Journal of the Korean Astronomical Society http://dx.doi.org/10.5303/JKAS.2016.49.1.9 49: 9 ∼ 18, 2016 February pISSN: 1225-4614 · eISSN: 2288-890X c 2016. The Korean Astronomical Society. All rights reserved. http://jkas.kas.org
Euclid ASTEROSEISMOLOGY AND KUIPER BELT OBJECTS Andrew Gould1, Daniel Huber2,3,4, and Dennis Stello2,4 1Department of Astronomy Ohio State University, 140 W. 18th Ave., Columbus, OH 43210, USA [email protected] 2School of Physics, University of Sydney, NSW 2006, Australia; dhuber,[email protected] 3SETI Institute, 189 Bernardo Avenue, Mountain View, CA 94043, USA 4Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark Received April 27, 2015; accepted December 20, 2015 Abstract: Euclid, which is primarily a dark-energy/cosmology mission, may have a microlensing component, consisting of perhaps four dedicated one-month campaigns aimed at the Galactic bulge. We show that such a program would yield excellent auxilliary science, including asteroseismology detections for about 100000 giant stars, and detection of about 1000 Kuiper Belt Objects (KBOs), down to 2–2.5 mag below the observed break in the KBO luminosity function at I 26. For the 400 KBOs below the break, Euclid will measure accurate orbits, with fractional period errors∼ . 2.5%. Key words: astrometry — gravitational microlensing — stars: oscillations — Kuiper belt
1. INTRODUCTION 2. EUCLID CHARACTERISTICS In two earlier papers, we pointed out that WFIRST microlensing observations toward the Galactic bulge We adopt Euclid survey characteristics from Penny et would automatically yield a treasure trove of asteroseis- al. (2013), with some slight (and specified) variations. mic (Gould et al. 2015) and Kuiper Belt Object (KBO) In each case, for easy reference, we place the corre- (Gould 2014) data. These papers contained detailed sponding assumed WFIRST parameters in parenthe- analytic calculations that permit relatively easy scaling ses. Mirror diameter D = 1.2m (2.4 m); pixel size p = 0.3′′ (0.11′′), detector size 8k 8k (16k 16k), me- to other missions and experiments. One very relevant × × mission is Euclid, which is presently scheduled to be dian wavelength λ =1.7 µm (1.5 µm), photometric zero launched in 2020. Unlike WFIRST, Euclid does not yet point Hvega = 23.5 (26.1), exposure time 52s (52s), ef- have a microlensing component, but such a component fective background (including read noise, Hvega,sky = −2 − − is being actively discussed. 20.0 arcsec , and dark current) 209 e (341 e ), full 5 5 Based on a most naive assessment, Euclid would pixel well nmax = 10 (10 ), and single read time appear to be much less effective in extracting non- tsingle = 2.6 s (tsingle = 2.6s). In fact, Penny et al. microlensing science from microlensing data than (2013) do not specify a read time, so we use the same WFIRST. Euclid has 1/2 the telescope diameter of value as for WFIRST to simplify the comparison. Also, WFIRST, 2.7 times larger linear pixel scale, 1/4 as Penny et al. (2013) adopt an exposure time of 54 s, but many visits to each target, and 1/2 the angular area we use 52 to again simplify the comparison. Finally, 16 of the survey. Penny et al. (2013) list nmax = 2 , but this appears However, as we show, such naive assessment would be to be in error. In any case, since the detector is very quite wrong since Euclid will be able to detect oscilla- similar to WFIRST, these two numbers should be the tions in giants that are about 0.8 mag brighter than for same. WFIRST, i.e., to about 0.8 mag above the red clump. The most striking of these differences is the 2.6 mag Hence, it will obtain asteroseismic measurements for (factor 11) difference in zero points because only a fac- roughly 100 000 stars. tor 4 is accounted for by the difference in mirror sizes. For KBOs, Euclid benefits by having an optical chan- The two other principal factors are shorter bandpass nel in addition to its primary infrared (IR) channel, (1.4–2.0 µm vs. 1.0–2.0 µm) and lower peak throughput even though it is expected that the optical exposure (47% vs. 77%), with the latter primarily due to the fact time will be 3 times smaller than for the IR. The op- that Euclid employs a dichroic beamsplitter whereas tical observations gain substantially from their smaller WFIRST does not, and (very secondarily) WFIRST has point spread function (PSF) as well as the fact that IR-optimized gold mirrors, which Euclid does not. KBOs (unlike stars) do not suffer extinction. Hence, Euclid will also be a powerful probe of KBOs. In our calculations we first consider simple 52 s expo- sures, but later take account of the fact that five such Corresponding author: A. Gould exposures will be carried out in sequence over 285 s. ∼ 9 10 Gould et al.
