Kepler Microlens Planets and Parallaxes
aa r X i v : . [ a s t r o - ph . E P ] J un Kepler
Microlens Planets and Parallaxes
Andrew Gould , Keith Horne ABSTRACT
Kepler ’s quest for other Earths need not end just yet: it remains capable ofcharacterizing cool Earth-mass planets by microlensing, even given its degradedpointing control. If
Kepler were pointed at the Galactic bulge, it could conducta search for microlensing planets that would be virtually non-overlapping withground-based surveys. More important, by combining
Kepler observations withcurrent ground-based surveys, one could measure the “microlens parallax” π E for a large fraction of the known microlensing events. Such parallax measure-ments would yield mass and distance determinations for the great majority ofmicrolensing planets, enabling much more precise study of the planet distribu-tions as functions of planet and host mass, planet-host separation, and Galacticposition (particularly bulge vs. disk). In addition, rare systems (such as planetsorbiting brown dwarfs or black holes) that are presently lost in the noise wouldbe clearly identified. In contrast to Kepler ’s current primary hunting groundof close-in planets, its microlensing planets would be in the cool outer parts ofsolar systems, generally beyond the snow line. The same survey would yield aspectacular catalog of brown-dwarf binaries, probe the stellar mass function in aunique way, and still have plenty of time available for asteroseismology targets.
Subject headings: gravitational lensing: micro — planetary systems
1. Introduction
The
Kepler satellite has found more than 3000 planetary candidates, the overwhelmingmajority of which are real planets (Batalha et al. 2013). To give one example of the newparameter space probed,
Kepler has discovered 231 “Earth-radius” planets (within 25% of Department of Astronomy, Ohio State University, 140 W. 18th Ave., Columbus, OH 43210, USA;[email protected] SUPA, University of St Andrews, School of Physics & Astronomy, North Haugh, St Andrews, KY169SS, UK; [email protected]
Kepler has detected planets only by the transit method, and asa result it is highly biased toward close-in planets. For example, the median period of the“Earth-radius” sample is 5.2 days, and the maximum period is 69 days.Here we propose to apply
Kepler to characterizing much colder planets in the outerparts of their solar systems, using the microlensing technique (Gaudi 2012). We show thatalthough
Kepler is not optimally designed for this task, it can be competitive with existingand under-construction ground-based surveys in terms of finding planets.However, what
Kepler would add that is fundamentally new would be microlens paral-laxes for a large fraction of microlensing events, including almost all of those with planetarysignals (whether detected by
Kepler or from the ground). In the great majority of cases,such parallax measurements would enable determination of the host mass and distance, andthus also the planet mass, which would greatly enhance the value of both groups of planets.The requirements of a
Kepler microlensing survey are well-matched to the limitationson its performance due to loss of pointing stability. In order to be an effective transit-searchtool,
Kepler had to monitor ∼ stars. Given data-transmission constraints, this impliedrelatively long (30 min) integrations on each star, which in turn required high pointing sta-bility. However, to be an effective microlensing-planet tool, it need only observe ∼ stars.Hence the same data-transmission constraints are compatible with much shorter exposures.The photometric requirements of microlensing planet searches are very different from Kepler ’s transit survey. Planetary deviations are typically tens of percent, compared to . . I .
16, comparedto V .
16 for the transit survey. Microlensing events typically last a few weeks to months.They are usually quickly identified from the ground, but
Kepler would have to be notified ofthese identification to conduct its search. Planetary deviations due to Jupiter-mass planetstypically last one day, while those due to Earth-mass planets typically last about one hour.Hence somewhat shorter cadences are needed than
Kepler ’s traditional 30 min in order toget full sensitivity to the lowest-mass planets.The photometric requirements for microlensing parallax measurements are substantiallyless restrictive than for finding planets because the parallax signal extends over the entireevent, not just a few hours or days. This is important: it means that even if the photometricchallenges prove too difficult to find a large number of planets on its own,
Kepler ’s maincontribution of precise characterization of ground-based planets can remain intact. 3 –
2. Observation Strategy
At present, roughly 2000 microlensing events are discovered per year by the OpticalGravitational Lens Experiment (OGLE ) and Microlensing Observations for Astrophysics(MOA ) collaborations. The overwhelming majority of these are found in a region thatcould fit in a single pointing of the 105 deg Kepler camera. Thus, the first element ofthe strategy would be simply to point
Kepler at this field, when permitted by its 55 ◦ Sunexclusion angle. Whenever a new microlensing event was found (from Earth), it would beadded to the list of
Kepler targets. Most events are detected at least several days beforethey do anything interesting, so such “uploads” of new targets could be grouped in batches,if necessary. Microlensing events could also be removed from the list when they returned tobaseline.We note that
Kepler is in a P = 372 . . ◦ yr − and hence is now roughly 1 month (0.5 AU) behind Earth. This is an excellent positionto create a large baseline for “parallactic viewing” while still having a strongly overlapping“bulge season” with Earth.
