OGLE-2016-BLG-1045: A Test of Cheap Space-based Microlens Parallaxes

The microlensing event OGLE-2016-BLG-1045 occurred on a source that lies at (α, δ)_J2000 = (17^h36^m5119, −34°32'397), which corresponds to the Galactic coordinates (l,b) = (354255, −1386). The Optical Gravitational Lensing Experiment (OGLE-IV: Udalski et al. 2015a) found this event and then the Early Warning System (Udalski et al. 1994; Udalski 2003) of the OGLE-IV survey announced the event on 2016 June 9. The observations were made with the 1.3 m Warsaw telescope in the I-band channel of a 1.4 square-degree camera located at the Las Campanas Observatory in Chile.

The event was highly magnified, implying that a planetary companion to the lens could probably be detected if it exists. Hence, a follow-up observation team called the Microlensing Follow-Up Network (μFUN: Gould et al. 2006) observed this event to capture any anomalies that might be produced by a planet. Auckland observatory, a μFUN member located in New Zealand, made the observations with a 0.4 m telescope using a number 12 Wratten filter (which is similar to R band). The Auckland observations successfully covered the peak of the event. This peak coverage did not reveal an anomaly in the light curve due to a planetary lens system. However, the good coverage of the peak provided a chance to detect the finite source effect, which enters the determination of the angular Einstein ring radius, i.e., ρ_* = θ_*/θ_E. The finite source effect can provide a mass–distance relation, $M/{D}_{\mathrm{rel}}=({c}^{2}/4G)\,{\theta }_{E}^{2}$, where the ${D}_{\mathrm{rel}}\equiv {({D}_{{\rm{L}}}^{-1}-{D}_{{\rm{S}}}^{-1})}^{-1}$ is the relative distance between distances to the lens (D_L) and the source (D_S), M is the lens mass, c is the speed of light, and G is Newton's constant.

Other μFUN observations exist in H band taken at the Cerro Tololo International Observatory in Chile with the 1.3 m SMARTS telescope (CTIO). These CTIO data were not included in the final models because of the similar coverage to the KMTNet data, but were used for the color–magnitude diagram (CMD) analysis of the event (see the Appendix).

The Korea Microlensing Telescope Network (KMTNet: Kim et al. 2016) also observed this event. Three identical 1.6 m telescopes located in the Cerro Tololo International Observatory in Chile (KMTC), the South African Astronomical Observatory in South Africa (KMTS), and the Siding Spring Observatory in Australia (KMTA) observed this event with the I-band channel of their 4 deg² cameras. The KMTNet observations provided overall coverage of the light curve.

The observed data sets were reduced by each group using their own pipelines and difference-imaging analysis packages: ((OGLE-IV (DIA): Alard & Lupton 1998; Wozniak 2000), (μFUN and KMTNet (pySIS): Albrow et al. 2009)).

This event was "secretly" chosen as a target of the 2016 Spitzer Microlensing Campaign on 2016 June 16 (UT 20:30) based on the possibility that the event could be highly magnified. The event was later claimed as a "subjective" target on 2016 June 18 (UT 16:34) once the event was observed to be moderate to high magnification (see Yee et al. 2015a for more details on different types of event selection). The observations began on 2016 June 18 (UT 9:56) and ended on July 8 (UT 2:43). The Spitzer Space Telescope took 24 total data points over 20 days with the 3.6 μm channel (L band) of the IRAC camera. The Spitzer data were reduced with point response function photometry (Calchi Novati et al. 2015b).

In Figure 1, we present light curves of the event observed from ground and space. We also present the best-fit model light curves and their residuals, which is the (−, +) case presented in Table 2. The ground-based light curve shows a symmetric Paczyński curve (Paczyński 1986) with a smooth peak feature, which implies that the event was produced by a single lens affected by the finite source effect. The Spitzer observations only partially covered the light curve. However, Han et al. (2017), Shin et al. (2017), and Wang et al. (2017) already showed that it is possible to accurately measure the SPRX even though the space-based observations are fragmentary. Thus, for this event, using the Spitzer observations and the finite source effect, it is possible to measure the microlens parallax and the angular Einstein ring radius, which yield the properties of the isolated lens. We note that there exists a systematic trend in the Spitzer observations. The origin of this trend is unknown. However, several publications that used the Spitzer data with a similar trend (e.g., Poleski et al. 2016; Shin et al. 2017; Shvartzvald et al. 2017; Zhu et al. 2017) concluded that the trend is not likely to affect determinations of their models. In this case, the trend is milder than those in the previous publications.

