Discovering habitable Earths, hot Jupiters and other close planets with microlensing
aa r X i v : . [ a s t r o - ph . S R ] D ec Discovering habitable Earths, hot Jupiters and other closeplanets with microlensing
R. Di Stefano
Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138
ABSTRACT
Searches for planets via gravitational lensing have focused on cases in whichthe projected separation, a , between planet and star is comparable to the Ein-stein radius, R E . This paper considers smaller orbital separations and demon-strates that evidence of close-orbit planets can be found in the low-magnificationportion of the light curves generated by the central star. We develop a protocolto discover hot Jupiters as well as Neptune-mass and Earth-mass planets in thestellar habitable zone. When planets are not discovered, our method can beused to quantify the probability that the lens star does not have planets withinspecified ranges of the orbital separation and mass ratio. Nearby close-orbitplanets discovered by lensing can be subject to follow-up observations to studythe newly-discovered planets or to discover other planets orbiting the same star.Careful study of the low-magnification portions of lensing light curves shouldproduce, in addition to the discoveries of close-orbit planets, definite detec-tions of wide-orbit planets through the discovery of “repeating” lensing events.We show that events exhibiting extremely high magnification can effectively beprobed for planets in close, intermediate, and wide distance regimes simply byadding several-time-per-night monitoring in the low-magnification wings, pos-sibly leading to gravitational lensing discoveries of multiple planets occupyinga broad range of orbits, from close to wide, in a single planetary system.
1. Introduction
Microlensing planet searches have been directed toward discovering planets in orbitswhose size is comparable to the size of the Einstein radius, R E , of the central star. Herewe study the detectability of planets in much closer orbits. This is necessary, because wenow know that many planets are in such close orbits . We demonstrate that ground basedsurveys for lensing events can detect a wide range of close-orbit planets, including “hotJupiters” orbiting sun-like stars, and even Earth-mass planets in the habitable zones of Mdwarfs. We also discuss the role of space missions.We were led to the study of close-orbit planets by work to determine whether or notplanets in the habitable zones of nearby stars could produce detectable lensing signatures(Di Stefano & Night 2008). The standard planetary-lensing scenario is most sensitive toplanets with α = a/R E in the range 0 . − , where a is the orbital separation (Mao& Paczy´nski 1991; Gould & Loeb 1992). In many cases, this corresponds to the region http://exoplanet.eu/catalog.php would be able to detect planets orbiting within thehabitable zones of nearby M dwarfs (Di Stefano & Night 2008). Here, we demonstrate theimportance of lensing by close-orbit planets for a broader range of systems: the planetsmay be in the habitable zone, closer in, or farther out. The planetary systems may bewithin a few tens of pc or much farther away. The host stars may be M dwarfs, browndwarfs, or much more massive stars.In § § §
2. Evidence for Close-Orbit Planets
As this text is being written, 696 planets are listed in the
Interactive Extra-solarPlanets Catalog . Most of these planets ( ∼ ∼ a , is comparable in size to the Einstein radius, R E , of the centralstar: 0 . < α < , with α = a/R E . To determine the effect this has on planet discovery,we treat each exoplanet central star as a potential lens, and determine the value of α foreach of the discovered planets. To do this calculation, we need to compute the value of R E for each central star. R E = 1 .
01 AU (cid:20)(cid:16) M ∗ M ⊙ (cid:17) (cid:16) D L
125 pc (cid:17) (cid:16) − D L D S (cid:17)(cid:21) , (1)To compute R E , we need the star’s mass, M ∗ , and its distance D L from us. The valueof R E also depends on the distance, D S , to the source that would be lensed. Because mostof the central stars do not have a bright source located directly behind them, the value of D S is not determined. To compute R E we must therefore make some assumptions aboutthe ratio D L /D S . Here we first consider lenses for which D L /D S <<
1; this allows us toignore the last factor in Equation 1. The physical meaning of this assumption is that weare focusing on nearby planetary systems or on planetary systems with source stars locatedmuch farther from us. We will mention a second case in §
4, with ( D S − D L ) /D S << , generally corresponding to lens stars very close to the lensed source.Making these assumptions, we can compute α = a/R E , for each known exoplanetplanet whose semimajor axes a has been measured. Figure 1 shows the results for all http://exoplanet.eu/catalog.php a, M ∗ , M pl , D L , and the orbital period, P. We havedefined q = M pl /M ∗ . The two dashed lines in each panel enclose the region 0 . < α < α inthis range, planets will be discovered for only a small fraction. This is because the pathof the source behind the lens must be favorable for planet discovery and also because thesensitivity and sampling cadence must be well suited to planet discovery.It is therefore advantageous to extend lensing planet searches to both smaller andlarger values of α. Larger values have been considered in some detail (Di Stefano & Scalzo1999a, 1999b; Han 2009). We will discuss them briefly in §
4. The major part of this paperis devoted to studying planet detection of smaller values of α. From Figure 1, we see that, judging by the planetary systems already known, smallervalues of α are expected for a wide range of stellar masses, planet masses, and distances D L . One variable that displays a trend with α is the orbital period, P . P = 71 . (cid:18) α (cid:19) (cid:18) M ∗ M ⊙ (cid:19) (cid:18) D L
