Short-duration lensing events: II. Expectations and Protocols
aa r X i v : . [ a s t r o - ph . E P ] D ec Short-duration lensing events: II. Expectations and Protocols
Rosanne Di Stefano
Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138
ABSTRACT
Ongoing microlensing observations by OGLE and MOA regularly identifyand conduct high-cadence sampling of lensing events with Einstein diametercrossing time, τ E , of 16 or fewer days. Events with estimated values of τ E ofone to two days have been detected. Short duration events tend to be generatedby low-mass lenses or by lenses with high transverse velocities. We computethe expected rates, demonstrate the expected ranges of parameters for lensesof different mass, and develop a protocol for observing and modeling short-duration events. Relatively minor additions to the procedures presently usedwill increase the rate of planet discovery, and also discover or place limits onthe population of high-speed dim stars and stellar remnants in the vicinity ofthe Sun.
1. Introduction
The work described in this paper is inspired by the success achieved by the OGLE(Udalski 2003) and MOA (Bond et al. 2001) teams in discovering and monitoring eventsof short duration. Short-duration events now constitute a significant fraction of all eventcandidates. For example, approximately 9% of 654 recent events listed by the OGLEteam on its
Early Warning Site (EWS; http://ogle.astrouw.edu.pl/ogle3/ews/ews.html) have Einstein-diameter crossing times shorter than 8 days. Twenty-seven percent of theseshort events have τ E < τ E < τ E is an estimate of the time during which the gravitational magnificationwould exceed 1 .
34 for close approaches between the source and lens. Current observingprograms are sensitive enough to detect ongoing events when the magnification is just afew percent, increasing the effective event duration by a factor that can be as large as3 . These programs are therefore able to call alerts on ongoing events with small valuesof τ E . Just as alerts on events deemed likely to produce caustic crossings inspire world-wide “follow-up” with more frequent observations (see Griest & Safizadeh 1998), alerts onshort-duration events can be accorded high priority. The goal is to collect enough data topermit detailed model fits. For heavily blended events, or events in which the peak magnification is low, the enhancement factorwill be smaller (Di Stefano & Esin 1995). and hypervelocity objects. In the compan-ion paper we explored the benefits of selecting for intensive study events of short duration(Di Stefano 2009). Here we develop strategies that can help monitoring programs to dis-cover planets, brown dwarfs, and hypervelocity objects.In § >
100 short-duration events should be detected. Thesecan be followed in real time, maximizing the science return from each. In §
2. The Rate of Short-Duration Events2.1. Estimates
The populations producing short-duration events are comprised of objects that are notyet well-studied. Predictions of the rates at which they should produce events thereforehave large uncertainties. We have used a simple and straightforward approach whichallows the effects of each assumption to be traced, so that adjustments can be easily madeif necessary.We assumed that the majority of lensing events presently detected are generated bystars, and that the majority of stellar-lens events are generated by M dwarfs. If the rate atwhich M-dwarf events are detected is R , then the rate of detected events caused by othertypes of lenses can be obtained by scaling according to lens mass, transverse velocity, andspatial density. This was carried out in Di Stefano 2008b for brown dwarfs and stellarremnants. Those calculations produced the relative rates of events shown in column 1 ofTable 1, for brown dwarfs, white dwarfs, neutron stars, and black holes.Main sequence stars more massive than M dwarfs also cause lensing events. To com-pute the rate at which they generate events, relative to the rate at which M dwarfs generate Any planet-mass object not bound to a star will be referred to as a free-floating planet in the rest ofthe text. Some such low-mass objects may have been formed in and ejected from planetary systems, andothers may have been formed in isolation. . M ⊙ produce approximately as manyevents as stars of lower mass, even though the mass function declines with increasing mass.Because, however, more massive stars are more luminous than M dwarfs, events generatedby them can be more difficult to detect, even if image differencing is employed. We there-fore assume that, while events by both nearby and more distant M dwarfs contribute to theevents discovered by the OGLE and MOA teams, only more distant stars of larger masscontribute. We therefore took the total rate of detectable lensing by stars to be 1 . R , withstars more massive than M dwarfs contributing 1 / √ M ∗ P i √ q i , where the sum is over the number of wide-orbit planets, and q i = M planet,i /M ∗ .Here we will assume that the rate of planet-lens events is 15% the rate of M dwarf events.The rate at which the outer planets of our solar system generate events, compared to therate of events due to a star of 0 . M ⊙ is roughly 0 . . Many stars appear to have planetsmore massive than Jupiter, however, and some of these appear to be in very wide orbits .Free floating planets are likely to add a significant contribution. Although the value of15% is uncertain, the true rate seems likely to lie within a factor of 2 of it.As the companion paper demonstrates, planet-lens events are likely to be the shortestevents. They are therefore more likely to be missed by present-day monitoring teams. InTable 1 we assume that the efficiency for detecting events by planet-mass lenses is only halfthat for longer-lasting events. As we will discuss in §
4, however, some future programs willimprove the efficiency of detecting short-duration events. In Table 1 we therefore include acolumn for “future” monitoring programs which achieve equally good efficiencies for shortand long events.Among high-velocity objects, we have included computations only for neutron stars,because we can make rough but reasonable estimates based on the inferred galactic popu-lation of neutron stars. Other high-velocity objects can be expected to supplement thesenumbers; these include dim halo stars and remnants of runaway and hypervelocity stars.