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The Spitzer observations were not taken with the idea of "cheap-SPRX" in mind. In fact, because the peak magnification was relatively unconstrained when the observations were scheduled, many similar events were observed based on the chance that one of them would be high-magnification (so, these observations cannot be considered "cheap"). Nevertheless, the resulting observations contain what would be obtained for a "cheap-SPRX" campaign, i.e., the Spitzer observations exist near the peak of the ground-based light curve and also exist near the baseline. Hence, this event can serve as an excellent testbed to perform a practical test of the cheap-SPRX idea.

We test the accuracy of the cheap-SPRX method by considering three different Spitzer data sets, which we refer to as the "Actual," "Realistic," and "Idealized" cases. These data sets differ in the amount of information they contain (most to least). We first consider the two extremes, which are the "Actual" case defined by the current experiment and the "Idealized" GY12 case. For the "Actual" case, we use all observed Spitzer data (24 points). From this case, we can obtain the actual SPRX measurement that can be used as a reference to compare with the measurements derived from the other cases. For the "Idealized" case, considering the ideal situation proposed by GY12, this represents the minimum amount of data necessary for the cheap-SPRX idea to work. For this case, we generate two artificial data points using the Spitzer data and the best-fit model light curve. One is located at the exact ground-based peak (HJD' = 7559.201) and the other is located at the baseline (HJD' = 7900.000). For the "Realistic" case, we choose two actual data points near the ground-based peak (HJD' = 7559.172 and 7559.482) because it is almost impossible to take an image at the exact peak time in realistic situations. In addition, we use the last point (HJD' = 7577.613) observed by Spitzer, which is located near the baseline. Based on these selected Spitzer data, we can obtain a measurement of the cheap-SPRX under realistic conditions. In Figure 2, we present light curves of the cases that clearly show the space-based observations used for the test.

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Based on the three cases, we conduct modeling to measure the SPRX value of each case. For the modeling, we use six parameters: (t₀, u₀, t_E, ρ_*, π_E, and Φ). Among them, three basic parameters (t₀, u₀, and t_E) describe the light curve produced by a single lens and a point source. These basic parameters are closely related to each other: t₀ is the time at the peak of the light curve; u₀ is the impact parameter, i.e., the separation between the center of the Einstein ring and the position of the source at time t₀; t_E is the crossing-time of the Einstein ring. Another parameter ρ_* is the angular source radius (θ_*) normalized by the angular Einstein ring radius (θ_E), ρ_* ≡ θ_*/θ_E, which describes the finite source effect. The last two parameters (π_E and Φ) describe the SPRX, which differs from the conventional way of describing the microlens parallax vector ${\boldsymbol{\pi }}$ (normally consisting of north (π_E,N) and east (π_E,E) components). In our parameterization (see also Bennett et al. 2008),

$$\begin{eqnarray}&&{{\boldsymbol{\pi }}}_{{\rm{E}}}=({\pi }_{{\rm{E}},N},{\pi }_{{\rm{E}},E})\to ({\pi }_{{\rm{E}}}\cos {\rm{\Phi }},{\pi }_{{\rm{E}}}\sin {\rm{\Phi }}).\end{eqnarray} \tag{ 6 }$$

The Φ angle is allowed to vary over the full possible range [−π, +π].²³ In addition, there are flux parameters (F_S and F_B) for each data set that describe the fluxes of the source and blend, respectively, which are fit linearly for each model. We note that the model flux for each data set, i, is derived from ${F}_{\mathrm{obs},i}(t)=A(t){F}_{S,i}+{F}_{B,i}$, where the A(t) is the model magnification as a function of time. Using these parameters, we search for the best-fit model with the minimum χ² between the observed and modeled light curves using a Markov Chain Monte Carlo (MCMC) χ² minimization (the details of our MCMC sampling method are described in Dunkley et al. 2005). To find the global minimum of the model parameters, especially the SPRX parameter (π_E), we initially conducted a grid search over π_E and Φ using the 200 × 200 grid points. The grid search results are the same as those of the MCMC simulations.

