The Star Blended with the MOA-2008-BLG-310 Source Is Not the Exoplanet Host Star

Most of the known exoplanets have been discovered by the Doppler radial velocity method (Mayor & Queloz 1995; Butler et al. 2006) or by the transit method (Pollacco et al. 2006), with the largest number coming from the Kepler exoplanet transit mission (Burke et al. 2006; Borucki et al. 2010). Despite the large number of exoplanets discovered, our knowledge about the distribution of exoplanets by these methods is limited by the selection effects of these methods. Most of these planets are hot or warm planets at small orbital (≤1 au) separation from their host stars, and the planets in wider orbits are generally more massive than Saturn. Microlensing is the only method that is sensitive to the low-mass planets at orbital separations larger than the snow line. According to the core accretion theory of the planet formation (Lissauer 1993), the planet formation process is most efficient beyond the snow line (Kennedy et al. 2006; Lecar et al. 2006), where the protoplanetary disk is cold enough for ices to condense. This gives a higher density of solid material that can coagulate to start the planet formation process. Hence, the microlensing method allows us to study the demographics of the planetary systems in the favored planetary birthplace, beyond the snow line.

Gravitational microlensing is the method for detecting the exoplanets with masses as low as an Earth mass (Bennett & Rhie 1996) at a distance ∼1–8 kpc from Earth. Since the technique does not depend on the light from the exoplanet or its host star, it is very effective in detecting planets that orbit very faint stars, including planets orbiting stars in the Galactic bulge. A number of planets with a host star probably in the bulge have already been discovered, including OGLE-2005-BLG-390 (Beaulieu et al. 2006), OGLE-2008-BLG-092 (Poleski et al. 2014), OGLE-2008-BLG-355 (Koshimoto et al. 2014), MOA-2008-BLG-310Lb (Janczak et al. 2010), MOA-2009-BLG-319 (Miyake et al. 2011), MOA-2011-BLG-262 (Bennett et al. 2014), MOA-2011-BLG-293 (Batista et al. 2013), and OGLE-2014-BLG-1760Lb (Bhattacharya et al. 2016).

To date, ∼50 planets have been discovered using microlensing, and 30 of these have been used to derive the exoplanet mass ratio function (Suzuki et al. 2016), which describes the occurrence rate of planets as a function of their mass ratio, q, and separation in Einstein radius units. To extend these results to find exoplanetary mass as a function of the host star mass and galactocentric distance, we must determine the planet and host star masses and their distance from Earth. For most planetary microlensing light-curve events we obtain the planet–host star mass ratio, the separation in Einstein radius, and the source radius crossing time, t_*, which leads to a determination of the angular Einstein radius, ${\theta }_{E}$. This is not enough information to measure the host star and planet masses, but we can estimate them with a Bayesian analysis using a Galactic model, with the assumption that the exoplanet mass function does not depend on the mass or distance of the host star. The masses of the planets and host star can be determined for events that include a measurement of the microlensing parallax effect (Gould 1994, 1995, 1999; Gaudi et al. 2008) or by detecting the lens in the high-resolution follow-up images (Bennett et al. 2006; Batista et al. 2015; Bennett et al. 2015). In some cases, both microlensing parallax and high-resolution follow-up imaging can provide mass measurements by independent methods (Gaudi et al. 2008; Bennett et al. 2010, 2016; Beaulieu et al. 2016). High-resolution image analyses of planetary microlensing events have already yielded the mass measurements or upper limits for OGLE-2003-BLG-235 (Bennett et al. 2006), OGLE-2006-BLG-071Lb (Dong et al. 2009b), OGLE-2005-BLG-169Lb (Batista et al. 2015; Bennett et al. 2015), OGLE-2007-BLG-368 (Sumi et al. 2010), MOA-2007-BLG-192 (Kubas et al. 2012), MOA-2008-BLG-310 (Janczak et al. 2010), MOA-2011-BLG-262Lb (Bennett et al. 2014), MOA-2011-BLG-293 (Batista et al. 2013), OGLE-2012-BLG-0026 (Beaulieu et al. 2016), OGLE-2012-BLG-0563Lb (Fukui et al. 2015), OGLE-2012-BLG-0950Lb (Koshimoto et al. 2016), and MOA-2013-BLG-605Lb (Sumi et al. 2016). In this paper we take a second look at the exoplanetary microlensing event MOA-2008-BLG-310 with two epochs of Hubble Space Telescope (HST) imaging taken in 2012 and 2014. While the discovery paper presented excess starlight at the position of the source with VLT adaptive optics (AO) imaging, our HST images allow us to determine whether the star or stars responsible for this excess flux have a lens–source relative proper motion that is consistent with the lens (and planetary host) star.

The MOA (Microlensing Observations in Astrophysics) group identified the microlensing event MOA-2008-BLG-310 on 2008 July 6 and 2 days later issued a high-magnification alert. The peak of this event and its planetary anomaly was meticulously covered by μFUN (Microlensing Follow Up Network) in CTIO I, V, H and μFUN R and Bronberg unfiltered passbands. MiNDSTEp and the PLANET collaboration also took data in the I band. This event was also observed using the VLT/NACO AO system on 2008 July 28, and these data showed additional H-band flux on top of the source, suggesting a possible detection of the planetary host star. However, as Janczak et al. (2010) pointed out, it is also possible that the excess flux could be due to a companion to the source, a companion to the lens, or an unrelated star. There was no ambiguity in the interpretation of the excess flux for planetary microlensing event OGLE-2005-BLG-169 because the high angular resolution follow-up observations from HST and Keck were able to demonstrate the lens–source relative proper motion and measure the host star flux in four passbands (Batista et al. 2015; Bennett et al. 2015). In this paper, we present a similar analysis of planetary microlensing event MOA-2008-BLG-310. It was observed by HST/WFC3-UVIS in the V and I bands in 2012 and 2014, 3.62 and 5.59 yr after the peak, respectively. In this paper we present the analyses of these HST images to confirm that the star blended with the source is not the planetary host star.

The paper is organized as follows: Section 2 presents light-curve modeling of a slightly different data set than used for the the discovery paper owing to a re-reduction of MOA light-curve data that corrected for systematic errors due to differential refraction. Sections 3 and 4 determine the source color and angular radius from the parameters presented in Section 2. HST follow-up image analyses with different point-spread function (PSF) fits are discussed in Section 5. Sections 5.1–5.3 explore the results of fitting single-star, dual-star, and triple-star PSFs. Section 9 presents the upper limit of the lens brightness and proceeds with the calculations of the lens properties.

A single microlens event light-curve model uses three nonlinear parameters: t₀—the time of peak magnification; u₀—the minimum separation between the source and the lens in Einstein radius units; and t_E—the Einstein radius crossing time. The Einstein radius is given by ${R}_{E}=\sqrt{(4{GM}/{c}^{2}){D}_{S}x(1-x)}$, where $x={D}_{L}/{D}_{S}$ and D_L and D_S are the lens and source distances, respectively. (G and c are the gravitational constant and speed of light, as usual.) There are also two linear parameters for each data set: the source flux, f_s, and the blend flux, f_bl. To fit a binary microlens model, we need three additional nonlinear parameters: q, the lens mass ratio, s, the projected separation between the lens masses measured in the Einstein radius units, and θ, the angle between the source trajectory and the lens axis. Also, binary events often have caustic or cusp crossings, which resolve the angular size of the source, so we need to model the finite source effects with the source radius crossing time, ${t}_{\star }$. In the years since the original paper on this event (Janczak et al. 2010), we have found that it is possible to improve the photometry for many events by removing trends due to air mass, differential refraction, and seeing that are observed in the data before and after the microlensing event. The removal of these systematic error trends can sometimes modify the best-fit t_E and f_s values, so we thought it prudent to use the new photometry. In this case, the detrended photometry resulted in slightly different parameters, but no large change in t_E or f_s was seen.

We modeled MOA red (R + I), CTIO SMARTS I, H, μFUN Auckland R, μFUN Bronberg unfiltered, and PLANET Canopus I-band data using the ${\chi }^{2}$ minimization recipe of Bennett (2010) to find the best-fit wide ($s\gt 1$) and close (s < 1) models, as shown in Table 1. These $s\leftrightarrow 1/s$ degenerate models (Griest & Safizedah 1998) are due to the usual high-magnification separation degeneracy as noted by Janczak et al. (2010). Once the best-fit models were identified, we ran several Markov chain Monte Carlos (MCMCs; Verde et al. 2003) to determine the distribution of parameters that are consistent with the light-curve measurements. The uncertainties are given by the rms variations over the MCMC links for each parameter, as shown in Table 1. The methods of error bar renormalization and the calculation of the limb-darkening effects are similar to those of Janczak et al. (2010). The difference between the models presented in Table 1 and those of the discovery paper is not noticeable in a light-curve plot.