3. ASTEROSEISMOLOGY However, whereas this regime applies to stars H < 3.1. Bright-Star Photometry 11.6 for WFIRST (x-intercept of lower curve of mid- dle panel in Figure 1 of Gould et al. 2015), this From Equation (16) of Gould et al. (2015), the frac- break occurs at a brighter value for Euclid. To eval- tional error (statistical) in the log flux F (essentially, uate this offset, we rewrite Equation (3): Npixel = 2 2 magnitude error) is kfNreadtsingle(D p/λ) A(rDp/λ). Hence, the offset (at fixed r) between the break points on the WFIRST and θ σ(ln F ) = [πn (3 Q)]−1/2r−1 ; r unsat Euclid diagrams is max − unsat unsat ≡ p (1) ∆H = 2.5 log [fDλ/p] /[fDλ/p] =2.8 (8) − E H where p is the pixel size, θunsat is the radius of the closest unsaturated pixel (in the full read), nmax is the That is, this boundary occurs at H = 11.6 ∆H = full well of the pixel, 8.8, which is significantly brighter than the majority− of potential asteroseismic targets. However, inspection of −1 −2/3 −1/3 Q = N + 3(2Nread) , (2) read that Figure shows that the same scaling (σ FH ) applies on both sides of the H = 11.6 “boundary”.∝ and N is the number of non-destructive reads. read The reason for this is quite simple. As we consider When comparing WFIRST and Euclid, the most im- fainter source stars F , the saturated region of course portant factor is r . To determine how this scales H unsat continues to decline. Since the Airy profile at large with mirror size, pixel size, exposure time t, throughput − radii scales as A(x) x 3, the radius of this region f, and mean wavelength λ, we adopt a common PSF ∝ scales as r F 1/3. Hence, the number of semi- function of angle θ, A(Dθ/λ), which is scaled such that unsat ∝ H dx2πxA(x) 1. Then, if a total of K photons fall saturated pixels (each contributing nmax to the total ≡ 2/3 onR the telescope aperture, the number falling in a pixel photon counts) scales FH , which implies that the ∝ − centered at θ is fractional error scales F 1/3. This breaks down only ∝ H 2 2 at (or actually, close to) the point that the central pixel npixel = p KfA(Dθ/λ)(D/λ) (3) is unsaturated in a full read (at which point the er- −1/2 Setting npixel = nmax, we derive ror assumes standard FH scaling). That is, in ∝ − the case of Euclid, the F 1/3 scaling applies to about −1 2 2 2 H A [nmaxλ /(KfD p )] H < 14.8 ∆H = 12.0, which is still toward the bright runsat = . (4) pD/λ end of potential− targets.
