up paper (M. Vicket al, in preparation).Finally, we want to point out that in its present form,our timing analysis method works best for binary lenseswith q not very different from unity. It is predicated onthe assumption that the oscillatory features repeat twiceper orbit. The fact that period detectability does notchange much from q = 1 . q = 0 . q forthe unequal-mass system and found that the detectabil-ity decreases by a factor ∼ q = 0 .
25 and dropsbelow 20% for q = 0 .
1. It is clear that in the planetaryregime (i.e., when q (cid:28)
1) a factor of 2 in Eqs. (15) and(16) does not apply, since the magnification pattern isno longer even remotely symmetrical (Di Stefano 2012).Fortunately, such systems produce visibly different lightcurve morphologies characterized by very spiky rather than sinusoidal features (see Di Stefano & Esin 2014, forsome examples of light curves produced by low- q binarylenses), and so can be flagged. We are now working onways to extend our method to such low- q systems.We would like to thank Christopher Night for signifi-cant contributions to an earlier version of this work andthe anonymous referee for constructive comments. Thiswork was supported in part by support from NSF AST-1211843, AST-0708924 and AST-0908878 and NASANNX12AE39GAR-13243.01-A.systems.We would like to thank Christopher Night for signifi-cant contributions to an earlier version of this work andthe anonymous referee for constructive comments. Thiswork was supported in part by support from NSF AST-1211843, AST-0708924 and AST-0908878 and NASANNX12AE39GAR-13243.01-A.