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Osculating Versus Intersecting Circles In Journal of the Korean Astronomical Society https://doi.org/10.5303/JKAS.2019.52.4.121 52: 121 ∼ 131, 2019 August pISSN: 1225-4614 · eISSN: 2288-890X Published under Creative Commons license CC BY-SA 4.0 http://jkas.kas.org O SCULATING VERSUS INTERSECTING CIRCLES IN SPACE-BASED MICROLENS PARALLAX DEGENERACIES Andrew Gould1,2 1Max-Planck-Institute for Astronomy, K¨onigstuhl 17, 69117 Heidelberg, Germany 2Department of Astronomy Ohio State University, 140 W. 18th Ave., Columbus, OH 43210, USA [email protected] Received May 16, 2019; accepted July 20, 2019 Abstract: I investigate the origin of arc degeneracies in satellite microlens parallax πE measurements with only late time data, e.g., t > t0 + tE as seen from the satellite. I show that these are due to partial overlap of a series of osculating, exactly circular, degeneracies in the πE plane, each from a single measurement. In events with somewhat earlier data, these long arcs break up into two arclets, or (with even earlier data) two points, because these earlier measurements give rise to intersecting rather than osculating circles. The two arclets (or points) then constitute one pair of degeneracies in the well-known four-fold degeneracy of space-based microlens parallax. Using this framework of intersecting circles, I show that next-generation microlens satellite experiments could yield good πE determinations with only about five measurements per event, i.e., about 30 observations per day to monitor 1500 events per year. This could plausibly be done with a small (hence cheap, in the spirit of Gould & Yee 2012) satellite telescope, e.g., 20 cm. Key words: gravitational microlensing 1. INTRODUCTION and D⊥ is the two dimensional (2-D) vector offset from In his original paper on space-based microlens parallax Earth to the satellite projected on the sky (approxi- measurements, Refsdal (1966) already noted that they mated as a constant during the observations). The first were subject to a discrete four-fold degeneracy. Two ob- component is then along this direction and the second servatories, one on Earth and one on a satellite, would is perpendicular to it. The four-fold degeneracy arises each see a single-lens single-source (1L1S) microlens- from the fact that only the magnitude (but not the sign) ing event, characterized by three Paczy´nski (1986) pa- of u0 can generally be inferred from the light curve. See Figure 1 from Gould (1994). rameters (t0, u0, tE), but these parameters would differ due to their different viewpoints. Here t0 is the time The great majority of subsequent theoretical work of maximum magnification, u0 is the impact parame- on space-based microlens parallax (and it degenera- ter normalized to the Einstein radius θE, and tE is the cies) took place within the context of events for which Einstein timescale, there were reasonably complete light-curve measure- ments from both Earth and the satellite, so that in θE 2 tE ≡ ; θE = κMπrel, (1) particular it was possible to measure (t0, u0)sat. For ex- µgeo ample, while Refsdal (1966) had suggested observations from a second satellite to break the four-fold degener- µ where M is the mass of the lens, (πrel, geo) are the acy, Gould (1995) argued that this might be possible lens-source relative (parallax, proper motion) and κ ≡ 2 −1 from a single satellite because the velocity difference 4G/c AU ≃ 8.14 mas M⊙ . In more modern language between the two observatories would yield differences (Gould 2000, 2004; Gould & Horne 2013), the microlens in tE that would allow one to distinguish among the parallax vector, ′ four values of ∆β±,±, where the first subscript refers to πrel µgeo the sign of u0,⊕ and the second to u0,sat. This was soon πE ≡ , (2) shown to be substantially more efficient for microlens- θE µgeo ing events toward the ecliptic poles (Boutreux & Gould could be determined from the inferred offset in the Ein- 1996) than toward the ecliptic (Gaudi & Gould 1997). stein ring A key issue in these early years appeared to be AU ′ ′ πE = (∆τ , ∆β ); (3) the much greater difficulty in measuring u0 compared D⊥ to t0 for 1L1S light curves. This arises from the fact where that the derivative of the microlensed flux with respect to only one parameter (t0) is odd (antisymmetric) in ′ t0,sat − t0,⊕ ′ ∆τ = ; ∆β = u0,sat − u0,⊕, (4) time, while there are four with derivatives that are even tE (symmetric) in time (u0, tE, fs, fb). Here (fs, fb) are the Corresponding author: A. Gould source flux and blended flux. Hence, u0 is strongly cor- 121 122 Gould related with other parameters while t0 is not. Gould corresponding offset in the Einstein ring usat: (1995) already recognized that the interplay of discrete and continuous degeneracies in the direction orthogo- nal to D⊥ was a major issue for space-based paral- ∆fsat 