125 pc (cid:16) − D L D S (cid:17)(cid:19) (2)Thus, smaller values of α are associated with shorter orbital periods. In the next section wewill study the geometry of the isomagnification contours of close-orbit planetary systems,and will find that there is a small region exhibiting deviations from the point-lens form atlarge distances ( u > R E ) from the center of mass. This region rotates around the centerof mass at the orbital period. The region of deviation can therefore rotate into the pathof the source track, increasing the probability of detection.If v is the relative transverse motion, the proper motion is µ = 0 . ′′ yr − (cid:18) v kms (cid:19) (cid:18)
125 pc D L (cid:19) (3)The Einstein angle is θ E = 0 . ′′ (cid:20)(cid:16) M ∗ M ⊙ (cid:17) (cid:16)
125 pc D L (cid:17) (cid:16) − D L D S (cid:17)(cid:21) , (4)Define τ E, to be the time taken for the source-lens separation to change by an angle equalto the Einstein angle. For nearby lenses, τ E, ≈ θ E /µ , and its value can be comparableto the value of P, when α is small. For, example, a solar mass lens at 125 pc will have τ E, ≈
88 days if v = 20 km s − . If the detection limit is 2%, the event may be detectableduring the time taken to cross through 6 R E . For a range of lens masses and distances,several orbits may occur during a lensing event.
3. Close-Planet Magnification Geometry and Light Curves3.1. The Role of α When the projected distance between planet and star is significantly smaller than R E , then at distances larger than R E , the system is well approximated by a point-lens of total 4 –mass equal to the sum of the stellar and planet masses. Nevertheless, there are smalldeviations from the point-lens form. To study these deviations we consider lenses withplanets, each planet characterized by the mass ratio q = M pl /M ∗ , and the separation α. We begin by considering the magnification geometry for a fixed value of q by computingthe magnification around each of a sequence of concentric rings, centered on the center ofmass. If the system were a point mass, we would expect that each ring of radius r wouldhave a single magnification, A ( r ), whose value would be equal to ( r + 2) / ( r √ r + 4). Foreach ring, we computed the difference, ∆ , between the maximum and minimum value ofthe magnification. Deviations from symmetry are associated with values of ∆ that differfrom zero. The top panel of Figure 2 shows ∆ as a function of r for a lens with q = 0 . α, and the peak occurs at r = R α = α − α. The black peak on the rightcorresponds to α = 0 . . The value of α increases by 0 .
05 for each peak toward the left,to α = 0 . α near or above 0 .
5, lensing by planetarysystems has been well studied; the black curve ( α = 0 .
5) shows that there are deviationsfrom the point-lens form larger than 1% over a wide range of values of r . For smaller α, however, the deviations in the region r > q . In fact, the locations of the peaks would be identical. The width of the peaks wouldbe larger (smaller) for larger (smaller) values of q .To see why non-linear effects become evident at the star-planet separation r = R α = α − α > , consider the image geometry for the simplified case of a point source, located adistance R α from a point lens. If the x axis connects the lens and source, then at a value of x equal to − α, there will be a negative parity image of the source. When a planet happensto lie near this point, its influence on the total magnification will be enhanced.Now consider a planetary system with the center of mass at the origin and the planetat x = − α. There are two tiny caustics located along the circle of radius R α ; one appearsat a positive value of y and one appears at a negative values of y. The caustics themselvesare too small to play a significant role, but they serve as a convenient way to locate theregions in the lens plane within which the magnification deviates from the point-lens form.When the source happens to lie behind one of these “perturbed” regions, the light curvewill exhibit features that signal the presence of the planet. q As shown in § α determines the distance from the center of massof the region with isomagnification perturbations. In this subsection we show that thesize of these perturbed regions is determined by the mass ratio, q . The alterations in theisomagnification contours are shown in the bottom-right panels of Figures 3, 4, and 5.With α = 1 / q ,which is 1 . × − , . × − , and 1 . × − in Figures 3, 4, and 5, respectively. Thesefigures demonstrate that the size of the perturbed regions is smaller for smaller values of q . The figures for the two smallest values of q correspond to a Neptune-mass planet 5 –and an Earth-mass planet, respectively, orbiting a star with 0 . M ⊙ . In each case, thelower-right panel zooms in on the perturbed region, to reveal that, in a small region of theannulus around R α , isomagnification contours from larger values of r are pulled in to smallervalues of r , with the contours from smaller r pushed out on either side. When a source withlarger transverse than radial speed passes behind this region, the magnification will deviateupward from the point lens form, then downward and up again before descending back tothe point-lens value. As shown in the top panel of all three figures and in the portionof the light curve shown in the lower left-hand panel of each figure, this characteristic“up-down-up-down” form of the light curve is exhibited when both α and q are small.The isomagnification contours in the lower-right-hand panels for the Neptune-massand Earth-mass planets exhibit small closed curves, which enclose caustics. The causticsare tiny and their positions are not marked here; in fact, the caustics do not play an impor-tant role in the light curve deviations. The light curve deviations are dominated instead bythe more subtle affects associated with the perturbations of the low-magnification isomag-nification contours. Nevertheless, the positions of the caustics, which can be vanishinglysmall, provide a convenient way to measure the size of the perturbed region.We define ∆ y c to be the straight-line distance between the tiny caustics discussedabove, expressed in units of R E . We compute a normalized separation, ∆ Y norm , by dividing∆ y c by C α = 2 π R α , the circumference of a circle of radius R α . Consider the bottom panel ofFigure 2. The variable along the vertical axis is the logarithm of the normalized separation,∆ Y norm ; it is plotted against log ( q ) . There are 5 colored curves for values of α rangingfrom 0 .
