A Metric and Optimisation Scheme for Microlens Planet Searches
aa r X i v : . [ a s t r o - ph . E P ] J a n Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 29 October 2018 (MN L A TEX style file v1.4)
A Metric and Optimisation Scheme for Microlens PlanetSearches
Keith Horne , Colin Snodgrass , Yianni Tsapras , SUPA Physics and Astronomy, University of St.Andrews, North Haugh, St.Andrews KY16 9SS, Scotland, UK.([email protected]). European Southern Observatory, Alonso de Cordova 3107, Casilla 19001, Vitacura, Santiago 19, Chile.([email protected]). Las Cumbres Observatory Global Telescope Network, 6740B Cortona Dr, Suite 102, Goleta, CA, 93117, USA. Astrophysics Research Institute, Liverpool John Moores University, Twelve Quays House, Egerton Wharf, Birkenhead, CH41 1LD, UK.([email protected]).
Accepted . Received ; in original form
ABSTRACT
OGLE III and MOA-II are discovering 600-1000 Galactic Bulge microlens eventseach year. This stretches the resources available for intensive follow-up monitoringof the lightcurves in search of anomalies caused by planets near the lens stars. Weadvocate optimizing microlens planet searches by using an automatic prioritizationalgorithm based on the planet detection zone area probed by each new data point.This optimization scheme takes account of the telescope and detector characteristics,observing overheads, sky conditions, and the time available for observing on each night.The predicted brightness and magnification of each microlens target is estimated byfitting to available data points. The optimisation scheme then yields a decision onwhich targets to observe and which to skip, and a recommended exposure time foreach target, designed to maximize the planet detection capability of the observations.The optimal strategy maximizes detection of planet anomalies, and must be coupledwith rapid data reduction to trigger continuous follow-up of anomalies that are therebyfound. A web interface makes the scheme available for use by human or robotic ob-servers at any telescope. We also outline a possible self-organising scheme that maybe suitable for coordination of microlens observations by a heterogeneous telescopenetwork.
Key words: gravitational lensing, planetary systems, methods: observational
Gravitational microlensing reveals stars and planets thatmagnify the light from a background source star (Mao& Paczynski 1991). The wide-field OGLE III ⋆ (Udalski,et al. 2003) and MOA II † surveys of Galactic Bulgestarfields discover ∼ − t E ∼
30 ( M ⋆ / . M ⊙ ) / days as the intervening M ⋆ ∼ . − M ⊙ lens star crosses near the line of sight. A planet near the lensstar acts as a smaller lens, smaller by a factor ( m p /M ⋆ ) / .When appropriately placed, the planet can produce a briefbut easily detectable flash or dip in the lightcurve. Such ⋆ ∼ ogle † planet anomalies last t p ∼ m p /m J ) / days, thus a fewdays for Jupiters or a few hours for Earths. The probabil-ity that the planet is detectable is P det ∼ . m p /m J ) / for “cool planets” in the “lensing zone”, a ∼ . − R E ∼ − M ⋆ / . M ⊙ ) / AU (Gould & Loeb 1992).When a planet anomaly is well sampled by observations,its duration, timing, and shape determine the mass ratio, q = m p /M ⋆ and the orbit size a relative to the Einstein ringradius R E . Roughly speaking, the planet anomaly’s duration t p sets the mass ratio, q ∼ ( t p /t E ) , and its time t , relativeto the event peak at t , measures the projected planet-starseparation a sin θ/R E ∼ ( t − t ) /t E . The lens star’s distance D and mass M ⋆ , when constrained by the event timescale t E ,are initially uncertain to factors ∼
3. Several methods usingfinite-source effects, parallax, and proper motion can furtherconstrain the lens geometry to establish m p , a and M ⋆ withhigher accuracy (Gould 2009). For example, high-resolution c (cid:13) Horne, Snodgrass imaging several years after the event can detect the lens starflux, colour and proper motion (Bennett, Andreson, Gaudi1996).The m / dependence of Einstein ring sizes makes mi-crolensing more sensitive to low-mass planets than othermethods. The microlens signature of an Earth-mass planetis brief, a few hours, but can be strong enough for easydetection (Bennett & Rhie 1996; Dominik, et al. 2007) pro-vided one is observing the right star at the right time. Witha detection probability P det ∼ . m p /m J ) / (Gould &Loeb 1992), a dedicated survey monitoring ∼ eventswith < ∼ η ⊕ cool Earths,if each lens star has η ⊕ of them. While this level of moni-toring has not yet been achieved, significant constraints onthe abundance of large cool planets were established (Gaudi,et al. 2002; Tsapras, et al. 2003; Snodgrass, Horne, Tsapras2004) even before the first secure microlens planet detection; η Jup < µ FUN ‡ . The high-magnification events are often identified a few days in ad-vance, permitting the rapid mobilisation of many telescopesto cover the peak of the lightcurve as intensively as possible.The PLANET § Collaboration (Albrow, et al. 1998) deploysa network of small ground-based telescopes to achieve quasi-continuous coverage of the most promising events. This ef-fort has been joined by RoboNet ¶ (Burgdorf, et al. 2007;Tsapras, et al. 2009), using three 2 m robotic telescopes. Amuch larger robotic telescope network is being laid out byLCOGT k in the next few years (Tsapras, et al. 2009). Withthe prospect of this network of 24 0.4m and 18 1.0 m robotictelescopes contributing to microlens planet searches, auto-mated strategies will be increasingly important to effectivelyorganise the follow-up observations.The OGLE-2002-BLG-055 lightcurve has one good datapoint that is 0.6 mag high. While this could be a planetanomaly (Jaroszynski & Paczynski 2002), undersamplingprevents adequate characterisation of this event (Gaudi &Han 2004). In the first secure characterisation of a microlensplanet, the lightcurve of OGLE-2003-BLG-235/MOA-2003-BLG-053 exhibits two fold caustics separated by 7 days, at-tributed to lensing by a ∼ . m J planet (Bond, et al. 2004).The 2005 season revealed three microlens planets, ∼ m J OGLE-2005-BLG-071Lb (Udalski, et al. 2005; Dong, et al.2009), ∼ m ⊕ OGLE-2005-BLG-390Lb (Beaulieu, et al.2006), and ∼ m ⊕ OGLE-2005-BLG-169Lb (Gould et al. ‡ ∼ microfun/ § http://planet.iap.fr ¶ http://robonet.lcogt.net/ k Las Cumbres Observatory Global Telescope. http://lcogt.net ∼ . M ⊙ ) host star (Gaudi,et al. 2008). From the 2007 season, MOA-2007-BLG-192(Bennett, et al. 2008) appears to be a brown dwarf with a ∼ m ⊕ planetary companion. Other planets from 2007 arenot yet published. It appears reasonable on present evidenceto expect increasing numbers of microlens planet discover-ies, leading to detection of cool Earth-mass planets within afew years, provided the capabilities for intensive monitoringof OGLE III and MOA II events continues to improve.This paper develops an optimal strategy for reactive mi-crolens planet searches that may help to increase the planetdiscovery rate, particularly with dedicated telescope net-works. Section 2 briefly reviews microlens lightcurves to de-fine notation and establish a few results for later use. Section3 employs numerical integrations and scaling laws to quan-tify the detection zone area that we propose as the metric ofsuccess for a microlens planet search. Section 4 develops theoptimal observing strategy. Section 5 discusses several prac-tical issues, and outlines a possible self-organising schemebased on continuously varying target priorities, that may besuitable for coordinating microlens observations by a hetero-geneous telescope network. Section 6 summarises the mainresults, and describes our web interface to the PLOP (PlanetLens OPtimisation) algorithm. During a microlensing event, light from a background sourcestar reaches the Earth along paths that bend toward anintervening lens star. With perfect alignment of the observer,lens and source, the observer sees the background star as anEinstein ring of angular radius θ E centred on the lens. Alight ray with impact parameter R bends toward the lensmass M by a small angle α = 2 S L R , (1)where S L = 2 G M L /c