Binary Source Microlensing Event OGLE-2016-BLG-0733: Interpretation of a Long-term Asymmetric Perturbation

Adopting the parametrization of Jung et al. (2015) for the description of a standard binary-lens light curve, we search for a binary-lens solution by following the method of Jung et al. (2015), one of the well-established modeling procedures developed based on the map-making method (Dong et al. 2006).²⁶ Figure 2 shows the ${\rm{\Delta }}{\chi }^{2}$ surface of $(s,q)$ parameter space derived from the initial grid search. Here s is the projected binary separation (normalized to the angular Einstein radius of the lens system, ${\theta }_{{\rm{E}}}$) and q $(={M}_{2}/{M}_{1})$ is the mass ratio between the lens components, and they are divided into 100 × 100 grids in the range of $-1.0\lt \mathrm{log}\,s\lt 1.0$ and $-4.0\lt \mathrm{log}\,q\lt 1.0$, respectively. From the grid search, we find one local minimum. By refining the local solutions, it is found that the best-fit binary-lens model has two masses with a separation of $s\sim 1.28$ and a mass ratio of  $q\sim 0.004$, which would make the companion a planetary-mass object projected near the Einstein radius of its host.

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In Table 1, we present the best-fit parameters of the binary-lens solution along with the ${\chi }^{2}$ values per dof. The lens system induces two sets of caustics, where one is located near the barycenter of the binary lens and the other is located away from the barycenter between the planet and the host. In Figure 3, we show the light curve of the binary-lens solution in the region of the anomaly and the corresponding geometry of the lens system. We note that the source trajectory in the upper panel is aligned so that the data points marked on the trajectory match those in the lower panel.

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Table 1.  Lensing Parameters

Parameters	Binary-lens	Binary-source
${\chi }^{2}$/dof	12955.8/12347	12411.8/12347
${t}_{\mathrm{0,1}}$ ($\mathrm{HJD}^{\prime}$)	7501.570 ± 0.004	7501.374 ± 0.005
${t}_{\mathrm{0,2}}$ ($\mathrm{HJD}^{\prime}$)	⋯	7507.804 ± 0.116
${u}_{\mathrm{0,1}}$	0.013 ± 0.001	0.015 ± 0.001
${u}_{\mathrm{0,2}}$	⋯	0.365 ± 0.033
${t}_{{\rm{E}}}$ (days)	27.838 ± 0.847	14.428 ± 0.602
s	1.281 ± 0.007	⋯
q (10−3)	3.912 ± 0.258	⋯
α (rad)	0.051 ± 0.001	⋯
${\rho }_{* }$ (10−3)	8.317 ± 0.392	⋯
${q}_{F,{RI}}$	⋯	2.078 ± 0.201
${q}_{F,I}$	⋯	1.853 ± 0.177

Note. ${\mathrm{HJD}}^{\prime }=\mathrm{HJD}\,-$ 2450,000.

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The fit in Figure 3 appears quite good. However, there exists some residuals near the exit of the planetary caustic $7517\lt \mathrm{HJD}^{\prime} \lt 7520$. In order to fit the residuals, we additionally test the models considering both the microlens parallax (Gould 1992, 2004) and the orbital motion of the lens (Dominik 1998). From this, we find that the introduction of higher-order effects still does not provide a fully acceptable fit. Furthermore, the estimated projected energy ratio between kinetic and potential energy ${(\mathrm{KE}/\mathrm{PE})}_{\perp }=5.77$, in contrast to the requirement that ${(\mathrm{KE}/\mathrm{PE})}_{\perp }\lt 1.0$ to be a bound system (Dong et al. 2009), indicates that the model is physically unrealistic.