10 to 0 .
33; these curves are almost indistinguishable. Moving to wider orbits, thegreen curve for α = 0 .
40 can be distinguished from the others, but it is close to them.All in all, there is very little alpha dependence, indicating that the linear dimensions ofthe perturbed area depend primarily on the value of q . The curves for small α and small q are well approximated by the equation: log (∆ Y norm ) = 0 . log ( q ) − . . Thus, thephysical separation, expressed in units of R E , can be expressed as a product of a factorthat depends only on α and one that depends only on q : ∆ y c = 2 π R α ∆ Y norm ( q ) . Figures 4 and 5 clearly show that the perturbed region is larger than the distancebetween the centers of the closed curves, which is an approximate measure of the separation∆ y c between caustics. Let L ( α, q ) represent the linear dimensions of the perturbed region,expressed in units of R E . On an empirical level, the size of the region is determined bythe size of the smallest deviations that can be reliably detected for any given observationalscheme. If deviations like those shown in the light cures in the top panels of Figures 3through 5 are detectable, then we find, empirically, that L ( α, q ) ≈ . y c . For the three cases shown in Figures 3, 4, and 5, the linear dimensions, L ( α, q ) , are approximately 0 . R E , . R E , and 0 . R E , respectively. In the absence of orbitalrotation, the event rate would be proportional to these linear dimensions. The eventdurations would be equal to the time taken for the relative lens and source positions to We note that, in order to generate these particular light curves, we used face-on circular orbits. Thegeneral theory applies to orbits of all orientations and eccentricity; the value of α and the geometry of theisomagnification contours are then time dependent. Nevertheless, if the time duration of the deviationsfrom the point-lens form is primarily determined by the value of the orbital period, the basic shape ofindividual deviations will have the same characteristics as shown here. L ( α, q ) . Let this time be denoted by T transverse .T transverse = 21 . (cid:16) L ( α, q )0 . (cid:17)(cid:16) R E (cid:17)(cid:16)
20 km / s v (cid:17) (5)In most cases, however, the events will be significantly shorter, because orbital motion playsan important role. In addition, orbital motion increases the likelihood that a detectableevent will occur. In the case in which T transverse > P, the probability that a detectableevent will occur is unity. Particularly in cases with α < . , the orbital period can be comparable to or evenshorter than T traverse the time taken for the source to traverse a distance L ( α, q ) . In suchcases, the orbital motion is very likely to rotate the perturbed region in front of the source.The probability is T traverse ( q ) /P orb when T traverse ( q ) < P orb and is unity otherwise. Whenthe probability is larger than unity, deviations repeat on a time scale roughly equal to P orb .If we are monitoring the system frequently enough to catch a deviation in progress, ourchance of seeing the deviation can be 100% if the planet exists.The time duration of a deviation from the point-lens form is T dev = L ( α, q )2 π R α P orb = 2 . P orb (0 . Log ( q ) − . (6)In the cases shown in Figures 3, 4, and 5, this produces deviations of durations 1 . .
56 days, and 0 .