2 Then noting that K = kfNreadtsingleD , where k is a 3.2. Bright-Star Astrometry constant, we obtain From Gould et al. (2015) A−1[n λ2/(kft N D4p2)] r = max single read (5) p unsat pD/λ σ(θ)= , (9) 3πnmax ln(1.78Nread +0.9) Finally, noting that in the relevant range, a broad-band p in the saturated regime. Hence, using the same param- Airy profile scales A(x) x−3, we find a ratio of Euclid eters (for a single Euclid sub-exposure) (E) to WFIRST (W) unsaturated∝ radii,
1/3 σ(θ)E runsat,E (NreadDλf/p)E =2.73. (10) = 0.42. (6) σ(θ)W runsat,W (NreadDλf/p)W ≃ Then taking account of the fact that Euclid has five In making this evaluation we note that (DE/DW ) = such sub-exposures, we obtain σ(θ)E /σ(θ)W = 1.22, 0.50, (pE/pW ) = 2.73, (λE /λW ) = 1.13, and i.e., Euclid is very similar to WFIRST. 0.4(23.52−26.1) 2 (fE/fW ) = 10 /(DE/DW ) = 0.365. The However, as in the case of photometric errors, the last number may be somewhat surprising. It derives boundary of the regime to which this applies is H < 8.8 mainly from WFIRST’s broader passband (1 µm vs. for Euclid (compared to H < 11.6 for WFIRST). And, 0.6 µm and its gold-plated (so infrared optimized) mir- more importantly, for astrometric errors, the functional ror. The other ratio of factors is form of errors does in fact change beyond this break (see Figure 1 of Gould et al. 2015). 3 QE − =1. (7) The reason for this change in form of astrometric 3 Q − W errors can be understood by essentially the same argu- Therefore, we conclude that in the regime that the cen- ment given for the form of photometric errors in the tral pixel is saturated in a single read, a sequence of 5 previous section. In the regime between saturation of Euclid exposures (requiring about 285 s including read- the central pixel in one read and Nread reads, the re- 1/3 out) yields a factor 1.06 increase in photometric errors gion r < runsat FH dominates the astrometric sig- compared to a single WFIRST exposure (lasting 52 s). nal. For astrometric∝ signals, each pixel contributes to Euclid Asteroseismology and KBOs 11
Table 1 the following assumptions about the Euclid microlens- Fundamental properties and simulation parameters for ing observations. First, they assume an 18 minute ob- Euclid simulations. serving cycle (compared to 15 minutes for WFIRST). Parameter 2437965 2425631 2836038 Second, they assume four 30-day observing campaigns (compared to six 72-day campaigns for WFIRST). Fi- Teff (K) 4356 4568 4775 nally, we adopt a specific “on-off” (bold, normal) sched- log g (cgs) 1.765 2.207 2.460 ule of (30,124,30,335,30,181,30) days. This schedule [Fe/H] (dex) 0.43 −0.10 0.33 has been chosen to be consistent with the Euclid sun- R(R⊙) 24.94 16.44 11.28 exclusion angle and to have two campaigns in each of MH (mag) −3.14 −2.38 −1.60 AH /AKp 0.45 0.45 0.45 the spring and autumn (useful for parallaxes) but is σE (mmag) 0.483 0.685 0.984 otherwise arbitrary. Reference P14 P14 C14 The area under the spectral window function in Fig- ure 1 is about 2.4 times larger than for WFIRST due Numbers in the first line are KIC IDs. References: P14 = Pin- sonneault et al. (2014); C14 = Casagrande et al. (2014). to the fact that the campaigns are a factor 72/30=2.4 times shorter. Nevertheless, the FWHM of this enve- lope is still only about 500 nHz, which is far less than the (S/N)2 as n /r2, and therefore the entire region max the ν 8 µHz for the brightest star shown in Fig- contributes max ure 2. Hence,∼ it is only for extremely bright stars that 2 runsat S rdr 1/3 the width of this envelope will degrade the measure- 2 = ln runsat; runsat FH (11) ment of νmax. N ∝ Z1 r ∝ Figure 2 is qualitatively similar to Figure 4 from Hence, between these two limits (11.6
1.0 Kepler Euclid 108 Kepler 106 104 0.5 102 Teff=4356K 100 log(g)=1.77 ) 0.0 -1 1 10 100 1 10 100 Hz 6
µ 10 Power
2 105 104 103 -0.5 102 WFIRST 101 0 Teff=4568K 10 log(g)=2.21 Euclid 10-1 -1.0 1 10 100 1 10 100
