Ground-based Parallax Confirmed by Spitzer: Binary Microlensing Event MOA-2015-BLG-020

In a microlensing event, companions to the primary lens object can be detected via their perturbations to the single-lens light curve (Mao & Paczynski 1991; Gould & Loeb 1992). From such perturbations, dimensionless parameters can be derived that are related to the binary system, such as the binary mass ratio q and the projected separation s (Gaudi & Gould 1997). Here, s is the instantaneous angular separation between the two components normalized to the angular Einstein radius

$$\begin{eqnarray}&&{\theta }_{{\rm{E}}}\equiv \sqrt{\kappa {M}_{{\rm{L}}}{\pi }_{\mathrm{rel}}},\end{eqnarray} \tag{ 1 }$$

where ${M}_{{\rm{L}}}$ is the total lens mass, and

$$\begin{eqnarray}&&\kappa \equiv \displaystyle \frac{4G}{{c}^{2}\mathrm{au}}\approx 8.14\displaystyle \frac{\mathrm{mas}}{{M}_{\odot }};\quad {\pi }_{\mathrm{rel}}\equiv \mathrm{au}\left(\displaystyle \frac{1}{{D}_{{\rm{L}}}}-\displaystyle \frac{1}{{D}_{{\rm{S}}}}\right).\end{eqnarray} \tag{ 2 }$$

Here, ${\pi }_{\mathrm{rel}}$ is the lens-source relative parallax, and ${D}_{{\rm{L}}}$ and ${D}_{{\rm{S}}}$ are the distances to the lens and the source, respectively.

Although statistical conclusions can be drawn from measurements of q and s, the physical properties of the lens system, such as ${M}_{{\rm{L}}}$, are of more interest. By far, the most popular way to convert from microlensing observables to physical quantities is to combine the measurements of ${\theta }_{{\rm{E}}}$ and the microlensing parallax, ${\pi }_{{\rm{E}}}\equiv {\pi }_{\mathrm{rel}}/{\theta }_{{\rm{E}}}$. Then,

$$\begin{eqnarray}&&{M}_{{\rm{L}}}=\displaystyle \frac{{\theta }_{{\rm{E}}}}{\kappa {\pi }_{{\rm{E}}}};\quad {\pi }_{\mathrm{rel}}={\pi }_{{\rm{E}}}{\theta }_{{\rm{E}}}.\end{eqnarray} \tag{ 3 }$$

There are several ways to measure ${\theta }_{{\rm{E}}}$ (see a short summary given in Zhu et al. 2015), but the most common is to use the finite-source effect, which is the deviation in the light curve from the point-like source model due to the extended nature of the source star (Yoo et al. 2004).

For most published binary events, the microlensing parallax parameter ${\pi }_{{\rm{E}}}$ is measured through the annual parallax effect, in which Earth's acceleration around the Sun introduces deviations from rectilinear motion in the lens-source relative motion (Gould 1992). This method generically assumes that the lens (or lens system) and the source (or source system) are, or can be treated as, not undergoing acceleration. For binary lens events, each component is under acceleration by the other, and this so-called lens orbital motion effect can be confused with the annual parallax effect (Batista et al. 2011).

Therefore, it is important to understand the validity of the annual parallax method for binary-lens events in practical use. This can be done by observing the binary system after the event, either photometrically (Dong et al. 2009; Bennett et al. 2010) or spectroscopically (Boisse et al. 2015; Yee et al. 2016). Another way is to measure ${\pi }_{{\rm{E}}}$ via the satellite parallax method (Refsdal 1966; Gould 1994). This is done by observing the same microlensing event from at least two well-separated locations; the difference between the light curves from these locations informs the parameter ${\pi }_{{\rm{E}}}$. The microlensing parallax measured in this way is then determined independently from the orbital motion effect, and thus can be used to test the annual parallax method.