During the modeling process, we consider the limb-darkening (LD) of the source star. We adopt LD coefficients for observed passbands from Claret (2000) based on the spectral source type determined by the CMD analysis (described in the Appendix). In addition, we rescale the errors of observations to enforce χ²/dof ≃ 1 using the equation e_new = k(e_old) where k, e_new, and, e_old are the error rescaling factor, rescaled errors, and original errors, respectively. The error rescaling process has been done based on the best-fit model, i.e., the (−, +) case. We note that, in the case of the OGLE-IV data, the observational errors are calibrated using a correction procedure that is described in Skowron et al. (2016), before applying the error rescaling process based on the best-fit model. In Table 1, we present these LD coefficients and error rescaling factors for modeling.

Table 1.  Limb-darkening Coefficients and Error Rescaling Factors

Observations	Γλ	k
OGLE (I)	0.5103	0.913
Auckland (R)a	0.6583	2.370
KMTC (I)	0.5103	1.116
KMTS (I)	0.5103	1.501
KMTA (I)	0.5103	1.446

Note.

^aWe use a modified LD coefficient for Auckland observations, ${{\rm{\Gamma }}}_{R}\,=({{\rm{\Gamma }}}_{R}+{{\rm{\Gamma }}}_{V})/2=(0.61118+0.7048)/2=0.6583$ because the Auckland observatory used a 12 Wratten filter having a flat transmission between 540 and 700 nm). Thus, the filter is similar to the mean value of R and V bands. Note that we did not use a Γ_L because it plays no role for the Spitzer observations.

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We also incorporate the color-constraint, (I − L) = 3.800 ± 0.020, which provides an independent constraint on the model. The constraint is determined using I-band ground observations (OGLE-IV) and L-band space observations (Spitzer) based on the CMD analysis. To incorporate the (I − L) color-constraint, we introduce ${\chi }_{\mathrm{penalty}}^{2}$ described in Section 3.2 of Shin et al. (2017). The ${\chi }_{\mathrm{penalty}}^{2}$ increases the χ² when the fitted (I − L) color of the model is different from the constraint. In particular, the ${\chi }_{\mathrm{penalty}}^{2}$ increases strongly when the difference between the fitted color and the constraint is larger than 2σ.

In Table 2, we present the best-fit parameters for each case (Actual, Realistic, and Idealized). For each case, we find that two degenerate solutions exist due to the "four-fold degeneracy" (Refsdal 1966; Gould 1994). In principle, the four-fold degeneracy has four solutions, (+, +), (+, −), (−, +), and (−, −) (denoted according to the convention described in Zhu et al. 2015), which are caused by different pairs of source trajectories (seen from ground and space) going through a similar lensing magnification pattern. This degeneracy can be divided into two categories by its origin (GY12 and references therein). The first (denoted by the first ±sign in this paper) is related to the relative positions of the Earth and satellite, whether they lie on the same or opposite sides of the lens. The other (denoted by the second ±sign in this paper) is related to the different possible source trajectories as seen from Earth, i.e., whether they pass on the left or right sides of the lens. The former degeneracy can affect the magnitude (π_E) of the ${{\boldsymbol{\pi }}}_{{\rm{E}}}$, while the latter degeneracy can only affect the direction of the ${{\boldsymbol{\pi }}}_{{\rm{E}}}$, which is less interesting in this test of the cheap-SPRX idea. The four-fold degeneracy can sometimes be resolved (e.g., Yee et al. 2015b; Udalski et al. 2015b; Han et al. 2016, 2017; Chung et al. 2017; Shin et al. 2017). For this event, we find that there exist only two solutions, (−, +) and (+, +), based on the grid search process. The other two solutions, (−, −) and (+, −), are merged with the (−, +) and (+, +) solutions, respectively. The reason that the four solutions are merged into only two solutions for this event is that u_0,Spitzer ∼ 0. For model parameters of each solution, uncertainties are determined based on the 68% confidence intervals of the MCMC chains.