In fact, there are two important clues that this solution is not correct. At first sight, these appear to be independent, but are actually closely related. The first is that the normalized source size is very large ${\rho }_{* }=8\times {10}^{-3}$ considering that the source is a dwarf. This would lead to a very small Einstein radius of ${\theta }_{{\rm{E}}}\sim 0.05$ mas and a very small lens-source relative proper motion of $\mu ={\theta }_{{\rm{E}}}/{t}_{{\rm{E}}}\sim 0.7\,\mathrm{mas}\,{\mathrm{yr}}^{-1}$, where ${t}_{{\rm{E}}}$ is the Einstein timescale. As discussed in Penny et al. (2016), where they presented the combination of μ and ${\theta }_{{\rm{E}}}$ for Galactic microlensing events using the model of Henderson et al. (2014), these values are a priori extremely unlikely, though not impossible: they could be generated by a very slow lens at a distance of ${D}_{\mathrm{LS}}\sim 20\,\mathrm{pc}{(M/{M}_{\odot })}^{-1}$ in front of the source.

What raises this suspiciously large ${\rho }_{* }$ to the level of implausibility is that it appears to be "driven" by the need to match basically smooth data to an intrinsically "bumpy" model. Because the source crossing time ${t}_{* }\equiv {\rho }_{* }{t}_{{\rm{E}}}\sim 5\,$ hr is similar to the duration of an observing night, it is appropriate to bin these data by observatory and by day. Figure 4 compares these binned data to the best-fit model and also to the same model but with a more typical value of ${\rho }_{* }=3\times {10}^{-3}$. This shows that the data are much smoother than the model for a typical ${\rho }_{* }$ and are still considerably smoother than the model for the best-fit ${\rho }_{* }$. We therefore are led to consider what other physical effects might generate this "planetary" anomaly.

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We consider the possibility that the long-term deviation may be caused by a single-lens event with a binary source. In case of a binary-source event, the lensing magnification corresponds to the superposition of the two single-source events generated by the individual source stars (Griest & Hu 1992; Han 2002), i.e.,

$$\begin{eqnarray}&&A=\displaystyle \frac{{A}_{1}{F}_{1}+{A}_{2}{F}_{2}}{{F}_{1}+{F}_{2}}=\displaystyle \frac{{A}_{1}+{q}_{F}{A}_{2}}{1+{q}_{F}},\end{eqnarray} \tag{ 2 }$$

where A_i denotes the lensing magnification of each source star with a flux F_i and ${q}_{F}={F}_{2}/{F}_{1}$ is the flux ratio between the two source stars. In contrast to a single-source magnification, the total magnification A is wavelength dependent because the different colors between two sources can induce a color change during the course of the lensing phenomenon.

Under the approximation that the transverse speeds of the two sources with respect to the lens are the same, the single-source magnification is related to the lens-source separation normalized to the angular Einstein radius, u_i, by

$$\begin{eqnarray}&&{A}_{i}=\displaystyle \frac{{u}_{i}^{2}+2}{{u}_{i}\sqrt{{u}_{i}^{2}+4}};\qquad {u}_{i}(t)={\left[{\left(\displaystyle \frac{t-{t}_{0,i}}{{t}_{{\rm{E}}}}\right)}^{2}+{u}_{0,i}^{2}\right]}^{1/2},\end{eqnarray} \tag{ 3 }$$

where ${t}_{0,i}$ is the moment of the closest lens approach to each source star, ${u}_{0,i}$ denotes the separation between the lens and individual sources at ${t}_{0,i}$, and ${t}_{{\rm{E}}}$ is the timescale required for the source to cross the angular Einstein radius (Einstein timescale). Therefore, one needs six principal parameters $({t}_{\mathrm{0,1}},{t}_{\mathrm{0,2}},{u}_{\mathrm{0,1}},{u}_{\mathrm{0,2}},{t}_{{\rm{E}}},{q}_{F})$ for the description of a standard binary-source event. Since the flux ratio varies depending on the passband, one must add additional flux ratio parameters for each of the observed passbands. We note that, due to the unknown sign of ${u}_{0,i}$, the binary-source interpretation suffers a similar degeneracy to the well-known satellite parallax degeneracy (Refsdal 1966; Gould 1994), resulting in four degenerate solutions that are generally denoted by $(+,\,+)$, $(-,\,-)$, $(+,\,-)$, and $(-,\,+)$, where the two signs in the parenthesis indicate the sign of ${u}_{\mathrm{0,1}}$ and ${u}_{\mathrm{0,2}}$, respectively.²⁷