3. Unique Impact: Microlensing Parallaxes
The observational strategy outlined above would accomplish two aims: measure the“microlens parallax” of a large fraction of events and detect planets in a subset. We willargue below that the planet-finding capability is comparable but not qualitatively supe-rior to ground-based capabilities. Hence, we focus first on what is unique about a
Kepler microlensing survey: parallaxes. π E ? The magnitude of the microlens parallax, π E is simply the lens-source relative parallax, π rel , scaled to the angular Einstein (1936) radius θ E π E = π rel θ E ; θ = κM π rel ; κ ≡ Gc AU = 8 . M ⊙ (1) http://ogle.astrouw.edu.pl/ogle4/ews/ews.html/ https://it019909.massey.ac.nz/moa/ π E /π E = µ /µ . The significance of a parallax measurement is that if θ E is also measured, then one canimmediately derive M and π rel , M = θ E κπ E , π rel = AU D L − AU D S = θ E π E , (2)Since the source distance D S is usually known quite well, measuring π rel immediately givesthe lens distance D L .While in general it is quite difficult to measure θ E , such measurements are almost alwayspossible in planetary lensing events. This is because the source must pass over or near a“caustic” caused by the planet if the planet is to be detected. The lightcurve deviation istherefore a function of ρ ≡ θ ∗ /θ E , where θ ∗ is the angular source size, which means that ρ can almost always be measured from the lightcurve of planetary events. Since θ ∗ can beroutinely measured from the source color and magnitude (Yoo et al. 2004), θ E = θ ∗ /ρ canalso be measured.Hence, microlens parallax is a Rosetta Stone for planetary microlensing events, turningwhat was initially thought to be a purely statistical technique (Gould & Loeb 1992) intoindividual planet-mass and distance measurements. To date, the overwhelming majority of microlens parallax measurements have relied onobserving lightcurve deviations induced by the accelerated motion of Earth during the event(Gould 1992; Alcock et al. 1995; Poindexter et al. 2005). However, because most microlens-ing events are short compared to the time required for Earth to move a radian (yr/2 π ∼
58 d),such “orbital” parallax measurements are quite rare. Another approach is to simultaneouslyobserve the event from two locations on Earth (Hardy & Walker 1995; Holz & Wald 1996),but since the projected Einstein radius ˜ r E ≡ AU /π E is typically several AU, this is onlypractical for extreme magnification events A & routinely return microlens parallaxes is to combine 5 –observations from a satellite at O (AU) from Earth and so enable simultaneous observa-tions from two locations separated by a distance that is comparable to ˜ r E (Refsdal 1966;Dong et al. 2007). Kepler therefore possesses two tremendous advantages for a microlens parallax survey:it is already in solar orbit and it can observe essentially all ongoing microlensing eventssimultaneously.