The numbers of events predicted in Table 1, though approximate, provide guidancefor what we can expect from existing and future data sets. Column 4 of Table 1 shows thatwe can expect 124 short events among each 1000 events (Column 3). We therefore expectthat, at present, ∼
12% of all detected events should have τ E <
16 days. This is roughlyconsistent with the data on the OGLE team’s web site, although we don’t know the fractionof the posted events actually associated with lensing. Nevertheless, it is likely that existingdata sets, which include more than 4000 event candidates, contain evidence for roughly 500short-duration events. Seventy percent of these are likely to have durations in the range8 −
16 days. Of these, ∼
80% were caused by brown dwarf lenses, and the remainder byhigh-velocity stellar remnants. The majority of the shorter events were caused by planets. http://exoplanet.eu/ §
3. Learning about the Lens3.1. Lens Location and Finite-Source Size
For each type of lens defined by a given mass range, the selection of short-durationevents corresponds to a selection of lenses in one of two distance regimes. This is becausethe equation for D L is quadratic, generally admitting two solutions, D + L and D − L .D ± L = (cid:18) D S (cid:19) (cid:20) ± s − h τ E .
68 d v kms i h D S i h M ⊕ M i(cid:21) (1)For lenses that are brown dwarfs, planets, and neutron stars, D L is shown as a functionof v in Figures 1 through 3, respectively. D − L can be in the range of tens or hundredsof pc. When the lens system is this close to us, the probability of being able to detectit is large. As discussed in the companion paper, the degeneracy inherent in lensing cantherefore often be broken. In fact, for a given lens, there may be several different ways ofmeasuring the some key quantities, such as the mass of the planet-lens (see, e.g., § D L = D + L . Nearby lenses producing short-durationevents can be identified as planets, brown dwarfs, or stellar remnants. If we assume thatstellar populations in the dense source systems contain similar populations, we can predictthe distributions of values of τ E and also of values of θ E . ( θ E is more likely to be measuredfor D L = D + L ). Comparisons between the predicted and observed distributions will allowus to test models.Note, in addition, that for a given lens mass and speed, the requirement that D L bereal, places an upper limit on the Einstein diameter crossing time. τ E < .
34 d (cid:18) kms v (cid:19) (cid:20) MM ⊕ (cid:21) (cid:20) D S (cid:21) (2) 5 –Thus, for the values of v expected for planets, Earth-mass planets (dwarf planets) canproduce only events shorter than about a day (an hour), while events generated by Jupiter-mass planets should have τ E less than roughly 18 days. In order for higher-mass lenses(such as brown dwarfs) to produce short events, they must be very nearby or else very closeto the source (Figures 1 and 3). Although there are fewer lenses in these small volumes,each of the nearby lenses has a relatively high probability of generating an event becausevalues of both θ E and ω tend to be high (Di Stefano 2008b). In many cases, particularly for mesolenses, it is possible to productively follow severaldifferent lines of evidence to learn more about the lens system. In the companion paperwe considered free-floating planets. Below we consider examples of brown dwarfs, boundplanets, and high-velocity stars.