14 days, which compare well with values of T dev shown in the bottom-leftpanels of each figure. The first exoplanet to be discovered orbiting a sun-like star was 51 Peg b (Mayor &Queloz 1995), a planet with m sin ( i ) ≈ . M J in a 4-day orbit. At present, there are 155known exoplanets with semimajor axis smaller than 0 . m sin ( i ) between0 . M J and 10 M J . These planets are generally referred to as “hot Jupiters”. The value of m sin ( i ) is larger than (1 M J , M J , M J ) in (96 , ,
31) cases, respectively.The planet corresponding to Figure 3 is a hot Jupiter. Its mass is equal to that ofJupiter, and it orbits a star of 0 . M ⊙ ; α = 1 /
3. With D L = 25 pc; D S = 8 kpc, wefind R E = 0 . a = α R E = 0 .
13 AU, and the orbitalperiod is 20 days. With a transverse speed of 20 km/s, the time taken to cross R E is τ E, = 35 days. The duration of the deviations is just over a day, and the deviationsrepeat. This example demonstrates that hot Jupiters can be found through their influenceon the low-magnification portion of lensing light curves. Note that if, for the same lensstar and orbit, the planet had a mass of (3 M J , M J , M J ), then T dev would be (1 . . . all orbital inclinations, incontrast to transit studies. Furthermore, lensing provides a direct measure of the lens mass,at least in cases in which the mass of the central star can be determined. Furthermore, aswe show below, lensing searches for hot Jupiters can be effective for nearby stars, allowingdetailed follow-up studies, and also for distant stars.An important issue is whether the discovery of hot Jupiters, or placing reliable limitson their presence around lens stars, can be accomplished on a regular basis. To answerthis, we consider a condition sometimes used to define the boundary between hot Jupitersand planets farther out: a < . α R E < . D S = 8 kpc). If the centralstar is a low-mass dwarf, with M ∗ ∼ . M ⊙ , then the condition above holds for all valuesof D L . For stars of 0 . M ⊙ , . M ⊙ , . M ⊙ , . M ⊙ , the condition holds for D L < . D L > . . . . In both Figures 4 and 5, the central mass is 0 . M ⊙ , and D L is 50 pc. This yields, R E ∼ .
32 AU. As in Figure 3, α = 1 / . The separation between planet and star is ∼ . − .Figures 4 and 5 show the light curve, the characteristic form of the deviations from thepoint-lens case, and the perturbations of the isomagnification contours for the Neptune-mass and Earth-mass planets, respectively. The shorter duration of the deviations forthe lower mass planets means that higher-cadence sampling would be required to fullyresolve them. Nevertheless, the general up-down-up-down form is clear in all three cases.Furthermore, the light curves shown in the top panels of both figures indicate that theorbital period may be recoverable in cases such as these.If planets are not uncommon in the habitable zones of their stars, then studyingthe low-magnification portions of lensing light curves for evidence of planets in α < . α would be in the habitable zone. Their resultsindicate that close-orbit planets in the habitable zone could be detected for a wide rangeof values of D L . For example, α is smaller than 0 . ∼ . M ⊙ star, with 800 pc < D L <
4. Successful Observing and Analysis Strategies
Lensing associated with close orbit planets is a new frontier. Fortunately, the ongoingmonitoring programs can allow us to begin exploring this frontier in the immediate future.Below we summarize the relevant features of lensing associated with close-orbit planets. Every light curve can be used to either discover close-orbit planets or else to place quan-tifiable limits on the presence of planets orbiting the lens in close orbits. This is because theregion in which the deviations occur are low-magnification regions, and every detectablelensing event exhibits low magnification, whatever peak magnification it achieves. When the magnification is A = 1 + δ, the corresponding value of α is α = 0 . δ (8)Thus, every interval of the low-magnification part of every light curve can be studied toeither discover or place limits on planets at a specific projected separation α. The higherthe precision of the photometric measurements, the smaller the values of α we can probe. For each value of α, the value of q determines the size of the region over which pertur-bations of a given magnitude are detectable. L ( α, q ) = 2 . ξ (cid:20) π (cid:16) α − α (cid:17)(cid:21) [0 . Log ( q ) − . , (9) L ( α, q ) is expressed in units of the Einstein radius, and the value of ξ depends on thephotometric sensitivity and frequency of sampling. The value of L ( α, q ) can be fairlylarge. For example, for α = 0 .