6 -1.0 -0.5 0.0 0.5 1.0 Power Density (ppm 10 Frequency (µHz) 105 104 Figure 1. Spectral window function for a typical Kepler time 103 2 series (top panel, black) and after degrading the time series 10 101 Euclid WFIRST 0 Teff=4775K to a typical duty cycle expected for and 10 log(g)=2.46 (bottom panel, blue, red). Note that while the Euclid peak 10-1 is much broader than for Kepler the power is still contained 1 10 100 1 10 100 µ within ∼ 0.5 µHz, which is substantially narrower than most Frequency ( Hz) spectral features of interest for most stars. Figure 2. Power spectra of Kepler observations (left panels) and simulated Euclid observations (right panels) for three larger amplitude oscillations and smaller photometric red giants in different evolutionary stages: high-luminosity red giant (top panels), 0.8 mag above red clump (middle errors, although for the most luminous giants the fre- panels) and red clump star (bottom panels). Red lines show quency resolution for a typical Euclid observing run will the power spectra smoothed with a Gaussian with a full- be too low to resolve ∆ν. width half-maximum of 2∆ν. Estimated stellar properties are given in the left panels, with a more complete description 3.5. Role of Euclid Parallaxes given in Table 1. In Gould et al. (2015), we argued that WFIRST par- allaxes could help resolve ambiguities in the measure- sess how well WFIRST could detect and characterize ment of ∆ν due to aliasing. That is, from color-surface Kuiper Belt Objects (KBOs), including orbits, bina- brightness relations, one approximately knows the an- rity, and radii. He employed a Cartesian parameteriza- gular radius, which combined with the measured paral- tion, first introduced by Bernstein & Khushalani (2000) lax, gives the physical radius R. By combining R, ν , max to facilitate numerical calculations and KBO recovery and general scaling relations, one can approximately from followup observations, and then further developed predict ∆ν, and so determine which of the alias-peaks by Gould & Yee (2013) to facilitate analytic estimates should be centroided to find a more precise value. of survey properties and their resulting measurements. However, this does not work for Euclid. From Equa- Here we apply these analytic formulae to Euclid. tion (13) and the fact that there are a total of 10000 In contrast to WFIRST, Euclid will observe in two observations, it follows that the parallax errors∼ are channels simultaneously, using an optical/IR dichroic beam-splitter. Because the microlensing targets will be ln 10 σ(π) = 8 µas (H 8.8)+1 (8.8
Kepler Euclid WFIRST because the full duration of the 5 co-added exposures is about 5.5 times longer than the WFIRST exposure. However, due to the shorter duration of the observing window and the fact that this window has one end roughly at quadrature (and the other 30 days 25 toward opposition), the typical relative velocities of the KBO and satellite are only about 5 km s−1, which cor-
Hz) −1
µ responds to 10 µas min , which is still too small to substantially∼ smear out the PSF, even in 5 min ex-
20 posures. ∼ Although the area of the Euclid field is only about
Frequency ( half the size of the WFIRST field, the fact that the 15 campaign duration is only 40% as long together with the very slow mean relative motion of the KBOs (pre- vious paragraph) implies that this is even less of an issue than for WFIRST, which Gould (2014) showed 10 was already quite minor. We therefore ignore it. 1.0 As with WFIRST, we assume that only 90% of the 0.8 observations are usable, due to contamination of the 0.6 0.4 others by bright stars. This balances the opposite im- 0.2 pact of two effects: larger pixels increase the chance 0.0 that a given bright star lands on the central pixel, 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 while brighter Hzero decreases the pool of bright stars Frequency mod 2.57 µHz that can contribute to