However, it also faces challenges. Some of these are related to its relatively large pointspread function (PSF), which we will discuss below. But one challenge is rooted in the natureof space-based parallax measurements: degeneracy. As already noted by Refsdal (1966) anddiscussed more thoroughly by Gould (1994), space-based parallax measurements are subjectto a four-fold discrete degeneracy. Basically, Earth and satellite see the same event, butdisplaced in the Einstein ring, and so having different peak times t and different impactparameters u . The microlens parallax, is then given essentially by π E = AU D ⊥ , sat (∆ τ, ∆ β ); ∆ τ ≡ t , sat − t , ⊕ t E ; ∆ β ≡ u , sat − u , ⊕ , (3)where D ⊥ , sat is the Earth-satellite separation projection onto the plane of the sky and t E is the Einstein timescale. The problem is that while t is uniquely determined from thelightcurve, u is a signed quantity whose magnitude is measured but not its sign. Thus, π E can take on four values depending on the signs of u as seen from Earth and the satellite.However, since the mass depends only on the magnitude of π E , only a two-fold degeneracyis really of major interest. That is, do u , sat and u , ⊕ have the same or opposite signs? Or,equivalently: is the source seen projected on the same or opposite side of the lens as seenfrom the two observatories? See Figure 1, and also Figures 1 and 2 from Gould (1994).Gould (1995) showed that this degeneracy could be broken because the timescales ofthe events as seen from Earth and the satellite are slightly different, and this difference isa function of ∆ β . Gaudi & Gould (1997) then investigated how well this degeneracy couldbe broken for events seen toward the Galactic Bulge. Their assumptions were far moreconservative than those likely to apply to Kepler observations. First, they considered anarrow-angle pointed mission (rather than a wide-angle survey), in which the observationsof each target would be limited to a relatively few epochs, whereas
Kepler observations wouldbe continuous. Second, at the time it was believed that the source flux in the space-filtercould not be accurately determined from the ground-based lightcurve, whereas subsequently 6 –Yee et al. (2012) have shown that this indeed is possible to at least 1% precision. See alsoGould (2013) and Yee (2013).Undoubtedly, there will be microlensing events discovered that are so faint that theparallax degeneracy will not be broken. However, few planets are likely to be found in suchfaint events. Moreover, depending on the geometry of the event, it is sometimes not necessaryto actually break the degeneracy to derive good mass estimates (e.g., if | ∆ τ | ≫ | ∆ β | ).
4. Planetary Science with
Kepler
Microlens Parallaxes
At present, most microlensing planet detections return θ E and hence the product M π rel = θ /κ , but not the mass and distance separately. Hence, these quantities are estimated onlystatistically for most events. The estimates make use of Galactic models together with vari-ous pieces of information, such as the geocentric lens-source relative proper motion µ = θ E /t E and upper limits on the lens flux from blended light. But generally these estimates are ac-curate to only a factor of two, and of course can be radically incorrect in cases of unusual orunexpected systems. In particular, there is only one planet out of about 30 detected to datethat is known to be in the Galactic bulge with good confidence, even though the majority oflenses are in the bulge. Hence, it is very difficult to disentangle the distributions of planetsas functions of controlling properties, such as planet mass, host mass, distance from host,and Galactic position.In one fell swoop, Kepler could resolve all of these uncertainties, and it could do so forthe several dozen planets per year that will be discovered with current and in-constructionexperiments. For example, standard core-accretion theory predicts a dip in the planet massfunction between Neptunes and Jupiters. By sharpening the mass resolution of microlensplanets,
Kepler could directly test this prediction for ice and gas giants found beyond thesnow line, which is presumably their birth place.Not only would this increased precision be of direct use in better understanding theplanets that microlensing is discovering, it would also put them “on the same playing field”as the planets, mostly much closer to their hosts, discovered by other techniques. Kepler
Cold Planets
In addition to measuring microlens parallaxes,
Kepler observations will probe a virtuallyindependent set of microlensing planets from those detected from the ground. This is becauseit is displaced from Earth by D ⊥ , sat / ˜ r E in the Einstein ring, which will typically be of order 7 –10%. Since planetary perturbations are usually much smaller than this, planets detectedfrom Earth will generally not be detected by Kepler and vice-versa.Here, we estimate the general competitiveness of
Kepler relative to the Korea Microlens-ing Telescope Network (KMTNet, Poteet et al. 2012) which is the largest ground-based mi-crolensing experiment currently under construction. KMTNet will have three 1.6m tele-scopes, each with a 4 deg field of view, located in Chile, South Africa, and Australia. Like Kepler , therefore, it is in principle capable of near-continuous coverage for the fraction ofthe year when the Sun is well away from the bulge. KMTNet will cycle through four fields,observing each for 2 min out of 10. Considering weather at these sites, it will have a com-bined duty cycle of perhaps 2/3. In addition,