1 Asat = 1 + ; usat = v2 − 1 . f u −2 ! laxes because it seemed to require very high signal-to- s,sat u 1 − A u sat noise ratio space-based light curves, which are intrinsi- t q cally expensive. He noted that if the space and ground (5) cameras had nearly identical responses, then this issue Then, in the approximation u0,⊕ → 0, the magnitude of could be largely resolved. This is because fs would the parallax vector is simply πE = (AU/D⊥)usat. There be known to be the same a priori, which would allow is then no information at all about the direction (φπ) ′ ∆β = (u0,sat − u0,⊕) to be measured much more pre- of πE, but this direction is not needed to determine the cisely than either impact parameter separately. How- main properties of the lens, i.e., its mass M = θE/κπE ever, this was believed to be extremely difficult even and lens-source relative parallax πrel = θEπE. for optical observations and essentially impossible for Of course, this requires that fs,sat be known, which the only photometric telescope then planned for solar in the previous conception required a good-coverage, orbit, namely SIRTF (a.k.a., Spitzer), whose shortest high-precision, space-based light curve. However, in the wavelength (3.6 µm) was essentially unobservable from meantime, Yee et al. (2012) had established that mi- the ground. crolensing source fluxes of sparsely covered light curves Pressed by M. Werner (1998, private communica- could be determined from color-color relations linked to tion) to find a solution to this problem that could be well covered light curves. Hence, Gould & Yee (2012) applied to Spitzer, Gould (1999) developed the idea of suggested that these relations be applied to space-based combining separate one-dimensional (1-D) parallax in- observations as well. formation from Earth and Spitzer to yield robust 2- Subsequently, Shin et al. (2018) demonstrated that D microlens parallaxes. That is, according to Equa- this approach works in practice. In particular, their Figure 3, which shows a circle nearly centered on the tion (3), the component of πE along D⊥ could be well ′ origin (excellent measurement of πE, no information on measured even if u0,sat (and so ∆β ) was not. There- fore, if there were additional 1-D information from the φπ) was a major inspiration for the present work. ground (not parallel to D⊥), then a relative handful of For 2014–2019, there were (or will be) major Spitzer space-based measurements (enough to measure t0,sat) microlens parallax campaigns toward the Galac- would be sufficient. tic bulge. During the first (pilot) year, the focus was on obtaining “full-coverage” light curves from Spitzer, in In fact, Gould et al. (1994) had already pointed particular capturing the peak, in order to demonstrate out that the annual parallax effect (Gould 1992) could the feasibility of the method. See, for example, Fig- measure the component of π parallel to Earth’s in- E ure 1 from Yee et al. (2015a) and compare to Figure 1 stantaneous acceleration at t0, even when the orthog- 1 of Gould (1994). However, in subsequent years, the cri- onal component was essentially unmeasurable. Thus, teria for event selection were substantially relaxed in unless Earth’s acceleration at t is closely aligned with 0 pursuit of the goal of measuring the Galactic distribu- D⊥, the two 1-D parallaxes (each by itself almost use- tion of planets (Yee et al. 2015b). In particular, events less) could be combined to yield a 2-D parallax. This led were frequently chosen even if the Spitzer observations to a proposal for target-of-opportunity observations to- were likely to begin well after peak. As discussed above, ward the Magellanic Clouds (where these two directions such light-curve fragments cannot by themselves yield are generally not aligned) and resulted in a successful useful information about (t , u ) . However, it was an- measurement based on just four Spitzer epochs (Dong 0 0 sat ticipated (and subsequently confirmed, Calchi Novati et et al. 2007). al. 2015) that fs,sat can be derived from color-color rela- The extremely high cost (hence low expected num- tions (provided that fs,⊕ is well measured from Earth). ber) of space-based measurements led Gould & Yee Nevertheless, despite the fact that there are now (2012) to suggest a radically different idea for “cheap several hundred Spitzer light curves that begin after space-based microlens parallaxes”. This required two peak, there has not yet been a systematic study of what special conditions. First, the event must be relatively is the character of the parallax information that is ac- high-magnification as seen from Earth (u0,⊕ ≪ 1). tually garnered from these light curves. Rather, Spitzer Second, it must be observed from the satellite at a and ground-based data are generally combined in a sin- time tsat ≃ t0,⊕.
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