The four-fold degeneracy introduces a two-fold degeneracy in the measurement of the source plane Einstein radius ${\hat{r}}_{{\rm{E}}}$, defined by

$$\begin{eqnarray}&&{\hat{r}}_{{\rm{E}}}={D}_{{\rm{S}}}{\theta }_{{\rm{E}}}={r}_{{\rm{E}}}\displaystyle \frac{{D}_{{\rm{S}}}}{{D}_{{\rm{L}}}}.\end{eqnarray} \tag{ 4 }$$

Here ${r}_{{\rm{E}}}={D}_{{\rm{L}}}{\theta }_{{\rm{E}}}$ is the physical Einstein radius, ${D}_{{\rm{S}}}$ and ${D}_{{\rm{L}}}$ are the distance to the source and the lens, respectively. The physical projected separation between the sources, ${d}_{{\rm{s}}}$, is related to ${\hat{r}}_{{\rm{E}}}$ by

$$\begin{eqnarray}&&{d}_{{\rm{s}}}={\hat{r}}_{{\rm{E}}\pm }{\left[{\left(\displaystyle \frac{{\rm{\Delta }}{t}_{0}}{{t}_{{\rm{E}}}}\right)}^{2}+{\rm{\Delta }}{{u}_{0}}^{2}\right]}^{1/2},\end{eqnarray} \tag{ 5 }$$

where ${\rm{\Delta }}{t}_{0}=| {t}_{\mathrm{0,1}}-{t}_{\mathrm{0,2}}|$, and ${\rm{\Delta }}{u}_{0}=| {u}_{\mathrm{0,1}}\pm {u}_{\mathrm{0,2}}|$. Therefore, if the separation ${d}_{{\rm{s}}}$ is measured from follow-up spectroscopic observation of the sources, the Einstein radius is determined with two possible values of ${\hat{r}}_{{\rm{E}}\pm }$ depending on ${\rm{\Delta }}{u}_{0}$ (Han & Gould 1997).

The transverse speeds of the two sources can be different when the orbital motion of the binary source is substantial ("source orbit"). In this case, the positions of the individual sources are described by the summation of the rectilinear motion of the barycenter of the binary source (CM) and the orbital motion of the sources with respect to the CM. Similar to the effect of the binary-lens orbital motion (Dominik 1998; Jung et al. 2013), the "source orbit" effect also causes the angle between the trajectory of the CM and the binary-source axis ${\alpha }_{{\rm{s}}}$ and the projected separation between the source components ${s}_{{\rm{s}}}$ (normalized to ${\theta }_{{\rm{E}}}$) to change in time. To first-order approximation, the source orbital motion can be parameterized as ${{ds}}_{{\rm{s}}}/{dt}$ and $d{\alpha }_{{\rm{s}}}/{dt}$, which represent the change rate of the binary-source separation and the orientation angle, respectively. Then, the binary-source separation and the orientation angle at time t are

$$\begin{eqnarray}&&{s}_{{\rm{s}}}^{\prime }={s}_{{\rm{s}}}+\displaystyle \frac{{{ds}}_{{\rm{s}}}}{{dt}}(t-{t}_{\mathrm{ref}}),\qquad {\alpha }_{{\rm{s}}}^{\prime }={\alpha }_{{\rm{s}}}+\displaystyle \frac{d{\alpha }_{{\rm{s}}}}{{dt}}(t-{t}_{\mathrm{ref}}),\end{eqnarray} \tag{ 6 }$$

where ${t}_{\mathrm{ref}}$ is the reference time for the binary-source orbital motion. The positions of the two sources on the lens plane are thus