The Spitzer microlensing campaigns utilize the Spitzer Space Telescope to measure ${\pi }_{{\rm{E}}}$ via the satellite parallax method for hundreds of microlensing events (e.g., Calchi Novati et al. 2015a; Udalski et al. 2015; Yee et al. 2015a; Zhu et al. 2015). Of the several published binary events, OGLE-2015-BLG-0479, is found to have inconsistent ${\pi }_{{\rm{E}}}$ from the annual parallax and satellite parallax methods, and this inconsistency can be well explained by the full orbital motion of the lens system (Han et al. 2016). In the case of OGLE-2015-BLG-0196 and OGLE-2016-BLG-0168, the annual parallax effect is confirmed by the satellite parallax method (Han et al. 2017; Shin et al. 2017).

In this paper, we present the analysis of Spitzer binary event MOA-2015-BLG-020. This is the second published case in which the annual parallax effect agrees with the satellite parallax effect. We summarize the ground-based and space-based observations in Section 2, describe the light curve modeling in Section 3, and derive the physical properties of the binary system in Section 4. In Section 5, we review all of the published microlensing binaries in which the annual parallax effect has been confirmed or contradicted by other methods, and discuss the implications.

2.1. Ground-based Alert and Follow-up

2.2. Spitzer Follow-up

3.1. Initial Solution Search

The light curve of event MOA-2015-BLG-020 suggests that it is a typical binary microlensing event (see Figure 1). In the standard terminology (i.e., binary event without parallax and lens orbital motion effects), the following seven parameters are used for characterizing a binary light curve: the time of the closest approach between the source and the binary lens (gravitational) center, t₀; the impact parameter normalized by the Einstein radius, u₀; the event timescale, ${t}_{{\rm{E}}};$ the source size normalized by the Einstein radius, ρ; the projected separation between the binary components normalized to the Einstein radius, s; the binary mass ratio, q; and the angle between the binary-lens axis and the lens-source relative motion, α. There are two further flux parameters that describe the source flux $({F}_{{\rm{S}}}^{j})$ and the blending flux $({F}_{{\rm{B}}}^{j})$ for each observatory j that translate the magnification A to the observed flux at given time t_i

$$\begin{eqnarray}&&{F}^{j}({t}_{i})={F}_{{\rm{S}}}^{j}\cdot A({t}_{i})+{F}_{{\rm{B}}}^{j}.\end{eqnarray} \tag{ 4 }$$

These flux parameters are found for each data set using linear fit. We use the the advanced contour integration code, VBBinaryLensing,⁴⁶ to compute of the binary lens magnification $A({t}_{i})$. This code includes a parabolic correction in Green's line integral, and can automatically adjust the step size of the integration based on the distance to the binary caustic, in order to achieve a desired precision in magnification. See Bozza (2010) for more details.

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We start with a grid search on the ground-based data alone for the possible binary solution (or solutions). The grid search is conducted on parameters (log s, log q, log ρ, α), with 16 values equally spaced between −1 ≤ log s ≤ 1, −3 ≤ log q ≤ 0, −4 ≤ log ρ ≤ 0, and 0° ≤ α ≤ 360°, respectively. For each set of (log s, log q, log ρ, α), we find the minimum ${\chi }^{2}$ by going downhill on the remaining parameters $({t}_{0},{u}_{0},{t}_{{\rm{E}}})$.

The global minimum is found at log s ∼ 0 (s ∼ 1), log q ∼ −0.6 (q ∼ 0.25), log ρ ∼ 2, and α ∼ 220°, and there is no other locus on this grid that has similar ${\chi }^{2}$.

We then refine the solution by performing Markov Chain Monte Carlo (MCMC) analysis around the initial solution found by the previous grid search, which employs the emcee ensemble sampler (Foreman-Mackey et al. 2013).

3.2. Inclusion of Microlensing Parallax Effect

The microlensing parallax effect has to be taken into account in order to simultaneously model the ground-based and space-based data. This effect invokes two additional parameters, ${\pi }_{{\rm{E}},{\rm{N}}}$ and ${\pi }_{{\rm{E}},{\rm{E}}}$, which are the northern and eastern components of the parallax vector ${{\boldsymbol{\pi }}}_{{\rm{E}}}$.

We try to constrain ${{\boldsymbol{\pi }}}_{{\rm{E}}}$ based on ground data alone, and by simultaneously modeling ground and Spitzer data, in order to check the consistency between annual parallax and satellite parallax effects. This check is possible here because MOA-2015-BLG-020 occurred relatively early in the season and had a fairly long timescale.