contamination. We also assume that 10% of the observing time is spent on other bands. Figure 3. Echelle´ diagram for KIC 2425631 (corresponding These will provide important color information but are to middle panel of Figure 2) for Kepler (left) and simulated Euclid (right) data. The abscissa is the frequency offset rel- difficult or impossible to integrate into initial detection ative to the beginning of each order, which are separated algorithms. Hence the total number of epochs per cam- paign is N = 1920, i.e., a factor 3 smaller than by (an adopted) large-frequency spacing ∆ν = 2.57 µHz. cam ∼ Hence, for example, at the frequency of maximum power for WFIRST. νmax ∼ 20 µHz, the ordinate corresponds to the eighth or- der. In the Kepler ´echelle, the three mode degrees l = 1, 2, 0 4.2. Correction to Gould (2014) Formula: from left to right are easily discerned, which is not true of Number of Trials the Euclid ´echelle. However, the Euclid ´echelle does display clear vertical “streaking” which is the signature that the To determine the minimum total SNR required for a adopted ∆ν = 2.57 µHz is the correct one. detection, we must determine the number of trials. In principle, we should be able to simply scale from Equa- tion (18) of Gould (2014). However, this equation con- first note that the Euclid point spread function (PSF) tains a significant error in the “radial velocity” term, is slightly more undersampled than that of WFIRST which must be corrected. We then take this opportu- by a factor (0.30/0.11) (1.2/2.4) (1.5/1.7) = 1.2. nity to rederive the entire formula in order to put it in We therefore adopt an effective× sky× background of 6B, − more general form. where B = 209 e is the effective background in a sin- Our guiding assumption is that the predictions of a gle pixel (compared to 3B for the oversampled limit). given trial must match the orbit of a KBO to within half That is, H = 23.5 2.5 log(6B/52) = 20, i.e., 1.7 sky − a pixel for all epochs. There are six phase space coor- mag brighter than for WFIRST. Then, following Equa- dinates, two for transverse position at the midpoint of tion (3) of Gould (2014), we estimate the astrometric observations, two for proper motion (µ) at this epoch, precision of each measurement as one for distance (r), and one for radial velocity (vr), σpsf 212 mas both at the same epoch. σast = √2 = , (15) 2 SNR SNR The first (position) term implies 4A/θpixel initial po- sitions, where θpixel is the size of a pixel and A is the i.e., exactly twice the value for WFIRST (because the angular area of the region probed. The second (proper mirror is two times smaller). Here, the signal-to-noise 2 motion) term requires π(µmax∆t/θpixel) trials for each ratio (SNR) of 5 co-added consecutive observations is position, where µmax is the maximum search radius (rel- given by ative to a KBO on a circular orbit), and ∆t is the du- SNR = 100.4(Hzero−H), H = 24.8. (16) ration of continuous observations. zero For observations near quadrature, the accelerated Comparing this to Equation (1) of Gould (2014), we see motion of Earth causes a maximum displacement 2 that Hzero is 3.5 mag brighter than for WFIRST. of ∆θ = (1/2)(Ω∆t/2) Π where Π = AU/r and Before continuing, we remark on the issue of smeared Ω = yr/2π. Hence, a total of 2∆θ/θpixel = 2 images, which is potentially more severe for Euclid than (1/4)(Ω∆t) ∆Π/θpixel independent searches must be 14 Gould et al. conducted for each position/proper motion being con- where Ncam is the number of observations during a sidered. Here ∆Π is the range of parallaxes being single-season observing campaign. Hence, the limiting probed. In principle, one must consider that this num- magnitude is H . 