The first examples we consider are nearby brown dwarf lenses. It is important to notethat two particularly exciting candidate brown-dwarf events have been detected within6 months of each other, both with τ E <
16 days. The first of these was detected serendip-itously by an amateur astronomer, A. Tago, searching for novae. The Tago event wasthe first lensing event discovered in a sparse field, through monitoring not specifically de-signed to find evidence of lensing. The lensed source is an ordinary A0 star located a kpcaway, clearly indicating that the lens is nearby. The event was of short duration, with theEinstein-diameter crossing time estimated to be in the range 10 −
15 days (Fukui et al.2007) and an estimated peak magnification of about 50 . Gaudi et al. (2008) find that themost likely explanation for this event is lensing by a nearby brown dwarf with a propermotion greater than or approximately equal to 20 mas yr − . Because the source star wasso bright, Fukui et al. (2007) were able to verify one of the fundamental properties oflensing predicted by Einstein’s theory, that the spectrum is unchanged by lensing. In thiscase, we still know little about the lens, in spite of its apparent proximity. During thecourse of the next decade, however, the lens and lensed source should separate by a largeenough angle that they can be resolved by JWST, which is scheduled to be launched withinthe next 5 years. The IR sensitivity of JWST, combined with its angular resolution willallow it to detect the lens, if the lens is a brown dwarf. The angular separation achievedat the time of the observation will provide a value of ω. The combination of ω and τ E willyield the value of θ E . If the flux and spectrum of the brown dwarf allow the photometricparallax to be determined, the brown dwarf’s mass will be derived. Otherwise, a sequenceof additional images will determine the geometric parallax, thereby allowing us to measurethe lens mass.Six months after the Tago event, a second short-duration event that was most likelycaused by a brown dwarf was observed. This event had a peak magnification larger than1000. Both parallax and finite-source-size effects were detected, allowing the mass (0 . ± . M ⊙ ) and distance to the lens (525 ±
40 pc) to be determined. The transverse velocity 6 –of the lens is 113 ±
21 km s − (Gould et al. 2009).The discovery of these two brown dwarf events is remarkable because each event had ahigh magnification, and therefore required a very small distance of closest approach. Suchevents are therefore rare. Each of the two events therefore represents a large number ofadditional brown-dwarf events of short duration. It would be difficult to use the detectionof these two events to formulate a realistic estimate of the total number of short-durationbrown-dwarf-lens events presently expected. Nevertheless, these detections add plausibilityto the estimates we have made above, based on rate calculations and the combined OGLEand MOA detection rates.Figure 1 demonstrates that brown-dwarf events with τ E in the range of 8 −
16 days cantake place at distances greater than a hundred pc for velocities in excess of ∼
50 km s − .Note in addition, that the total volume, hence the number of possible lenses and the rateof lensing by any given population of lenses, increases with distance from us. (See, e.g.,Di Stefano 2008a, 2008b for details.) Therefore the largest number of brown-dwarf lensesgenerating 8 − >
50 km s − and be located at distanceslarger than 100 pc. This is consistent with the events observed to date. Consider a planetary system in which one planet serves as a lens, producing a short-duration event with a measured value of τ E . Suppose that the star orbited by the planetlens is detected. Suppose further, that a sequence of high-resolution measurements allowsthe geometric parallax, proper motion, and Einstein angle of the star to be measured (see,e.g., Di Stefano 2009). The combination of D L and θ E, ∗ produces a high-precision valueof the gravitational mass, M ∗ , of the star. In general, the stellar mass is estimated basedon spectral and flux information. A direct measurement of the gravitational mass allowsstellar models to be tested. In some cases, it may be possible to conduct subsequent transitor radial-velocity studies to measure the gravitational mass of the star in a second way,i.e., by studying the orbit of the planet that served as a lens and/or the orbits of otherplanets. Thus, for some stars orbited by planet lenses, we may be able to compare thegravitational mass measured via lensing with the gravitational mass measured via orbitaldynamics.Up to this point in our discussion, the planet has played only a peripheral role: (1) itproduced a photometric event that alerted us to the possibility of measuring astrometriclensing by the star, and (2) it alerted us to the presence of at least one planet orbitingthe star, thereby motivating subsequent transit and/or radial-velocity studies. The lensingevent can of course teach us a good deal about the planet.First, if finite-source-size effects are detected, then θ E,planet can be directly measured.The distance to the planet is, to high precision, the same as the