Table 1.  Microlensing Model Parameters

Parameter	Units	Best Fit		MCMC Averages
		Close	Wide	No Constraint	Lens Brightness Constraineda
tE	days	10.22	10.29	10.27(0.27)	10.27(0.24)
t0	HJD −2,450,000	4656.39	4656.39	4656.39(0.00011)	4656.39(.00011)
u0	10−3	3.26	3.18	3.21(0.09)	3.21(0.09)
s	⋯	0.93	1.08	1.04(0.07)	1.04(0.07)
θ	rad	1.95	1.93	1.94(0.02)	1.94(0.02)
q	10−4	3.29	3.49	3.38(0.28)	3.38(0.28)
${t}_{\star }$	days	0.054	0.055	0.055(0.00011)	0.055(0.00011)
χ2	⋯	6616.88	6618.93	⋯	⋯

Note.

^aThe lens brightness constraint is based on the triple-star PSF fits discussed in Sections 5.3 and 9.

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The models listed in Table 1 yield the source brightness in the CTIO SMARTS I and H bands as ${I}_{\mathrm{CTIO}}=18.93\pm 0.03$ and ${H}_{\mathrm{CTIO}}=21.47\pm 0.03$. We use OGLE-III and VVV magnitudes as the calibrated magnitudes in the visible and infrared bands, respectively. We calibrate the CTIO I-band reference image to the OGLE-III catalog (Szymański et al. 2011) using 248 isolated stars of brightness ${I}_{{\mathrm{OGLE}}_{\mathrm{III}}}\lt 16.0$. We obtained the following calibration relation:

$\begin{eqnarray}{I}_{\mathrm{CTIO}} & = & {I}_{{\mathrm{OGLE}}_{\mathrm{III}}}+0.057998{(V-I)}_{{\mathrm{OGLE}}_{\mathrm{III}}}\\ & & -0.596221\pm 0.02.\end{eqnarray} \tag{ 1 }$

The CTIO H-band reference image is matched with the VVV catalog (Saito et al. 2012), and 139 bright isolated stars are cross-identified to obtain the H-band calibration relation:

$\begin{eqnarray}&&{H}_{\mathrm{CTIO}}={H}_{2\mathrm{MASS}}+3.781\pm 0.004.\end{eqnarray} \tag{ 2 }$

The uncertainties in Equations (1) and (2) are given by rms/$\sqrt{N}$, where N represents the number of stars used. Equation (2) yields a best-fit source magnitude of ${H}_{2\mathrm{MASS}}=17.69\pm 0.03$.

Our aim is to derive ${I}_{{\mathrm{OGLE}}_{\mathrm{III}}}$ from Equation (1), but we do not have a measurement of ${(V-I)}_{{\mathrm{OGLE}}_{\mathrm{III}}}$. There is only a single CTIO V-band measurement that is magnified, and that measurement has a 25% uncertainty. So, we cannot get an accurate source color measurement from the CTIO V-band data. Therefore, we choose an iterative method to determine the source color with the help of the color–color relations of Kenyon & Hartmann (1995). This iterative method utilizes ${I}_{\mathrm{CTIO}}$, ${H}_{\mathrm{CTIO}}$, and Equations (1) and (2). The steps of this method are described in detail below and in Figure 1. We use the extinction values of $E(V-I)=0.822$, A_I = 1.013, and A_V = 1.835 from Nataf et al. (2013). From the knowledge of A_I and A_V, using Cardelli et al. (1989), we find A_H = 0.16. The extinction-corrected H-band source magnitude is ${H}_{0}\,={H}_{2\mathrm{MASS}}-{A}_{H}=17.53\pm 0.03$. Also we use Equation (2) providing the source magnitude ${H}_{2\mathrm{MASS}}=17.69\pm 0.03$ and Equation (1) for this iterative method.

• Step 1. We assume ${(V-I)}_{{\mathrm{OGLE}}_{\mathrm{III}}}=1.45$.
• Step 2. With the value of ${I}_{\mathrm{CTIO}}$ and ${(V-I)}_{{\mathrm{OGLE}}_{\mathrm{III}}}$, we determine ${I}_{{\mathrm{OGLE}}_{\mathrm{III}}}$ from Equation (1).
• Step 3. With ${(V-I)}_{{\mathrm{OGLE}}_{\mathrm{III}}}$ from step 1 and ${I}_{{\mathrm{OGLE}}_{\mathrm{III}}}$ from step 2, we determine ${V}_{{\mathrm{OGLE}}_{\mathrm{III}}}={I}_{{\mathrm{OGLE}}_{\mathrm{III}}}+{(V-I)}_{{\mathrm{OGLE}}_{\mathrm{III}}}$.
• Step 4. From ${V}_{{\mathrm{OGLE}}_{\mathrm{III}}}$ in step 3, A_V = 1.835, and ${H}_{0}\,=17.53\pm 0.03$, we find the extinction-corrected ${(V-H)}_{0}$.
• Step 5. This extinction-corrected ${(V-H)}_{0}$ gives a new value of the extinction-corrected ${(V-I)}_{0}$ using the color–color relation of Kenyon & Hartmann (1995).
• Step 6. We add $E(V-I)$ to this value to get the new $(V-I)$ for our next iteration.

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In the next iteration we use this new $(V-I)$ and repeat steps 2–6, and we continue this iteration over $(V-I)$ until its value converges. The method is shown in Figure 1, and the values for each iteration round are shown in Table 2. This method converges to the source color ${(V-I)}_{{\mathrm{OGLE}}_{\mathrm{III}}}=1.64\pm 0.06$ and the source magnitude ${I}_{{\mathrm{OGLE}}_{\mathrm{III}}}\,=19.43\pm 0.04,{V}_{{\mathrm{OGLE}}_{\mathrm{III}}}=21.07\pm 0.07$. With the extinction values given above, we find an extinction-corrected magnitude of ${I}_{s0}=18.41\pm 0.04$ and an extinction-corrected color of ${(V-I)}_{s0}=0.82\pm 0.06$. This is somewhat redder than the ${(V-I)}_{s0}=0.69$ claimed by the discovery paper (Janczak et al. 2010), but it is consistent with the estimate of ${(V-I)}_{s0}=0.75\pm 0.05$ based on a spectrum measured at high magnification (Bensby et al. 2013). Using the source color from Bensby et al. (2013), we obtain I- and V-band magnitudes of the source of 19.44 ± 0.04 and 21.01 ± 0.07, respectively. These values are consistent with the values obtained in our iteration method. According to Kenyon & Hartmann (1995), the source star is most likely a G5–K0 star.

Table 2.  Deriving Source Color Using Iteration from Kenyon & Hartmann (1995) Color–Color Relation^*

Iteration	V − I	${I}_{{\mathrm{OGLE}}_{\mathrm{III}}}$	${V}_{{\mathrm{OGLE}}_{\mathrm{III}}}$	${(V-H)}_{\mathrm{ext}\_\mathrm{corr}}$	${(V-I)}_{\mathrm{ext}\_\mathrm{corr}}$	New $(V-I)$
Number		Equation (1) + ${I}_{\mathrm{CTIO}}$
1	1.450	19.44	20.89	1.53	0.76	1.582
2	1.582	19.43	21.02	1.65	0.79	1.611
3	1.611	19.43	21.04	1.68	0.8	1.622
4	1.622	19.43	21.05	1.69	0.81	1.631
5	1.631	19.43	21.06	1.70	0.815	1.637
6	1.637	19.43	21.07	1.71	0.82	1.642
7	1.642	19.43	21.07	1.71	0.82	1.639
8	1.639	19.43	21.07	1.71	0.82	1.640

Note. * Refer to Figure 1 for details. The color of the star was not readily available in Cousins I and Johnson V from observations; hence, this iteration method was adopted.