Table 2.  The Best-fit Model with Degenerate Solutions of Each Case

Case	Actual		Realistic		Idealized
Parameter	(−, +)	(+, +)	(−, +)	(+, +)	(−, +)	(+, +)
${\chi }_{\mathrm{total}}^{2}/{N}_{\mathrm{data}}$	1368.70/1372	1368.99/1372	1344.89/1351	1345.04/1351	1343.83/1350	1343.95/1350
${\chi }_{\mathrm{Ground}}^{2}/{N}_{\mathrm{data}}$	1345.01/1348	1345.09/1348	1344.77/1348	1344.77/1348	1343.83/1348	1343.95/1348
${\chi }_{{\text{}}{Spitzer}}^{2}/{N}_{\mathrm{data}}$	23.69/24	23.90/24	0.12/3	0.27/3	0.00/2	0.00/2
${\chi }_{\mathrm{penalty}}^{2}$	0.017	0.075	0.000	0.010	0.003	0.014
(I − L) [3.80]	3.797	3.794	3.800	3.802	3.799	3.803
t0 (HJD')	7559.201 ± 0.001	7559.201 ± 0.001	7559.201 ± 0.001	7559.201 ± 0.001	7559.202 ± 0.001	7559.202 ± 0.001
u0 (10−2)	$-{1.308}_{-0.042}^{+0.033}$	${1.314}_{-0.044}^{+0.036}$	$-{1.318}_{-0.037}^{+0.041}$	${1.318}_{-0.044}^{+0.033}$	$-{1.312}_{-0.044}^{+0.033}$	${1.309}_{-0.033}^{+0.044}$
tE (days)	${11.981}_{-0.098}^{+0.064}$	${11.963}_{-0.084}^{+0.088}$	${11.950}_{-0.083}^{+0.084}$	${11.947}_{-0.083}^{+0.088}$	${11.956}_{-0.094}^{+0.073}$	${11.952}_{-0.088}^{+0.076}$
ρ* (10−2)	${3.186}_{-0.026}^{+0.033}$	${3.190}_{-0.030}^{+0.030}$	${3.195}_{-0.030}^{+0.027}$	${3.195}_{-0.033}^{+0.028}$	${3.194}_{-0.029}^{+0.030}$	${3.193}_{-0.028}^{+0.031}$
πE	${0.355}_{-0.006}^{+0.004}$	${0.352}_{-0.005}^{+0.006}$	${0.355}_{-0.008}^{+0.005}$	${0.350}_{-0.006}^{+0.008}$	${0.365}_{-0.015}^{+0.004}$	${0.346}_{-0.004}^{+0.014}$
Φ (radian)	${1.291}_{-0.062}^{+0.165}$	${1.353}_{-0.066}^{+0.167}$	${1.210}_{-0.284}^{+0.381}$	${1.178}_{-0.177}^{+0.458}$	${0.341}_{-1.185}^{+0.955}$	${0.407}_{-1.188}^{+1.141}$
${\pi }_{{\rm{E}},{\text{}}E}$	${0.341}_{-0.012}^{+0.012}$	${0.344}_{-0.011}^{+0.013}$	0.332	0.323	0.122	0.137
${\pi }_{{\rm{E}},{\text{}}N}$	${0.098}_{-0.059}^{+0.027}$	${0.076}_{-0.058}^{+0.028}$	0.125	0.134	0.344	0.317
${F}_{{\rm{S}},\mathrm{OGLE}}$	${1.370}_{-0.010}^{+0.014}$	${1.373}_{-0.013}^{+0.012}$	${1.375}_{-0.012}^{+0.012}$	${1.375}_{-0.013}^{+0.011}$	${1.374}_{-0.011}^{+0.014}$	${1.374}_{-0.012}^{+0.013}$
${F}_{{\rm{B}},\mathrm{OGLE}}$	$-{0.032}_{-0.014}^{+0.010}$	$-{0.034}_{-0.012}^{+0.012}$	$-{0.036}_{-0.012}^{+0.011}$	$-{0.037}_{-0.012}^{+0.012}$	$-{0.035}_{-0.014}^{+0.010}$	$-{0.036}_{-0.013}^{+0.011}$
${F}_{{\rm{S}},{Spitzer}}$	${45.257}_{-0.932}^{+0.919}$	${45.212}_{-0.870}^{+1.004}$	${45.518}_{-1.061}^{+0.953}$	${45.622}_{-1.199}^{+0.902}$	${45.439}_{-0.909}^{+1.124}$	${45.618}_{-1.127}^{+0.973}$
${F}_{{\rm{B}},{Spitzer}}$	$-{6.528}_{-0.993}^{+0.941}$	$-{6.430}_{-1.178}^{+0.819}$	$-{8.032}_{-1.094}^{+0.963}$	$-{8.179}_{-1.046}^{+1.174}$	$-{6.674}_{-1.190}^{+0.844}$	$-{6.854}_{-1.039}^{+1.062}$

Note. HJD' = HJD-2450000.0. The N_data after each χ² value indicates the number of data points that are used for the modeling. We note that the π_E,E and π_E,E are not modeling parameters. These are calculated from the modeling parameters, π_E and Φ (see Equation (6)). We do not present the errors of π_E,N and π_E,E for Realistic and Idealized cases because these errors are meaningless: only the error in π_E has meaning.