The annual parallax effect leads to two discrete solutions arising from the ±u₀ degeneracy (e.g., Smith et al. 2003; Poindexter et al. 2005). In the case of MOA-2015-BLG-020, we find that the two solutions have ${\rm{\Delta }}{\chi }^{2}\geqslant 100$ because of strong annual parallax effect, indicating that the $+{u}_{0}$ solution is strongly favored over the $-{u}_{0}$ solution. We show in Figure 2 the 3-σ constraints on ${{\boldsymbol{\pi }}}_{{\rm{E}}}$ based on ground-based data alone.

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We then take into account the satellite parallax effect in order to include Spitzer data. We extract the geocentric locations of Spitzer during the entire season from the JPL Horizons website,⁴⁷ and project them onto the observer plane. The projected locations are then oriented and rescaled according to a given ${{\boldsymbol{\pi }}}_{{\rm{E}}}$ to work out Spitzer's view of the microlensing geometry.

We include the $I-[3.6\,\mu {\rm{m}}]$ color constraint on the source star to better constrain the parallax parameters, considering the simple monotonic falling behavior of the Spitzer light curve. This has been demonstrated to be effective in single-lens cases (e.g., Calchi Novati et al. 2015a; Zhu et al. 2017). The $I-[3.6\,\mu {\rm{m}}]$ color constraint comes from putting the measured V − I source color into the $I-[3.6\,\mu {\rm{m}}]$ versus V − I relation, which is derived based on a nearby field stars of similar colors (see Calchi Novati et al. 2015b for more details). This yields $I-[3.6\,\mu {\rm{m}}]=3.18\pm 0.05$ mag. During the MCMC process, we calculate $I-[3.6\,\mu {\rm{m}}]$ from the flux parameters (which are found using a linear fit), and reject any values that are $\gt 3\sigma$ away from the central value.

The constraints on ${{\boldsymbol{\pi }}}_{{\rm{E}}}$ from the simultaneous modeling of ground and Spitzer data are also shown in Figure 2. We note that these are the constraints from annual parallax and satellite parallax signals together. In order to separate constraints from these two types of parallaxes, we assume that the posteriors of the ground-only and ground+Spitzer fits are multivariate Gaussians, and derive the covariance matrix of the satellite parallax part (see the Appendix and Gould 2003) The derived matrix has a determinant that is statistically consistent with zero, which indicates a strong correlation between ${\pi }_{{\rm{E}},{\rm{N}}}$ and ${\pi }_{{\rm{E}},{\rm{E}}}$. This is because the Spitzer observations occasionally only measure a one-dimensional parallax component (see Shvartzvald et al. 2015 for a detailed discussion). We nevertheless proceed with the standard procedure and compute the difference between the annual parallax and the satellite parallax measurements, and find ${\rm{\Delta }}{\chi }^{2}=11$. Although this would formally indicate a probability of ${e}^{-11/2}=0.4 \%$, it is actually well within the systematic uncertainty that the ground-based data can introduce.⁴⁸ Therefore, the small ${\chi }^{2}$ indicates a good agreement between the annual parallax and the satellite parallax.

Notice that we took into account the four-fold parallax degeneracy (often denoted as (++), (+,−), (−,+), and (−,−)). Two of these are eliminated since the $-{u}_{0}$ solution is excluded by the ground-based data, while the (+,−) degeneracy from Spitzer is also eliminated.

3.3. Inclusion of Binary Lens Orbital Motion Effect

We then introduce the lens orbital motion effect into the light curve modeling. Despite the degeneracy between orbital motion and parallax, as we will see below, in this case this has no effect on the measured values or uncertainties of the parallax. We introduce orbital motion into the models in two different ways. First, we use the linear orbital motion approximation, which involves two parameters, ${\text{}}d\alpha /{\text{}}{dt}$ and ${\text{}}{ds}/{\text{}}{dt}$. For the binary system to remain bound, we also impose the constraints on the projected kinetic to potential energy ratio (Dong et al. 2009). Second, we also include z and ${\text{}}{dz}/{\text{}}{dt}$ in addition to ${\text{}}d\alpha /{\text{}}{dt}$ and ${\text{}}{ds}/{\text{}}{dt}$, in order to account for the full Keplerian motion of the binary system. Here z and ${\text{}}{dz}/{\text{}}{dt}$ quantify the binary separation (normalized to the Einstein radius) along the line of sight and its time derivative, respectively. See Skowron et al. (2011) for the conversion between these phase-space parameters and Keplerian parameters. This conversion requires an input of the source angular size ${\theta }_{\star }$ in order to set the absolute physical scale, and we use ${\theta }_{\star }=23.9\,\mu \mathrm{as}$ (see Section 4). Once the Keplerian parameters are derived, we check the orbital period P and reject any solution with ${P}_{\mathrm{orb}}\geqslant 200\,\mathrm{years}$, in order to avoid the influence of systematics in the data.

