WFIRST Ultra-Precise Astrometry I: Kuiper Belt Objects
aa r X i v : . [ a s t r o - ph . E P ] A ug Journal of the Korean Astronomical Society http://dx.doi.org/10.5303/JKAS.2014.00.0.1 : 1 ∼
99, 2014 March pISSN: 1225-4614 / eISSN: 2288-890X c (cid:13) http://jkas.kas.org WFIRST U LTRA -P RECISE A STROMETRY
I: K
UIPER B ELT O BJECTS
Andrew Gould
Department of Astronomy Ohio State University, 140 W. 18th Ave., Columbus, OH 43210, USA [email protected]
Received —; accepted —
Abstract:
I show that the
WFIRST microlensing survey will enable detection and precision orbit de-termination of Kuiper Belt Objects (KBOs) down to H vega = 28 . ∼
17 deg .Typical fractional period errors will be ∼ . × . H − . with similar errors in other parametersfor roughly 5000 KBOs. Binary companions to detected KBOs can be detected to even fainter limits, H vega = 29, corresponding to R ∼ . D ∼ H ∼
23, binarycompanions can be found with separations down to 10 mas. This will provide an unprecedented probeof orbital resonance and KBO mass measurements. More than a thousand stellar occultations by KBOscan be combined to determine the mean size as a function of KBO magnitude down to H ∼
25. Currentground-based microlensing surveys can make a significant start on finding and characterizing KBOs usingexisting and soon-to-be-acquired data.
Key words: astrometry – Kuiper belt – gravitational microlensing
1. I
NTRODUCTION
Kuiper Belt Objects (KBOs) provide an extraordi-nary probe of the origin and history of the Solar Sys-tem. When Pluto was discovered by Clyde Tombaugh(Slipher, 1930a,b) and was then found to be in a 3:2resonance with Neptune, it was hardly guessed that itwas only the largest of a vast class of such objects. Sub-sequent discovery of KBOs in 2:1 resonance, in variouskinematic and composition subclasses, of binary KBOs,and of a break in the size distribution at R ∼ . WFIRST microlensing survey will, without any ad-justment, yield a KBO survey that is both substantiallydeeper and three orders of magnitude wider and moreprecise than existing deep KBO surveys based on
HubbleSpace Telescope (HST) data.
2. KBO
S IN M ICROLENSING D ATA
Microlensing surveys are conducted toward the denseststar fields on the sky and therefore would appear atfirst sight to be the worst place to carry out a searchfor KBOs. In particular, confusion in the identificationof moving objects is known to be an extremely strongfunction of the density of the stellar background (e.g.,Monet et al. 2003).Surprisingly, microlensing fields are actually the bestplace to conduct deep searches for moving objects,KBOs in particular. First, contrary to naive expec-tation, there is no problem of confusion at all. Mi-crolensing searches are conducted on a series of differ-
Corresponding author:
Andrew Gould ence images , which are constructed by first forming adeep (essentially noiseless) reference image from manyindividual images, aligning each image geometrically tothe reference image, convolving the reference image tothe point spread function (PSF) of each image, align-ing each image photometrically to the reference image,and finally subtracting each image from the referenceimage (Alard & Lupton, 1998). The result is a virtuallyflat image with essentially all stars removed and onlyphoton and detector noise left. The only exceptions arestars (or other objects) that have changed relative tothe reference image, either changed their brightness orchanged their position, and a very small number of arti-facts due to difficulties in subtracting extremely brightand/or saturated stars. Hence, microlensing fields arenot crowded at all: they are essentially blank, muchmore so than high-latitude fields.Second, the noise properties of every pixel (as a func-tion of both position on the sky and position withinthe detector array) are understood essentially perfectly.This is because there are hundreds, thousands, or eventens of thousands of images of the same field. In thecase of ground-based surveys, these encompass the fullrange of observing conditions, while for space-based sur-veys, the images are taken under constant conditions (orrather conditions that vary much less than is relevant tofaint objects).Third, microlensing surveys carry out hundreds orthousands of observations during a single season (dur-ing which KBOs remain mostly within the microlensingfields). While each individual exposure is not particu-larly deep, the fact that the images are essentially blankmeans it is straightforward to combine many observa-tions to detect faint KBOs. In particular, while theKBO will occasionally “land” at the position of a rel-atively bright star, the precise knowledge of high noise1
Andrew Gould at this location (derived from photometric deviations onhundreds of other images – see previous paragraph) willautomatically “suppress” this observation within the en-semble. That is, KBO searches will automatically bedominated by the highest signal-to-noise ratio (SNR)observations.Fourth, because the Galactic Center is only about 6 ◦ south of the ecliptic, microlensing surveys automaticallyprobe regions of the sky on and near the ecliptic.Finally, the high density of stellar background is ac-tually an advantage because it gives rise to occultationsthat can be used to measure the size distribution.To date, no one has yet suggested, let alone explored,the possibility of KBO searches in microlensing fields.Here, I focus on the potential of future space based sur-veys, but the same principle applies to ground-baseddata. In particular, the Optical Gravitational Lens Ex-periment (OGLE) survey already has more than 4 yearsof data covering about 5 deg with a cadence of roughly30 observations per night during peak microlensing sea-son, with typical seeing slightly more that 1 ′′ , whilethe new Korean Microlensing Telescope Network (KMT-Net) survey is expected to cover 16 deg with a cadenceof about 120 observations per night with only slightlyworse seeing (counting only Chile and South Africa tele-scopes).While no KBO searches have been conducted us-ing microlensing data, OGLE did search for KBOsin the 2500 deg of the southern Galactic plane(Shephard et al., 2011). Because there were only threeexposures, each 180 s (for the overwhelming majorityof fields), the survey reached only R = 21 .
6. Highlyillustrative is their “rediscovery” of Pluto at roughly( l, b ) = (13 , −
2) in a dense stellar field (their Figure 3),which becomes essentially blank (except for Pluto) inthe difference image (their Figure 4).
3. A
STROMETRY F ROM
A M
ICROLENSING S URVEY
A space-based microlensing survey will almost automat-ically return high-quality astrometric data. I discussthis potential within the framework of the proposed
WFIRST survey, but the same principles could be ap-plied to any mission of this type. I adopt the followingparameters for the microlensing component of the pro-posed
WFIRST mission.1) 10 contiguous, 0 .
28 deg Galactic bulge fields2) six ∆ t = 72 day continuous “campaigns” , each cen-tered at quadrature (March or September)3) 15 min cycle time consisting of ten 52s exposures4) 90% of exposures in broad H band, 10% in a nar-rower band, e.g., Y
5) 2.4m telescope, θ pix = 110 mas pixels,6) one photon per second at H vega = 26 . B = 341 total “counts” per pixel in read noise, zodi-acal light and dark current per 52s integrationThe diffraction limited point spread function (PSF)has FWHM= 275 mas, implying an equivalent Gaussianwidth σ psf = FWHM / .