is the Schwarzschild radius of thelens. The point-mass gravitational lens has strong sphericalaberration, the effective focal length being f = Rα = R S L . (2)If D L and D S are the observer-lens and observer-source dis-tances, respectively, the lens formula of geometric optics is1 D L + 1 D S − D L = 1 f = 2 S L R . (3)Solving for R gives the radius of the Einstein Ring, R E = (2 S L D S X (1 − X )) / , (4)where X ≡ D L /D S is the lens/source distance ratio, 0 1. The angular radius of the Einstein ring is θ E = R E D L = (cid:16) S L D S − XX (cid:17) / . (5) c (cid:13) , 000–000 ptimising Microlens Planet Searches For M L = M ⊙ , D L = 5 kpc, and D S = 10 kpc, the EinsteinRing radius R E ≈ θ E ∼ . u ± = u ± (cid:0) u + 4 (cid:1) / , (6)giving two image positions, u + = θ + /θ E > u − = θ − /θ E < u = θ/θ E for the unlensed source. Note for future referencethat u ± = T ± B , (7)where T = u + u − = u + 2 and B = u − u − = u (cid:0) u + 4 (cid:1) / .Fig 1 shows the Einstein ring and trajectories of thetwo images during a microlensing event. On the lens planeperpendicular to the line of sight, we define cartesian coor-dinates x and y with the origin at the lens star, the sourcestar moving in the + x direction and crossing the + y axis atclosest approach. In units of θ E , the source-lens separationis u = (cid:0) u + u x (cid:1) / , (8)with u the separation at closest approach, and u x = µ ( t − t ) θ E = ( t − t ) t E , (9)where µ is the relative proper motion, t is the time of closestapproach, and the event timescale, t E = θ E µ rel , (10)is the time to cross the radius of the Einstein ring.The image-lens separations satisfy u u − = 1, so that,as seen in Fig 1, the major image at u + is always outsidethe Einstein ring, while the minor image at u − remains in-side. The major image slides “over the top” of the Einsteinring, while the minor image traces a loop inside the ring.Both images become brighter as they approach the Einsteinring. Each point on the disc of the source star maps to acorresponding lensed position on the image. The images arethus stretched in azimuth by a factor u ± /u and squashed inradius by d u ± / d u . With surface brightness conserved, thenet magnification arising from the increased solid angle is A ± = (cid:12)(cid:12) u ± u (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) d u ± d u (cid:12)(cid:12)(cid:12) = 12 u dd u (cid:2) u ± (cid:3) = u ± u − u − = T ± B B = A ± . (11)The image magnifications satisfy A + = 1+ A − , and the totalmagnification is A ≡ A + + A − = u + u − u − u − = TB = u + 2 u (cid:0) u + 4 (cid:1) / . (12)Since u changes with time, this defines a characteristic point-source point-lens (PSPL) lightcurve (Fig 1). Power-law ap-proximations (Fig. 2) for large, intermediate, and small u are Figure 1. Top panel shows snapshots of the unlensed backgroundstar and of the two distorted images of it that appear on oppositesides of the lens during a micro-lensing event. The major imagepasses over the top of the Einstein ring, while the minor imageexecutes a loop inside the Einstein ring. The two images becomecompressed in radius but elongated in azimuth as they approachthe Einstein ring, resulting in a net magnification. This producesthe characteristic lensing lightcurves shown in the bottom panel,where dashed lines show the separate lightcurves of the two im-ages, A + ( t ) and A − ( t ), and the solid line is their sum, A ( t ). Notethat A + ( t ) = 1 + A − ( t ). A ( u ) − u − u > ∼ . ,u − / . < ∼ u < ∼ . ,u − u < ∼ . . (13)Note that A ( u ) has the inverse: u = (cid:16) h(cid:0) − A − (cid:1) − / − i(cid:17) / . (14) For a point source at high magnification, as u → A → /u becomes formally infinite, corresponding to the formationof an Einstein ring of infinite magnification and infitesimalthickness. This point-source approximation breaks down,however, when the source star’s finite size becomes impor- c (cid:13) , 000–000 Horne, Snodgrass Figure 2. The point-source point-lens (PSPL) magnification A ( u ) compared with the power-law approximations defined inEqn 13. tant. In Fig.1, curvature of the highly magnified images is al-ready evident. At still higher magnifications, the major andminor images extend farther in azimuth, eventually touch-ing each other and merging to form an Einstein ring of fi-nite width. The magnification remains finite due to the fi-nite source size (Bennett & Rhie 1996; Dominik 1998). Thesource star’s angular radius θ S = R S /D S becomes compara-ble to the angular radius θ E of the Einstein ring, at u S ≡ θ S θ E = R S (2 S L D S X (1 − X )) / = 0 . (cid:16) R S R ⊙ (cid:17) (cid:16) M L M ⊙ (cid:17) − / (cid:16) X − X (cid:17) / . (15)Finite-source effects set in at high magnification, A > ∼ /u S ,thus (for M L ∼ . M ⊙ ) at A > ∼ 500 for a main-sequencesource star with R S ∼ R ⊙ , or already at A > ∼ R S ∼ R ⊙ .Finite source effects are important for planet anomalieswhen the source star’s angular radius θ S exceeds that of theplanet’s Einstein ring (Bennett & Rhie 1996). Since θ p = q / θ E , the finite source effect is important when q < ∼ (cid:16) θ S θ E (cid:17) ∼ × − (cid:16) R S R ⊙ (cid:17) (cid:16) M L M ⊙ (cid:17) − (cid:16) X − X (cid:17) . (16)Since m ⊕ = 3 × − M ⊙ , this is m p = q M L < ∼ . m ⊕ (cid:16) R S R ⊙ (cid:17) (cid:16) X − X (cid:17) . (17)For large source stars the anomaly from a small planetcan be smeared out and diluted in amplitude, rendering itundetectable. On this basis, detection of Earth-mass planetsis more favourable with main sequence source stars (Bennett& Rhie 1996), though a detectable ( ∼ A planet near the lens star acts like a small defect in thegravitational lens. If the planet is well away from the twoimage trajectories, the light it deflects does not reach Earth.In this case the planet has no measurable effect on thelightcurve and thereby evades detection. However, if theplanet is close to one of the image trajectories, its grav-ity can significantly perturb the bundle of light rays thatwould otherwise reach Earth. This distorts the image andchanges the magnification to produce a brief anomaly in thelightcurve. The magnification curve A ( t ) for a star+planetlens deviates by a factor 1 + δ ( t ) from the correspondingPSPL magnification curve A ( t ): A ( t ) = A ( t ) (1 + δ ( t )) . (18)This defines the planet anomaly δ ( t ), which depends onthree additional parameters: the mass ratio q , and the co-ordinates, x and y , of the planet’s projected position on thelens plane.The planet anomaly may be brief but large. The planet’sEinstein ring radius is r p ≡ R E q / . (19)The planet anomaly may be large when one of the sourceimages passes closer to the planet than r p , provided thesource is not much larger than r p . The duration of the planetanomaly is roughly the time it takes the image to cross thediameter of the planet’s Einstein ring, t p ≡ t E q / . (20)Detecting the planet requires data points in the lightcurveof sufficient accuracy and at the right time to detect theanomaly produced as the image passes by the planet. We define the “detection zone” as the region on the lensplane ( x , y ) where the lightcurve anomaly δ ( t, x, y, q ) is largeenough to be detected or ruled out with high confidence bythe observations. For N data points with fractional accuracy σ i at times t i , the detection zone is defined by N X i =1 (cid:16) δ ( t i , x, y, q ) σ i (cid:17) > ∆ χ , (21)for some detection threshold ∆ χ . This detection thresholdmust be set high enough so that noise affecting the obser-vations does not produce false triggers at an unacceptablyhigh rate. ∆ χ in the range 25 to 100 corresponds to a 5 σ to 10 σ deflection in the lightcurve if the anomaly is confinedto a single data point.Fig. 3 highlights the detection zones for a planet withmass ratio q = 10 − derived from a lightcurve A ( t ) withmaximum magnification A = 5. Data points uniformlyspaced in time sample the lightcurve with an accuracy σ = (5 /A / )%, and the detection criterion is ∆ χ > c (cid:13) , 000–000 ptimising Microlens Planet Searches Figure 3. Detection zones on the lens plane indicate the regionswhere