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The results of two modelings with different treatment of the orbital motion are given in Table 1; the triangle diagram of the fitting parameters for the full orbital model is shown in Figure 3. The source trajectories and caustics of the full orbit motion are shown in Figure 4. This solution has slightly worse ${\chi }^{2}$ than the linear orbital motion solution, even though the former has two more free parameters. This is because the linear orbit solution (with the ratio of the perpendicular kinetic energy to the potential energy, $\beta ={{KE}}_{\mathrm{perp}}/{PE}=0.145$) has preferentially long orbital periods (${P}_{\mathrm{orb}}\gg 200$ years) for the binary system, which are not allowed in the full orbit solution. Nevertheless, the microlensing parameters (especially the microlens parallax) are stable regardless of whether and how the lens orbital motion is included.

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Table 1.  Best-fit Parameters

Parametersa	Ground+Spitzer			Ground-only
	Full Orbit	Linear Orbit	Parallax Only	Full Orbit	Linear Orbit	Parallax Only	Without Parallax
${\chi }^{2}/$dof	4003.5/4028	4000.8/4030	4222.4/4032	3940.1/3967	3936.4/3969	4162.0/3971	5907.2/3973
$\mathrm{log}\,s$	0.0962 ± 0.0003	0.0963 ± 0.0003	0.0940 ± 0.0002	0.0962 ± 0.0003	0.0963 ± 0.0003	0.0940 ± 0.0002	0.08767 ± 0.00009
$\mathrm{log}\,q$	−0.684 ± 0.002	−0.683 ± 0.002	−0.703 ± 0.001	−0.685 ± 0.002	−0.685 ± 0.002	−0.703 ± 0.001	−0.7477 ± 0.0007
u0	0.0716 ± 0.0008	0.0715 ± 0.0008	0.0798 ± 0.0004	0.0711 ± 0.0008	0.0715 ± 0.0008	0.0799 ± 0.0004	0.09109 ± 0.00015
t0	7145.48 ± 0.03	7145.46 ± 0.03	7145.72 ± 0.03	7145.48 ± 0.03	7145.51 ± 0.03	7145.71 ± 0.03	7145.66 ± 0.01
tE (days)	63.56 ± 0.06	63.55 ± 0.06	62.94 ± 0.06	63.74 ± 0.07	63.61 ± 0.08	62.94 ± 0.06	64.69 ± 0.03
$\mathrm{log}\,\rho$	−1.745 ± 0.002	−1.744 ± 0.002	−1.757 ± 0.002	−1.746 ± 0.002	−1.745 ± 0.002	−1.757 ± 0.002	−1.7969 ± 0.0009
α (deg)	219.12 ± 0.08	219.14 ± 0.08	218.27 ± 0.05	219.18 ± 0.08	219.14 ± 0.09	218.28 ± 0.05	217.43 ± 0.03
${\pi }_{{\rm{E}},{\rm{N}}}$	−0.223 ± 0.004	−0.223 ± 0.005	−0.220 ± 0.006	−0.212 ± 0.006	−0.211 ± 0.010	−0.218 ± 0.006	⋯
${\pi }_{{\rm{E}},{\rm{E}}}$	−0.007 ± 0.002	−0.006 ± 0.002	0.003 ± 0.002	−0.010 ± 0.002	−0.010 ± 0.002	0.003 ± 0.002	⋯
$\tfrac{{ds}}{{dt}}$ (year−1)	0.212 ± 0.015	0.217 ± 0.017	⋯	0.240 ± 0.017	0.225 ± 0.018	⋯	⋯
$\tfrac{d\alpha }{{dt}}$ (rad/year)	0.383 ± 0.014	−1.197 ± 0.038	⋯	0.371 ± 0.017	−1.197 ± 0.040	⋯	⋯
z	7.03 ± 3.17	⋯	⋯	4.78 ± 4.52	⋯	⋯	⋯
$\tfrac{{dz}}{{dt}}$ (year−1)	0.037 ± 0.258	⋯	⋯	0.277 ± 0.270	⋯	⋯	⋯
V − I	2.875 ± 0.003	2.874 ± 0.003	2.867 ± 0.003	2.876 ± 0.003	2.874 ± 0.003	2.868 ± 0.002	2.860 ± 0.002