35 = 75 mas. Due to the slightlyundersampled PSF, I assume a total background light of 9 B , i.e., about 1.5 times larger than the 4 π ( σ psf /p ) B appropriate to the oversampled limit. This leads to anequivalent “sky” of H sky = 26 . − . ∗ log(9 B/
52) =21 .
7. Since this paper will concern only sources well be-low this limit, I restrict further consideration to “belowsky” sources. For these, the signal-to-noise ratio (SNR)is given bySNR = 10 . H zero − H ) ; H zero = 26 . N cam = 5600 (2)images, distributed roughly uniformly in time over eachcampaign. I assume that each of these has astrometricprecision σ ast = √ σ psf SNR = 106 masSNR . (3)The images will be dithered and the KBOs will moverelative to the stellar background, so the astrometricand photometric errors of the N cam = 5600 images willbe essentially uncorrelated. Because of the very largenumber of images, the central limit theorem thereforeimplies that Gaussian statistics strictly apply.
4. W
ORLD C OORDINATE S YSTEM
In order to fit KBO orbits over the ∼ ◦ trajectories thatthey traverse during a 72-day campaign, the WFIRST astrometric frame must be calibrated to at least the pre-cision of the KBO measurements. Since GAIA will mea-sure positions, parallaxes, and proper motions for sev-eral million stars in the
WFIRST microlensing fields, itappears at first sight that this issue will be taken care of“automatically”. While this ultimately proves to be thecase, there are some subtleties, which I now address.The central difficulty is that
WFIRST will “satu-rate” (see below) at H vega ∼
14, while the majorityof GAIA stars will be bulge sub-giants and giants with( V − H ) > . E ( V − H ) ∼ .
5. For such stars to avoid saturationin
WFIRST , they must be
G >
19 in the GAIA band(which is near V ). While GAIA will observe stars to sig-nificantly fainter magnitudes over most of the sky, it willhave a substantially brighter limit in the bulge due todata-rate limitations. Hence, it is not immediately ob-vious that the WFIRST frame can be tied to the GAIAframe.The best frame-tying stars are foreground G dwarfs,with ( V − H ) ∼ . M V ∼
5. I assume A V =0 . − and A V /A H = 6. Such stars reach the WFIRST saturation limit at D ∼ . H ∼
14 and V ∼ .
2, while at D ∼ . H ∼
15 and V ∼ .
5. These
BOs from WFIRST Astrometry G band magnitudes 16 . < G < . ranging from (32 , ,
22 yr − ) µ as to(54 , ,
39 yr − ) µ as.To set up a reference frame, all we are really con-cerned with is how well the positions of these stars willbe known roughly T = 10 yr after the midpoint of theGAIA mission when WFIRST is launched. (
WFIRST proper motions of these overlap stars will be measuredfar better than those of GAIA, so we are concerned withoverlap at the beginning of the
WFIRST mission, notits midpoint.) Hence, the positions will be known to aprecision of 200–400 µ as.The surface density of such stars on the sky is N = n ( D − D ) / − , where n = 0 .
01 pc − is the number density of G dwarfs at the mean distance.These overlap stars must be used for two types ofalignment. First, the 10 WFIRST fields must be alignedwith each other. Second, the “pixel frame” of thecamera must be calibrated to an externally determinedangular scale. Since each frame has 0 . × ∼ µ as / √ ∼ µ as. For purposes of internal align-ment, all the overlap stars observed in the 10 fields canbe used, implying a density of ∼ − = (18 ′′ ) − .At what point do each of these two calibration is-sues become the limiting factor? Each KBO spendsroughly 25 days (or 1800 observations) inside a field,which implies that a 9 µ as calibration error becomes anissue when the individual astrometric error falls below9 µ as √ µ as. This occurs at SNR = 280, i.e., H = 20, far brighter than any KBO of practical inter-est. Similarly, the KBO remains in the neighborhoodof a calibration star for 18 ′′ , corresponding to about 25observations, so calibration would become an issue atSNR = 60 corresponding to H = 21 .
7, still much toobright to be of concern.I note that
WFIRST microlensing fields will bedithered, so that 40,000 images of each field will per-mit high-precision calibration of the pixel scale on the18 ′′ scale of the typical separations between calibratingstars.Finally, I return to the question of the WFIRST sat-uration limit. The arrays go non-linear at about halfthe full well of 100,000 photo-electrons. In the broad H filter no more than 30% of photons land in a sin-gle pixel. The pixels are read out every 2.6 s. Thesenumbers lead to a conservative saturation estimate of H = 26 . − . / (0 . ∗ .
6) = 14 .
1. The ques-tion of what is “saturated” depends strongly on the ap-plication. If one is interested in precise individual pho-tometric measurements, then probably several reads arerequired, not just the single read in the above calcula-tion. However, if one is interested solely in determin-ing the astrometric center of the PSF to, say, 1% ofa pixel, it is doubtful whether it is actually necessarythat the central pixel be unsaturated, let alone in thelinear regime, since the position can be centroided fromother pixels. Further, one could restrict attention to the subset of observations that land near the corner of fourpixels. Thus, it is quite plausible that the saturationlimit for astrometry could go 1 mag or more brighterthan H = 14. However, the above calculation showsthat for present purposes, such a detailed investigationis unnecessary.
5. P
RECISION OF O RBIT D ETERMINATION
Assuming that a KBO is identified in the data, how wellcan its orbit be measured? To answer this question, wemust first ask how long it will stay within the field ofview. The first point is that at semi-major a ∼
40 AU,and hence period P = ( a/ AU) / yr ∼
250 yr, a KBOwill move about 1.4 deg from one year to the next,and hence will typically not return to the microlensingfields the next year. On the other hand, since the fieldsare observed at quadrature when Earth is acceleratingtransverse to the line of sight at A ⊕ = 4 π AU yr − ∼ . − day − while the KBO will be moving of or-der v ⊥ ∼ − , the net projected relative motion ofEarth against the KBO will be∆ x ∼ sin (cid:18) π (∆ t/ v ⊥ / AU)yr (cid:19) AU ∼ .