26.5. ber may be less than one, in which case of course a minimum of unity must be imposed. In practice, how- 4.3. Euclid Optical Observations of KBOs ever, for experiments with WFIRST and Euclid, this To compare optical with IR observations, we first note number easily exceeds unity. that the field of view is the same, but with pixels that Finally, to evaluate the vr contribution, we should are smaller by a factor 3. This means that the optical consider how two orbits differ in projection if they have data are nearly critically sampled (100 mas pixels and identical θ0, µ0, and Π0, but differ in ∆vr. This is just 180 mas FWHM). The “RIZ” band is centered in I 2 2 band and has a flux zero point of Ivega = 25.1. As ∆vr∆t/2 (1/2)(Ω∆t/2) AU ∆vrΩ AU ∆θ = = (∆t)3 noted above, band-width constraints limit the number r r 16r2 (17) of optical images that can be downloaded (in addition In contrast to the previous term, we actually do need to to the primary IR images) to one per hour (for each of evaluate the conditions under which this term is greater the three fields). Taking into account the read noise of 4.5 e−, a sky than θpixel/2. That is, whether the predicted number −2 of trials background of I = 21.0 arcsec , and the nearly critical 2 sampling, we find the analog of Equation (16) to be 2∆θ ∆vr,maxΩ AU 3 Nvr = = (∆t) 2 0.4(Izero−I) θpixel 8r θpixel SNR = 10 , Izero = 27.4 (22) − ∆v θ 1 r,max pixel for 270 s exposures. Even allowing for the fact that = 37 −1 4kms 110 mas there are only 30% as many I-band observations and − r 2 ∆t 3 that the typical color of a KBO is I H 1, this (18) − ∼ ×40 AU 72 days still represents an improvement of 27.4 24.8 1 + 2.5 log(0.3)=0.3 magnitudes relative to H−band.− Thus, is greater than unity. In Equation (18), we have nor- the I-band observations will overall provide more in- malized this calculation to WFIRST assumptions, in formation than H band, and we will henceforth focus particular the ∆t = 72 day campaigns. We have also primarily on these. We note however that H band data −1 −1 adopted a full range of ∆vr = 4kms (i.e., 2kms ) will be quite useful, both to measure KBO colors and ± for the radial velocity search. Hence, there are two confirm marginal detections, regimes An important benchmark for understanding the rel- −5 2 ative sensitivity of WFIRST and Euclid is that at the A θ ∆µ 13 pixel max KBO luminosity function break (R = 26.5, I = 26.0, Ntry,0 = 4 10 2 ′′ × deg 110 mas 12 /day H = 25.1), we have SNR 2.5 for WFIRST and ∼ ∆Π ∆t 4 SNR 3.6 for Euclid. (19) For∼ fixed field size and ∆t, (and taking account of ×1/40 day the fact that Nvr ,IR 1 the number of trials scales or −6 ≃ 22 2.9 Ntry θpix, i.e., Ntry 10 , a factor 10 higher for Ntry = Ntry,0 Nvr (20) ∝ ∼ × the optical than the IR. This implies that reaching the Before proceeding, we note that, by chance, Equa- theoretical detection limit will require roughly 1025.6 tion (20) is just a factor 2 larger than the result reported floating point operations (FLOPs). While this is 3.7 by Gould (2014) for WFIRST parameters. Since this orders of magnitude lower than WFIRST, it is not triv- is well below the uncertainty in the estimates of com- ially achieved (Gould 2014). For the moment we evalu- putational efficiency a decade from now, the results of ate the detection limit assuming that it can be achieved that paper are essentially unchanged. and then qualify this conclusion further below. Assum- For Euclid, with its shorter (∆t = 30day) cam- ing that 10% of observations are lost to bright stars and paigns and larger (300 mas) pixels, N r 1. Hence, v ≃ then solving Equation (21) yields SNR& 0.39, i.e., a de- Equations (19) and (20) both yield the same result, tection limit of I . 