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In Figure 3, we present the SPRX distributions of each case. The distributions are constructed from the MCMC chains. These distributions clearly show the consistency of the SPRX measurements. We present two types of distributions. One type of distribution is presented according to the conventional parameters, (π_E,E, π_E,N), which are calculated from the MCMC parameters as π_E,E = π_E sin Φ and ${\pi }_{{\rm{E}},N}={\pi }_{{\rm{E}}}\cos {\rm{\Phi }}$. The other is the (π_E, Φ) distribution, which can be used to directly check the accuracy of the magnitude of the SPRX measurement.

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From the modeling of the actual case, we obtain the SPRX measurements for the (−, +) and (+, +) cases: ${\pi }_{{\rm{E}}}={0.355}_{-0.006}^{+0.004}$ and ${0.352}_{-0.005}^{+0.006}$, respectively. We find that the magnitudes of the SPRX values between the (−, +) and (+, +) solutions of the actual case are consistent to well within 1σ. Based on the actual SPRX measurements, we can compare the other test cases of the cheap-SPRX idea to check the accuracy of the cheap-SPRX measurements. For the realistic case, we find that the SPRX measurements of both degenerate solutions, ${0.355}_{-0.008}^{+0.005}$ and ${0.350}_{-0.006}^{+0.008}$, are consistent with those of the actual case to within 1σ. For the idealized case, the measurements, ${0.365}_{-0.015}^{+0.004}$ and ${0.346}_{-0.004}^{+0.014}$, are consistent to within ≲1σ using the idealized-case errors.

Based on the SPRX measurements, we can determine the properties of this isolated lens by combining it with the angular Einstein ring radius (θ_E = θ_*/ρ_*), where θ_* is the angular source radius determined from the CMD analysis (described in the Appendix) and ρ_* is determined from the finite source effect. We determine the angular Einstein ring radius as

$$\begin{eqnarray}&&{\theta }_{{\rm{E}}}=0.244\pm 0.015\,\mathrm{mas}.\end{eqnarray} \tag{ 7 }$$

In Figure 4, we present distributions of physical properties of the lens for each case. the lens mass (M_L) and the lens distance (D_L) are determined from MCMC parameters as

$$\begin{eqnarray}&&{M}_{{\rm{L}}}=\displaystyle \frac{({\theta }_{* }/\kappa )}{{\rho }_{* }{\pi }_{{\rm{E}}}},\,\,\kappa =8.144\,\mathrm{mas}\,{M}_{\odot }^{-1},\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{D}_{{\rm{L}}}=\displaystyle \frac{\mathrm{au}}{({\pi }_{{\rm{E}}}/{\rho }_{* }){\theta }_{* }+{\pi }_{S}},\,\,{\pi }_{S}=\displaystyle \frac{\mathrm{au}}{{D}_{S}},\end{eqnarray} \tag{ 9 }$$

where D_S is the distance to the source estimated from Nataf et al. (2013). For this event, the estimated D_S is ∼8.87 kpc. We find that both properties are consistent to within 1σ across all cases. In fact, the uncertainty in the properties is dominated by the uncertainty of the θ_* determination. Quantitatively, the uncertainty of the SPRX measurement is <3% compared to the ≥6% uncertainty in θ_*. Thus, we find that the accuracy of the SPRX measurement based on the cheap-SPRX idea is sufficient to accurately determine the properties of the isolated object. The isolated lens of this event is a low-mass stellar object with M_L ∼ 0.08 ± 0.01 M_⊙, which is located at ∼5.02 ± 0.14 kpc from us.²⁴

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The cheap-SPRX idea assumes that an observed light curve seen from space, e.g., the Spitzer observations, resembles a single-lensing light curve. However, if the lens is a binary and there are only two observations from the spacecraft, it will not be possible to determine from the space-based observations alone whether these are affected by the binary or whether the single-lens assumption is sufficient. Indeed, if the binary is not detected in the ground-based data, an anomaly in the space-based data due to a binary would go undetected. Then, the magnification computation to measure the cheap-SPRX may be inaccurately determined due to the effect of the binary-lensing perturbation on the light curve. As a result, a violation of the single-lensing assumption can, in principle, yield an incorrect measurement of the cheap-SPRX when a second mass exists.