71 AU , (4)which corresponds to 1.0 deg. Hence, of order 60% ofthe KBOs will remain in the field for the duration of thecampaign. I will initially restrict attention to this sub-group and reserve discussion regarding the remainder toSection 10.To estimate the precision of the orbital parameters, Iapproximate Earth as being in a circular orbit and ap-proximate the KBO physical motion during the periodof observation as uniform. The deviations from both as-sumptions are slight and, what is more important, deter-ministic with respect to the adopted parameters. For ex-ample, the acceleration of the KBO is given directly byits distance and position on the sky. Thus, making theseassumptions only slightly changes, but vastly simplifiesthe “trial functions” and hence renders tractable theerror estimates while not significantly impacting theseestimates. I follow Gould & Yee (2013) in making theinitial estimates in Cartesian phase-space coordinates(instantaneous positions and velocities) rather than thetraditional orbital invariants. Of course, actual fits todata will use Kepler invariants, but the Cartesian ap-proach is more closely matched to short timescale ob-servations and therefore facilitates both deeper under-standing and simpler results. The implications for Ke-pler invariants are then easily derived.The KBO then has motion x ( t ) = x + v t , describedby six parameters ( x , v ) = ( r , x ⊥ , v r , v ⊥ ). I adopt themidpoint of the campaign as the origin of time. Forconvenience, these six parameters can be re-expressedas (Π , θ , ν ⊥ , ν r ), θ ≡ x r ; ν ⊥ ≡ v ⊥ Ω r ; Π ≡ AU r ; ν r ≡ v r Ω r . (5)where Ω = 2 π yr − is Earth’s orbital frequency. The WFIRST fields are only a few degrees from the ecliptic,
Andrew Gould and for simplicity I assume that the KBO is directly onthe ecliptic.The first four of these parameters in Equation (5) areessentially direct observables, i.e., the position and in-stantaneous (normalized) proper motion of the KBO atthe zero point of time. The last two pose the main chal-lenge. Since these are derived entirely from the motionof the KBO within the ecliptic plane, I restrict atten-tion to these two dimensions (radial and 1-D transverse).Then the equation for the angular position θ ( t ) is θ ( t ) = x + v ⊥ t − AU(cos Ω t − r + v r t − AU sin Ω t = θ + ν ⊥ Ω t + 2Π sin (Ω t/ ν r Ω t − Π sin Ω t . (6)Since there are four parameters to be determined, weshould expand to third order in time θ ( t ) = X i =0 a i (Ω t ) i (7)where a = θ a = ν ⊥ + θ Z a = 0 .
5Π + ν ⊥ Z + θ Z a = − . ν r + 0 . + Π ν ⊥ − ν ⊥ ν r + ν r ν ⊥ + θ ( Z − Π /
6) (8)and Z ≡ Π − ν r . To a good approximation the fourcoefficients are well represented by their leading terms a = θ a → ν ⊥ a → . a → − . ν r . (9)For a uniform set of N observations over time ∆ t , thecovariance matrix for these four coefficients is given by(e.g., Gould 2004), c ij = σ N cam (Ω∆ t ) − ( i + j ) ˜ c ij ˜ c ij = / −
15 00 75 0 − −
15 0 180 00 −
420 0 2800 . (10)Therefore, the errors in the three angular variables ofinterest (i.e., excluding θ ) are σ ( ν ⊥ )Π / = r N (Ω∆ t ) − Π − / σ ast ,σ (Π)Π = r N (Ω∆ t ) − Π − σ ast ,σ ( ν r )Π / = r N (Ω∆ t ) − Π − / σ ast , (11)where in each case I have normalized to a relevantphysical scale. For typical parameters (Ω∆ t ∼ . ∼ / > ν r completely dominate. Trans-lating to physical variables, we obtain σ ( v ⊥ ) v ⊕ Π / = r N (Ω∆ t ) − Π − / σ ast → . × − SNR (cid:18) N (cid:19) − / (cid:18) ∆ t
72 d (cid:19) − (cid:18) r
40 AU (cid:19) / ,σ ( r ) r = r N (Ω∆ t ) − Π − σ ast → . × − SNR (cid:18) N (cid:19) − / (cid:18) ∆ t
72 d (cid:19) − (cid:18) r
40 AU (cid:19) ,σ ( v r ) v ⊕ Π / = r N (Ω∆ t ) − Π − / σ ast → . × − SNR (cid:18) N (cid:19) − / (cid:18) ∆ t
72 d (cid:19) − (cid:18) r
40 AU (cid:19) / . (12)These results imply that the orbit-parameter error ellip-soid is essentially a 1-dimensional structure. In Carte-sian space, this one dimension is associated with a singleparameter: v r . After transforming to Kepler coordi-nates, all parameters inherit this error in v r but in ahighly correlated way. For example, for roughly circularorbits, the period error σ ( P ) is σ ( P ) P = 32 σ ( a ) a ≃ σ (KE)KE ≃ v r v ⊕ Π / σ ( v r ) v ⊕ Π / , (13)where KE is the kinetic energy. Hence, for typical values v r ∼ . v ⊕ Π / , the fractional period error is somewhatsmaller than the last expression in Equation (12). Inthe next section, I will show that the theoretical limitfor finding KBOs is near SNR ∼ /
7. Thus, even at thislimit, the period precision is of the order of 1.5%. Atthe break in the KBO luminosity function, R ∼ .
5, soroughly H ∼ . ∼ .
5, the period error is σ ( P ) /P ∼ .
6. F
INDING
KBO
S IN THE D ATA
WFIRST may well be in geosynchronous orbit. In prin-ciple, this would add information to the measurementsof r and v r . However, as I show below, this added infor-mation plays an insignificant role except in the marginsof parameter space and was therefore ignored in Sec-tion 5. A geosynchronous orbit would also somewhat When making comparisons to optical measurements, I adopt R − H = 1 .
4, which is about 0.3 mag redder than the Sun.Doressoundiram et al. (2008) (Figures 2 and 3) show that thereis a clustering of KBOs near ( V − I, J − H ) ∼ (0 . , . ∼ (0 . , . V − I ). My adopted value R − H = 1 .
4, which is meant only to give an indication of morecommonly used optical magnitudes, is representative of thesepopulations. Note, however, at least some KBOs have optical-infrared colors very similar to the Sun. For example Orcus has V − J = 1 . ± .
04, compared to ( V − J ) ⊙ = 1 . BOs from WFIRST Astrometry
WFIRST field has N pix = 2 . / (110 mas) =3 . × pixels. Thus, for KBOs withSNR > ∼ r N pix SNR = 6 . , (14)it is possible to comfortably identify KBO candidateswithout fear of massive contamination by noise spikesusing a simple two-dimensional (2-D) search over theimage. At this limit, there may be some contamination,but this could easily be vetted by examining successiveimages. There are two points to make about Equa-tion (14). First, it assumes Gaussian statistics. Thismay not be valid for the case of a 2-D search. How-ever, Equation (14) primarily serves as an entry pointto the much larger (4-D, 5-D, and 6-D) searches that Idescribe below, for which Gaussian statistics are valid.I therefore ignore this complication. Second, the SNRappears on both sides of the equation, meaning that theequation must be solved self-consistently. This poses noactual difficulties, since it appears inside a rather largelogarithm factor on the rhs, but does call for an explicitremark.Next, I consider 4-D searches over position (2-D, asabove) and proper motion (2-D). I consider a search ina “proper motion” circle µ = 12 ′′ day − (relative to aKBO in circular motion at r = 40 AU. By conducting a4-D search, I am implicitly assuming that the other twoCartesian phase-space coordinates (Π and ν r ) are “notimportant”. Explicitly, this assumption means that theparallax differences among the KBOs being searchedlead to angular displacements of < < Π = 0 .