25 and q = 0 . , L ( α, q ) = 0 . . This is the radius of theannulus around R α within which the perturbations are potentially detectable. The sourcemust pass through this annulus both on the way in toward higher magnifications and as itemerges from the higher-magnification region. Orbital motion increases the probability of detection. In the case considered in point 3(just above), the total time spent in this annulus would be ∼ . τ E, . If, e.g., the Einsteinradius crossing time is 30 days, the source would spend more than 10 days crossing theannulus. We would have a very good chance of detecting deviations for hot Jupiters withorbital periods smaller than 10 days, because the perturbed region would rotate into thepath of the source one or more times. The deviations from the point-lens form will have a magnitude that can be easilycomputed by using the formula for L ( α, q ). The point-lens magnification will be that for u = R α = α − α. The upward and downward deviations will have magnitudes approximatelycorresponding to the point lens magnifications at R α ± L ( α, q ) . Thus, for each α , we cancompute the range of magnifications expected during a deviation for each q , and determinehow large q would have to be in order for a planet to produce detectable deviations,given the quality of the observations. Alternatively one can decide whether more sensitivephotometric observations should be taken, in order to be able to detect a planet with aparticular value of q , hence planetary mass. For each value of α, the value of q determines the duration of the deviation. If we 9 –assume that orbital motion dominates, then. T dev = L ( α, q )2 π R α P orb = 2 . ξ P orb [0 . Log ( q ) − . (10)In fact, for close orbit planets, orbital motion is likely to dominate for all but stars withexceptionally large proper motion. Consider a solar-mass star at 125 pc. Equation 4 tellsus that, if the lensed source is in the Bulge, θ E ≈ R α is approximately equal to3 , and if the orbital period is ∼
70 days, then the orbital angular speed of the deviation is ∼ . ′′ yr − , larger than the angular speeds of all but a handful of stars. Nearby Lenses:
If the lens star lies within a kpc or so, it is likely to be detectable. It maybe catalogued, perhaps even by the monitoring programs that search for evidence of lensing.We have found, e.g., that ∼
8% of all lensing event candidates have 2MASS counterparts,many likely to correspond to the lens (McCandlish & Di Stefano 2011), while >
10% ofthe lenses producing the events detected by the monitoring programs are predicted to liewithin about a kpc (Di Stefano 2008a, 2008b). Thus, we may know the spectral type ofthe lens and be able to estimate its mass and distance from us. We may even know itsproper motion. This information, combined with the Einstein angle crossing time, allowsus to determine the total lens mass and distance. The wide range of other informationpotentially derivable from fits to the lensing light curve, may allow us to also determinethe planet’s mass and key features of its orbit. Thus, if we do find a planet, we can learna great deal about it, including its gravitational mass, from the lensing observation. Inaddition, because it is nearby, follow up studies to learn more about this planet and tosearch for others orbiting the same star may be possible. On the other hand, if we do notfind evidence of a planet, we can place quantifiable limits on the presence of planets witha well-defined range of properties orbiting a star of known type.Given the importance of what we can learn about planets orbiting nearby stars, itis important to identify those events with counterparts that may be nearby stars. Thus,in addition to conducting automated searches through catalogs for possible counterpartsto lensing events, we can employ
Virtual Observatory (VO) capabilities to scan existingimages of the area within which the lensing event occurs. By identifying nearby lens stars,we can direct resources toward that subset of events whose study is most likely to beproductive through planet discovery or, alternatively, through providing opportunities toplace meaningful limits on the presence of planets. Distant Lenses:
Events associated with close-orbit planets may also be produced when D L is large, particularly when ( D S − D L ) /D S << . In such cases, we may not be ableto detect the central star. We therefore may not have any specific information about itsmass or distance from us. The value of τ E, , fit from the light curve, provides a relationconnecting M, D L , D S , and the transverse speed. Beyond this, we may have to resort tostatistical arguments based on the distribution of stars in the Galaxy, to provide furtherconstraints. The deviations may exhibit periodicity, allowing us to estimate the orbitalperiod, or the fits to the deviations, combined with other light curve information, mayotherwise allow us to determine approximate values of α and q .For large D L , the Einstein angle can be small, comparable in size to the magnifica-tion features associated with deviations. This means that finite-source-size effects can be 10 –important. Finite source size can play a negative role by softening and diminishing theshort-duration deviations associated with the presence of planets. Thus, finite-source-sizeeffects may make it more difficult to identify the effects of close-orbit planets orbiting dis-tant stars. If, however, the deviations remain detectable, then the alteration in their shapeproduced by finite-source-size may allow us to derive the value of θ E . While this will notentirely break the degeneracy, it does give an extra relation connecting M and D L (as-suming that D S is known, at least approximately). It is therefore important to include, inthe fits to deviations in the low-magnification portions of the light curve, finite-source-sizeeffects. Blending:
When the source star contributes only a fraction of the baseline light, then themeasured magnification (i.e., the ratio between the light received portion of the event andthe baseline light) during the low-magnification portion of the event is actually A measured =1 + f δ. If, therefore, we are not aware of the blending, we will underestimate the value of δ, hence α. It is therefore important to include the effects of blending in the light curvefits (Di Stefano & Esin 1995). If the event is studied to search for planets as it occurs,then it is worthwhile observing it in several filters as it occurs, to determine the amountof baseline light that is lensed, as a function of wavelength.
The simple points listed above lead to an important conclusion: every light curvecan be used to either place limits on the presence of possible close-orbit planets or elseto discover them.
Furthermore, a relatively straightforward procedure can be employed toachieve these goals. We begin by discussing the case of catalogued events and then considerwhat can be learned from ongoing events.