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From the dereddened source magnitude and color, we obtain the source radius using the relations from Boyajian et al. (2014). From the fit parameters, ${t}_{\star }=0.055$ days and t_E = 10.27 days, and from the knowledge of ${\theta }_{\star }$, ${\theta }_{E}$ is calculated from ${\theta }_{E}={\theta }_{\star }/{t}_{\star }\times {t}_{E}$ and ${\mu }_{\mathrm{rel}}={\theta }_{\star }/{t}_{\star }={\theta }_{E}/{t}_{E}$:

$\begin{eqnarray}{\mathrm{log}}_{10}(2{\theta }_{\star }) & = & 0.53665+0.072703(V-H)-0.2H\\ {\theta }_{\star } & = & 0.719\pm 0.023\,\mu \mathrm{as},\end{eqnarray} \tag{ 3 }$

$\begin{eqnarray}{\mathrm{log}}_{10}(2{\theta }_{\star }) & = & 0.53026+0.16595(I-H)-0.2H\\ {\theta }_{\star } & = & 0.743\pm 0.054\,\mu \mathrm{as}.\end{eqnarray} \tag{ 4 }$

The discovery paper has ${\theta }_{\star }=0.76\pm 0.05\,\mu \mathrm{as}$ and ${\mu }_{\mathrm{rel}}\,=5.1\pm 0.3$ mas yr⁻¹. The relative proper motion ${\mu }_{\mathrm{rel}}$ mentioned in this section is the geocentric relative proper motion. We derive the weighted average of ${\theta }_{\star }=0.723\pm 0.021\,\mu \mathrm{as}$. The corresponding ${\mu }_{\mathrm{rel}}=4.85\pm 0.15$ mas yr⁻¹.

The uncertainty is calculated from the standard error over MCMC chains. The uncertainty arises from the 0.06 mag uncertainty in the source color and the 0.06 mag uncertainty in the source magnitude. There are also 2.4% and 7% uncertainties in the source star angular diameter arising from the scatter about the source star size–color relations in Equations (3) and (4) of Boyajian et al. (2014). Equation (3) is not included in Boyajian et al. (2014); it was obtained through a private communication with T. S. Boyajian. She reports a 2% scatter of measurements around the fit in the $V-H,H$ relation. Kervella et al. (2004) report a 1.12% uncertainty in the ${\theta }_{* }$ value for their $V-H,H$ relation. The difference is that Boyajian et al. (2014) just report the scatter in the fit, while Kervella et al. (2004) subtract the scatter due to the photometric measurement uncertainties. Hence, the Boyajian et al. (2014) method is very conservative, and the Kervella et al. (2004) method should be more accurate as long as the photometry error bars are accurately estimated. If the photometry error bars are overestimated, then the error bars on the relation will be underestimated. Because of this controversy, we prefer to use the more conservative approach of Boyajian et al. (2014). There is also uncertainty due to ${t}_{\star }$ and t_E (see Table 1).

The event MOA-2008-BLG-310 was observed with the HST Wide Field Camera 3 Ultraviolet Visible (WFC3-UVIS) instrument on 2012 February 22 as part of the program GO 12541 with a second epoch of observations on 2014 February 09. Both observation epochs used both the F814W and F555W passbands (which are HST versions of I and V bands). In each epoch, eight images were taken with the exposure times of 70 and 125 s for F814W and F555W filters, respectively. (Hereafter, we refer to the F814W and F555W passbands as the HST I and V bands.) To obtain this many short dithered exposures, it was necessary to read out only a small 1k × 1k subset of each image. Each WFC3-UVIS pixel subtends approximately 40 mas on a side. These dithered images were reduced and stacked following the methods described in Anderson & King (2000, 2004). Fifteen bright isolated stars with $1.0\,\leqslant {(V-I)}_{{\mathrm{OGLE}}_{\mathrm{III}}}\leqslant 2.1$ and I < 17.0 are cross-identified and matched between HST stack images and stars from the OGLE-III catalog (Szymański et al. 2011) to obtain the following calibration relations for the 2012 epoch:

$\begin{eqnarray}{I}_{{\mathrm{OGLE}}_{\mathrm{III}}} & = & 29.078375+{I}_{{\text{}}{HST}}+0.004528\\ & & \times {(V-I)}_{{\text{}}{HST}}\pm 0.02023\\ {V}_{{\mathrm{OGLE}}_{\mathrm{III}}} & = & 30.593088+{V}_{{\text{}}{HST}}-0.065837\\ & & \times {(V-I)}_{{\text{}}{HST}}\pm 0.03233.\end{eqnarray} \tag{ 5 }$

Similarly for the 2014 epoch, 14 stars with $1.0\,\leqslant {(V-I)}_{{\mathrm{OGLE}}_{\mathrm{III}}}\leqslant 1.9$ and I < 17.0 are matched to obtain these calibration relations:

$\begin{eqnarray}{I}_{{\mathrm{OGLE}}_{\mathrm{III}}} & = & 29.054552+{I}_{{\text{}}{HST}}-0.024727\\ & & \times {(V-I)}_{{\text{}}{HST}}\pm 0.023431\\ {V}_{{\mathrm{OGLE}}_{\mathrm{III}}} & = & 30.563614+{V}_{{\text{}}{HST}}-0.092588\\ & & \times {(V-I)}_{{\text{}}{HST}}\pm 0.043662.\end{eqnarray} \tag{ 6 }$

The uncertainties are due to PSF shapes and the linear fits.

The images in each band are reduced with correction for charge transfer efficiency losses using a method developed specifically for WFC3-UVIS based on the algorithm described in Anderson & Bedin (2010) for ACS. The images are taken in a custom dither pattern that ensured the most uniform possible pixel-phase coverage for the eight exposures. We adopted the first exposure of each passband as the reference image. The we measured the stars in this image and in all the other images with a library PSF and corrected them for distortion. We then used the positions of the stars common to each exposure and the reference exposure to define a six-parameter linear transformation from each frame into the reference frame. This allowed us to transform the location of each pixel into the reference frame, both for the purposes of stacking and for the purposes of using them as simultaneous constraints in modeling the scene. We stack all these individual images into a single frame (Fruchter & Hook 2002), or stack image, for each passband and identify the target object in the stack images using the NACO VLT high-resolution image presented in the discovery paper (Janczak et al. 2010), which was identified based on the location found in difference images taken near peak magnification. Next, we select about 23 isolated stars within a 150-pixel radius of the target object and with a color similar to the target. We use these stars to do another round of mapping between different frames. The goal of this step is to generate more precise local coordinate transformations from each frame to the reference frame and to derive a more accurate PSF for the target. We extract the pixels in the vicinity of the target from each exposure and transform their locations into the reference frame. We use affine transformations between pixel coordinates for this step. Then we solve for the effective PSF (Anderson & King 2000) appropriate for the target object.

The main problem in dealing with HST images relative to ground-based images is that HST images are undersampled. This means that a large portion of the flux of a star falls inside one pixel. As a result, several different PSF models can provide equally good fits to the data (Anderson & King 2000). The pixels are too wide to sample all of the information that the telescope is delivering to the detector. This is not a big problem for determining total flux using aperture photometry as long as stellar images are reasonably isolated. But it is a big problem for astrometry. The degeneracy in the PSF fits yields degeneracies in the positions of stars, making it impossible to measure precise stellar positions. PSF fitting photometry routines designed for ground-based data, like DAOPHOT (Stetson 1987) and DOPHOT (Schechter et al. 1993), are not designed to deal with undersampled data, and they are ill-equipped to deal with intrapixel-scale detector sensitivity variations, which can be important for undersampled data. Theoretically, one might attempt to deal with undersampling by separately determining the instrumental PSF (iPSF) and the subpixel-scale response function, but Anderson & King (2000) point out that it is simpler and more accurate to deal with the effective PSF (ePSF), which is the convolution of the iPSF and the detector response,

$\begin{eqnarray}&&{\psi }_{E}={\mathfrak{R}}\times {\psi }_{{\imath }},\end{eqnarray} \tag{ 7 }$

where ${\mathfrak{R}}$ and ${\psi }_{{\imath }}$ are the intrapixel sensitivity function and iPSF, respectively. We selected about 19 isolated stars within a 150-pixel radius and within 0.1 mag color and 0.5 mag V-band brightness of the target star. These stars were used to build the ePSF. We used all the individual images of these stars to determine the effective PSF following the method of Anderson & King (2000, 2004). We select a pixel region centered on each of the PSF-contributing stars and divide it into a grid with a quarter-pixel interval. The ePSF is evaluated on these grid points, and the intermediate points are interpolated using cubic spline interpolation. The value of the ePSF is the fraction of a star's light that should fall in a pixel centered on the specified coordinates. Hence,