03. The acceleration ofEarth then leads to a differential pixel displacement of∆ θ Π = 12 (Ω t ) Π = 8 θ pix (cid:18) ∆ t day (cid:19) (15)due to parallax. Thus, a 4-D search is restricted to∆ t < µ ∆ t and hence a total number N µ = π (cid:18) µ ∆ tθ pix (cid:19) = 5300 (cid:18) ∆ t (cid:19) (16)of searches at each of N pix = 3 × pixels, for a totalof N try = N pix N µ = 1 . × searches. In nine hours,there are approximately N im = 30 images. Hence thisyields a SNR threshold ofSNR > ∼ N − / r N try SNR − ln N im = 1 . . (17) To dig to lower SNR, one must probe over longer du-rations, which requires 5-D or 6-D searches. To findthe boundary, I adopt a radial-velocity search range∆ ν r = 1 /
300 corresponding to ± − at r = 40 AU.Thus the radial velocity becomes important at ∆ t ∼ θ pix / (∆ v r ΠΩ) ∼ t ) for proper motions, (∆ t ) for parallax,and (∆ t ) for radial velocity), implies a search total of N try = 7 × (cid:18) ∆ t day (cid:19) , (18)implying a maximum search total for ∆ t = 72 day of N try = 1 . × . Applying Equation (17), with N im =5600, yields a threshold SNR > ∼ .
15, and so a theoreticallimit of H vega = 28 .
2, or roughly R ∼ .
6. Recall fromEquation (13) that even such extreme below-sky KBOswould have orbital parameter errors of order 1.5%.However, reaching this theoretical limit will be no pic-nic. One could convolve all the images with the PSF(e.g., Shao et al. 2014; Gould 1996), so that each searchwould require only ∼ N im floating point operations,and thus a total of ∼ × operations. This shouldbe compared to the ∼ floating point operations persecond (FLOPS) of a current graphics processing unit(GPU). One might imagine assigning 10 GPUs to thistask for a year, but this would only enable 3 × op-erations, which is a factor q ∼ . × short of whatappears to be needed.I will argue immediately below that this computa-tional shortfall can probably be bridged by a combina-tion of several factors. However, it is useful to ask howan arbitrary shortfall q would impact the depth of thesurvey. From Equation (18) the number of trials scales N try ∝ (∆ t ) , while the number of computations foreach trial is linear in ∆ t . Hence, a shortfall factor q can be compensated by reducing ∆ t by a factor q / =8( q/ . × ) / . Naively, this leads to an increase inthe SNR limit by a factor q / = 2 . q/ . × ) / .In fact, taking account of the impact on the logarith-mic factor due to the smaller number of trials in Equa-tion (17), the actual degradation is a factor 2.6, i.e., alimit SNR = 0 .
38, corresponding to
H < .
1. Notethat while the KBO is found using a restricted subset ofthe data, the full data set can still be used to estimateorbital parameters, so that the estimates of precisiongiven in Section 5 remain valid.For the survey to reach its maximum potential (i.e., 1mag deeper than the above limit) requires either greatercomputing power or better search algorithms. Sincethe data will not be available for at least a decade,we should fold in a “Moore’s Law” factor of 30 (as-suming doubling time of 2 years), but this is still only ∼ operations. Since the number of operationsscales ∝ (∆ t ) , this seems to permit analysis of dataintervals ∆ t < ∼
16 days, so only reaching SNR > ∼ .
32 andlimiting magnitude H vega = 27 . Andrew Gould
For example, one could begin by restricting the searchto ∆ t = 16 days as above, but initially cull out tra-jectories with ∆ χ >
28. This would capture exactlyhalf of all ultimately recoverable KBOs (i.e., those withSNR > . h ∆ χ i = 28, whileat the same time suffering noise-spike contamination of“only” ∼ exp( − / / √ ∼ − . Now, of course,this would still result in ∼ noise spikes, but thesecould be vetted fairly efficiently, as follows. From Equa-tion (11), σ (Π) = 4 × − and σ ( µ ) = 11 mas day − . Infact, it is easily shown that the proper-motion error inthe direction perpendicular to the ecliptic is smaller by p /
75 = 0 .
4. Therefore, even allowing for a 3 σ rangefor these two quantities, the total number of searchesrequired for each such “preliminary candidate” is only ∼ . If the procedure were repeated on 4 indepen-dent subsamples, it would recover 1 − − ∼
94% of allwith SNR > .
15. That is, almost full recovery with ∼ × FLOPS rather than ∼ × requiredfor a brute-force search. That is, this algorithmic im-provement, by itself, pushes down the magnitude limitby ∆ H ∼ (2 . /
12) log(1000) ∼ . > . H vega < . R < .
6, withperiod errors σ ( P ) /P < ∼ . H vega < . R < .
5, with period errors σ ( P ) /P < ∼ . In the above treatment, the emphasis was on findingextremely faint KBOs, with SNR substantially belowunity in individual exposures. To this end, it was nec-essary to restrict the search space to orbital parametersthat are in some “expected” range, partly to limit thefrequency of noise spikes, but mainly to make the searchcomputationally tractable. However, it will also be pos-sible to relax essentially all limits on orbital parametersdown to H ∼ .
1, which is at the detection thresholdSNR ∼ . < . ∼ H ∼
26, i.e., about 1 magbelow the break. If this search found significant num- bers of objects in unexpected (e.g., retrograde) orbits,then further searches could be fine-tuned to find fainterKBOs in similar orbits.