Note.

^aWe use ${\theta }_{* }=23.9\,\mu \mathrm{as}$ in our full orbit model.

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We estimate the angular size of the source following the standard procedure (Yoo et al. 2004). First, we measure the centroid of the red clump in the OGLE $(V-I,I)$ color–magnitude diagram (CMD) of the stars within 2' × 2' of our event (see Figure 5). By using stars in the box $1.8\lt V-I\lt 2.4$ and $15.5\lt I\lt 16.5$, we find the centroid of the red clump to be (V − I, I)_RC = (2.11 ± 0.05, 16.05 ± 0.11). This, when combined with the instrumental color and magnitude of the source star (V − I, I)_S,OGLE = (2.87, 14.34), yields an offset of Δ(V − I, I)_OGLE = (V − I, I)_S,OGLE − (V − I, I)_RC,OGLE = (0.76, −1.71). After applying the correction of the non-standard V band of OGLE-IV (Udalski et al. 2015; Zhu et al. 2015) Δ(V − I)_JC = Δ(V − I)_OGLE × 0.92 = 0.70 (in which "JC" represents the standard Johnson-Cousins system) and adapting the intrinsic color and magnitude of the clump (V − I, I)_RC,0 = (1.06, 14.56) (Bensby et al. 2013; Nataf et al. 2013), we find that the intrinsic color and magnitude of source is (V − I, I)_S,0 = Δ(V − I, I)_JC + (V − I, I)_RC,0 = (1.76, 12.85).

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To determine the source angular size, we employ the color-surface brightness relation of giant stars from Kervella et al. (2004), and finally find

$$\begin{eqnarray}&&{\theta }_{\star }=23.9\pm 1.0\ \mu \mathrm{as}.\end{eqnarray} \tag{ 5 }$$

Therefore,

$$\begin{eqnarray}&&{\theta }_{{\rm{E}}}=1.329\pm 0.049\ \mathrm{mas}.\end{eqnarray} \tag{ 6 }$$

Then we obtain the total mass of the system using Equation (3):

$$\begin{eqnarray}&&M=0.731\pm 0.034\,{M}_{\odot }.\end{eqnarray} \tag{ 7 }$$

Combined with the mass ratio from our fit, we find that the lens system is a binary of $0.606\,{M}_{\odot }$ and $0.125\,{M}_{\odot }$. The lens-source relative parallax is ${\pi }_{\mathrm{rel}}=0.296\pm 0.017\,\mathrm{mas}$, indicating a disk binary. Under the assumption that the source is at the same distance as the red clump centroid at this location (${D}_{{\rm{S}}}=8.8\,\mathrm{kpc}$, Nataf et al. 2013), the distance to the lens is

$$\begin{eqnarray}&&{D}_{{\rm{L}}}=2.44\pm 0.10\,\mathrm{kpc}.\end{eqnarray} \tag{ 8 }$$

The lensing binary is vertically about 120 pc away from the Galactic plane, likely from the thin disk.

The binary components are separated in projection by

$$\begin{eqnarray}&&{r}_{\perp }=4.04\pm 0.23\ \mathrm{au}.\end{eqnarray} \tag{ 9 }$$

We summarize the physical parameters in Table 2.