$\begin{eqnarray}&&{P}_{{ij}}={z}_{* }{\psi }_{E}(i-{x}_{* },j-{y}_{* })+{s}_{* },\end{eqnarray} \tag{ 8 }$

or conversely,

$\begin{eqnarray}&&{\psi }_{E}(i-{x}_{* },j-{y}_{* })=({P}_{{ij}}-{s}_{* })/{z}_{* },\end{eqnarray} \tag{ 9 }$

where ${x}_{* },{y}_{* }$ is the position of the star and P_ij is the observed pixel value of the pixel centered at (i, j). The parameters s_* and z_* are the background flux and the the total flux of the star, respectively. We start with the total flux of the star measured by aperture photometry as z_* and the centroid of the star as the position ${x}_{* },{y}_{* }$. For the background flux we calculated the flux between 8.5 and 13 pixels from the center of the star. The ePSF is computed with an iterative procedure. In each iteration, we determine the ePSF for each of those 19 stars in each individual image. Then we average the ePSFs of each star from all the images combined. It is this step of averaging of the ePSFs from all the dithered images that helps to overcome the undersampling problem. Now we use this average ePSF to get a best fit of the star by minimizing:

$\begin{eqnarray}&&\chi 2=\displaystyle \sum _{i,j}{w}_{{ij}}{[{P}_{{ij}}-{s}_{* }-{z}_{* }{\psi }_{E}(i-{x}_{* },j-{y}_{* })]}^{2}.\end{eqnarray} \tag{ 10 }$

We do not fix the value of (${x}_{* },{y}_{* }$) in this minimization procedure, so it yields a new set of values for star positions, which is used to calculate the effective PSF from Equation (8) in the next iteration step. This iteration procedure converged to our final ePSF after six iterations.

Once we have determined our ePSF model, we are ready to fit our target object with two-star models, so that both the source and lens stars can be included. We also consider three-star models in cases (like this one) where the two-star models do not provide a good fit to the properties of the source and lens stars. Our model for the total star flux distribution changes from ${z}_{* }{\psi }_{E}(i-{x}_{* },j-{y}_{* })$ for a single star to ${f}_{1}{\psi }_{E}(i-{x}_{1},j-{y}_{1})\,+(1-{f}_{1}){\psi }_{E}(i-{x}_{2},j-{y}_{2})$ for the dual-star model. This increases the number of model parameters by three, for the brightness and coordinates of the second star. The parameter f₁ denotes the ratio of the star-1 brightness to the total stellar brightness of both stars, so $(1-{f}_{1})$ is the fractional brightness of the second star. The parameters ${x}_{1},{y}_{1}$ and ${x}_{2},{y}_{2}$ are the positions of the two stars. We have two different strategies for these ePSF fits. For the simplest, two-star models, we start with a simple grid search that gives us a list of χ² values for all parameter sets that fall on the parameter grid. This is robust, but inefficient. It is much more efficient to use the MCMC method, which avoids highly unlikely parameter choices. We use the MCMC method to fit ${x}_{1},{y}_{1},{x}_{2},{y}_{2},{f}_{1}$ in order to minimize the following χ² for each individual image:

$\begin{eqnarray}\chi 2 & = & \displaystyle \sum _{i,j}{w}_{{ij}}[{P}_{{ij}}-{s}_{* }-{f}_{1}{\psi }_{E}(i-{x}_{1},j-{y}_{1})\\ & & +{(1-{f}_{1}){\psi }_{E}(i-{x}_{2},j-{y}_{2})]}^{2}.\end{eqnarray} \tag{ 11 }$

The "chain" of solutions generated with the MCMC serves as a probability distribution of parameters that we use to determine the uncertainties on the PSF fit parameters.

The R.A. and decl. of the stack images are related to x and y positions using Equations (12) and (13) for 2012 and 2014 data, respectively:

$\begin{eqnarray}{\rm{R}}.{\rm{A}}. & = & 2.56\times {10}^{-2}(x-558.4)+3.08\\ & & \times {10}^{-2}(y-623.2)+966819.47\\ \mathrm{Decl}. & = & 3.05\times {10}^{-2}(x-558.4)-2.56\\ & & \times {10}^{-2}(y-623.2)-125199.89,\end{eqnarray} \tag{ 12 }$

$\begin{eqnarray}{\rm{R}}.{\rm{A}}. & = & 2.53\times {10}^{-2}(x-498.6)+3.09\\ & & \times {10}^{-2}(y-635.1)+966818.03\\ \mathrm{Decl}. & = & 2.95\times {10}^{-2}(x-498.6)-2.38\\ & & \times {10}^{-2}(y-623.2)-125201.95.\end{eqnarray} \tag{ 13 }$

The R.A. and decl. are expressed in arcseconds. From these relations it is clear that the x and y pixel positions of lens and source are different in different frames. But the relative separation between the lens and the source is independent of the frames. Each pixel is ∼40 mas. We will be using separations in x and y pixel coordinates in this paper.

5.1. Single-star PSF Fit

5.2. Dual-star PSF Fits

In dual-star PSF fits, we expect to fit and detect the source and the lens. The source magnitudes in I and V bands are 19.43 ± 0.04 and 21.07 ± 0.07, respectively. The geocentric lens–source relative proper motion is 4.81 ± 0.15 mas yr⁻¹. Since the light-curve data of this event were very well covered (Janczak et al. 2010), there is a very small scope of the light-curve model to be incorrect. Following the relative proper motion, we expect to see a lens- source separation of about 17.4 ± 0.4 and 27.4 ± 0.7 mas in the first and second epochs, respectively. We can claim the detection of lens and source if and only (a) the brightness of one star is consistent with the brightness of the source in each passband and (b) the separation measured between the two stars is consistent with the predicted lens–source separation in each epoch.

5.2.1. Unconstrained Best Fit

For the dual-star ePSF models there are total of six parameters to fit. These are the pixel positions of the two stars in x and y and the total flux and the flux ratio between the two stars. The dual-star models were run with both the grid search and MCMC methods, which yielded essentially identical results. These are presented in the top section of Table 3. Note that the χ² values for each fit were initially somewhat larger than the values reported in this table, which were initially estimated on the basis of Poisson noise and read-out noise in the individual HST images. However, it is reasonable to presume that there is an additional uncertainty due to imperfections in the ePSF models. Therefore, we renormalize the uncertainties to give ${\chi }^{2}/\mathrm{dof}=1$ for each passband. The numbers of pixels fitted in the 2014 I and V bands are 207 and 202, and the correction factors for the 2014 I and V bands are 1.73 and 1.53, respectively. Similarly, the numbers of pixels fitted in the 2012 I and V bands are 200 and 199, while the correction factors are 1.42 and 1.27, respectively. This is similar to the procedure applied to light-curve modeling for virtually all planetary microlensing events (Bennett et al. 2008).

Table 3.  List of Dual-star Fits

Dual-star Fit	Year	Filter	Magnitude		Separation	Predicted	Separation Star 2–Star 1		χ2
			Star 1	Star 2	(mas)	Sep (mas)	$\bigtriangleup$x	$\bigtriangleup$y
Best fit	2012	I	19.84(0.15)	20.31(0.24)	14.1(3.2)	17.4(0.4)	9.5(2.6)	10.1(2.6)	194.3
		V	21.37(0.18)	21.74(0.56)	15.2(2.9)	17.4(0.4)	6.7(2.5)	13.1(2.1)	193.1
	2014	I	19.86(0.14)	20.28(0.18)	12.2 (3.3)	27.4(0.7)	10.6(1.8)	4.7(2.8)	201.1
		V	21.47(0.22)	21.64(0.27)	11.6(3.7)	27.4(0.7)	10.3(2.2)	6.2(3.1)	195.9
Source-flux-constrained	2012	I	19.47(0.05)	21.35(0.29)	16.6(2.1)	17.4(0.4)	11.2(1.2)	12.4(1.6)	204.2
		V	21.11(0.12)	22.26(0.38)	16.1(2.9)	17.4(0.4)	9.2(2.1)	11.6(2.1)	199.2
	2014	I	19.45(0.05)	21.43(0.31)	14.1(2.1)	27.4(0.7)	12.4(1.2)	6.1(1.2)	210.9
		V	21.11(0.11)	22.31(0.43)	13.5(2.4)	27.4(0.7)	11.6(1.2)	7.6(2.1)	200.9
Source-flux and separation-constrained	2012	I	19.46(0.04)	21.41(0.21)	17.4(3.4)	17.4(0.4)	11.4(2.5)	12.7(2.3)	214.5
		V	21.09(0.13)	22.28(0.28)	17.3(3.6)	17.4(0.4)	6.5(2.8)	13.6(2.2)	206.2
	2014	I	19.38(0.06)	22.18(0.27)	26.9(7.7)	27.4(0.7)	20.3(7.1)	14.5(3.1)	233.9
		V	21.02(0.09)	22.66(0.48)	26.8(9.3)	27.4(0.7)	23.8(8.3)	12.5(4.2)	218.7

Note. This table presents magnitudes calibrated to the OGLE-III scale. The separation in column (6) is from the dual-star MCMC fits, and the predicted separation in column (7) is from the light-curve fitting.