7. B
INARY C OMPANIONS AND M ASS M EASUREMENTS
Regardless of the exact limit achievable for an ab ini-tio search for KBOs, it is possible to reliably identifybinary companions to all those that are found downto H vega ∼
29, significantly fainter than the theoreti-cal limit for isolated KBOs. Detection of such binarycompanions will lead to mass estimates and mass mea-surements of the parent KBO.The reason that the search for companions can godeeper than the search for primaries is that the searchspace is smaller. The first task is therefore to quantifythe size of the search space.I parameterize the semi-major axis a c , and hence thecharacteristic angular separation of the companion by θ c ≡ a c a = η (cid:18) MM ⊙ (cid:19) / = η (cid:18) ρρ ⊙ (cid:19) / DD ⊙ ≃ η DD ⊙ (19)where D and ρ are the primary KBO effective diameterand density respectively, and where I have approximated( ρ/ρ ⊙ ) / ∼
1. To be bound (within the Hill sphere), η < ∼
1. This sets a strict upper limit on the separation.In fact, to remain bound over the lifetime of the solarsystem, the companion must be substantially closer, sothis limit is conservative and is therefore robust againstsomewhat denser or less reflective KBOs. Hence, θ c ∼ η × . ′′ ( D/
10 km) is constrained to be within a fewarcsec near the magnitude limit . The relative propermotion of the primary and companion are then of order∆ µ ∼ η − / θ c ΩΠ / ∼ η − / D
10 km (cid:18) a
40 AU (cid:19) − / (20)That is, near the detection limit (for primaries), onecan search for companions simply by looking for non-moving objects (i.e., those that move much less than1 pixel relative to the primary during a 72-day cam-paign) within a few arcsec of the primary, i.e., ∼ trials for each primary. If we estimate that there will be ∼ KBO-primary detections, then SNR > ∼ .
07 is re-quired to avoid noise-spike contamination. This impliesa flux limit H vega ∼ .
0, plausibly corresponding to adiameter D ∼ . σ (∆ µ ) = r N σ ast ∆ t →
35 mas yr − SNR . (21) When converting from magnitudes to diameters, I adopt analbedo of 0.04 in R band and a “typical” distance of 40 AU,which yields 11 km at R = 29 . H vega = 28 . BOs from WFIRST Astrometry D = 29 km(SNR) / and a = 40 AU, we expect∆ µ ∼
100 (SNR) / η − / mas yr − . (22)This implies, very roughly, that such proper motions canbe detected for η < ∼ (SNR) , i.e., for all bound compan-ions at SNR > ∼ ∼ − , corresponding to about0 . ◦ yr − . Thus, one could use WFIRST itself (in itssurvey mode) to make brief (e.g., one day) surveys ofthe fields to which the KBOs were drifting one to sev-eral years after and/or before the discovery campaign inorder to better characterize the orbits. The utility andcharacteristics of such observations could be much bet-ter assessed after logging the discoveries from the firstcampaign.
Most KBO binaries detected to date have compan-ions within ∼ R < ∼
24) primaries in current samples(Doressoundiram et al., 2008). Many of these compan-ions are also quite close, with a median separation nearthe
WFIRST pixel size p = 110 mas.There are two methods of detecting such close com-panions that are of relatively comparably brightness:“orbiting” centroid of light and extended images.There are three requirements to detect center of lightmotion: First, the actual separation must obey θ c < ∼ p .Otherwise, given the undersampled PSF, the compan-ion would be directly detectable. Second, the companionmust complete a large fraction of an orbit. Otherwise,the centroid of light motion will simply be absorbed intothe KBOs orbital parameters with respect to the Sun.To be moderately conservative, and for simplicity, I in-terpret this requirement as completing one orbit during∆ t = 72 days. Third, the signal must be sufficientlyhigh to distinguish centroid motion from noise spikes.Figure 1 shows that the fractional displacement of thecenters of light and mass (assuming same density andalbedo) peaks broadly 7% < ∼ f cl < ∼
9% for 0 . < ∆mag < .
2. Assuming that the rms projected amplitude is √ σ sig-nal is required for detection, this implies a minimumSNR for each of the 5600 measurements ofSNR > ∼ r pf cl θ c > ∼ . pθ c (cid:18) f cl . (cid:19) − , (23)which corresponds to H < . R < ∼ . ∆ mag F r a c t i ona l C en t e r o f L i gh t O ff s e t ( % ) f cl Figure 1.
Offset of binary center of light from center ofmass, as a fraction of physical separation under the as-sumptions that the two components have equal density andequal albedo. There is a broad peak 0 . < f cl < . . < ∆mag < . From the definition of η , the boundary P c =∆ t = 72 days implies η = ( a/ AU) − ( P c / yr) / =0 . a/
40 AU) − . Hence, imposing θ c = p inEquation (19) at this period yields D = pD ⊙ /η =88 km( a/
40 AU), which corresponds to H ∼ . R ∼ .
1) at a ∼
40 AU.These two calculations show that there is considerableparameter space for detection of this effect. In Figure 2,I show the SNR for light centroid motion as a functionof binary separation θ c for a range of KBO brightnessesfrom H = 23 to H = 25. The curves are displayed onlyfor periods P < ∆ t = 72 days and so “cut off” at theright for the fainter KBO tracks. Note in particular thatat H = 23 (roughly R = 24 .
4, so almost completely en-compassing the region of present studies), the separationthreshold is θ c = 11 mas,It is important to remark that this method is criti-cally dependent on excellent overall WFIRST astrome-try. For example, the ∼ H = 23 KBOs at the detection limit in Figure 2 wouldnot be detectable without the excellent world coordinatesystem described in Section 4.For truly equal-mass (and equally reflecting) KBOs,the center of mass and center of light will be identical,implying identically zero offset between mass and light.For small differences, f cl = 0 . Andrew Gould θ c (mas) S NR = ( ∆ χ / ( li gh t c en t r o i d ) Figure 2.
Signal-to-noise ratio [(∆ χ ) / ] for orbiting KBOsat a range of separations that are less than the WFIRSTpixel size θ c < p = 110 mas. The KBO brightness rangesfrom H vega = 23 to H vega = 25 as indicated. Tracks endto the right when the orbital period P = ∆ t = 72 days,the duration of an observing campaign. Longer period or-bits would have substantially lower signal. The calculationsassume f cl = 8% (see Fig. 1) and make a variety of assump-tions listed in the text, including D ( H vega = 24 .
6) = 58 km, a = 40 AU, and rms projected separation equal to θ c / √ for the KBOs in the magnitude range shown in Figure 2will be fairly deep. For example, at H vega = 25, thecombined image will have SNR = 200 and will be re-solved at subpixel scales because it is composed of 5600dithered images.
8. KBO S
IZE E STIMATES FROM O CCULTATIONS
One of the major advantages of KBO searches in densemicrolensing fields is the large number of occultationsthat are automatically observed. These occultations canyield statistical information on the relation between ef-fective diameter and reflected light, i.e., the albedo. Inprinciple, these occultations might give rise to astromet-ric noise, but I show that this effect is negligible.