Table 2.  Physical Parameters of the Binary Lens System MOA-2015-BLG-020

Parameters
${\theta }_{{\rm{E}}}$ (mas)	1.329 ± 0.049
${\pi }_{\mathrm{rel}}$ (mas)	0.296 ± 0.017
M1 (${M}_{\odot }$)	0.606 ± 0.028
M2 (${M}_{\odot }$)	0.125 ± 0.006
Distance to lens (kpc)	2.44 ± 0.10
Projected separation (au)	4.04 ± 0.23
Geocentric proper motion (mas ${\mathrm{yr}}^{-1}$)	7.64 ± 0.28

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We analyzed the binary-lensing event MOA-2015-BLG-020, which was observed both from the ground and from the Spitzer Space Telescope. The light curve from ground-based observations significantly deviated from the lensing model based on the rectilinear lens-source relative motion, and we measured the microlensing parallax from the analysis of the deviation. The measured parallax was confirmed by the Spitzer data, showing the consistency between annual parallax and satellite parallax effects. By additionally measuring the angular Einstein radius from the analysis of the resolved caustic crossing, the mass and distance to the lens are determined. We find that the lens is a binary composed of two low-mass stars located in the Galactic disk. In our analysis, we find that the linear orbit model and full orbit model can fit the data almost equally well. Although the full orbit parameters z and ${\text{}}{dz}/{\text{}}{dt}$ are not well constrained, we report lens physical parameters based on the full orbit model because of its physical foundation (Han et al. 2016).

The binary-lensing event MOA-2015-BLG-020 is peculiar in one aspect. Usually, the light curve for a caustic-crossing binary event has one sharp rise and one sharp fall during the caustic entrance and exit, respectively, producing a "U"-shaped light curve. However, for this binary event, the caustic exit gracefully merges with the cusp crossing (see Figure 4), which causes the absence of the sharp decline in the ground-based light curve. This is almost a direct result of lens orbital motion. Note that this sharp decline is predicted to be present in the space-based light curve, but it is not observable by Spitzer due to various observational constraints (see Udalski et al. 2015 for details).

In Table 3, we summarize the published microlensing binaries in which the parallax parameters detected from ground (through annual parallax effect) are tested by other methods. The parallax parameters based on ground data in OGLE-2011-BLG-0417 and OGLE-2015-BLG-0479 seem to be inconsistent with the results of independent checks. Although there is still the possibility that the parallax detected by microlensing is wrong, it is believed that the radial velocity measurements for OGLE-2011-BLG-0417 do not test the parallax model. The reason is that the blended light of OGLE-2011-BLG-0417 is not the lens because it is brighter than predicted flux from the lens. The inconsistency in OGLE-2015-BLG-0479 is expected because of the strong lens orbital motion effect. Indeed, it is because of this inconsistency that the full Keplerian parameters of the binary in OGLE-2015-BLG-0479 can be well constrained (Han et al. 2016).

Table 3.  Summary of Events that have Annual Parallax Tested by Other Measurements

Event Name	Confirmed?	(${\pi }_{{\rm{E}},{\rm{N}}}$, ${\pi }_{{\rm{E}},{\rm{E}}}$)	Mass Ratio(s)	${t}_{{\rm{E}}}$ (day)	t0(HJD')	u0	${I}_{{\rm{S}}}$	${\rm{\Delta }}{\chi }_{\mathrm{parallax}}^{2}$	${\rm{\Delta }}{\chi }_{\mathrm{orbit}}^{2}$	References
High-resolution Imaging
OGLE-2005-BLG-071	Yes	($-{0.30}_{-0.28}^{+0.24}$, -0.26 ± 0.05)	$1.3\times {10}^{-4}$	71.1	3480.7 (04/20)	0.0282	19.5	$\lt 100$	⋯	1, 2
OGLE-2006-BLG-109	Yes	(−0.316 ± 0.013, 0.139 ± 0.006)	$(1.4,0.5)\times {10}^{-3}$	128.1	3831.0 (04/05)	0.00345	20.9	13.5	⋯	3, 4
OGLE-2007-BLG-349	Yes	0.204 ± 0.034a	$(3.4\times {10}^{-4},0.49)$	118	4348.7 (09/05)	−0.00198	20.4	153	⋯	5
Radial Velocity:
OGLE-2009-BLG-020	Yes	(−0.022 ± 0.086, 0.149 ± 0.010)	0.273	76.9	4917.3 (03/26)	0.0613	16.4	⋯	26.3	6, 7
OGLE-2011-BLG-0417	No	(0.375 ± 0.015, −0.133 ± 0.003)	0.292	92.3	5813.3 (09/08)	−0.0992	⋯	2024	656	8, 9, 10
Satellite Parallax:b
OGLE-2014-BLG-0124	Yes	(0.018 ± 0.012, 0.108 ± 0.023)	$0.752\times {10}^{-3}$	131	6836.2 (06/27)	0.2099	18.6	⋯	⋯	11
OGLE-2015-BLG-0479c	Nod	(−0.06 ± 0.01, −0.11 ± 0.01)	0.81	86.3	7166.4 (05/23)	0.417	19.6	⋯	43.5	12
OGLE-2015-BLG-0196	Yes	(0.198 ± 0.016, 0.100 ± 0.009)	0.99	96.7	7115.5 (04/03)	−0.037	15.9	2527.6	24.4	13
OGLE-2016-BLG-0168	Yes	(0.382 ± 0.022,0.057 ± 0.011)	0724	97.0	7492.5 (04/13)	−0.201	19.5	159.4	115.04	14
MOA-2015-BLG-020	Yes	(−0.223 ± 0.005, −0.007 ± 0.002)	0.207	63.6	7145.5 (05/03)	0.0716	14.3	1983.5	218.9	This work