Many of the more than 8500 candidate events already discovered are well-enoughsampled at low magnification to provide fertile hunting grounds for close-orbit planets.Not all of the candidate events correspond to lensing events, but those with acceptablelens-model fits should be considered as strong candidates. The fit provides an estimateof τ E, , which relates the total lens mass to D L , D S , and v. The fit also provides a valuefor the blending parameter, f, the fraction of the baseline light provided by the lensedsource. Although multiple values of f can be consistent with the data, the degeneracy canbe lifted if the peak magnification is higher than about 3 (Wozniak & Paczynski 1997).The degeneracy is also broken if the event is observed in a variety of wavebands, even justa few times, or if we have information about other sources of light along the direction tothe event, such as the lens itself.To search for close-orbit planets, we must search the low-magnification portions of thelight curve for any upward or downward deviations from the point-lens form. The value
11 –of δ in the region containing the deviation provides an estimate of α. If there are severalpoints per deviation, a model fit can provide an estimate of q. Whatever the numbers ofpoints per deviation, we search for signs of periodicity in the wings of the light curves.Repeating signatures of close-planet lensing are not exactly periodic (Di Stefano & Esin2011), but it is possible to introduce a correction to extract the correct period (Gao et al.2011). An interesting feature of the near-periodicity, is that it is a transient phenomenon,occurring in the wings of a light curve. That is, if it is due to a close-orbit planet, it is nota long-term property of the baseline, nor is it necessarily exhibited throughout the event.An orbital period for the portion of the light curve corresponding to a particular value of α (hence the projected angular separation), connects the systems mass with true separation.The combination of these tests provides a great deal of information about the massof the lens system, the mass of the planet, and the size and orientation of the planetaryorbit. These quantities are all expressed in terms of D L , D S , and v . The value of D S is usually known approximately, because the source is likely to be located in the densestellar field being monitored, often the Bulge, but sometimes the Magellanic Clouds orM31. Galactic models can be used to construct a probability distribution for the valuesof D L and v. If, however, the lens is a catalogued star, then estimates of the values of D L , v, and M may already be known. Alternatively, observations taken several years afterthe event can resolve the separation between lens and source, especially in those cases inwhich the lens happens to be nearby. This type of study has already been done for theevent MACHO-LMC-5, for which an HST image taken 6 years after the event was able toprovide a photometric parallax and measure the proper motion, allowing the gravitationalmass of the lens to be determined (Alcock et al. 2001a).When there is no sign of deviations or of deviations that repeat, then it is possibleto place limits on the orbital period and the value of q of any planet that might be inclose orbit with the lens star. This is because it is possible to estimate the length oftime the magnification is close to δ. This tells us the duration T transverse of the intervalwhen deviations caused by a planet with α = 0 . δ would have been detectable. Fororbital periods shorter than T transverse , there would be a chance to detect deviations causedby the planet at least once. Thus, by studying the frequency of sampling during thistime, we can determine the duration T dev,min of the shortest deviation to which we wouldhave been sensitive. This allows us to compute the smallest value of q to which theobservations would be sensitive. To quantify limits on the presence of planets, we canrun a Monte Carlo simulation in which we model the planetary system, generate largenumbers of light curves, and compute the fraction of all planets within some range ofmasses, orbital separations, orientations, and eccentricities would have been discovered,given the frequency and sensitivity of the observations. The present discovery rate of candidate lensing events is roughly 1500 per year. Thesensitivity to low-magnification is good, as witnessed by the fact that events with estimatedpeak magnification smaller than 10% are regularly identified. Whereas for events that havealready finished, we must rely on whatever data has already been collected, for ongoing 12 –events we have opportunities to collect as much data as would be needed to discover anyclose orbit planets. Fortunately, significant improvements in detection efficiency can beachieved with relatively modest changes in the observing plan.The key improvement would be to ensure regular sampling of the baseline. If, e.g.,we want to be able to catch any up-down-up-down deviation that lasts for at least 12 − ∼ − − − . R E < α < . R E . We will therefore refer to the procedure we suggest asa moderate increase in monitoring. Nevertheless, with more than 100 events occurring atany given time, it is unrealistic to think that this type of program can be carried out foreach. This means that we must select events for special attention. Criteria that could beuseful include the following. High peak magnification:
After a handful of points have been collected as the eventrises from the baseline, it is possible to begin to predict the peak magnification, A peak . Values of A peak greater than about 3 make it easier to reliably determine the blendingparameter from the light curve fit. Since this is important to determining the value of δ in the low-magnification wings, it makes sense to devote special attention to eventswith predicted high values of the magnification. Moderate monitoring, like that describedabove, can be started while the light curve is still on the rise, after the first 5 −
10 pointsabove baseline have been obtained. It is especially important that the modest increase inmonitoring frequency continue during the decline to baseline to ensure that, at least on oneside of the light curve, we have ideal time coverage. Note that extreme high-magnificationevents are already selected for intensive monitoring near peak (Griest & Safizadeh 1998).We suggest that these events receive, in addition, monitoring that is not so intensive butwhich supplements what is normally done at present, to ensure that deviations near baselinewould be detected. Transient periodicity in the wings:
A nearly periodic signal that becomes detectable asthe light curve begins to depart from baseline may be a signature of close-orbit planets.To identify such light curves, checks for periodicity could be made in a sliding window.Windows with a range of sizes should be considered, since the characteristic size L ( α, q ) ofthe region within which perturbations can be detected is not known a priori . Counterpart that could be the lens or lensed source:
If there is a counterpart in acatalog, or else if images of the region reveal evidence for a possible counterpart, it couldbe that the counterpart is the lens or the lensed source. It is necessary to check thatthe association between the position of the event and the possible counterpart is likelyto be real, which can be accomplished with a Monte Carlo simulation (McCandlish & 13 –Di Stefano 2011). Determining whether the counterpart is the lens (making it possible tosearch for nearby planets) or the lensed source (possibly making it easier to measure themagnification, especially if the baseline is bright) can be accomplished through measuringthe blending parameter. Bright baseline:
Whether or not there is an identified counterpart, a bright baseline maysignal that either the lensed source or else the lens itself is bright. In the first case is idealfor the detection of deviations, and smaller telescopes may be able to play an importantrole in monitoring the event. In the latter case, we may have an opportunity to test forthe presence of nearby planets.