a planet with mass ratio q = m/M = 10 − is detectedwith ∆ χ > 25. The lightcurve A ( t ) has maximum magnification A = 5, and the accuracy of the measurements is σ = (5 /A / )%.Each data point probes for planets close to the two images ofthe background star. The detection zone areas scale roughly asΩ ≈ R (2 A − q / ( σ ∆ χ ). t i is perturbed by δ > σ = (25 /A / )%. Improving theaccuracy of the data or increasing the mass of the planetenlarges the size of the detection zone. If the data points in the lightcurve are widely spaced, as theyare in Fig. 3, then the detection zones arising from differentdata points are well isolated from each other. We may thenevaluate numerically the area Ω of the detection zone thatis carved out by each data point. This quantifies the planetdiscovery potential of each data point.Fig. 4 shows a close-up of the detection zones definedby this criterion. The region displayed is chosen in advancefrom rough estimates and is used for numerical evaluation ofthe dectection zone area Ω. The cases shown illustrate howthe detection zones shrink and change shape as the magnifi-cation A ( u ) declines with increasing lens-source separation u . The detection zone shapes are complicated. At small u and high A they bear some resemblance to 4-leafed cloverswith radial and azimuthal lobes straddling the image posi-tions. With increasing u , decreasing A , the azimuthal lobesof the major image detection zone collapse radially. The ra-dial lobes then merge radially to form a circular detectionzone as u → ∞ . On the minor image detection zone, the ra-dial lobes merge and vanish, leaving two isolated azimuthallobes that shrink and vanish. It will be helpful to understand how planet detection zoneareas scale with the accuracy of the data, the source magni-fication, and the mass of the planet. If accurate scaling lawscan be found, we may then avoid long numerical calculationsto determine the detection zone area. In this section we de-velop a useful analytic formula, and test it against detailednumerical integrations.Consider first a planet located quite far from the lensstar, affecting the major image at a time well before or wellafter the stellar lensing event, when u + >> A + ≈ 1. Inthis case the planet and star act as independent lenses, anda significant anomaly occurs when the major image sweepspast the position of the planet. If z is the separation betweenthe planet and the major image, the planet magnifies themajor image by a factor A ( u p ), where u p = z/r p , and r p = q / R E is the planet’s Einstein ring radius.A data point with fractional uncertainty σ can detectthe anomaly δ when δ = A ( u p ) − > σ (cid:0) ∆ χ (cid:1) / . (22)This criterion corresponds to a circular detection zonearound the major image at the time of the observation. Theradius of the detection zone is found by solving Eqn. (22)for u p and hence z = r p u p . Using Eqn. (14) to invert A ( u p ),the area of the detection zone isΩ πr = (cid:16) zr p (cid:17) = h (cid:0) − (1 + δ ) − (cid:1) − / − i . (23)Using the approximations in Eqn. (13), the correspondingapproximations for the detection zone area areΩ πr = (2 /δ ) / δ < ∼ . , (3 δ ) − . < ∼ δ < ∼ ,δ − δ > ∼ . (24)Fig. 5 indicates that the middle approximation predictsfairly accurately the detection zone area for anomalies in therange 0 . < δ < 3. For ∆ χ = 25 this range correspondsto fractional uncertainties 0 . < σ < . 6, quite appropriatefor CCD data, givingΩ R E = π q δ . (25)Now move the planet closer to the lens star. The planetacts as a defect in the stellar lens, and the resulting detectionzone can have a quite complicated shape (Fig. 4). We guessthat the detection zone area may scale roughly with themagnification A . However, we find that Ω ∝ A − R ≈ π q δ (2 A − . (26) c (cid:13) , 000–000 Horne, Snodgrass Figure 4. Planet detection zones for mass ration q = 10 − near the major and minor image positions for data points at magnifications A = 100, 5, 1.5, and 1.1. The detection zones are symmetric around the image positions in ( θ ,ln u ) coordinates. For u → ∞ and A → Figure 5. The detection zone area Ω scales with the area πr of the planet’s Einstein ring, and decreases with the size of thelightcurve anomaly δ that can be detected by the data. The ap-proximations (dashed lines) are those defined in Eqn. 24. Fig. 4 shows that the detection zones are roughly sym-metric as a function of ln u and θ rather than x and y . Itmay therefore be more appropriate to express detection zoneareas using a d θ d ln u metric, evaluating w = Z P (det | u, θ ) d u d θu , (27)rather thanΩ R = Z P (det | u, θ ) u d u d θ = Z P (det | x, y ) d x d y . (28)The d θ d ln u metric may be appropriate from a secondperspective. Exo-planet orbits should have random orienta-tions, so the planets should be uniformly distributed in θ . If their orbit size distribution is also roughly uniform in log a ,then the planet distribution on the lens plane will be roughlyuniform in ln u , and the planet detection probability will de-cline to zero long after the peak of an event, rather thanreaching a positive asymptotic value. In effect the d θ d ln u metric recognizes a detection zone with area Ω as more likelyto include a planet, and therefore more valuable to us, whenit is measured at small u and probes a larger range of log u .For small detection zones the two metrics are related by w ≈ Ω R u ≈ π q δ (cid:16) A − u (cid:17) . (29)But we must be more careful to treat separately the majorand minor image detection zones, surrounding the images at u + and u − respectively. This gives w ± ≈ π q δ F ± ( A ) , (30)where for the major image at u + F + ( A ) ≡ A + − u + ) = 2 TB ( T + B ) = A (cid:16) A − A + 1 (cid:17) / , (31)and for the minor image at u − F − ( A ) ≡ A − ( u − ) = 2 B = ( A + 1) (cid:16) A − A + 1 (cid:17) / . (32)In deriving the above expressions, we used Eqns. (7), (11)and (12) to write A ± = T ± B B , ( u ± ) = T ± B ,T = u + u − = u + 2 = 2 A (cid:0) A − (cid:1) / ,B = u − u − = u (cid:0) u + 4 (cid:1) / = 2 (cid:0) A − (cid:1) / . (33) c (cid:13) , 000–000 ptimising Microlens Planet Searches Figure 6. The magnification, detection zone area, image velocity,and detection zone crossing time are shown for an event withpeak magnification A = 5. The dashed and dash-dot curves arefor the major and minor images respectively, except for the imagevelocity panel where the dashed curve is d θ/ d t while the dash-dotcurve is d ln u/ d t for the major image. The total detection zone area, summing the detection areasof both images, is w = π q δ F ( A ) = π q F ( A )3 (cid:0) ∆ χ (cid:1) / σ (ln A ) , (34)with σ (ln A ) the fractional accuracy in measuring A , ∆ χ the threshold for planet detection, and F ( A ) ≡ F + ( A ) + F − ( A )= 2 B (cid:16) T + BT + B (cid:17) = (2 A + 1) (cid:16) A − A + 1 (cid:17) / . (35)The functions F ± ( A ), for the separate images and F ( A ) forthe total detection zone area are plotted in Fig. 6 for anevent with A = 5. Fig. 7 compares numerically-integrated detection zone areaswith the analytic result in Eqn. (34). The figure shows how w depends on A − q = 10 − , 10 − and Figure 7. The detection zone areas w in the d θ d ln u metricincrease with magnification A , increase with mass ratio q , anddecrease with the size of the lightcurve anomaly δ that can bedetected by the data. The fast analytic approximation (dotted)defined in Eqn. (34) is compared with more exact numerical inte-grations. Dotted curves in the lower panel give fractional contri-butions of the major and minor images, in the analytic approxi-mation. − , and for three accuracies, δ = σ (ln A ) (cid:0) ∆ χ (cid:1) / = 5,10 and 20%.The analytic approximation clearly captures the mainscaling, w ∝ q A/δ , for 2 < A < < A < ∼ F + ≈ F − ≈ (2 A − / 2, the two images contributing roughlyequally. At intermediate magnifications, 2 < A < 10, theanalytic result is low by up to ∼ A -dependentbias could be reduced by adjusting the formula for F ( A ) inEqn. (35). However, the accuracy is already sufficient for ourpurposes in the range 2 < A < A > θ and in ln u , thesaturation in θ should set in when w ∼ (2 π q A/ δ ) > ∼ π ,i.e. A > ∼ π δq . (36) c (cid:13) , 000–000 Horne, Snodgrass We enter a new regime in which the azimuthally-merged de-tection zone may continue to expand in ln u but is saturatedin θ . The slope should drop to w ∝ ( q A/δ ) / , i.e. w (cid:16) A > ∼ δq (cid:17) ≈ π (cid:16) π q A δ (cid:17) / . (37)These expectation are roughly consistent with the behaviourin Fig. 7.Note that in the very-high magnification regime finitesource effects will also become important, altering the re-lationship between u and A with u > u S , as discussed inSec. 