Notes.

^aBennett et al. (2016) used a different parameterization, with which the amplitude of the parallax vector was constrained. ^bThe microlens parameters are all from ground-based detection unless otherwise stated. The color, magnitude, and ${\rm{\Delta }}{\chi }^{2}$ are from the combination of ground and satellite data sets. ^cParameters of OGLE-2015-BLG-0479 are from the analysis of the combination data set. ^dThat the full orbital motion is detectable in this event is noticeable based purely on the ground-based data. Therefore, this inconsistency is expected.

References. 1. Udalski et al. (2005), 2. Dong et al. (2009), 3. Gaudi et al. (2008), 4. Bennett et al. (2010), 5. Bennett et al. (2016), 6. Skowron et al. (2011), 7. Yee et al. (2016), 8. Shin et al. (2012), 9. Boisse et al. (2015), 10. Santerne et al. (2016), 11. Udalski et al. (2015), 12. Han et al. (2016), 13. Han et al. (2017), 14. Shin et al. (2017).

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For the remaining events, the microlensing parallax parameters from the annual parallax effect are confirmed by additional observations (high-resolution imaging, radial velocity, or satellite parallax). As expected, they all have relatively long timescales (${t}_{{\rm{E}}}\gtrsim 60$ days), and the majority of them peaked (as seen from ground) either early (before May) or late (after August) in the microlensing season. Among these events, half are confirmed by ground observations (adaptive optics or radial velocity) and they all have a small impact parameter u₀, while those confirmed by satellite observations could have larger u₀ (modest magnification). This highlights Spitzer's power to discover parallax events for moderately magnified events. The abundance of stellar binaries with $-1\leqslant \mathrm{log}q\leqslant 0$ is roughly uniform as a function of $\mathrm{log}q$ (${\text{}}{dN}/{\text{}}{dq}\propto {q}^{-1}$), consistent with Trimble (1990), although we caution that the number of events is small and no selection effects have been taken into account. As more and more lenses with definite masses are determined from microlensing, it will be very interesting to study these statistics much more carefully in the future.