28.4. Considering that the break N = 8 1018. This is 5.6 orders of magnitude try × in the KBO luminosity function is R 26.5 and that smaller than the corresponding number for WFIRST. typically R I 0.45, this is about∼ 1.5 mag below Hence, the challenges posed by limitations of comput- the break. Hence,− ∼Euclid optical observations will be a ing power that were discussed by Gould (2014) are at powerful probe of KBOs. most marginally relevant for Euclid. Therefore, we ig- nore them here. 4.4. Detections Then to find the limiting SNR at which a KBO can The main characteristics of the Euclid optical KBO be detected, we solve Equation (17) from Gould (2014), survey can now be evaluated by comparing to the re- i.e., sults of Gould (2014). The total number detected up N to the break (and also the total number detected per SNR & N −1/2 2 ln try ln N =0.21, (21) cam r SNR − cam magnitude between the break and the faint cutoff) will Euclid Asteroseismology and KBOs 15 be (4/6)/2.1 = 32% smaller than for WFIRST (Fig- For resolved companions, the situation for Euclid is ure 4 from Gould 2014) because there are 4 campaigns essentially identical to WFIRST. That is, the resolu- (rather than 6) and the survey area is a factor 2.1 times tion is the same, and gain in sensitivity from reduction smaller. in trials is nearly the same. Therefore, Euclid can de- Thus, there will be a total of 400 KBOs discovered tect binaries down to I . 29.3. Note that because of that are brighter than the break and 530 per magnitude the relatively small number of trials, this limit is in- fainter than the break. In fact, this distribution is only dependent of whether the computational challenges to known to be flat for about 1.5 mag. Euclid observations reaching the I = 28.4 detection limit can actually be will reach about 2.4 mag below the break, provided that achieved. the computational challenges can be solved. However, Once the companion is detected, its proper motion as we discuss in Section 4.8, even if they cannot be (relative to the primary) can be measured with a pre- solved, this will pull back the magnitude limit by only cision (Equation 21 of Gould 2014) a few tenths. Hence, Euclid will discover at least 800 KBOs down 24 σ 250masyr−1 σ(∆µ)= ast = , (25) to the point that the KBO luminosity function is mea- r N ∆t SNR sured, and perhaps a few hundred beyond that. where σast = 105 mas is the Gaussian width of the PSF. 4.5. Orbital Precision This is a factor 7 larger error than for WFIRST. Con- ∼ To determine the precision of the orbit solutions, we sidering that single-epoch SNR for Euclid is 3.6/2.5 = first note that the PSF is almost exactly the same size 1.44 better than for WFIRST, Equation (22) of Gould (both mirror and observing wavelength are half as big). (2014) becomes ∆µ 83(SNR)1/2η−1/2 where η repre- Hence, Equation (12) from Gould (2014) remains valid: sents the binary separation∼ relative to the Hills-sphere radius. Therefore, a 3-sigma detection of the proper σ(vr) motion requires η . (SNR/4.3)3. Hence, for example, v⊕√Π at the break, η . 0.6. Since for KBOs at the break (di- −3 −1/2 −3 5/2 ameter D 50 km), and for orbits with a 40 AU, the 3.8 10 N ∆t r ∼ ′′ ∼ = × Hills sphere is at roughly 7 , this still leaves plenty of SNR 5600 72d 40 AU room for detections with proper motion measurements. − − 0.15 N 1/2 ∆t 3 r 5/2 Such measurement can be used to statistically constrain = . (23) the masses of the KBOs. However, the main problem SNR 650 30d 40 AU is that most observed KBO companions are at much closer separations, indeed too close to be resolved. Here vr is the instantaneous KBO radial velocity, r is the KBO distance, Π = AU/r, v⊕ is Earth’s orbital Gould (2014) therefore investigated how well these velocity, and N is the number of contributing images. could be detected from center-of-light motion. This re- Gould (2014) argued that because vr was by far the quires that the companion be separated by less than a worst measured KBO Cartesian coordinate, all orbital- pixel (otherwise not unresolved), the period be shorter parameter errors would scale with this number, and in than the duration of observations (otherwise center- particular for the period P (his Equation 13), of-light internal motion cannot be disentangled from center-of-mass motion around the Sun), and high- σ(P ) 3v