2.2. As we neglect both effects in our analysis, ourscaling law applies only up to a maximum magnification A < ∼ δ/q ∼ A < 2, Fig. 7 shows that theanalytic formula over-predicts detection zone areas by fac-tors of up to ∼ 3. The structure is independent of q but de-pends on δ and A , due to the complicated structure of the de-tection zones as seen in Fig. 4. In this regime, A ≈ − u − , F + ≈ u − , F − ≈ u − , and F ≈ u − . With F − ≈ F + ,the analytic formula gives the minor image twice the detec-tion area of the major image. The F + formula has correctasymptotic behaviour at both high and low magnifications,so the problem is with the F − formula. In fact at low mag-nifications the radial lobes of the minor image merge anddisappear, as seen in the right two columns of Fig. 4. Thiscuts the total detection zone area by a factor of about 3when A drops below a threshold, A − ≈ u < ∼ δ . (38)As shown in Fig. 8, a fairly successful attempt to repair thisdeficit is w = π q δ (cid:16) A − A + 1 (cid:17) / ( A + (1 + A ) C ( x )) , (39)where x ≈ δ/ ( A − ≈ u δ/ 2, and C ( x ) ≈ max h , − x x i (40)cuts off the minor image contribution at the appropriatethreshold. However, this makes F − depend on δ as well as A . The cadence of observations aiming to detect planet-likeanomalies should ideally be matched to the time it takesfor the images to cross the detection zones, so that eachmeasurement probes for planets in an independent region ofthe lens plane, rather than overlapping with the region al-ready sampled by previous measurements. This will clearlydepend on the size of the detection zones, and on the speedat which the images move on the lens plane.If we approximate the width of the detection zone asthe square root of its area, then the detection zone crossingtimescale is t ± ≈ (cid:16) w ± v (cid:17) / ≈ (cid:16) q F ± ( A ) δ v (cid:17) / , (41)where w ± = q F ± ( A ) /δ is the detection zone area in thed θ d ln u metric, and v is the corresponding image speed, Figure 8. As in Fig. 7 but using the approximation in defined inEqn. (39) rather than Eqn. (34). v = (cid:16) d θ d t (cid:17) + (cid:16) d ln u ± d t (cid:17) = ˙ θ + (cid:16) ˙ u ± u ± (cid:17) . (42)We show below that v is actually the same for both images,they move at the same speed on the θ vs ln u plane. Theimage speed and planet detection timescale are plotted inFig. 6 for an event with A = 5.To evaluate the azimuthal velocity ˙ θ , note that bothimages sweep around in θ at the same rate as the unlensedsource position. With x = ( t − t ) /t E and y = u , the un-lensed source position moves with˙ x = 1 t E , ˙ y = 0 . (43)Differentiate θ = arctan yx to find˙ θ = ˙ x d θ d x = − ˙ x yx + y = − u u t E . (44)Evaluating the ln u ± image velocities is more involved,but leads to a simple result. First, differentiate u = x + y to find˙ u = ˙ x d u d x = ˙ x xu = xu t E . (45)Next, differentiate ( u ± ) = ( T ± B ) / u ± u ± = ˙ T ± ˙ B u ± ) = u ˙ u (1 ± A )2 ( u ± ) , (46)where we have differentiated Eqn. (33) to find˙ T = 2 u ˙ u , ˙ B = 2 u ˙ u A . (47) c (cid:13) , 000–000 ptimising Microlens Planet Searches Then, since A = T /B , and ( u ± ) = ( T ± B ) / u ± u ± = u ˙ u ( B ± T ) B ( T ± B ) = − u ˙ uB = − xt E B . (48)Notice that both images have the same velocity in ln u , aswell as in θ , so that the image velocity v is the same for bothimages.Substituting Eqns. (48) and (44) into (42), the imagespeed on the θ, ln u plane is( v t E ) = u u + x B = 1 + 4 u /u u + 4 = u + 4 u u (cid:0) u + 4 (cid:1) . (49)The image speed is plotted in Fig. 6 for an event with A =5. The azimuthal velocity ˙ θ dominates near the peak, andthe radial velocity d ln u/ d t dominates in the wings of thelightcurve. For large u , the image speed varies as v t E ≈ /u . The maximum speed v t E = 1 /u is reached at thepeak of the event, at t = t , where u = u .The planet anomaly crossing timescale is given by (cid:16) t ± t E (cid:17) = w ± ( v t E ) = q F ± ( A ) δ u + 41 + 4 u /u . (50)Crossing times for the major and minor images are shownin the lower panel of Fig. 6. The crossing time is slightlylarger for the minor image, by a factor ( F − /F + ) / =(( A + 1) /A ) / (see Eqns. (31) and (32)). The crossing timeat first rises due to the increasing size of the detection zone,and then drops to a minimum at the peak, where the max-imum image velocity v = 1 / ( t E u ) is reached. This mini-mum crossing time, at the peak of the event, is t ± (0) = t E u (cid:16) qδ (cid:16) A + 12 ∓ (cid:17)(cid:17) / (cid:16) A − A + 1 (cid:17) / ≈ t E (cid:16) qδ A (cid:17) / . (51)Here the final approximation, using F ± ≈ A and u ≈ A − ,holds to 15% or better for A > ∼ q = 10 − , in a typical eventwith t E = 30 d. For a peak magnification A = 5, asin Fig. 6, we have u = 0 . F + = 5 (4 / / = 4 . F − = 6 (4 / / = 4 . 9. For a good data point with σ (ln A ) = 1%, and a detection threshold at ∆ χ = 100,the smallest detectable planet-like anomaly deviates by δ = σ (ln A ) (cid:0) ∆ χ (cid:1) / = 0 . 1. The crossing time for the majorimage is t + (0) = 30 × . (cid:16) . . (cid:17) / (cid:16) (cid:17) / = 1 . . (52)For an Earth-mass planet and a typical lens mass M L ≈ . M ⊙ the mass ratio is q = 10 − , and the crossing time is t ± (0) ≈ . h (cid:16) t E 30 d (cid:17) (cid:16) q − (cid:17) / (cid:16) δ . (cid:17) − / (cid:16) A (cid:17) − / . (53) In this section we consider how an observer might try tooptimize a microlens planet search. We assume that the observer has many targets to choose from. This is a goodassumption because MOA II and OGLE III are finding ∼ − Because detection zone areas scale as w ∝ σ − , e.g.Eqn. (34), a critical issue is the accuracy of photometricmeasurements that can be achieved, and the rate at whichthat accuracy improves with exposure time. We assume thatthe data analysis is close to optimal, so that photon count-ing statistics dominate the noise budget. Thus CCD readoutnoise, cosmic ray hits, and other noise sources are neglectedin comparison with the Poisson noise from detected star andsky photons. The signal-to-noise ratio then increases as thesquare-root of the exposure time, σ (ln A ) = (cid:16) τ ∆ t (cid:17) / , (54)where ∆ t is the exposure time, and τ is the exposure timerequired to reach a signal-to-noise ratio of 1.The parameter τ controls the exposure time needed toobtain information on the current magnification A . It de-pends on the telescope collecting area, the detector sensi-tivity and bandwidth, on the brightness of the magnifiedsource star, and the degree of dilution of its photons by skybackground and by other stars that are blended with it. In-cluding these three sources of Poisson noise, τ = f ⋆ + f B + f sky f ⋆ , (55)where f ⋆ , f B and f sky are the number of detected photonsper unit time from the magnified source star, from the lensstar and other stars blended with the source star, and fromthe sky, respectively. We elaborate these three Poisson noisesources below.For a star of magnitude m ⋆ and spectral energy distri-bution f λ ( λ ), we observe thru the atmospere with transmis-sion T ( λ ), with a detector effective area A eff ( λ ). The photondetection rate can be evaluated precisely as f ⋆ = Z f λ ( λ ) d λh ν A eff ( λ ) T ( λ ) , (56)or approximately as f ⋆ ≈ f ( λ ) T ( λ ) 10 − . m ⋆ . (57) c (cid:13) , 000–000 Horne, Snodgrass The photon detection rate from Vega (magnitude 0) is f ( I ) ≈ 500 s − (cid:16) A eff cm (cid:17) (cid:16) ∆ λ ˚A (cid:17) , (58)for a telescope with mean effective area A eff over a band-width ∆ λ near the I band, where