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These solutions and their uncertainties show that in both the epochs, neither star's brightness matches the source magnitudes of ${I}_{S}=19.43\pm 0.04$ and ${V}_{S}=21.07\pm 0.07$. The source is brighter than star 1 by 2.6σ and 2.8σ in the I band and by 1.6σ and 1.7σ in the two epochs of V-band data, so the total significance of the difference between the star 1 flux and the source flux is about 4.2σ when all four measurements are considered. Thus, these solutions are not consistent with the microlensing light-curve data, which require a source brighter than star 1. We attribute this difference to minor problems with the PSF model, which are overcome in the next section, which describes source-flux-constrained fits.

5.2.2. Source-flux-constrained Fits

To find a solution consistent with the source magnitude from our light-curve models, we add a constraint on the flux of the source star in our MCMC chains. The constraint is imposed by adding a term of the form $\exp [{({f}_{1}-{f}_{s})}^{2}/(2{\sigma }_{{f}_{s}}^{2})]$ to each model ${\chi }^{2}$ calculation. As discussed in Section 2, the parameter f_s represents the source flux and ${\sigma }_{{f}_{s}}$ is its uncertainty. This uncertainty comes from the light-curve models, the calibration relations (1), (5), and (6), and the source star color determination described in Section 4. The parameter f₁ represents the source flux from the dual-star PSF model, because we identify star 1 with the source. Our constraint term forces the source brightness to be consistent with the light-curve models. The results of these constrained MCMC runs are shown in Table 3. The source brightness constraint raises χ² by an amount ranging from 5.0 to 9.9. These values are a bit above what one would expect from Gaussian statistics, but we expect that this is due to inadequacies in the PSF model.

The predicted lens–source separations are 17.4 ± 0.4 mas and 27.4 ± 0.7 mas in 2012 and 2014, respectively. In the 2012 epoch, 3.62 yr after peak magnification, the separation between the two stars is ∼1σ away from the predicted lens–source separation. In the 2014 epoch, 5.59 yr after peak magnification, this separation is at least 7σ away from the predicted value. This is not consistent with the assumption that the excess flux is due to the lens star.

5.2.3. Source-flux- and Lens–Source-separation-constrained Best Fit

The results of the source-flux-constrained fit show that the separation between the source and the blend star is not consistent with the predicted lens–source separation. Hence, the excess flux blended with the source is not dominated by the lens star. As a further check on this blend–lens star model, we have performed fits with both the source flux and the lens–source separation fixed.

For the source-flux- and separation-constrained fit, we add the extra term $\exp [-{({f}_{1}-{f}_{s})}^{2}/{(2{\sigma }_{{f}_{s}}^{2})-({s}_{12}-{s}_{\mathrm{lc}})}^{2}/(2{\sigma }_{{s}_{\mathrm{lc}}}^{2})]$ to the χ² for each link in the MCMC. The parameters f₁, f_s, and ${\sigma }_{{f}_{s}}$ refer to the flux of star 1, the source flux, and the uncertainty in the source flux from the light-curve models, while the parameters s₁₂, ${s}_{\mathrm{lc}}$, and ${\sigma }_{{s}_{\mathrm{lc}}}$ refer to the star 1–2 separation, the lens–source separation predicted by the light-curve model, and the uncertainty in the light-curve model separation prediction. The first term in the exponential is the source flux constraint, as mentioned previously. The second term in the exponential similarly is the lens–source separation constraint. This term forces the fit separation between stars 1 and 2 in the image fits to match, within the measurement uncertainty, the light-curve model prediction of the lens–source relative proper motion, ${\mu }_{\mathrm{rel}}$. The lens–source separation, ${s}_{\mathrm{lc}}$, is determined from the light-curve model by ${s}_{\mathrm{lc}}={\mu }_{\mathrm{rel}}{\rm{\Delta }}t$, where Δ t is the time interval between the microlensing event peak and the time of the HST observations. We should note that the separation measured between the two stars in the HST frame, s₁₂, is directly related to the relative proper motion, ${\mu }_{\mathrm{rel}}$, in the heliocentric frame, while the ${\mu }_{\mathrm{rel}}$ value determined from the light-curve model, described in Section 4, is determined in a "geocentric" reference frame that that moves with a constant velocity that matches Earth's velocity at the peak of the microlensing event. The relation between the relative proper motion in these two reference frames is given by the equation (Dong et al. 2009a)

$\begin{eqnarray}&&{{\boldsymbol{\mu }}}_{\mathrm{rel},{\rm{H}}}={{\boldsymbol{\mu }}}_{\mathrm{rel},{\rm{G}}}+\displaystyle \frac{{{\boldsymbol{v}}}_{\oplus }{\pi }_{\mathrm{rel}}}{\mathrm{au}},\end{eqnarray} \tag{ 14 }$

where ${{\boldsymbol{v}}}_{\oplus }$ is the projected velocity of Earth relative to the Sun at the time of peak magnification. The relative parallax ${\pi }_{\mathrm{rel}}\equiv 1/{D}_{L}-1/{D}_{S}$ is related to the lens mass by the following relation:

$\begin{eqnarray}&&{\pi }_{\mathrm{rel}}=\displaystyle \frac{{c}^{2}}{4G}{\theta }_{E}^{2}\displaystyle \frac{\mathrm{au}}{{M}_{L}},\end{eqnarray} \tag{ 15 }$

where D_L and D_S are the distances to the lens and source, respectively. This implies that

$\begin{eqnarray}&&{{\boldsymbol{\mu }}}_{\mathrm{rel},{\rm{H}}}={{\boldsymbol{\mu }}}_{\mathrm{rel},{\rm{G}}}+\displaystyle \frac{{{\boldsymbol{v}}}_{\oplus }{c}^{2}{\theta }_{E}^{2}}{4{{GM}}_{L}}.\end{eqnarray} \tag{ 16 }$

From Equation (14), we see that the difference between the heliocentric and geocentric proper motions is minimized when the relative parallax, ${\pi }_{\mathrm{rel}}$, is small, i.e., when ${D}_{L}\approx {D}_{S}$. From Equation (16), we can also see that this corresponds to a large lens mass. Due to the relatively small angular Einstein radius, ${\theta }_{E}$, the light-curve model for this event predicts a low lens mass, unless the lens is quite close to the source. This implies that if the lens star dominates the blend flux, then we have a large lens mass, M_L, and small relative proper motion, ${\pi }_{\mathrm{rel}}$. This provides us an easy way to deal with the fact that the transformation to heliocentric coordinates, Equation (14), depends on the direction of the lens–source relative motion. We simply assume the lens mass derived from the lens–blend assumption in the discovery paper (Janczak et al. 2010) and then include the direction uncertainty as a contribution to the uncertainty in $| {{\boldsymbol{\mu }}}_{\mathrm{rel},{\rm{H}}}|$. This yields $| {{\boldsymbol{\mu }}}_{\mathrm{rel},{\rm{H}}}| =4.98\,\pm 0.31$ mas yr⁻¹. This value of $| {{\boldsymbol{\mu }}}_{\mathrm{rel},{\rm{H}}}|$ is used to constrain the star 1–2 separation in the source-flux- plus lens–source-separation-constrained fit.

The results of these fits are summarized in Table 3. The uncertainties for the values in this table are the root square of the distributions in the MCMC. For the magnitudes, we also include the uncertainties in the calibration relations. The results in this table indicate that the 2014 epoch fits show a significant increase in χ², when the lens–source separation constraint is added, with increases of ${\rm{\Delta }}{\chi }^{2}=23.0$ and ${\rm{\Delta }}{\chi }^{2}=17.8$ in the I and V bands, respectively. The increases in the χ² values for the 2012 data are smaller (${\rm{\Delta }}{\chi }^{2}=10.3$ and ${\rm{\Delta }}{\chi }^{2}=7.0$ for I and V, respectively), as expected because the lens–source separation was smaller in 2012. Thus, we conclude that the blend flux is not dominated by flux from the lens star.