σ(v ) enough (& 7 sigma) detection to distinguish from noise r r . (24) P ≃ v⊕Π1/2 v⊕Π1/2 spikes. From Equation (23) of Gould (2014) for such detec- Hence, at fixed SNR (per observation), Euclid period tion, we have errors are a factor 40 larger than for WFIRST. The pri- − mary reason for this is the factor 2.4 shorter observing 2 p p f 1 SNR & 7 & 4.85 cl , (26) campaign, which enters as the third power. In addition, r650 f θ θ 0.08 there are about 7 times fewer observations, which enters cl c c as the square root. At the break (I 26), we have from ∼ where θc is the binary projected separation and fcl is Equation (22), SNR 3.6. Following Gould (2014), we the ratio of center-of-light to binary motion, which has ≃1/2 adopt vr 0.2v⊕Π , and derive σ(P )/P 2.5%. a broad peak 0.07 . f . 0.09 (Gould 2014). This ∼ ∼ cl This means that orbital precisions for KBOs below the corresponds to I < 25.6, i.e., almost a half mag above break will be good enough to determine orbital families, the break. but in most cases not good enough to detect detailed The requirement that the period obey Pc < ∆t im- −1 2/3 subtle structures. plies that η = (a/AU) (Pc/yr) = 0.0047. Hence, imposing θ = p (maximum unresolved orbit) yields a 4.6. Binaries c KBO diameter D = pD⊙/η = 145km, corresponding Microlensing-style observations can detect KBOs to I 23.5, i.e., 2.5 mag above the break. through two distinct channels: resolved companions These∼ analytic estimates imply that detection of bi- and unresolved companions detected from center-of- naries from light-centroid motion is much more difficult light motion Gould (2014). than with WFIRST, a conclusion that is confirmed by 16 Gould et al.
4.8. Computational Challenges 25.6 40 As mentioned in Section 4.3, a total of 10 FLOPs would be required to reach the detection limit of I = 28.4. By contrast, Gould (2014) argued that it would be straightforward today to carry out 1023.5 FLOPs per 30 year, and that perhaps in 10 years Moore’s Law might raise this number to 1025. This would leave a shortfall of q = 102.1 or q = 100.6, in the two cases, respec- I=23 tively. If these shortfalls could not be overcome, then according to the argument given in Section 6 of Gould
(light centroid) 20 (2014), this would lead to a cutback of the magnitude 1/2 ) 2 limit by ∆I (1.25/(n + 1) log q, where n =4 or n =7 ∼ ∆χ according to whether Equation (19) or Equation (20) from Section 4.2, above, is used. Since for Euclid op- 10 tical observations n = 7 (Section 4.3), we have 0.3 or SNR=( 0.1 mag in the two cases, respectively. Note that in both cases, the limit would still be beyond the current I=25 limit where the luminosity function is measured (i.e., 0 1.5 mag beyond the break). Hence, even if such com- 0 20 40 60 80 100 putational challenges cannot be overcome, this will not θ c (mas) compromise Euclid’s capability to explore new regimes of KBO parameter space. Figure 4. Signal-to-noise ratio [(∆χ2)1/2] for orbiting KBOs at a range of separations that are less than the Euclid optical 4.9. Comparison to the Deep HST Search pixel size θc < p = 100 mas. The KBO brightness ranges from I = 23 to I = 25 as indicated. Tracks end to the right It is of interest to compare the corrected formulae when the orbital period P = ∆t = 30 days, the duration of presented here with the practical experience of Bern- an observing campaign. Longer period orbits would have stein et al. (2004), who searched for KBOs based on substantially lower signal. The calculations assume fcl = Ncam = 95 Hubble Space Telescope (HST) epochs taken 8% (see Figure 1 of Gould 2014). The region of partial over ∆t = 15days near quadrature, with a field size (broad) sensitivity I < 24 (I < 23) will have only about 25 2 A = 0.019 deg and pixel scale θpixel = 50mas. They (6) detected KBOs, so not many binaries will be detected searched current distances 25 AU
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