This work has been supported in part by the National Natural Science Foundation of China (NSFC) grants 11333003 and 11390372 (S.M.). This research uses data obtained through the Telescope Access Program (TAP), which has been funded by the Strategic Priority Research Program "The Emergence of Cosmological Structures" (grant No. XDB09000000), National Astronomical Observatories, Chinese Academy of Sciences, and the Special Fund for Astronomy from the Ministry of Finance. Work by W.Z., Y.K.J., I.G.S., and A.G. was supported by AST-1516842 from the US NSF and JPL grant 1500811. Work by J.C.Y. was performed in part under contract with the California Institute of Technology (Caltech)/Jet Propulsion Laboratory (JPL) funded by NASA through the Sagan Fellowship Program executed by the NASA Exoplanet Science Institute. This research has made use of the KMTNet system operated by the Korea Astronomy and Space Science Institute (KASI) and the data were obtained at three host sites of CTIO in Chile, SAAO in South Africa, and SSO in Australia. The OGLE Team thanks Drs. G. Pietrzyński and Ł. Wyrzykowski for their contribution to the collection of the OGLE photometric data over the past years. The OGLE project has received funding from the National Science Centre, Poland, grant MAESTRO 2014/14/A/ST9/00121 to A.U. Work by C.H. was supported by the grant (2017R1A4A1015178) of the National Research Foundation of Korea. This work makes use of observations from the LCO network, which includes three SUPAscopes owned by the University of St Andrews. The RoboNet programme was an LCO Key Project using time allocations from the University of St Andrews, LCOGT, and the University of Heidelberg together with time on the Liverpool Telescope through the Science and Technology Facilities Council (STFC), UK. This research has made use of the LCO Archive, which was operated by the California Institute of Technology, under contract with the Las Cumbres Observatory. The MOA project is supported by JSPS KAKENHI grant Nos. JSPS24253004, JSPS26247023, JSPS23340064, JSPS15H00781, and JP16H06287.

Software: corner.py (Foreman-Mackey 2016), emcee (Foreman-Mackey et al. 2013), VBBinaryLensing (Bozza 2010).

Let a and c denote a parallax measurement and its covariance matrix, and the subscripts "comb" and "ground" indicate quantities appropriate for the combined and ground-based data. For example, ${a}_{\mathrm{comb}}$ and ${c}_{\mathrm{comb}}$ are the measured parallax and its covariance matrix from our MCMC fits for the combined data. We follow Gould (2003) to derive the "Spitzer" parallax and the consistency between the ground and Spitzer parallaxes (expressed as ${\rm{\Delta }}{\chi }^{2}$) through the following steps. First, we introduce several quantities for later use

$$\begin{eqnarray*}{b}_{\mathrm{comb}} & = & {({c}_{\mathrm{comb}})}^{-1},{b}_{\mathrm{ground}}={({c}_{\mathrm{ground}})}^{-1},\\ {({d}_{\mathrm{comb}})}_{i} & = & \displaystyle \sum _{j}{({b}_{\mathrm{comb}})}_{{ij}}\times {({a}_{\mathrm{comb}})}_{j},\\ {({d}_{\mathrm{ground}})}_{i} & = & \displaystyle \sum _{j}{({b}_{\mathrm{ground}})}_{{ij}}\times {({a}_{\mathrm{ground}})}_{j}.\end{eqnarray*}$$

Then we calculate the corresponding quantities for the Spitzer parallax

$$\begin{eqnarray*}{b}_{\mathrm{spitzer}} & = & {b}_{\mathrm{comb}}-{b}_{\mathrm{ground}},{d}_{\mathrm{spitzer}}={d}_{\mathrm{comb}}-{d}_{\mathrm{ground}},\\ {c}_{\mathrm{spitzer}} & = & {({b}_{\mathrm{spitzer}})}^{-1},\\ {({a}_{\mathrm{spitzer}})}_{i} & = & \displaystyle \sum _{j}{({c}_{\mathrm{spitzer}})}_{{ij}}\times {({d}_{\mathrm{spitzer}})}_{j}.\end{eqnarray*}$$

Finally, we compute the difference between the annual and satellite parallax measurements

$$\begin{eqnarray*}&&{\rm{\Delta }}{\chi }^{2}=\displaystyle \sum _{{ij}}{({a}_{\mathrm{diff}})}_{i}\times {({b}_{\mathrm{diff}})}_{{ij}}\times {({a}_{\mathrm{diff}})}_{j},\end{eqnarray*}$$

where

$$\begin{eqnarray*}{a}_{\mathrm{diff}} & = & {a}_{\mathrm{ground}}-{a}_{\mathrm{spitzer}},\\ {c}_{\mathrm{diff}} & = & {c}_{\mathrm{ground}}+{c}_{\mathrm{spitzer}},\\ {b}_{\mathrm{diff}} & = & {({c}_{\mathrm{diff}})}^{-1}.\end{eqnarray*}$$