5.3. Triple-star PSF Fit for the Source, Lens, and an Additional Star

In the previous section, we showed that the extra flux on top of the source is not dominated by the lens star. This implies that there must be an additional star blended with the source that must contribute significant flux. If the lens star is very faint, then the source-flux-constrained fits may accurately describe the HST data. But in the more general case, there should be a total of three stars: the source, the lens, and another star that contributes most of the excess flux. Similarly, in the field of eclipsing binaries, the presence of a third body is referred to as the third light (Getley et al. 2017; Tavakkoli et al. 2017).

In our triple-star PSF fits, the source flux and the lens–source separation are constrained as in the dual-star fits discussed in Section 5.2.3, but we now add an additional star. This additional star is presumably either a companion to the source or the lens or else a nearby star. There is no constraint on the position of this star. The direction of the lens–source separation vector is also unconstrained. For this fit, three additional parameters were introduced: the two position coordinates for this third star and the flux ratio between this third star and the lens star. We maintain the same error bar normalization as discussed in Section 5.2.1. The results of these fits are shown in Table 4. The lens and the source positions from these MCMC runs are shown in Figure 2. The source positions form a clump in the center for both the I- and V-band fits, while the lens positions form an arc around it, as shown in Figure 2. We calculated the calibrated lens flux for each link in the MCMC chains. The lens flux distribution for the 2014 images in the I and V passbands is shown in Figure 3. The 2012 images show a lens flux distribution consistent with the 2014 results. The parameters from this fit are presented in Table 4. We use polar coordinates to describe the lens–source separation, since Figure 2 shows that the lens positions are largely distributed in an arc. The uncertainties in the lens–source separation direction and the lens's and additional star's brightness are high, largely because the data do not demand any light from the lens star. The source-flux-constrained two-star fits shown in Table 3 provide an acceptable fit to the data. Hence, the brightness and position of the lens star cannot be well constrained. The uncertainty in the lens star brightness increases the uncertainty in the brightness of the additional blend star. About 40 and 100 source and lens positions fall outside the central clump and arc distributions for the F814W and F555W images, respectively. The total numbers of links in the Markov chains used to create these figures were ∼29,000 and ∼18,500 for F814W and F555W, respectively, so these represent ≲0.5% of the distribution. These constrained fits are used in Section 5.3.1 to derive the upper limit of brightness on the lens and planetary host star.

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Table 4.  Triple-star Fit

Year	Filter	Magnitude			Lens–Source	Angle	${\chi }^{2}$
		Source	Lens	Blend Star	Separation (mas)
2012	I	19.62(0.05)	22.62(0.44)	22.11(0.91)	17.6(1.5)	232.9(22.2)	206.1
	V	21.11(0.07)	23.08(0.53)	22.71(0.83)	17.2(2.3)	84.2(49.7)	195.8
2014	I	19.58(0.04)	23.39(0.63)	21.93(0.77)	27.2(1.5)	74.7(38.9)	210.5
	V	20.94(0.08)	23.84(0.69)	22.35(0.78)	27.3(2.2)	68.8(57.6)	197.1

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5.3.1. Upper Limit Calculation on the Lens Brightness and Mass

Figure 3 shows the distribution of the I- and V-band lens star magnitudes from the 2014 source-flux- and lens–source-separation-constrained triple-star fits discussed in Section 5.3. We use these histograms to determine the upper limit on the lens brightness. In 2014 I band and V band, 99% of the lens magnitude distribution lies fainter than I > 22.15 and V > 23.41. The corresponding limits from the 2012 data are I > 21.27 and V > 22.02. These are weaker owing to the smaller lens–source separation in 2012.

The microlensing light-curve model provides the mass–distance relation,

$\begin{eqnarray}{M}_{L} & = & \displaystyle \frac{{c}^{2}}{4G}{\theta }_{E}^{2}\displaystyle \frac{{D}_{S}{D}_{L}}{{D}_{S}-{D}_{L}}=\displaystyle \frac{{c}^{2}}{4G}{\theta }_{E}^{2}\displaystyle \frac{\mathrm{au}}{{\pi }_{\mathrm{rel}}}=0.9823\,{M}_{\odot }\\ & & \times {\left(\displaystyle \frac{{\theta }_{E}}{1\mathrm{mas}}\right)}^{2}\left(\displaystyle \frac{x}{1-x}\right)\left(\displaystyle \frac{{D}_{S}}{8\,\mathrm{kpc}}\right),\end{eqnarray} \tag{ 17 }$

where $x={D}_{L}/{D}_{S}$. This relation can be combined with a mass–luminosity relation to obtain the mass and the distance of the host star and the planet following the method in Bennett et al. (2015). We use the empirical mass–luminosity relations of Henry & McCarthy (1993), Henry et al. (1999), and Delfosse et al. (2000). For ${M}_{L}\gt 0.66\,{M}_{\odot }$, we use the Henry & McCarthy (1993) relation; for $0.12\,{M}_{\odot }\lt {M}_{L}\lt 0.54\,{M}_{\odot }$, we use the Delfosse et al. (2000) relation; and for $0.07\,{M}_{\odot }\,\lt {M}_{L}\lt 0.10\,{M}_{\odot }$, we use the Henry et al. (1999) relation. In between these mass ranges, we linearly interpolate between the two relations used on the boundaries. We interpolate between the Henry & McCarthy (1993) and the Delfosse et al. (2000) relations for $0.54\,{M}_{\odot }\lt {M}_{L}\lt 0.66\,{M}_{\odot }$, and we interpolate between the Delfosse et al. (2000) and Henry et al. (1999) relations for $0.10\,{M}_{\odot }\lt {M}_{L}\lt 0.12\,{M}_{\odot }$. The 99% and 95% confidence level upper limits on the lens system parameters from 2014 I- and V-band images are listed in Table 5. Since a detectable lens star must be close to the source in the Galactic bulge, the dust in the foreground of the lens is similar to that of the source. Hence, we use the same extinction for the lens and the source stars. For this upper limit, we assume a source distance ${D}_{L}=8$ kpc. This limit implies that the exoplanet is likely to be a sub-Saturn-mass planet orbiting an M-dwarf star at a distance of ${D}_{L}\lesssim 7.8$ kpc toward the bulge.

Table 5.  Upper Limit Constraints on the Lens System Parameters

Parameter	Units	99% Confidence		95% Confidence
		I	V	I	V
Host star mass, M*	${M}_{\odot }$	0.64	0.73	0.61	0.72
Planet mass, mp	M⊕	72	82	69	81
Host star–planet 2D separation, a⊥	au	1.1	1.1	1.1	1.1
Lens distance, DL	kpc	7.8	7.8	7.8	7.8

Note.${I}_{L}\geqslant 22.15,{V}_{L}\geqslant 23.41$ (99% confidence). ${I}_{L}\geqslant 22.44,{V}_{L}\geqslant 23.62$ (95% confidence).

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The lens–source separation can also be detected by the color-dependent centroid shift method, which has been used for planetary events OGLE-2003-BLG-235 (Bennett et al. 2006) and OGLE-2005-BLG-071 (Dong et al. 2009b). This method is only effective if the source and lens have different colors so that the centroid of the blended lens+source flux will be different in the two passbands, I and V in our case. From the source-flux-constrained fit discussed in Section 5.2 and presented in Table 3, we see that that the ratio of the blend flux to that of the source is ∼0.85 and ∼0.78 in the I and V bands, respectively. First, let us consider the case of this source-flux-constrained model. It implies that the blend star is ∼13.8 mas away from the source. This implies that the centroid of the source and the blend star is 2.11 and 3.02 mas away from the source star in the I and V bands, respectively. Hence, the centroid shift is 0.91 mas in this case.

Now, let us consider the blend–lens situation, which has already been ruled out by the two-star fits. In this case the blend–source separation will be 27 mas. This implies a centroid shift of 4.12 and 5.91 mas away from the lens position in the I and V bands, respectively. So, in this case, the color-dependent centroid shift is 1.79 mas, in this blend–lens situation.

With these theoretical calculations in hand, we proceed to measure the color-dependent centroid shift for an additional test of the blend–lens scenario. We selected 20 isolated stars within a 200-pixel radius of the target. These stars are selected to be within 0.1 mag of the color and 0.5 mag of V-band brightness of the target star. We use these stars to measure the average centroid shifts of single stars between I- and V-band frames. Then, we measure the centroid shift of the target star relative to these reference stars. We also compare the centroid shift of the target star to the centroid shift of each of the reference stars in each individual image. The measured centroid shift of the target star is 0.45 ± 0.49 mas. The uncertainty is calculated from the rms of the centroid shifts of the reference stars. This measurement is within 1σ of the 0.91 mas centroid shift predicted by the flux-constrained two-star model, but it is 2.7σ less than the separation expected from the blend–lens model. So, the blend–lens model is also ruled out by this color-dependent centroid shift test.