distance to the centralstar. The combination of θ E,planet and D L measures the mass of the planet. With the grav-itational masses of both the planet and star measured, the mass ratio q can be computed.If, in addition, the projected orbital separation, a, between the central star and planetlens is less than roughly 3 . R E, ∗ , or if the event “repeats”, then q and a can both beestimated from a fit to the planet-lens light curve. The value of q so measured can be 7 –checked for consistency with the measured values of the planet’s and star’s gravitationalmasses.The projected orbital separation, a, measured from the light curve, can be relatedto a combination of the true orbital separation at the time of the event and the orbitalinclination: a = a true cos ( θ ) . The value of a true is related to the orbital speed at the timeof the event: v = 30 km / s × p ( M ∗ /M ⊙ ) (AU /a true ) . The proper motion of the planet can be measured from a combination of θ E,planet and τ E . With D L known, we can estimate the projected value of the planet’s orbital velocity, v cos ( θ ) , by comparing the values of ω planet and ω ∗ . Combining the equations for v and a, yields a value for the inclination of the orbit.This example illustrates that for nearby lenses, orbital solutions can be obtained, evenfor face-on orbits. Furthermore, there can be enough information to provide independentchecks on the values of some physical parameters. Even planets in very wide orbits canbe well studied with lensing, especially if there is a repeating event (Di Stefano & Scalzo1999b). When high-velocity stellar-mass objects are nearby, their Einstein angles are largeenough to be measured during the event by measuring the centroid shift in the lensedsource. In general, this leaves a degeneracy between D L and M . The degeneracy can beresolved if the lens is detected. Consider, for example, a halo dwarf star with M = 0 . M ⊙ and v = 180 km s − . If this lens is 34 . τ E = 5 days. Assuming that D S >> D L , The value of θ E would be 7 . ∼ ′′ yr − . With such a highangular speed. it is very likely that, if the background field is dense, additional lensingevents will occur over a time interval of ∼
10 years. Because the presence of the lensis already known, it is easier to identify future events with confidence, even if the angleof closest approach is larger. Once a second event is discovered, the general direction ofthe lens motion is known and future photometric and astrometric events are more easilypredicted. If a sequence of events is detected, the proper motion and parallax can becomputed by comparing the locations and times of the events. This means that even fordark high-speed lenses, mass measurements can be made. When the lens is a neutron star,it may radiate x-rays as it cools and/or accretes matter from the ISM; it may thereforebe catalogued as a weak x-ray source. It is worth noting that repeating events have beenobserved (Skowron et al. 2009). 8 –
4. Science, Strategies and Prospects4.1. Science Goals
Every event of short duration is likely to be associated with an interesting lens: alow-mass object, or a high-velocity mass. These events should therefore be high-prioritytargets for intense monitoring and follow-up, even when the value of the peak magnificationis modest. The choice of the range of values of τ E on which to spend the greatest resourcesmust be governed by the science goals of the investigators. While any set of short events is likely to include planet-lens events, the set of theshortest events, those with τ E less than roughly 4 − a > . R E, ∗ is larger than thenumbers in the narrow annulus within which the so-called “resonant” events are generated,the rate of planet discovery by lensing will increase when concerted efforts to discover andstudy short events are made. We note further that the wide-orbit planets are almostcertainly augmented by a significant number of free-floating planets.The potentially large contribution of wide-orbit planets to short and repeating lensingevents was noted more than a decade ago (Di Stefano & Scalzo 1999a, 1999b). At thattime, however, the set up of the monitoring and alert observing programs was not wellsuited to the study of short-duration and repeating events. A decade of advances in theobserving programs has put in place the ingredients needed to detect and identify short-duration and repeating events. Furthermore, the more sensitive photometry used todaywill play an important role in increasing the detection efficiency Di Stefano & Scalzo 1999a,1999b).In addition, as we note in the appendix, in the near future, short events could beginto be routinely identified in cases in which the planetary separation is small: a < . R E, ∗ . These events present the exciting possibility of using lensing to find nearby planets in thehabitable zones of their central stars (Di Stefano & Night 2008).In fact, as we have shown in the companion paper, choosing short-duration eventstends to select nearby lenses as well as lenses very close to the source star. The nearbyplanets that are identified through their actions as lenses can be among the best-studiedand will become touchstones in the field of planetary studies.