As I will show below, in contrast to the general astromet-ric measurements, which are below sky ( H vega < . H vega < (21 ,
22) is (0 . , . p − (i.e.,“per pixel”). I assume that sources can be reasonablyphotometered down to H vega , ∗ = 21, keeping in mindthat the “exact” location of the source (to a small frac-tion of a pixel) is known from the occultation itself to-gether with the known KBO orbit. This precise locationof the occulted star (together with subpixel resolutionfrom 40,000 images) makes it much easier to disentanglethe occulted star from significant blends.The transverse velocity of the KBO in the Earth framewill generally be dominated by reflex motion of Earth,which will have an rms value of v ∼
10 km s − duringthe 72 days of observations near quadrature. I will adoptthis as a typical value in my initial, simplified treatment.I begin by examining the case of KBOs with H = 24 . D ∼
58 km, and hence a maximum self-crossing timeof
D/v ∼ . H ∗ = 21, the (above-sky) photo-metric error is ∼ . / . ∼ σ level. This im-mediately raises the question of what is the thresholdat which we should say that an occultation has been“detected” in the face of (negative) noise spikes? Toevaluate this, we must first estimate the effective num-ber of “trials”. I conservatively estimate that H < . × π (0 . <
1% We mustconsider all 6900 observations (including Y -band and,of course, those landing near bright stars). Hence, therewill be a total of ∼
70 “trials” per KBO. As I will showshortly, we expect only about one real occultation fromthese 70 trials. This means that a 3 σ cut would yielda ∼
20% false positive rate. This would be acceptableif one carried out a very careful statistical study but tobe conservative, I adopt a 4 σ threshold, for which the(false positive)/(real occultation) rate is about 0.6%.Ignoring this threshold for the moment, the rate ofoccultations per KBO is N oc = N obs n Dvt exp a = 0 . N obs n . p − D
58 km v
10 km s − t exp
52 s (cid:18) a
40 AU (cid:19) − , (24)where n is the surface density of H ∗ <
21 stars. In
BOs from WFIRST Astrometry σ , while the adopted threshold 4 σ . This means thatexposures that begin more than half-way through an oc-cultation will fall below the threshold, but these will beexactly compensated by the ones that end during theoccultation but less than half way through it. Moregenerally, such edge effects will not exactly cancel andmust be taken into account. However, for the presentcase, Equation (24) does not require adjustment. Inaddition, KBOs that transit chords that are less thanhalf the diameter will fall below the detection thresh-old. This corresponds to 1 − p − (4 / ∼
14% inthe present case (approximating the projected form asa circular disk), but will vary for other parameters.In particular, this means that at H = 24 .
6, there areabout 0.5 occultations per KBO. As I discuss in Sec-tion 12, this implies that among the several thousandKBOs discovered, there will be over one thousand oc-cultations.Because the occultation time is short compared tothe exposure time, the flux decrement (combined withocculted-star flux estimate, and the known instanta-neous KBO proper motion and distance) directly yieldsthe chord length. The frequency of such occultationsgives one an estimate of the mean diameter (at fixedabsolute magnitude) while the peak of the chord-lengthdistribution gives another. Formally, these are respec-tively transverse and parallel to the direction of motion,but these are expected to be statistically identical. If thetypical detections are 8 σ , then the second measure willhave 8 times smaller formal error than the first, but theexistence of two independent measurements will providea useful check on the systematics.The KBO break at H = 25 . D , i.e., as 10 − . H . More-over, at about 1 mag fainter than the break, the fluxdecrement drops below the detection threshold (for fidu-cial parameters), so one is restricted to brighter (hencerarer) occulted stars and/or KBOs observed at timesthat they are moving more slowly relative to Earththan average. On the other hand, KBOs that are 1mag brighter than the break are 4 times less common(Bernstein et al., 2004). This is compensated by the fac-tor ∼ . The first point to note is that occulted stars are alignedwith the geometric center of the KBO much more closelythan the astrometric precision of individual measure-ments. Consider, for example, a KBO at 40 AU and H ∼ . D ∼
60 km.To be occulted, the star must be aligned to within D/ a ∼ ∼
25 mas. Hence, the only impacton measurement is the loss of net flux from the KBO dueto a “hole” caused by the lost light of the backgroundstar. Moreover, it is only the absolute value of the fluxchange at the KBO position that enters the astrometricprecision. That is, it is just as easy to centroid a “hole”as a “bump”, provided it is recognized that there is hole(i.e., that an occultation is taking place). Hence, it isreally only occultations within about a half magnitudeof the KBO brightness that have an adverse impact onastrometry.To evaluate the practical impact, let us consider twocases, one at ∼ H ∼ D ∼
75 km) and the other near the limit H ∼
28 ( D ∼
12 km). Since the KBOs are observed near quadrature,I adopt transverse relative velocities of v = 10 km s − ,which implies that the KBOs near the break have a self-crossing time of about 7 seconds, i.e., ∼ / t exp = 52 s. Hence, for an occultation toreduce the flux by an amount similar to the KBO bright-ness requires that the star has H ∗ ∼
22. The number ofstars within a half magnitude of this value is similar tothe total number brighter than it, which led (above) toan estimate of less than one occultation per KBO.If we now consider KBOs near the limit (i.e., 40 timesfainter), the duration of occultation is 40 / ∼ .
9. E
FFECTS OF G EOSYNCHRONOUS O RBIT If WFIRST is in geosynchronous orbit, this will havealmost no effect on the calculations in this paper. How-ever, it will have some modest benefits for the followupobservations proposed in the previous section.Geosynchronous orbit induces diurnal parallax of am-plitude Π geo = ǫ Π where ǫ ∼ / σ (Π) = r N σ ast ǫ (25)where N is the number of observations in some timethat is at least one day. Equating this to the sec-ond expression in Equation (11), one finds that orbitalparallax overtakes geosynchronous parallax at (∆ t ) Π =360 / ǫ / Ω − ∼ . t ) v r = (35 / / (∆ t ) Π ∼ . ∼ ǫ Π ∼ . ′′ .This implies an absolute minimum of about 20 pixels0 Andrew Gould f Q Q exit Q enter Figure 3.
Degradation factor Q for the error in the radial ve-locity measurement (by far the worst measured phase-spacecoordinate) as a function of f , the fraction of a campaignthat is spent outside (inside) the microlensing field by a KBOthat exits (enters) during the campaign. For low f , Q exit ismodest for those exiting, and this is compensated at high f by those entering. Only about 40% of all KBOs that areinitially in the field exit during a campaign, so the net ef-fect of exits/entrances is modest. Abscissa is f because thedistribution of field area is uniform in this quantity. in the parallax “direction” of the search space, contraryto the assumption of the “position and proper motion”search described at the beginning of Section 6. However,this simplified search was actually presented only fordidactic purposes. The real searches described furtheralong in that section already have a much larger par-allax footprint. Hence, diurnal parallax due to geosyn-chronous orbit has essentially no impact either on theprecision of measurement or the difficulty of searchingfor KBOs.