Kepler has a “white-light” response comparedto I -band filters that are needed from the ground. Taking account of the increase in bothsignal and noise implies a factor 1.6 advantage. Altogether, these factors give Kepler anadvantage of a factor 12.However,
Kepler has disadvantages as well, and these are overall stronger. First, itsaperture is smaller by a factor 1 . . Most important, its PSF is much larger. At best, theFWHM ∼ . ′′ , whereas average KMTNet seeing is likely to be ∼ . ′′ . Since almost allphotometry is likely to be below sky in either case, these two disadvantages combine to afactor (1 . × . / . ∼
17. Finally, the problems posed by field drift are difficult to estimatewithout detailed simulations. The exposures can be made short enough that this drift doesnot affect individual images, but the undersampled PSF, in conditions of crowded bulge fieldsis likely to increase the photometric noise beyond the above naive calculation. Thus,
Kepler will find fewer planets within the 16 deg probed by KMTNet, which contain the richestplanet hunting ground. By the same token, of course, a Kepler microlens planet searchwould fall far short of one by
WFIRST (Green et al. 2012; Spergel et al. 2013). However,
Kepler will also find planets in outlying regions, which are being surveyed by OGLE andMOA at lower cadence. And, as emphasized above, virtually all the planets that it does findwill be undetected from the ground.Nevertheless, this calculation shows that the planets found by
Kepler are not reasonenough, by themselves, for it to do a microlensing survey. Moreover, since it will be lookingonly at events found by others, it will be useless for finding free-floating planets (FFP), whichare a unique capability of microlensing (Sumi et al. 2011). Rather, its principal value is to It will, however, be useful for vetting the main contaminant of the FFP signal, stellar microlensingevents whose timescales are exceptionally short due to small π rel despite high mass M . If FFP events can bealerted to Kepler within ∼ Kepler because their parallaxes will be small π E = ( π rel /κM ) / . However, if they are due to planets, then Kepler will see no event at all because thelarge parallax puts the source well outside the Einstein ring from
Kepler ’s perspective
6. Other Microlensing Applications
Microlensing surveys are also a powerful probe of binaries, in particular low-mass bi-naries that are difficult or impossible to detect by other methods. For example, Choi et al.(2013) discovered two brown dwarf binaries, which obeyed the binding-energy floor foundpreviously using standard brown-dwarf search techniques, but at much lower mass and tighterseparation. That these binaries yielded mass measurements (and so could even be recognizedas brown dwarfs, not stars) was only due to the fact that they were unusually nearby (fewkpc) and so had large, easily measurable parallaxes. Like planetary events, binary eventsroutinely yield θ E , so that Kepler microlens parallaxes would give masses and distances forall binaries, and so sift out these brown-dwarf binaries, which are otherwise generally unrec-ognizable. Moreover, binaries, in contrast to planets, would often be detected by both
Kepler and ground observatories, which would provide detailed information on their orbits. Finally,while the point-lens events would not generally yield θ E (and so masses), their mass functioncould be studied statistically from a Kepler microlens parallax survey (Han & Gould 1995).
7. Non-microlensing Applications
The number of microlensing targets to be observed is not large, at most 2000 in a season,and not all must be observed all season. The exposure times must be fairly short becauseof
Kepler ’s degraded pointing stability, but it is unlikely that they need to be 100 timesshorter than
Kepler ’s traditional 30 min exposures. From a microlensing standpoint, thereare no drivers for exposures shorter than about 5 minutes. Thus, it is likely that microlensingtargets will absorb only a small fraction of the available data-transmission capability. Otherobjects, such as bright asteroseismology targets could therefore be observed. In particular,since short exposures are needed due to stability problems, one could target bright dwarfs,which have higher-frequency oscillations than the giant-star targets on which
Kepler hasconcentrated to date.We thank Scott Gaudi and Jennifer Yee for stimulating discussions. Work by AG wassupported by NSF grant AST 1103471 and NASA grant NNX12AB99G. KH is supportedby a Royal Society Leverhulme Trust Research Fellowship. 9 –
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This preprint was prepared with the AAS L A TEX macros v5.2.
11 –Fig. 1.— Illustration of four-fold degeneracy derived from comparison of
Kepler and groundbased lightcurves. Upper panel shows two possible trajectories of the source relative to thelens for each of
Kepler (red) and Earth (blue) observatories. Each set would give rise to thesame point-lens lightcurve in the lower panel (same colors), leading to an ambiguity in theEarth-