Events of 8 −
20 days are more likely to have been induced by brown dwarfs orfast moving stars. The discovery of nearby neutron stars would, by itself, be of greatscientific value. The same is true of hypervelocity stars, runaway stars, and their remnants.Gravitational mass measurements of neutron stars and other stellar remnants, as well asof brown dwarfs, would advance our knowledge of fundamental science. The feasibility ofmass measurements has been demonstrated in several cases (In addition to Gould 2009, 9 –see Alcock et al. 2001; Gould 2004; Gould, Bennett, & Alves 2004; Drake, Cook, & Keller2004; Nguyen et al. 2004.) The mass functions of brown dwarfs and stellar remnants havenot yet been well established. In addition, having a set of nearby objects would allow moredetailed studies of the relevant equations of state.
Because monitoring teams are already identifying short-duration events, identificationof planets, brown dwarfs, and high-velocity stars can start immediately. This is demon-strated by Gould et al. (2009), although for an event that had a duration on the longerend of “short-duration” events. Current programs could discover, every year, 45 −
69 shortevents caused by brown dwarfs, 23 −
36 short events caused by wide-orbit planets, 8 − ∼ ,
000 sq. degrees every three tofour nights. While the cadence of these programs is not ideal for the detection of shortevents, the efficiency for the detection of events caused by brown dwarfs, high-velocitystars, and the most massive planets will be high. Once identified by wide-field monitoringprograms, more frequent follow-up is required to fully characterize the light curve. Forevents of shorter duration, the efficiency of event detection and identification will be lower.If, however, as part of each independent scan, different filters are used in a sequence over atime interval during which the magnification is changing, even very short-duration eventscould be identified as targets for more frequent monitoring. The maximum value of theefficiency would be: f τ E / ∆ t, where f is a number in the range 0 . − , ∆ t is the timebetween distinct sequences of observations. In LSST and portions of the Pan-STARRsfields, ∆ t will be 3 − Korean Microlensing TelescopeNetwork (KMTNet) , which plans to use 3 telescopes across the Southern Hemisphere toprovide monitoring with 10-minute cadence of approximately 16 sq. degrees (Han 2009).As Figure 2 shows, KMTNet can provide excellent monitoring for Earth-mass planets. Itcould even discover dwarf planets with Einstein-diameter crossing times of less than anhour, although in such cases, follow-up by other observers would be required to obtaingood sampling. 10 –
Below we summarize the steps that can be taken in the immediate future by monitoringteams already in action. At the end of the section we discuss work that could be conductedwith archival data. • The first step is to increase the effective cadence, to increase the chances of identifyingevents of short duration. Cooperation among teams, e.g., the MOA and OGLE teamscould effectively increase the cadence of monitoring in selected fields.The philosophy of microlensing monitoring has been based on the premise that eventsare generated by lenses that are not known a priori.
For mesolenses, though, this is notalways the case. For example, when the lens producing a short-duration event happens tobe a planet in orbit with a nearby star, there is a good chance that the star has alreadybeen detected, especially when the region has been monitored over an interval of years. • To facilitate the identification of all events in which radiation from a component ofthe lens system can be detected, it is important for monitoring programs to developautomatic links to multiwavelength data bases that catalogue objects that might cor-respond to the lens. Stellar remnants may have been detected at X-ray wavelengths,brown dwarfs in the infrared, stars orbited by planets at optical and/or infrared. Wenote that, although this step is important for short events, it is also potentially veryuseful even for longer events. Whatever the event duration, this step will identifythose events caused by nearby lenses, facilitating high-precision measurement of thelens mass. Ideally, in addition to the catalogues, the monitoring teams should havequick access to the data taken at various wavelengths, since many nearby sourcesmay not have been catalogued. • When a short-duration event is caused by a planet orbiting a star, and when theplanet-star separation is less than roughly 3 . R E, ∗ , the presence of the star wouldalready have caused a gradual increase in the magnification, even before the short-duration event caused by the planet. Monitoring teams can test for such a priortrend at the start of what appears to be a short-duration event. • “Repeating” events in which a planet and its star serve as independent lenses areexpected in as many as ∼