most microlens observa-tions are taken (for the V band, Vega’s flux is 1000 ratherthan 500 photons cm − ˚A − s − ).For the source star, magnified by a factor A , f ⋆ = f S A T = f T − . m ⋆ . (59)Stars blended with the magnified source star contributePoisson noise to the measurement. The source flux f S andblend flux f B are normally measured by fitting observedlightcurves (corrected for atmospheric transmission) withthe model f ( t ) = f S A ( t ) + f B . (60)When using differential flux measurements ∆ f ( t ), obtainedby a difference image analysis, the reference flux added tothese is somewhat arbitrary. As a consequence, the blendflux f B arising from the lightcurve fit is also somewhat ar-bitrary, and can even be negative in some cases.The blend flux contributing to the Poisson noise in-cludes not only flux from the lens star, and any other starsthat are “exactly” coincident on the sky with the magnifiedsource star, but also stars that are close enough on the skyso that the point-spread functions overlap. If m i is the mag-nitude and θ i is the angular separation of star i from thetarget star, the blend flux contributing Poisson noise to themeasurement is f B (∆) = f ⋆ X i − . m i − m ⋆ ) e − ( θ i / ∆) . (61)This expression assumes a gaussian point-spread functionwith standard deviation ∆, and optimal extraction to mea-sure the target star flux. Note that the Poisson noise due toblended stars increases with the seeing. Although it is notyet done in practice, the specific dependence on seeing foreach microlens target can be evaluated in advance from agood-seeing image of the starfield, for example the OGLEor MOA finding-chart images made available for each event.Finally, the detection rate of sky background photonsoverlapping with the target star is f sky = f ∆ θ − . µ sky , (62)where µ sky is the magnitude of a square arcsecond of sky,and ∆ θ is the solid angle subtended by the photometricaperture (for aperture photometry) or by the point-spreadfunction of the star images (for psf-fitting photometry) insquare arcseconds. The sky brightness, including e.g. air-glow, zodiacal light and scattered moonlight, may be eval-uated using a sky model, e.g. (Krisciunas & Schaefer 1991;Patat 2003). The effective sky coverage of a gaussian point-spread function is∆ θ = 4 π ∆ = π W , (63)where ∆ is the standard deviation and W is the full-width athalf-maximum (FWHM) of the gaussian point-spread func-tion. Atmospheric seeing is usually reported in terms of W .Combining the above equations, we can rewriteEqn. (55) as τ = τ ⋆ + τ B + τ sky , (64)with τ ⋆ = 10 . m ⋆ − . T ( λ ) (cid:16) A eff m (cid:17) (cid:16) ∆ λ ˚A (cid:17) , (65) τ B = τ ⋆ X i − . m i − m ⋆ ) e − ( θ i / ∆) , (66)and τ sky = τ ⋆ π ∆ T ( λ ) 10 − . ( µ sky − m ⋆ ) . (67)These expressions make explicit how τ depends on the mag-nified source star brightness (magnitude m ⋆ ), on the nearbystars blended with the target (magnitude m i , separation θ i ), on capabilities of the telescope (effective area A eff ,bandwidth ∆ λ ), and on observing conditions (sky bright-ness µ sky , seeing ∆, atmospheric transmission T ). When thesky and blend fluxes are negligible, a 100s exposure with A eff = 1 m and ∆ λ = 10 ˚A reaches 1% accuracy at mag-nitude m ⋆ = 16 . The “worth” of an observation, from the perspective ofplanet hunting, is proportional to the area of the result-ing detection zone. Combining Eqns. (34) and (54), we seethat detection zone areas increase with the square-root ofthe exposure time, w = q F ( A ) δ = q F ( A ) (cid:18) ∆ tτ ∆ χ (cid:19) / ≡ g (∆ t ) / . (68)The proportionality constant g characterizes the “goodness”of observing this particular target, g ≡ q F ( A ) (cid:0) τ ∆ χ (cid:1) / . (69)These g values can be used to prioritise the events that areavailable at any given time. They depend on the propertiesof the event, characteristics of the telescope, and on thepresent observing conditions.The key point to note here is that the fractional mea-surement error decreases with the exposure time, σ ∝ ∆ t − / , and this expands the detection zone area as w ∝ ∆ t / . The detection zone area grows most rapidly at thebeginning of the exposure, with diminishing returns as theexposure progresses. For this reason at some point it be-comes advantageous to abandon observations of this targetin favour of moving on to a fresh target that has not yetbeen observed.Suppose that we are contemplating making observationsof N targets during an upcoming night in which we expect tohave available a total observing time t . How much exposureshould we devote to each target? For each target i we cancalculate the goodness factor g i . If we observe target i withexposure time ∆ t i , then t = N X i =1 ∆ t i (70)is the total observing time. The total worth of observing the N targets is c (cid:13) , 000–000 ptimising Microlens Planet Searches Figure 9. When observing N microlens events, for which theplanet detection zone areas w i grow with exposure time ∆ t i as w i = g i ∆ t i / , the total detection zone area W = P i w i is max-imised when the available exposure time is divided in proportionto g i . Optimisation for the case N = 2 is illustrated here. W N = N X i =1 g i (∆ t i ) / . (71)Given a fixed total observing time t , we can optimizethe exposure times by solving ∂W N /∂ ∆ t i = 0. For example,with N = 2 targets, as illustrated in Fig. 9, the total timeis t = ∆ t + ∆ t , and the sum of the detection zone areas is W N = g (∆ t ) / + g ( t − ∆ t ) / . (72)Maximizing W N gives0 = ∂W N ∂ ∆ t = g t ) / − g t ) / , (73)and thus ∆ t / ∆ t = ( g /g ) . Similarly, for the general caseof N targets, the optimal exposure times that maximize W N also satisfy ∆ t i ∝ g i , and are therefore given by∆ t i = (cid:16) g i G N (cid:17) t , (74)where G N ≡ N X i =1 g i . (75)If we adopt the optimal exposure times, substitutingEqn. (74) into (71) gives the total worth of the observationsas W N = G N t / . (76)This analysis suggests that the optimal strategy to max-imize the planet detection capability is to observe all avail-able targets, spending more time on the best targets, usingexposure times proportional to the square of the goodness,∆ t i ∝ g i . The optimal observer skips no targets. As we willsee, however, this conclusion is altered when we take accountof observing overheads. Figure 10. Illustration of the reduction in the net exposure timeand the corresponding degradation in planet hunting capabil-ity for a 100s observation accounting for a CCD readout time t read = 10s, for splitting the exposure into 3 sub-exposures toavoid saturation, and for a telescope slew time t slew = 30 s . In practice the CCD camera takes a finite time t read to readout, and the telescope takes a finite time t slew to slew fromone target and settle into position on the next. Typical read-out and slew times are t read ∼ − 20 s and t slew ∼ − n shorter exposures. Theseoverheads reduce the on-target exposure time accumulatedduring an observation time t to∆ t = t − t slew − n t read . (77)These overheads diminish the planet hunting capability ofthe observations, as illustrated in Fig. 10. We must allow forthese overheads when implementing an optimal observingstrategy.If the CCD exposure is too long, the target will saturate.If the CCD exposure is too short, the readout noise willdominate over sky noise and information will be lost. Theseconsiderations set the range that should be considered forthe CCD exposure time: t min < t exp < t max . (78)A total exposure longer than t max is accumulated by takinga series of n shorter exposures, where t − t slew t max + t read < n < t − t slew t min + t read . (79)For bright targets where t max , the longest exposure thatavoids saturation, is less than t min , the shortest exposurethat avoids readout noise domination, the need to avoid sat-uration must take precedence. Having n > n is decided, the duration of each ex-posure is t exp = ( t − t slew ) /n − t read . (80)Allowing for these overheads, the detection zone area be-comes w = g (cid:16) t exp t exp + t read (cid:17) / ( t − t slew ) / . (81) c (cid:13) , 000–000 Horne, Snodgrass