10. E
DGE E FFECTS
Up to this point, I have assumed that the KBOs remainin the field for the entire 72 day campaign. However,KBOs that are initially near the ecliptic East (West)edge of the field for a Spring (Autumn) campaign willmove off the field as Earth approaches the equinox andwill return into the field later on, whereas other KBOsthat initially lie just beyond the ecliptic West (East)edge will enter the field and then leave it. Now, as shownin Section 5, the majority of KBOs do in fact remain inthe field for the full 72-day campaign, and so to zerothorder one could in principle just ignore these edge ef-fects. However, here I show that these effects actuallyplay only a small role even at first order. Since the apparent motion of the KBOs is dominatedby reflex motion of Earth, I calculate the time spentwithin the field as though this were the only cause. Thenthe time spent inside (and outside) the chip is symmet-ric about the midpoint of the campaign. I first considerthe KBOs that begin within the field and parameterizethe time spent out of the field by f , i.e., they spend atime f ∆ t outside the field. I focus on the precision ofthe radial velocity measurement because it is by far theweakest of the six phase space coordinates. Repeatingthe integral that led to Equation (10) but excluding ob-servations during the time spent outside the field, f ∆ t ,yields Q ( f ) ≡ c (full) c (partial)= 1 − (25 / f (1 − . f + f ) + f − f . (26)This function declines monotonically with f , but there isa compensating effect of KBOs entering the other side ofthe field and spending time f ∆ t within the field. Sincethese entering KBOs are observed continuously, the cor-responding calculation is trivial, Q enter ( f ) = f . . (27)Finally, I note that one must account for the fact thatthe distribution of KBOs leaving (or entering) the fieldfor a time f ∆ t is not uniform in f but rather in dis-tance from the edge of the field at the midpoint of thecampaign, which scales ∝ f . That is, 75% of all KBOsthat leave (or enter) the field do so for more than halfthe campaign.To visualize these effects, I plot Q exit and Q enter ver-sus f in Figure 3. When taking account of both ef-fects, the maximum degradation factor is Q ∼ . f ∼ .
57. This factor is modest given the hugerange of KBO brightness being probed. Moreover, it iscompensated by the fact that twice as many KBOs areprobed at these distances from the edge of the field. Inbrief, the total area of the fields in all six campaigns,6 × . ∼ is a reasonable estimate of theeffective area of the KBO survey.
11. KBO L
OST AND F OUND
Much of the science that can be extracted
WFIRST
KBOs will be derived directly from the
WFIRST datathemselves. This includes the distribution of KBOs asa function of orbital parameters, colors, size, binarity,etc. However, there are a number of applications thatcould require re-observing a detected KBO one or manyyears later. For example, as mentioned in Section 7,one might want to obtain late-time observations of abinary companion in order to better estimate the cor-responding primary’s mass. As another example, onemight want to measure a late-time position of a KBOthat has a moderate-precision period in order to pre-cisely determine whether it was actually on, or simplynear a resonance.
BOs from WFIRST Astrometry
WFIRST data. For illustration, I focus onthose at the detection limit, for which I have estimatedperiod errors of σ ( P ) /P ∼ . a ∼
40 AU with 1.5%period error will lie somewhere along a “well-definedarc”, with a 1 σ position error along that arc of σ ( θ ) ∼ . / / ) × ◦ = 13 ′ . Considering that a con-servative search might look within ± . σ , this implies asearch along an arc of ∼ ◦ . On the other hand, as dis-cussed below Equation (11), the deviations orthogonalto this track are at least 300 times smaller than along it,implying a search width < ∼ ′′ . Given that the surfacedensity of KBOs at this limit is only ∼
100 deg − mag − ,there is only a small probability that there will be evenone other KBO of similar magnitude in the search zone.Because these KBOs are well below sky, re-detection willnecessarily require several epochs and therefore will au-tomatically return a proper-motion measurement thatwill almost certainly distinguish between the (possible)candidates in this zone.Brighter KBOs will, of course, be much easier to find.For example, at the break in the KBO luminosity func-tion ( H ∼ . . ′ × . ′′ .
12. E
XPECTATIONS B ASED ON P REVIOUS
KBOS
URVEYS
The
WFIRST microlensing survey will, without anymodification, probe KBOs down to H vega ∼ . ∼
17 deg and yield orbits with period pre-cision of 1.5% at the magnitude limit (and much betterat brighter magnitudes). For my adopted conversion R − H = 1 .
4, this corresponds to R ∼ .
6. How doesthis compare to previous deep surveys?Bernstein et al. (2004) searched 0 .
019 deg down to R ≤
29 (strictly, m ≤ .
2) using 22 ks exposureswith ACS on the
HST . They found three new objects.For the two of these with m >
28, they obtainedonly crude ( ∼ m > . m = 29, implying that the distri-bution is flat (or falling) beyond R = 28 .
5. Combiningtheir results with previous work at brighter magnitudes,they fit a relation that can be expressed as, d NdR d
Ω = 100mag deg . R − . ( R ≤ .
5) (28)and then a roughly flat distribution ∼
100 mag − deg − for 26 . < R < . ∼
70 deg − . Based onthis estimate, we can expect that WFIRST would detect4500–6500 KBOs. The Bernstein et al. (2004) relation(Equation (28)) scaled by 17 deg is shown in Figure 4. H (KBO) l og ( N u m be r / M ag )
22 24 26 280123 KBOsOccultations
Figure 4.
Predicted number of KBO detections (solid) andKBO occultations (bold) per magnitude that can be ex-tracted from the
WFIRST microlensing survey. The detec-tions are derived by multiplying the Bernstein et al. (2004)KBO luminosity function by 17 deg . The curve is dashedfor H > . H ∗ <
21 stars that canbe detected with at least 4 σ significance. The dashed part of the curve indicates an extension intothe unmeasured regime.
In Section 8, I presented a simplified calculation of theKBO occultation rate, in order to illustrate the basicphysics. In Figure 4, I show the results of a more de-tailed calculation including all the elements outlined inSection 8. In particular, I integrate over the full rangeof Earth-KBO relative velocities, which is mainly drivenby the changing reflex motion of Earth but also includesa small component due to intrinsic KBO motion. I con-tinue to demand 4 σ detections and consider only occul-tations of sources H ≤
21. The principal results fromthe simplified calculations are all confirmed.One point to further note is that at the bright end ofthe distribution shown in Figure 4 ( H = 21) there areabout four occultations per KBO. Although this appliesto only a half dozen objects per magnitude, it does holdout the hope that some shape information can be ex-tracted, particularly if high-resolution followup imagingcan determine the precise source location and thereforethe precise impact parameter of the occultation.2 Andrew Gould
Statistics on KBO companions are available primarilyfor relatively bright primaries R < ∼
24, corresponding to H < ∼ .