10% of events in which a star with a planetary systemserves as a lens. (See Di Stefano & Scalzo 1999b.) The most likely case is theone in which the planet lens producing the shorter event of the two events is theinnermost “wide” planet. Roughly half the time, the star should have produced thefirst event. Any short duration event that follows a stellar-lens event has a goodchance of having been produced by a planet in orbit with the star. It would beworthwhile to subject the declining portion of each stellar-lens event to monitoringcapable of quickly identifying subsequent short events. For planets with massescomparable to or larger than the that of the Earth, monitoring 1 − τ E of the return to baseline. An alert should becalled immediately when such deviations are detected. It is important to note that 11 –systematic monitoring of this type will either yield planet detections (the most likelyoutcome) or will allow meaningful limits on the structure of planetary systems to bederived. • High-resolution images can play an important role for a wide range of lenses producingshort-duration events. The possible uses of such imaging are described in § • When an ongoing event is identified as a possible short-duration event, an alertshould be called. Ground-based multiwavelength observations with good angularresolution, taken as the magnification increases, can establish the amount of blendingand determine if the short duration of the event is an artifact caused by blending. Ifso, monitoring for the reasons explored in this paper is no longer necessary. Eventsthat are genuinely short should be subject to intensive monitoring by a global networkof telescopes. In some cases, the monitoring programs may themselves sample thelight curve frequently enough to allow good model fits. Model fits to a well-sampledlight curve can determine the values of τ E , b , and discover whether finite-source-sizeeffects, the existence of a companion, and even parallax effects influence the lightcurve shape. The cadence of monitoring should be greatest near peak, even if themagnification is not high.Finally, we note that it is not always possible to follow each event as it unfolds. This iscertainly the case for events that have already occurred. Even future monitoring programs,with their improved detection efficiencies, may not identify or be able to follow all short-duration events in real time. In these cases, light curve fits to the short-duration event,including possible long-term deviations from baseline that could “frame” the short-durationevent, can be carried out. In addition, step 2 above (studying multiwavelength catalogsand data that cover the region around the event), can provide important information. Insome cases, follow-up HST imaging can also be used to test models, e.g., allowing the lensmass to be determined in some cases.
5. APPENDIX: Very Close Planets and the Habitable Zone
In the context of planetary systems, independent events of short duration were orig-inally predicted for planets in wide orbits ( a greater than roughly 1 . R E, ∗ ). Recently,however, close orbits ( a < . R E, ∗ ) were studied, and a second class of short-durationevent considered (Di Stefano & Night 2008).For small a , there are two regions in the lens plane where the effects of the planetare significant. The first is close to the central star. Events in which the source trackpasses close to the star therefore exhibit short-lived deviations from the point-lens form.As the value of a decreases below 0 . R E, ∗ , the region of perturbations caused by the planet 12 –becomes smaller. Consequently the deviations in the light curve are shorter-lived and arealso more easily washed out by finite-source-size effects. The second region is at a distanceof roughly 1 /a from the star, along the binary axis. (See Di Stefano & Night 2008). Thisregion has a small caustic, which has little effect on most light curves. There is a regionaround the caustic, however, in which the deviation from the low-magnification effects ofthe star can be as large as a few percent, lasting from hours to days.KMTNet will be very sensitive to these deviations. But even today’s monitoringprograms could find them. To do so would be important, because for nearby M dwarfs,orbital separations of 0 . − . R E, ∗ put the planet in the zone of habitability.The teams could call alerts on such events with a high level of confidence because ofthe presence of the star along the line of sight and also because a low-magnification effectsassociated with the star, frame the short-duration deviation caused by the planet. Thisis in exact analogy to the situation in which a short-duration event is caused by a planetwith 1 . R E, ∗ < a < . R E, ∗ . There are several new features of lensing in this close-planet regime. (1)
The deviation that signals the presence of the planet is not generally well approximatedby a point-lens model. (2)
The deviation that signals the presence of the planet occurs when the source is farfrom the central star ( ∼ /a ). (3) Because the planet is close to the star, orbital motion can be significant during theevent. This means that the regions of deviation are likely to swing into the path of thesource star. We have found that the rate of these potentially important events can becomparable to the rate of stellar-lens events exhibiting
A > . Acknowledgments:
This work benefited from discussions with Charles Alcock, ErinArai, Mary Davies, Nitya Kallivayalil, M.J. Lehner, Christopher Night, Brandon Patel,Frank Primini, Pavlos Protopapas, Rohini Shivamoggi, Kailash Sahu, and Takahiro Sumi.I would also like to thank the anonymous referee for useful comments. The research wasconducted under the aegis of NSF grants AST-0708924 and AST-0908878, and a grantfrom the Smithsonian endowment. I would like to thank the Aspen Center for Physics forits hospitality during the early phases of this work.