Figure 11. The number of microlens events to observe is takento maximise the total worth of the observations. With too manytargets, observing time is reduced by the time required to slewthe telescope from target to target. This optimisation of N is il-lustrated for 100 targets with an exponential distribution of good-nesses, and for different total amounts of telescope time. With lesstime fewer events are observed, and the total worth is reduced. The first bracket accounts for the reduction in on-target ob-serving time due to the CCD readout time. We can absorbthis term into the definition of g , g → g (1 + ( t read /t exp )) / , (82)as shown by the dashed curves in Fig. 10. The effect is tosuppress interest in observing targets that are so bright asto require inefficient observations with t exp < t read . A targettoo bright for efficient observations with a large telescopemay thus remain a prime target for smaller telescopes. Inthis way the scheme may serve well to coordinate observa-tions by a community with a variety of telescope types.The second bracket in Eqn. (81), allowing for the slewtime, delays the onset of detection zone growth while thetelescope is moving from one target to the next. We will seebelow that this term dictates which of the less promisingtargets to omit from the observing schedule. N Events If we try to observe too many of the ongoing events, we willspend all night slewing from target to target and no timeat all collecting photons from the targets. If the total timeavailable for observations is t , and slew time is t slew , then themaximum number of targets we can contemplate observingis N max = t/t slew . (83)If we observe N ≤ N max targets, the total worth of theobservations will be W N = G N ( t − N t slew ) / . (84)We would like to maximize W N . The first term G N increaseswith N , and the second term ( t − N t slew ) / decreases with N . Therefore W N has a maximum value for some N < N max .This is the number of targets that we should observe tomaximize the planet hunting capability of our observations. To make W N grow as fast as possible, sort the targetsand consider them in order of decreasing goodness, g N ≥ g N +1 . We should keep target N + 1 only if W N +1 > W N . Todecide whether or not to retain target N + 1, note that (cid:16) W N +1 W N (cid:17) = (cid:18) (cid:16) g N +1 G N (cid:17) (cid:19) (cid:16) − t slew t − N t slew (cid:17) . (85)Target N + 1 survives only if g N +1 > G N t / ( t − N t slew ) / = G N ( N max − N ) / , (86)where we have used N max = t/t slew .To illustrate this optimisation, Fig. 11 shows the resultof dividing time among N = 100 targets with an exponentialdistribution of goodnesses g i , for several different total avail-able observing times t , corresponding to N max = 20 , , ... .As available time t increases, the optimal strategy spendsmore time on each target, and also extends time to addi-tional lower-priority targets.To further illustrate, more realistically, we consider inFig. 12 the recommended observations from among 443OGLE events that were available on 2003 Aug 31. To de-cide on the observing strategy, we first fit a PSPL lightcurvemodel to the OGLE data on each event to evaluate the eventparameters. This results in predicted magnitudes and mag-nifications for each target on each night in question. Wenext evaluate the goodness factors g i for a telescope witheffective area A eff = 1 . , with a sky magnitude 19. Weassume a slew time t slew = 60s, a readout time t read = 10s,a maximum exposure time t exp < W N would increase monotonically if there were no slewtime. The solid curve shows show how W N at first increaseswith N but then decreases as slew time becomes important.For t = 1 . 5h (left panel of Fig. 12), the maximum numberof targets that could be observed is N max = t/t slew = 90,but the optimal sampling to maximise W N undertakes ob-servations of just N = 6. For t = 8 . 5h (right panel) all 443targets can be observed, but the maximum of W N occurs at N = 20.In the bottom panels of Fig. 12, the resulting lightcurvesare shown when this strategy is employed on every night.The area of the plot symbols are proportional to the observ-ing time allocated to each target. On most nights the opti-mal sampling spreads observing time over many targets. Ona few nights when one very high magnification event is avail-able, that target captures most or all of the recommendedobserving time. One bright target receives some attentioneven though its magnification is small. The observations in-clude fainter targets when more observing time is available.Targets fainter than the sky are seldom scheduled. We must emphasize that the scheme outlined above is de-signed to detect anomalies, not to characterise them. Ob-serving many targets for the recommended exposure timecan be advocated only so long as each new observation in-dicates that no significant anomaly is underway. It is there- c (cid:13) , 000–000 ptimising Microlens Planet Searches Figure 12. Optimal sampling of 443 OGLE events available on 2003 Aug 31. Exposure times are chosen to maximize the total planetdetection zone area, for a 2 m telescope with fixed total observing time 1 . 5h (left) and 8 . 5h (right) per night. Observing too many targetsis inefficient because of the 120s telescope slew time. Observing too few targets is inefficient because planet detection zone areas growonly as t / . On the resulting lightcurves (lower panels), the plot symbol areas are proportional to the allocated exposure time. On mostnights observing time spreads over many targets. On some nights one high-magnificaiton target captures most or all of the attention. fore best if a rapid reduction of each new observation can beundertaken with sufficient accuracy and reliability to checkeach new data point for consistency or otherwise with thePSPL model. This is feasible because only one or at most afew stars on each CCD image will be undergoing microlens-ing at a given time. The sub-image around the target of in-terest can be quickly reduced to measure its brightness withrespect to nearby comparison stars. In practice the real-timeimage-subtraction pipelines currently in use by PLANETand RoboNet can reduce each CCD image within a few min-utes of the end of the exposure.Whenever a significant anomaly is identified, the ob-server can temporarily suspend the anomaly-hunting strat-egy of observing many targets in sequence, returning to thetarget that offered up the anomalous data point. Additionalobservations of this target then aim to establish either thatan anomaly is in progress, or else to dismiss the false alarmcaused by unreliable data. If the return observations failto confirm the anomaly, then the anomaly-hunting obser-vations can resume. If the return observations confirm theanomaly, then continuous observations are initiated to clar-ify the nature of the anomaly, and an alert can be issued totrigger follow-up observations on other available telescopes. An implementation of this is the SIGNALMEN anomaly de-tector (Dominik, et al. 2007).By following this two-stage approach – prioritised multi-target anomaly hunting punctuated by episodes of anomalyconfirmation and characterisation – we can simultaneouslymaximize the opportunity to detect anomalies by observinga large number of targets, while retaining the ability to re-liably establish the nature of anomalies that we detect. Ifthe second-stage is omitted from the observing strategy, therisk is a series of single-point anomalies will be found whoseidentity cannot be securely established. When event parameters change significantly during a night,or when the image positions fall inside detections zones fromprevious observations, then the results derived in the previ-ous section are no longer strictly valid. Movement of the im-ages will increase, while overlap will decrease the detectionzone areas. We are nevertheless hopeful that the optimisa-tion scheme advocated above will still be helpful in guidingfollow-up observations.One way to cope with the more general situation isto employ a scheme with continually-evolving target prior- c (cid:13) , 000–000 Horne, Snodgrass Figure 13. Top panel shows 5 data points at times t i with accu-racy σ i = 5% on the decline of a lensing lightcurve with maximummagnification A = 20. Bottom panel shows the evolving prior-ity given to proposed new data points with accuracy σ = 1, 2,and 4% (top to bottom) when searching for planets with massratio q = 10 − (solid curves) and 10 − (dashed curves). Thepriority for an isolated data point is proportional to the planetdetection zone area, Ω ∝ q (2 A ( t ) − /σ , where A ( t ) the time-dependent magnification. The priority drops when the new ob-servation would probe for planets inside a detection zone alreadyestablished by previous observations. The priority then recoverson the timescale needed for images to cross the detection zones. ities. The highest-priority target is observed. The priorityof that target must then fall dramatically, since immediatere-observation would probe for planets inside the detectionzone just carved out. The priority should then recover in duecourse, as changes in the event geometry move the image po-sition outside of the detection zone. Such a scheme may beideal for fully automatic follow-up observations with robotictelescopes, but could also be used by human observers will-ing to follow directions from a computer programme.A dynamical priority scheme of this sort is illustratedin Fig. 13. The top panel of Fig. 13 shows a set of datapoints with accuracy σ i = 5% at times t i during the declineof an event with peak magnification A = 20. The lowerpanel shows the evolving priority given to a proposed newdata point at time t with accuracy σ = 1, 2, and 4% whensearching for planets with mass ratio q = 10 − and 10 − . The priority is evaluated numerically as the increase in de-tection zone area arising from the proposed new data point.The dips in priority evident in the lower panel of Fig. 13indicate the reduced planet hunting capability caused by theoverlap of detection zones when the new data point probesfor planets inside the detection zone of an earlier measure-ment. We see in Fig. 13 that the reduction is small for σ = 1% and substantial for σ = 4%. This is because the old σ = 5% data are important when the new data point is ofsimilar accuracy, but unimportant when the new data pointhas much higher accuracy. We see also in Fig. 13 that thepriority recovery time is faster for q = 10 − than for 10 − .This is plausible since detection zone sizes scale as q / . Therather irregular recovery arises from the complicated shapesof the detection zones (Fig. 4).In practise there will be not just a single previous mea-surement with accuracy σ , but rather a set of prior measure-ments at times t i with accuracies σ i . Noting that indepen-dent measurements combine optimally with 1 /σ weights,the net effect at time t of all prior measurements may beapproximated by using the scheme1 σ ( t ) = X i M [( t − t i ) /s i ] σ i , (87)where s i is an “expiration time” for the observation at time t i , and M ( x ) is a “memory function”, 1 for t = t i and de-creasing to 0 for t >> t i , effectively forgetting sufficientlyold observations. Possibilities for the memory function areGaussian or Lorentzian: M ( x ) = e − x / , M ( x ) = 11 + x . (88)The detection zone crossing, time worked out in Section. 3.5,provides a suitable expiration time s i .The new detection zone area grows more slowly dueto overlap with earlier zones. It is as if an exposure time t done has already been done to achieve the accuracy σ ( t ) =( τ /t done ) / . The new exposure time ∆ t then adds to t done ,increasing the detection zone area by∆ w = g (cid:16) (∆ t + t done ) / − t / (cid:17) . (89)with t done = τσ ( t ) = τ X i M [( t − t i ) /s i ] σ i . (90)Another relevant consideration is that slew times arenot equal for all targets. The slew time is zero for the cur-rent target, and for other targets should increase with theirangular distance from the current target. If we include theslew time, then the increase in detection zone area is∆ w = g (cid:16) (max [0 , ∆ t − t slew ] + t done ) / − t / (cid:17) . (91)If we require the exposure time to be not shorter than someminimum time, perhaps some multiple of the CCD readouttime, in order to have a reasonably high observing efficiency,then one compares the options of observing longer on thepresent target without slew time vs slewing to another tar-get. As t done increases on the current target, the potential forincreasing its detection zone area declines until it becomesbetter to slew to and expose on the next target.This scenario is illustrated in Fig. 14. Here three targets c (cid:13) , 000–000 ptimising Microlens Planet Searches Figure 14. The slew time is zero for the current target, but sig-nificant for two others. Observations of the current target shouldcontinue until the marginal improvement in detection zone areabecomes less for this target than for one of the alternatives, takingthe slew times into account. are considered. We are currently exposing on target 1, withslew times of 100s and 200s to reach targets 2 and 3. Weconsider a minimum exposure time of 40s. The circles showthe result of slewing to an alternative target and exposing forthe minimum time. In the top panel we have accumulated a1000s exposure on target 1. The circles for both alternativetargets are below the solid curve, so we should not slew. Inthe bottom panel we have accumulated a 2000s exposure ontarget 1, and this increase in t done reduces the slope of thesolid curve to such an extent that the circle on target 3 isnow just above it. At this point we should therefore decideto slew and expose on target 3, rather than remaining ontarget 1. This cycle may be iterated throughout the night.We expect an optimisation scheme based on approxi-mations like those described above to be helpful in decidinghow long to continue observing the present target, and whichtarget is the best one to observe next. We have not yet simu-lated this possibility in great detail, but outline the concepthere as a possible starting point for a self-organising schemethat may be suitable for coordinating optimal microlens ob-servations by a heterogeneous network of telescopes. Assum-ing rapid sharing of information among the telescope nodes,each telescope can independently decide which target to ob- serve next, taking into account prior observations made byall other telescopes, with their various times and accuracies. The event parameters t , t E , and A are often uncertain andcorrelated in the early stages of an event before the observa-tions have sampled both sides of the lightcurve peak. Withhighly uncertain event parameters, large errors may arisein the assigned target priorities. A frequent example occurswhen an early fit to the rising part of the lightcurve suggestsa very high magnification event that later turns out to beof only modest magnification. How will such uncertaintiesaffect our strategy?One happy aspect: the detection zone areas depend oncurrent values of the magnification A and star magnitude m ⋆ , rather than on the values at the peak of the lensingevent. This is helpful in the early stages of an event whenthe eventual peak magnification is still difficult to predict.On the other hand, in a highly-blended event the true mag-nification of the source star can be higher than the apparentmagnification.The event parameter uncertainties remaining after fit-ting the PSPL lightcurve model to the extant data pointscan be quantified, for example by using the parameter co-variance matrix or Markov-Chain Monte-Carlo techniques.The corresponding uncertainy in the event priority may thenbe taken into account using a Bayesian average over the pos-terior probability distributions.One may also contemplate giving priority to observa-tions aiming to reduce uncertainty in the event parame-ters. This secondary goal will then need to be traded-offin some satisfactory way with the primary goal of discov-ering planets. An anomaly found on the rise should attractattention, making it likely that accurate event parameterswill be nailed down by observations across and after