The source-flux-constrained dual-star fit to the 2014 data is not consistent with the blend–lens model because the measured source–blend separation is much smaller than the predicted separation between the source and lens stars, and our attempt to detect the color-dependent centroid shift tends to confirm this conclusion. Without a clear detection of the lens star, we cannot use mass–luminosity relations to get a precise distance and masses for the host star and planet, as we have previously done for OGLE-2005-BLG-169 (Batista et al. 2015; Bennett et al. 2015). We can still estimate the masses and distance to the lens system with Bayesian analysis, using the Galactic model of Bennett et al. (2014), but without a lens brightness measurement, this estimate has a significant dependence on our prior assumptions. We are interested in this event because it indicates the presence of an exoplanet, but we do not know whether the probability of hosting a planet similar to the detected planet depends on the host star mass or distance. The simplest assumption is that the probability that a lens star hosts a planet like MOA-2008-BLG-310Lb does not depend on the host star mass or distance. We certainly know that planets have been discovered orbiting a wide variety of stars, but we do not have strong information to indicate that the planet-hosting probability does not depend on the host star mass or distance. So, we are not sure how accurate this assumption is.

This Bayesian analysis indicates that the lens star has a median I-band magnitude of ${I}_{L}\sim 26.2$ with a 1σ range of 24.1–28.6 and a 2σ range of 23.0–36.1. Note that this 2σ upper limit of 36.1 is not a real magnitude. It is a magnitude that we assigned to brown dwarfs, which are too faint to detect. ${I}_{L}\sim 26.2$ is certainly too faint to detect. I ∼ 23.0 is enough to perturb the fit, while I ∼ 24.1 is starting to become a bit marginal. From our MCMC analysis (assuming that all stars are equally likely to host a planet), we find that 78% of the time the lens star is <1% of the source brightness (that is, ${I}_{L}\gt 24.58$). This implies that it is quite likely that the host star is too faint to have much influence on our models of the HST images. If so, the source-flux-constrained fit listed in Table 3 should provide a good model of this system.

The fit results in Table 3 indicate that the position of this blend star has consistent positions with respect to the source position in the two passbands in epoch. It is quite possible that the lens is too faint to contribute significantly to the I- or V-band flux of the target. This would be the case if the lens is a faint M dwarf or even a brown dwarf or white dwarf. If the lens star is too faint to be detectable, then there is likely to be a single blend star that dominates the excess flux on top of the source in the I and V bands. This blend star could be a companion to the source or the lens or an unrelated star. According to the discovery paper, the a priori probability of a blend star within 0.5 mag of the H-band magnitude of the blend star for each of these possibilities is ∼5%. The two epochs of observations yield weighted means of the separations of the blend star with respect to the source in I and V bands of Δx = 10.7 ± 1.0 mas, Δy = 12.1 ± 1.3 mas for 2012 data and Δx = 12.0 ± 0.8 mas, Δy = 6.5 ± 1.0 mas for 2014 data. This implies a relative proper motion of this blend star with respect to the source of ${{\boldsymbol{\mu }}}_{\mathrm{rel},{\rm{b}},{\rm{H}}}=(0.66\pm 0.65)\hat{x}-(2.84\pm 0.83)\hat{y}\,$ mas yr⁻¹ from Table 3. The magnitude of the relative proper motion for this star is ${\mu }_{\mathrm{rel},{\rm{b}},{\rm{H}}}=2.92\pm 0.83$ mas yr⁻¹, which is consistent with the proper-motion dispersion of the bulge stars (Kozlowski et al. 2006). Hence, this star can be an unrelated nearby bulge star. The implied separation of the source and blend stars is ≥129 ± 9 au, while the relative proper motion implies a blend–source relative velocity of ≥111 ± 31 km s⁻¹, which is much larger than the ∼5 km s⁻¹ escape velocity implied by the source–blend separation. Hence, the blend star cannot be a companion to the source if the lens star is faint.

The discovery paper indicates that the VLT/NACO AO data were taken 3.2 yr before the 2012 HST images, so we can use the ${{\boldsymbol{\mu }}}_{\mathrm{rel},{\rm{b}},{\rm{H}}}$ above to estimate the separation at the time of the high-resolution VLT/NACO AO images. We determine a source–blend separation of Δx = 8.6 mas, Δy = 21.2 mas at the time of the VLT images. So the source–blend separation was 22.9 mas, which is much smaller than the VLT image FWHM of 130 mas. This explains why this blend star was not resolved in the VLT/NACO AO observations in 2008.

Since the magnitude uncertainties for the blend star are large, its color can vary anywhere from spectral type A to K, considering 2σ uncertainties on the magnitudes. But this star has faint apparent magnitudes. So if this star is an A star, it must be located very far behind the Galactic bulge. But this would mean that it would be many scale heights below the Galactic plane, where the density of A stars is very low. So, it is reasonable to assume that this star is probably a G or K star residing in the Galactic bulge.

The final possibility is that this blend star is a companion to the lens star (Bennett et al. 2007; Gould 2016). The blend–source heliocentric relative proper motion is ${\mu }_{\mathrm{rel},{\rm{b}}}\,=2.92\,\pm 0.83$ mas yr⁻¹, which is 2.3σ smaller than the lens–source relative proper motion of ${\mu }_{\mathrm{rel}}=4.98\pm 0.31\,$mas yr⁻¹. Hence, a companion to the lens is marginally excluded, if the lens star has negligible brightness.

A companion to the source or lens might be possible if the lens star contributes significantly to the flux. In that case, we would need to use the triple-star fit instead of the source-flux-constrained two-star fit. As Table 6 indicates, the relative positions of the blend and source stars are not well constrained in these models.

We can also use the three-star fits to check whether the blend star is a companion or an ambient star in the case where the lens star flux is not negligible. But from Table 6 we find that the blend–source relative proper motion is ${{\boldsymbol{\mu }}}_{\mathrm{rel},{\rm{H}},{\rm{b}}}\,=(0.6\pm 4.8)\hat{x}-(3.0\pm 4.2)\hat{y}\,$mas yr⁻¹. Without a constraint that the lens star flux is negligible, the error bars on the blend–source relative proper motion are too large to constrain any of the possibilities: an ambient star or a companion to the source or lens.

In this paper we have developed a technique for fitting a two- or three-star PSF to the blended HST images consisting of a microlensed source, the lens star, and possible companions to the source and lens stars, following a procedure outlined by Bennett et al. (2007). We have constrained these fits with constraints on the source flux and the lens–source separation from the microlensing light-curve model. In our previous analysis of event OGLE-2005-BLG-169 (Bennett et al. 2015), the unconstrained best-fit solution was consistent with the source flux and the predicted lens–source separation in three different passbands, as well as subsequent Keck AO H-band images (Batista et al. 2015). Since the lens–source relative proper motion is determined by the planetary signature in the light curve, the confirmation of the separation predicted by the light curve is also a confirmation of the planetary interpretation of the light curve.

The confirmation of planetary signal was important for the OGLE-2005-BLG-169 event owing to the fragmentary character of the light curve over peak. But for the MOA-2008-BLG-310 event, the light curve is well covered (Janczak et al. 2010), allowing very little chance for incorrect modeling. Hence, we can expect to measure the relative lens–source proper motion more precisely and compare with the lens–source proper motion derived from the light-curve modeling. But the event MOA-2008-BLG-310 did not yield a confirmation of the blend–lens model. Instead, we found that the source-flux-constrained solutions were not consistent with the lens–source relative proper motion predicted by the light curve. The two epochs of HST data were taken in 2012, 3.62 yr after the event, and in 2014, 5.59 yr after the peak magnification. The expected lens–source separations at these two observing epochs are 17.4 ± 0.4 mas and 27.4 ± 0.7 mas for the 2012 and 2014 observations, respectively. The 2014 HST observations, in particular, were not consistent with the blend–lens model. The lens–source-separation-constrained models had a χ² increase of Δχ² = 40.8 compared to models, which only constrained the source flux. Hence, we have concluded that the blend flux, first identified by Janczak et al. (2010), is not due to the planetary host (and lens) star. This conclusion is strengthened by the lack of a detectable color-dependent centroid shift (Bennett et al. 2006). Therefore, we consider three-star models in order to constrain the brightness and mass of the planetary host star. In these fits, we constrain the source flux and separation, but not the direction of the lens–source separation vector.