Figure 1 demonstrates that planetary systems exhibit a wide range of separations.From the perspective of gravitational lensing, they range from “close” to “wide”. Theintermediate range, bounded by the dashed lines, is sometimes called the “resonant zone”.It is in this range that planet-lens light curves sometimes exhibit caustic crossings, andthis is the range that on which most lensing-event searches have concentrated. The workwe have done in this paper can significantly help with the discovery of just over 1 / α > . . The wide-orbit planets populate the upper portion of Figure 1. Di Stefano & Scalzo(1999b), showed that the probability that an event would “repeat”, with one portion dueto lensing by the star, and another short-duration portion showing evidence of the planet,could be as high as a few percent to about 10% . The type of moderately intensive moni-toring we have suggested for the wings of the light curve to discover close-orbit planets isalso ideally suited to the discovery and study of repeating events. The type of monitor-ing we suggest therefore provides opportunities to discover planets in two orbital ranges.Furthermore, because many wide-orbit planets are likely to be massive enough to produceevents that last a day or more, failure to detect planets is meaningful. If, for example,the Einstein-crossing time for a solar-mass planet is 30 days, then a >
2% deviation willtypically take 180 days; a >
2% deviation for a Neptune-mass planet would take about1 . Searches for close-orbit planets can begin with studies of existing data. The study ofthe archived light curves can provide evidence for planets. Whether or not such evidenceis found, these studies can place limits on the existence of such planets. Although thelimits might only extend across a limited range of values of α and q , they will be new andinteresting.As new data is taken, searches for close-orbit planets can be incorporated withoutmaking major changes. The few changes that will be useful are to conduct searches fortransient periodicity, and to select some events for the moderate increases in monitoringnear baseline that can improve the chances for discovering or placing limits on close-orbitplanets.Ongoing monitoring has already established that events can be identified, and thatbinary and planet-events can be found. If there is a new technical challenge in the studyof close-orbit planets, it is posed by the importance of the low-magnification portion of thelight curves. For close orbit planets we want to measure the value of δ = A − . Unidentifiedblending can interfere with these measurements. Yet, it is very interesting to consider casesin which blending may occur, because nearby stars which serve as lenses may contributelight to the baseline. Measurements of the blending parameter are therefore crucial. It isalso important to be aware of low-level variability in light from the stars along the line ofsight, including the source and, possibly, the lens. Stellar variability is, however, unlikelyto exhibit the form seen in the lower-left panels of Figures 3, 4, and 5.The search for close-orbit and wide-orbit planets can proceed simultaneously, usingthe same observing strategy and data. Fits must make sure to model planets in both typesof orbit. Of particular importance are extreme-magnification events. As pointed out above,we can use these events to search for evidence of close-orbit, resonant-orbit, and wide-orbitplanets occypying the same planetary system.Interestingly enough, there are other near-term opportunities to search for evidence ofclose-orbit planets. One of them is provided by a predicted close passage between the high-proper-motion dwarf star VB 10 and a background star. This event, slated to occur duringthe winter of 2011/2012, was predicted by Lepine & Di Stefano (2011). The signatures ofany planets that may be orbiting VB 10 are presented in Di Stefano et al. (2011). Theform of the prediction is a probability distribution of possible times and distances of closestapproach between VB 10 and the background star. For those cases in which the distanceof closest approach is smaller than approximately 60 mas, close-orbit planets could bedetected. For example, at 60 mas, a planet with an orbital separation of 0 .
012 AU and anorbital period of 1 . Kepler space mission will either discover or place limitson the existence of close-orbit planets.