References
Alcock, C., et al. 2001, Nature, 414, 617Bond, I. A., et al. 2001, MNRAS, 327, 868 https://it019909.massey.ac.nz/moa/Chambers, K. C., & Pan-STARRS 2004, American Astronomical Society Meeting Ab-stracts, 205,Di Stefano, R. 2009, arXiv:0912.1611Di Stefano, R. 2008a, ApJ, 684, 46Di Stefano, R. 2008b, ApJ, 684, 59 13 –Di Stefano, R., & Esin, A. A. 1995, ApJ, 448, L1Di Stefano, R., & Scalzo, R. A. 1999b, ApJ, 512, 579Di Stefano, R., & Scalzo, R. A. 1999a, ApJ, 512, 564Di Stefano, R. & Night, C. 2008, ArXiv e-prints, 0801.1510Drake, A. J., Cook, K. H., & Keller, S. C. 2004, ApJL, 607, L29Fukui, A., et al. 2007, ApJ, 670, 423Gaudi, B. S., et al. 2008, ApJ, 677, 1268Gould, A., et al. 2009, arXiv:0904.0249Gould, A., Bennett, D. P., & Alves, D. R. 2004, ApJ, 614, 404Gould, A. 2004, ApJ, 606, 319Griest, K., & Safizadeh, N. 1998, ApJ, 500, 37Near-Field Han, C. 2008, ApJ, 681, 806Han, C. 2009, private communicationSkowron, J., Wyrzykowski, L., Mao, S., & Jaroszy´nski, M. 2009, MNRAS, 393, 999Udalski, A. 2003, Acta Astron., 53, 291 14 –Table 1: Numbers of Short-Duration Events ( τ E <
16 days)Number Number Number NumberRelative of events of short events of events of short eventsLens Event per 1000 per 1000 per 1000 per 1000Type Rate detected detected detected detected
Present (1)
Present Future (2)
Future
M dwarfs 1 . (3)
465 5Other stars 0 . .
19 92 69 88 66White dwarfs 0 .
17 82 0 79 0Neutron stars 0 .
13 63 13 60 12Black holes 0 .
01 5 0 5 0Wide-orbit planets 0 . (4)
36 36 70 70
NOTES: (1)
Present refers to the number of events generated by each type of lens per 1000 eventsdetected by monitoring programs with the capability of OGLE III. (2)
Future refers to the numberof events generated by each type of lens per 1000 events detected by monitoring programs of thefuture which will be more sensitive to events with τ E < (3) We assume that roughly 1% ofall events generated by M dwarfs will have τ E <
16 days). These correspond to very fast transversevelocities or M-dwarf lenses very close to us. More massive stars that are close to us may be toobright to generate events detectable by the monitoring programs. (4)
We assume that only half ofthese are long enough to be detected by today’s monitoring programs, but that programs of thefuture, such as KMTNet (see § Fig. 1.— Logarithm of D L vs v for a lens with fixed mass (0 . M ⊙ ) in the brown-dwarf regime. The field of background sources was placed at a distance D S of 8 kpc.Each lens was given a randomly generated value of the transverse velocity, v, in the range25 −
125 km s − . We assumed that the Einstein-diameter crossing time was 1 day in theleftmost panel, increasing by a factor of two in each panel to the right. For each lens,there are two possible values of D L , D + L and D − L . We computed both and show the results.If, for D + L , the value of b θ E D S (with b = 0 .
1) was smaller than 10 R ⊙ , we assumed thatfinite-source-size effects could be detected and plotted the point in red. 15 – Fig. 2.— Each row consists of a set of 6 panels in which the logarithm of D L vs v isplotted for a lens with fixed mass. The mass chosen is that of the planet which labels therow. For each planet the range of interesting time scales is shown. Note how this range isdifferent for planets of different mass. If, for D + L , ( D − L ) the value of b θ E D S (with b = 0 . R ⊙ , we assumed that finite-source-size effects could be detected andplotted the point in red (green). See the caption of Figure 1 for additional details. 16 – Fig. 3.— Logarithm of D L vs v for a lens with fixed mass (1 . M ⊙ ). The Einstein diametercrossing time was 1 ..