We tried our triple-star MCMC fits on the event OGLE-2005-BLG-169. We found no additional solution other than the obvious degenerate solutions of dual-star fits—A. The lens star divided into two stars in the same position and B. There is a very faint third star in the middle of the source and the lens star. This faint third star has an upper limit of 8% and 2% of the total flux in I and V bands, respectively.

As mentioned in Section 7, we have performed a Bayesian analysis to estimate the lens system properties using the Galactic model of Bennett et al. (2014) assuming that all the stars and brown dwarfs have an equal probability of hosting a planet with the detected properties. We ran a series of Markov chains with a total of 662,000 links to determine the allowed distribution of lens parameters, including the lens brightness constraint from Section 5.3, and the results of this calculation are presented in Figure 4 and Table 7.

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Table 7.  Planetary System Parameters from Bayesian Analysis with Lens Flux Constraint

Parameter	Units	Mean Values and rms	2σ Range
Host star mass, M*	${M}_{\odot }$	0.21 ± 0.14	0.033–0.56
Planet mass, mp	M⊕	23.4 ± 17	3.7–64
Host star–planet 2D separation, a⊥	au	1.10 ± 0.17	0.73–1.42
Host star–planet 3D separation, a3D	au	1.61 ± 0.98	0.82–4.75
Lens distance, DL	kpc	7.7 ± 1.1	5.3–9.7
Lens magnitude, IL	Cousins I	26.2 ± 2.2	23.0–36.1

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The source star brightness, color, and radius for each model in the Markov chain are included in this calculation. The source star is assumed to be a bulge star distributed at a distance $5\,\mathrm{kpc}\leqslant {D}_{L}\leqslant 12$ kpc following a standard galactic model (Bennett et al. 2014). For each model, the source distance is chosen randomly from the microlensing-rate-weighted Galactic bulge distribution. We use empirical mass–luminosity relations (Henry & McCarthy 1993; Henry et al. 1999; Delfosse et al. 2000) and the mass–distance relations given in Equation (17) to determine the lens distance, D_L, the host star and the planet masses (M_* and m_p), the host star I-band magnitude, I_L, and the host star–planet projected separation, ${a}_{\perp }$. The uncertainties shown in Table 7 represent 1σ error bars. The mass of the host star (lens star) is approximately determined to be ${M}_{* }\,={0.21}_{-0.09}^{+0.21}\,{M}_{\odot }$, so it could be an M dwarf or a brown dwarf. The predicted magnitude of the lens is ${I}_{L}={26.2}_{-2.1}^{+2.3}$, with a 2σ range extending down to I_L = 23.0. Hence, unless the lens brightness is near this 2σ upper limit, it will be too faint to detect while blended with the ${I}_{S}=19.43\pm 0.05$ source star. The planet mass is ${m}_{p}={23}_{-10}^{+24}{M}_{\oplus }$, with a 2σ range of $3.7\,{M}_{\oplus }\lt {m}_{p}\lt 64\,{M}_{\oplus }$, so it could be a super-Earth- or sub-Saturn-mass gas giant. The distance to the lens system is more precisely determined, due to the relatively small angular Einstein radius, ${\theta }_{E}=0.29\pm 0.05\,$ mas. Our analysis predicts a lens system distance of ${D}_{L}=7.7\pm 1.1$ kpc. This implies that the lens system is very likely to be in the Galactic bulge, as claimed by the discovery paper (Janczak et al. 2010), although a super-Earth planet orbiting a brown dwarf in the disk cannot be ruled out.

As discussed in Sections 5.2 and 5.3, there are two possible solutions available. One is that the lens is too faint to influence the HST image models, and the extra flux is solely due to a star that is an unrelated nearby star. The second possibility is that the lens is bright enough (I ∼ 23) to influence the ePSF fit. In this case, with two additional stars, besides the source star, contributing flux, the uncertainty on the proper motion of the blend star is much larger. Hence, the blend star in the three-star fit can be either a nearby unrelated star or a companion to the source or the lens. The triple-star fit does provide an upper limit of the host star mass and therefore the planet mass. This implies that the exoplanetary system is a sub-Saturn-mass system located in the bulge orbiting around an M-dwarf star or a brown dwarf.

The method that we have presented in this paper is an important development in the effort to characterize exoplanets found by the microlensing method. We have shown that it is possible to determine whether excess flux at the location of a microlensed source star is due to the lens star of a planetary microlensing event. In the case of OGLE-2016-BLG-169, such measurements indicated that the blend star was the planetary host star (Bennett et al. 2015), but for MOA-2008-BLG-310, we have shown that the blend star is not the lens. To use our method, we need the finite angular radius and the predicted lens–source relative proper motion. Most planetary microlensing features resolve the finite angular size of the source star and predict the lens–source relative proper motion, ${\mu }_{\mathrm{rel}}$. Hence, this allows us to perform similar measurements on the high-resolution follow-up images of most microlensing planetary events. A few years after the microlensing event, when the lens has moved away from the source, the lens and source stars will be partially resolved. Then, the analysis of the high-resolution images of the partially resolved lens–source system will allow us to detect the lens (and planetary host star) and determine its mass and that of its planet. The method presented in this paper allows us to determine whether any excess flux is actually due to the host star.

This characterization of planetary systems discovered by microlensing is important because microlensing is unique in its sensitivity to the cold, low-mass exoplanets beyond the snow line, where other exoplanet detection methods are not so effective (Spergel et al. 2015). Microlensing has discovered sub-Saturn-, Neptune-, and super-Earth-mass planets such as OGLE-2005-BLG-169 (Gould et al. 2006), OGLE-2005-BLG-390 (Beaulieu et al. 2006), OGLE-2006-BLG-109 (Gaudi et al. 2008), OGLE-2007-BLG-349 (Bennett et al. 2016), OGLE-2007-BLG-368 (Sumi et al. 2010), MOA-2009-BLG-266 (Muraki et al. 2011), MOA-2009-BLG-319 (Miyake et al. 2011), MOA-2011-BLG-028 (Skowron et al. 2016), OGLE-2012-BLG-0724 (Hirao et al. 2016), OGLE-2012-BLG-0950 (Koshimoto et al. 2016), MOA-2013-BLG-605 (Sumi et al. 2016), and OGLE-2014-BLG-1760 (Bhattacharya et al. 2016). But the light-curve analysis of most planetary microlensing events yields only the planet–host star mass ratio and separation in Einstein radius units. More observations like the ones we have analyzed in this paper of MOA-2008-BLG-310 and those of OGLE-2005-BLG-169 (Bennett et al. 2015) will allow us to expand the current state-of-the-art analysis of exoplanet statistics beyond the snow line (Suzuki et al. 2016) to include the dependence of the exoplanet mass function on the host mass and distance.

The analysis we have presented here confirms the prediction (Bennett & Rhie 2002; Bennett et al. 2007) that such measurements should be possible for virtually all planets found by a space-based microlensing survey. In fact, this method is likely to be the primary method for determining planet and host star masses for exoplanets discovered by the WFIRST exoplanet microlensing program (Spergel et al. 2015). The advantage of a space-based microlensing survey is that the WFIRST observations themselves will provide the high angular resolution observations needed to detect the exoplanet host stars. However, the WFIRST fields will be more crowded than the fields of OGLE-2005-BLG-169 or MOA-2008-BLG-310, in part because WFIRST will observe in the infrared. Hence, the probability of blending by unrelated stars will be higher. Thus, it will be more important to distinguish lens from unrelated blend stars and companions to the source and lens. Therefore, it will be necessary to use the method developed in this paper to avoid errors in determining the host star brightness, mass, and distance. About 44% of G dwarfs have stellar companions, as do 26% of K and M dwarfs (Duchene & Kraus 2013). So stellar companions will be a common source of contamination in the attempt to identify host stars for planets discovered by the WFIRST microlensing program. With WFIRST it will be more the rule than the exception to have source, lens, and an additional stellar companion that is faint. Hence, getting the excess flux is probably not enough in the WFIRST era; we will have to follow a more careful investigation, like MOA-2008-BLG-310 follow-up analysis. Thus, the methods that we have developed in this paper are an important step forward in the development of the WFIRST exoplanet mass measurement method that will measure the demographics of cool and cold exoplanets.

Based on observations made with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute (STScI), which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are associated with program nos. 12541 and 13417. A.B. and D.P.B. were supported by NASA through grants NASA-NNX12AF54G and NNX13AF64G. I.A.B. was supported by the Marsden Fund of Royal Society of New Zealand, contract no. MAU1104.