However, high-magnification events (this is a basic assumption for applying the cheap-SPRX idea) are very sensitive to binary lenses. This implies that, for a high-magnification event, we can rule out very broad class binary-lens configurations because these would produce clear anomalies on the ground-based light curve. We perform a quantitative test to check the effect on the cheap-SPRX measurement caused by binary-lensing. The test is performed using the following procedures.

First, we separately conduct a binary-lens modeling with ground-based observations only. The best-fitting of this modeling yields a χ² threshold to exclude binary-lensing cases, which have noticeable anomalies. The best-fit model has ${\rm{\Delta }}{\chi }^{2}=({\chi }_{\mathrm{single}}^{2}-{\chi }_{\mathrm{binary}}^{2})=13.9$. Thus, we set the χ² threshold ${\chi }_{\mathrm{th}}^{2}=15.0$. This is the criterion for dividing simulated binary-lensing cases into two categories: ${\chi }^{2}\gt {\chi }_{\mathrm{th}}^{2}$ are the cases with anomalies that are detectable in the ground-based light curve, and ${\chi }^{2}\lt {\chi }_{\mathrm{th}}^{2}$ are the cases having nondetectable anomalies.

Second, we simulate binary-lensing cases with only ground-based observations using the Rhie method (Bennett & Rhie 1996; Rhie et al. 2000). In this procedure, the binary-lensing cases are simulated using a grid of the projected separation (s), mass ratio (q), and angle (α) of the source trajectory with respect to the binary-axis: $\mathrm{log}s=[-1.2,1.2]$, $\mathrm{log}q=[-5.0,1.0]$, and α = [0, 2π]. Each range of the grid is divided into 120 grid points (i.e., total 120³ binary-lensing cases are simulated). We adopt the other parameters, t₀, u₀, t_E, and ρ_*, from the actual (−, +) solution to produce an artificial data set of the binary-lensing case. For each binary-lensing case with the artificial ground-based data set, we calculate a χ² value by fitting with a finite-source single-lensing model.

Third, we can build two types of diagrams (Figure 5) using the simulated binary-lensing cases and the χ² threshold: one is the diagram showing the detection efficiency of this event, and the other is the diagram showing two categories of the binary-lensing cases at a specified mass ratio. From this diagram, we can extract a "boundary" with Δχ² = 15, which represents a kinds of extreme binary-lensing cases having nondetectable anomalies that may possibly affect the cheap-SPRX measurement. In Figure 5, we present an example of such diagrams at the q = 0.1 and their boundaries.

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Fourth, at these boundary cases, we can check the effect on the cheap-SPRX measurement caused by the hidden anomalies of the binary-lensing cases. To quantitatively check the effect, we set a criterion as

$$\begin{eqnarray}&&{\left|\displaystyle \frac{{A}_{\mathrm{binary}}^{{Spitzer}}}{{A}_{\mathrm{PSPL}}^{{Spitzer}}}-1\right|}_{\mathrm{peak},\oplus }\lt \displaystyle \frac{\sigma ({\pi }_{{\rm{E}}})}{{\pi }_{{\rm{E}}}},\end{eqnarray} \tag{ 10 }$$

where the ${A}_{\mathrm{binary}}^{{Spitzer}}$ and ${A}_{\mathrm{PSPL}}^{{Spitzer}}$ are magnifications of the Spitzer light curve at the ground-peak time (HJD' ∼ 7559.20) computed using binary-lens and single-lens models, respectively. The π_E and σ(π_E) are the cheap-SPRX measurement and its uncertainty adopted from the actual (−, +) case. This criterion shows how much an undetected anomaly due to binary-lensing could affect the magnification of the Spitzer light curve. If the criterion in Equation (10) is met, the inaccuracy in the magnification is less significant than uncertainties from other sources. Using this criterion, we check three cases of boundaries at q = 0.01, 0.1, and 1.0.

In Figure 6, we present the quantitative results of this test. We find that, for all cases along the boundary, the deviations between magnifications of the Spitzer light curve at the ground-peak are much smaller than the relative error of the SPRX that is actually measured. This implies that the binaries that do not give to detectable signals in the ground-based data also do not significantly affect the SPRX measurement. Hence, in this case, even if there exists an undetected binary-lens anomaly, we can still obtain an accurate SPRX measurement using the cheap-SPRX idea.

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