thepeak. For an event well past the peak, it may be too late foradditional observations to pin down uncertain event param-eters, or additional observations at critical stages may help alot to break the ambiguity between blending, magnification,and event timescale. Targets could be given reduced prior-ity when their event parameters are uncertain and there islittle prospect of improving them, or higher priority when acritical observation would help to nail down the uncertainprameters. This issue needs careful investigation.Our estimates of detection zone areas assume that theunderlying lens parameters are or will be well constrained byobservations outside the planet anomaly. When this is notthe case, then the actual detection zones will be smaller, be-cause the loose event parameters can shift the model towardthe anomalous data points that would otherwise be ableto detect or rule out planets. Fig. 15 illustrates this effect,where the reduction of detection zone areas is consideredfor the OGLE data on OGLE-2005-BLG-390. In this event,one OGLE data point occurs during a planet anomaly. Thereduction of detection zone areas is noticeable but not largeenough in cases such as this to be a serious problem for ouroptimisation scheme. It would be a more serious problem forevents with only a few measurements covering the magnifiedpart of the lightcurve. c (cid:13) , 000–000 Horne, Snodgrass Figure 15. Top panel: OGLE III observations of OGLE-2005-BLG-390. One data point occurs during a planet anomaly. Thepoint-source point-lens (PSPL) model fits 5 parameters, the peaktime t , peak magnification A , Einstein radius crossing time t E ,the source flux f S and blend flux f B . Middle panel: Greyscale rep-resentation of χ ( x, y ), moving a planet with mass ratio q = 10 − on the ( x, y ) lens plane, holding fixed the 5 PSPL parameters. The χ increases by 100 or more in the white areas, where the planet isruled out, and decreases by 100 or more in the black areas. Bot-tom panel: The χ ( x, y ) map re-fitting the 5 PSPL parametersfor each planet position, showing the smaller size of the resultingplanet detection zones. As our knowledge of the exo-planet distribution functionaccumulates, one might contemplate introducing a prior onthe parameters q , M ⋆ , and a in order to target the planetsearch toward particular types of stars or planets. For exam-ple, since t E ∝ M / ⋆ , fast events correspond on average tolower-mass stars. Similarly, since a ∝ t E u ± , larger orbits canbe targeted by observing slower events and observing longerafter the event peak. It is straightforward to tilt the searchtoward any specific parts of parameter space. However, atthis stage our knowledge of the cool planet distribution isso scant that it is probably premature to invest much effortinto such fine-tuning. OGLE III and MOA II are discovering 600-1000 GalacticBulge microlens events each year. This stretches the re-sources available for intensive follow-up monitoring of thelightcurves in search of planets near the lens stars. We ad-vocate optimizing microlens planet searches by using an au-tomatic prioritization algorithm based on the planet detec-tion zone area probed by each new data point. We evaluatedetection zone areas numerically and validate a plausiblescaling law useful for rough but rapid calculations. The pro-posed optimization scheme takes account of the telescopeand detector characteristics, CCD saturation, readout time,and telescope slew time, sky brightness and seeing, past ob-servations of microlensing events underway, and the timeavailable for observing on each night. The current brightnessand magnification of each target are estimated by extrapo-lating fits to previous data points. The optimal observingstrategy then provides a recommendation of which targetsto observe and which to skip, and a recommended exposuretime for each target, designed to maximize the planet de-tection capability of the observations. This must be coupledwith rapid data reduction to trigger continuous follow-upobservations whenever an anomaly is detected. It is hopedthat the algorithm will provide helpful guidance to follow-up observing teams, and may be a useful starting point foroptimising fully-robotic microlens planet searches. An implementation of this optimisation scheme, PlanetLens OPtimisation (PLOP or web-PLOP), can be found at (Snodgrass, et al.2008). This system was designed with two motivations: toprovide an optimal target list for the automated observingof the RoboNet project (Burgdorf, et al. 2007; Tsapras, etal. 2009), and also to provide such lists to human observersat any telescope. It is formed of two parts. First, a userinterface web form takes input for the telescope and observ-ing conditions parameters ( A eff , t slew , t read , µ sky , ∆ θ etc.)and the total available observing time, t . Secondly, a back-ground code keeps track of the current data on each event(from OGLE, MOA, RoboNet, and all teams that make dataavailable in real time), and produces a new PSPL fit when-ever new data arrives. The results from these fits give theevent parameters t E , A etc. that are used to predict the c (cid:13) , 000–000 ptimising Microlens Planet Searches magnification at the requested time of observation. Thesetwo sets of inputs allow the calculation of g i for each eventusing Eqn. (69), and therefore an optimal list of targets withsuggested exposure times for the requested telescope at therequested time. The list is then put out in either a machinereadable or sortable human friendly format. With RoboNet,this output controls the telescope, and new data is fed backinto the PSPL model to close the loop and give prioritiesthat are based on data just taken. For human observers atother sites, the output pages are customisable to displayany desired parameters along with the priority of each mi-crolensing event, and also show light-curves and detectionzone maps along with links to the finding charts and orig-inal OGLE and/or MOA pages for each. Although writtenfor RoboNet, this prioritisation tool is freely available andother microlensing observers are encouraged to make use ofit. Acknowledgements Keith Horne was supported by a PPARC Senior Fellowshipduring the early stages of this work. We thank Steve Kane,Martin Dominik, Scott Gaudi, and Pascal Fouque for helpfulcomments on early versions of the manuscript. REFERENCES Albrow, M., et al. 1998, ApJ, 509, 687.Beaulieu, J.-P., et al. 2006, Nature, 439, 437.Bennett, D. P., Andrerson, J, Gaudi, B. S., 2007, ApJ 660, 781.Bennett, D. P., Rhie, S. H. 1996, ApJ, 472, 660.Bennett, D. P., et al. 2008, ApJ, 684, 663.Bond, I. A., et al. 2004, ApJ, 606, 155.Burgdorf, M.J., et al. 2007, P&SS, 55, 582.Dong, S., et al. 2009, ApJ, submitted. (arXiv:0804.1354)Dominik, M. 1998, A&A, 333, L79.Dominik, M., et al. 2007, MNRAS, 380, 792.Gaudi, B. S., et al. 2002, ApJ, 566, 463.Gaudi, B. S., Han, C., 2004, ApJ 611, 528.Gaudi, B. S., et al. 2008, Science, 319, 927.Gould, A. 2009, in “The Variable Universe: A Celebration ofBodhan Paczynski”, ed. K.Stanek. ASP Conf. ???, 2009.(arXiv:0803.4324).Gould, A. & Loeb, A. 1992, ApJ, 396, 104.Gould, A. et al. 2006, ApJ, 644, 37.Griest, K. & Safizadeh, N. 1998, ApJ, 500, 37.Jaroszynski, M. & Paczynski, B. 2002, Acta. Astron., 52, 361.Krisciunas, K., Schaefer, B. E., 1991, PASP, 103, 1033.Mao, S. & Paczynski, B. 1991, ApJ, 304, 1.Patat, F. 2003, A&A 400, 1183.Rattenbury, N. J., Bond, I. A., Skuljan, J., Yock, P. C. M. 2002,MNRAS, 335, 159.Snodgrass, C., Horne, K., Tsapras, Y. 2004, MNRAS, 351, 967.Snodgrass, C., Tsapras, Y.P., Street, R., Bramich, D., Horne,K., Dominik, M., Allan, A. 2008, in “The Manchester Mi-crolensing Conference: The 12th International Conferenceand ANGLES Microlensing Workshop”, eds. E. Kerins,S. Mao, N. Rattenbury and L. Wyrzykowski, PoS(GMC8)056.(arXiv:0805.2159).Tsapras, Y., et al. 2003, MNRAS 343, 1131.Tsapras, Y., et al. 2009, AN, 330, 4. (arXiv:0808.0813)Udalski, A., 2003, Acta Astron. 53, 291.Udalski, A., et al. 2005, ApJ 628, 109. This paper has been produced using the Royal AstronomicalSociety/Blackwell Science L A TEX style file. c (cid:13)000