Kepler samples the light from each of its target starsevery 30 minutes. (Light from a small subset of the targets are sampled every minute.)
Kepler can detect photometric changes of roughly 50 parts per million for a star of twelfthmagnitude. This means that
Kepler can detect low-magnification events and can probethe low-magnification portion of all events. We have shown that there is a high probabilitythat
Kepler will observe ∼ a dozen low-magnification events when either a low-mass objectpasses in front of one of the 150 ,
000 target stars monitored by the mission, or when atarget star passes in front of a background source. In fact, target stars with the highestprobability of participating in lensing events are presently being monitored (PI: Di Stefano)In the future, studies like those needed to discover close-orbit planets will be conductedas part of ongoing monitoring programs. Planned programs, such as KNET, will be ideallysuited to the discovery of close-orbit planets. KNET is an ambitious monitoring programapproved for funding from the South Korean government. Observing from locations atseveral positions in the Southern Hemisphere, KNET will provide continuous coverage ofa large patch of sky, sampling each region with a cadence of 10 minutes.Gravitational lensing is becoming an effective tool for planet discovery. The workpresented here shows that some simple modifications in the way we monitor events havethe potential to increase the discovery rate by extending the range of our searches. Theprocedures we suggest are ideally suited to the discovery of close-orbit planets, and willalso discover wide-orbit planets. Undoubtedly, there will be challenges in implementingthese ideas, as there have been with every method of planet discovery. Within several yearshowever, it should be possible to regularly discover close-orbit planets both near and far. 16 –I would like to thank Ann Esin and James Matthews for conversations and for helpwith the figures. This work was supported in part by NSF under AST-0908878.
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17 –Known Exoplanets -3 -2 -1 0 1-2-1012 0 2 4 0 2 4 6LOG[P (days)] -4 -2LOG[q] 0 1 2 3 4
Fig. 1.— Each point corresponds to a known planetary orbit. In each panel, the valueof α is shown along with a second quantity. From left to right, the second quantity is theplanet’s mass, the stellar mass, the orbital period, the mass ratio M pl /M ∗ , and the distanceto the lens. Planets in orbits with values of α in between the two dashed lines are in the“resonant zone” and can be found by methods already employed. The systems below thelower dashed lines are the close-orbit planets we focus on in this paper. Those above theupper dashed line are wide-orbit systems. These can produce repeating events that willautomatically be discovered by searches for close-orbit planets. 18 – Fig. 2.—
Top panel: log (∆) vs log ( r ) for a set of planetary systems with q = 0 . α. α = 0 .
10 for the right-most curve and increases by 0 .
05 for eachcurve to the left. r is the distance from the center of mass; ∆ is the difference betweenthe maximum and minimum magnification around the ring of radius r . For a point lens,∆ = 0 . Bottom panel:
Normalized distance between small caustic vs log ( q ). The curveis multicolored, with each color corresponding to a different value of α : 0 .
10 (blue); 0 . .
20 (cyan); 0 .
25 (black); 0 .
33 (yellow); 0 .
40 (green). The fact that it is difficult toresolve these curves shows that there is little dependence on α.
19 –
Time /days0 100 200 300 400-3-2-10 Time /days245 246 247-1.8-1.6-1.4-1.2 Einstein Radii-4 -2 0 2 4-6-4-20246
Fig. 3.—
Jupiter-mass planet in orbit with a star of . M ⊙ , α = 1 / . Top panel: light curves. Each light curve corresponds to a different value of the distance of closestapproach: b = 2 / / Bottomleft:
Zoomed-in image of a single deviation.
Bottom right:
Isomagnification contoursassociated with the light curves in the top panel. 20 –
Time /days0 20 40 60 80-1.8-1.7-1.6-1.5-1.4 Time /days58.6 58.8 59 59.2-1.8-1.7-1.6-1.5-1.4 Einstein Radii2.5 2.6 2.7 2.8-0.2-0.100.10.2
Fig. 4.—
Neptune-mass planet in orbit with a star of . M ⊙ , α = 1 / . Top panel: light curves. Each light curve corresponds to a different value of the distance of closestapproach: b = 2 . .
05 in each subsequent curve.
Bottomleft:
Zoomed-in image of a single deviation.
Bottom right:
Isomagnification contoursassociated with the light curves in the top panel. 21 –
Time /days0 20 40 60 80-1.8-1.7-1.6-1.5 Time /days58.6 58.8 59 59.2-1.64-1.62-1.6-1.58-1.56 Einstein Radii2.5 2.6 2.7 2.8-0.2-0.100.10.2
Fig. 5.—
Earth-mass planet in orbit with a star of . M ⊙ , α = 1 / . Top panel: light curves. Each light curve corresponds to a different value of the distance of closestapproach: b = 2 . .
05 in each subsequent curve.
Bottomleft:
Zoomed-in image of a single deviation.