6. These have a median separation of ∼
100 mas,and are primarily of near-equal brightness, with a sub-stantial majority roughly uniformly distributed over1 > ∆mag >
0. About 22% of classical and 5% of otherKBOs have such companions (Doressoundiram et al.,2008). Essentially all analogs of these companions willbe found by
WFIRST for the bright KBOs in its field.If there are similar companions down to H = 24, thenthe total number of such binaries will be about 200. Asshown by Figure 2, at fainter magnitudes the close bi-naries become less accessible and then inaccessible for H >
25. Moreover, there is essentially no informationon the frequency of companions at these magnitudes.Hence, while the unexplored parameter space is fairlylarge, there is no reliable way to estimate the number ofcompanions in these regimes.
13. A
PPLICATION TO G ROUND -B ASED M ICROLENSING S URVEYS
There are several ground-based microlensing surveysthat could in principle be searched for KBOs, includ-ing the ongoing OGLE-IV and MOA-II surveys, as wellas the KMTNet survey, which is about to begin. HereI describe some relatively low-effort “entry points” intothese data sets and briefly sketch extensions that wouldprobe much deeper. The “entry point” searches could becarried out using a single night of data and would yieldof order a dozen KBOs. The extensions could reachwithin ∼ R ∼ .
5, thus multiplying this number several fold.The calculations below are based on the summariesprovided by Henderson et al. (2014) of the telescope,detector, and observing characteristics of the OGLEand KMTNet surveys. In particular I adopt a photo-electron rate ˙ γ = 4 . D/ . s − at I = 22, where D is the diameter of mirror, an ambient backgroundof I = 18 . − , and an effective PSF area ofΩ back = 1 . × π (FWHM) / ln 256. I also assume thatthe below-sky errors are 1.3 times larger than the photonnoise. As noted by Henderson et al. (2014), the I = 18 . ,with a cadence of 3 hr − , with exposures of about 100 s,using a 1.3m telescope. The fields can be observed for10 hours per night (so 30 observations) for about onemonth centered on the summer solstice. I assume thatone of these nights is clear, with low moonlight, and verygood seeing of FWHM ∼ ′′ , and that 90% of the ob-servations are not seriously affected by above-sky stars,cosmic rays, or pixel defects. Each (below-sky) obser-vation then has SNR = 10 . . − I ) . The observationstake place at opposition rather than quadrature (as for WFIRST ), which means one must consider a proper mo-tion rectangle of (30 km s − /
30 AU) × (3 km s − /
30 AU), which for 10 hours of observations and 1 ′′ seeing im-plies 300 searches per resolution element or a total of2 × searches. This requires a total SNR = 7,which in 30 × . I = 22 . R = 22 .
7. Thereare roughly 0.4 such KBOs per square degree, implyingthat a few would be expected in the 5 deg high-cadenceOGLE field. However, this could be repeated for eachof 5 years of OGLE-IV archival data, yielding about 10KBOs.KMTNet will have three 1.6m telescopes, two of whichwill be at excellent and somewhat overlapping sites inChile and South Africa. Its cadence at each will be6 hr − , with 120 s exposures over 16 deg . Combin-ing these and adopting a slightly worse mean seeingof 1 . ′′ (to account for the greater difficulty of coor-dinating observations from two sites), yields a simi-lar SNR = 10 . . − I ) . Then, taking account of thefour times greater number of exposures, the limit of de-tectability is I ∼ . R = 23 .
5. Thiswould yield roughly 1 KBOs per square degree or a totalof about 15.Once identified, these KBOs would yield excellent or-bits because microlensing fields cover about 100 deg over many years, albeit at lower cadence.Over a whole season, KMTNet would observe anygiven KBO roughly 10,000 times from Chile and SouthAfrica, of which 1/4 would be in good seeing, goodtransparency, and with low Moon background. As with WFIRST , the search space would be much larger thanfor a simple one-night search, so I adopt a similar∆ χ = 115 threshold. This leads to a detectability limitof I = 24 .
3, or R ∼ .
8, i.e., about 100 KBOs. Notethat in contrast to
WFIRST detections, one need notbe extremely rigorous about eliminating noise spikes atthe detection phase because these will be automaticallyvetted when the KBO is tracked to additional seasonsto measure its orbit.
14. C
ONCLUSIONS
Space-based microlensing surveys are an extremely pow-erful probe of KBOs basically because microlensing mo-tivates very high cadence observations over long timebaselines and fairly wide fields that happen by chanceto lie near the ecliptic. The very large number of imagesallows one to construct essentially noiseless (comparedto the individual images) templates, and so constructessentially blank images from the “crowded” fields viaimage subtraction.In particular, the
WFIRST microlensing survey, with-out any modification, can yield 4500-6500 KBOs downto H vega = 28 .
2. The last magnitude of such a searchrequires algorithmic and/or hardware development tocarry out the computations in a timely manner. How-ever, the more restricted search to H vega = 27 . BOs from WFIRST Astrometry
WFIRST , the period errors scale as σ ( P ) /P ∼ . × . H vega − H break ) , where H break = 25 . H vega ∼
29, regardless ofthe limit of the primary search. The limit is deeper be-cause the search space is smaller, implying fewer noisespikes. These binaries can provide statistical mass in-formation, or if followed up by additional observations,individual mass measurements.Binary companions with separations down to 0.1 pix-els (11 mas) can be found for roughly equal (but notexactly equal) masses for primaries H ≤
23 from theoffset between centers of mass and light, and for largersubpixel separations down to H ∼
25. Exactly (orvery nearly) equal-mass binaries at sub-pixel separa-tions can be detected from image elongation. Analogsto essentially all binaries currently being found ( R < ∼ θ c > ∼ . ′′ ) will be found by WFIRST , but it will alsoprobe a huge parameter space of binary companions thathas not yet been explored.A side benefit of the fact that microlensing searchesare carried out in the most crowded fields (prior to imagesubtraction) is the high probability of occultations. Onaverage, each KBO at the break will occult 0.4 stars with H ∗ <
21 (so reliably detected in the deep drizzled image)and with at least 4 σ detections. Over 1000 occultationsof detected KBOs will enable measurement of the KBOalbedo as functions of orbital properties and absolutemagnitude.Finally, using the same techniques outlined in thispaper, it should be possible to find roughly 100 KBOsusing current and soon-to-be-initiated ground-based mi-crolensing surveys. A CKNOWLEDGMENTS