$$\begin{eqnarray}&&{u}_{i}^{2}(t)={\left[\left(\displaystyle \frac{t-{t}_{0}}{{t}_{{\rm{E}}}}\right)\pm {s}_{{\rm{s}},{\rm{i}}}\cos {\alpha }_{{\rm{s}}}^{\prime }\right]}^{2}+{({u}_{0}\mp {s}_{{\rm{s}},{\rm{i}}}\sin {\alpha }_{{\rm{s}}}^{\prime })}^{2},\end{eqnarray} \tag{ 7 }$$

where ${s}_{{\rm{s}},1}={s}_{{\rm{s}}}^{\prime }{q}_{{\rm{s}}}/(1+{q}_{{\rm{s}}})$ and ${s}_{{\rm{s}},2}={s}_{{\rm{s}}}^{\prime }/(1+{q}_{{\rm{s}}})$ are the separation between the two sources and the CM, and ${q}_{{\rm{s}}}={M}_{{\rm{s}},2}/{M}_{{\rm{s}},1}$ is the mass ratio between the source components. Here the parameters $({t}_{0},{u}_{0},{t}_{{\rm{E}}})$ describe the relative motion between the lens and the CM. As a result, one requires nine principal parameters $({t}_{0},{u}_{0},{t}_{{\rm{E}}},{s}_{{\rm{s}}},{q}_{{\rm{s}}},{\alpha }_{{\rm{s}}},{{ds}}_{{\rm{s}}}/{dt},d{\alpha }_{{\rm{s}}}/{dt},{q}_{F})$ to describe the orbital motion of the binary source.

We separately test "standard" and "source orbit" models. In the "source orbit" model, we test four degenerate solutions resulting from the unknown sign of ${u}_{0,i}$. We investigate the solution of the parameters by a downhill approach. The initial values of the parameters are set based on the peak time, magnification, and duration of the event. We adopt the Markov Chain Monte Carlo method for the ${\chi }^{2}$ minimization of the downhill approach.

From the comparison between the "standard" and the "source orbit" models, it is found that the orbital motion of the binary source does not improve the fit significantly. The ${\chi }^{2}$ difference between the "standard" and each "source orbit" model is very low $({\rm{\Delta }}{\chi }^{2}\lt 1)$. Consequently, the measured values of two orbital parameters $({{ds}}_{{\rm{s}}}/{dt},d{\alpha }_{{\rm{s}}}/{dt})$ are consistent with zero in all four "source orbit" solutions. These imply that the light curve of the event does not have sufficient information for constraining the orbital parameters (see the Appendix of Han et al. 2016 for a detailed review). We therefore only consider the "standard" model.

We find that the binary-source interpretation provides a good fit to the observed light curve. In Table 1, we summarize the best-fit parameters of the binary-source solution. In Figure 5, we show the geometry of the relative lens-source motion (upper panel) and the model light curve (lower panel) of the binary-source solution. In the upper panel, the two lines with arrows represent the trajectories of the two source stars with respect to the lens (marked by "M"). We designate the source passing closer to the lens as the "primary" source and the other source as the "secondary" source. We find that the flux ratio (${q}_{F,{RI}}=2.0$ and ${q}_{F,I}=1.9$) is greater than unity, indicating that the source approaching closer to the lens is fainter than the source approaching farther from the lens. While binary-lens light curves are almost perfectly achromatic (after subtraction of the blended light), binary-source event light curves can be chromatic, implying that the magnification varies depending on the observed passband, and thus we present two model light curves corresponding to RI band (MOA) and I band (the other groups) data sets. According to the binary-source interpretation, the strong anomaly near the peak was produced when the fainter source star approached close to the lens and the anomaly in the declining wing was produced when the brighter source approached.

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By comparing the binary-lens and binary-source interpretations, we find that the binary-source model better explains the observed light curve. The binary-source model improves the fit by ${\rm{\Delta }}{\chi }^{2}\sim 544$ compared to the binary-lens model. In Figure 6, we present the cumulative distribution of ${\rm{\Delta }}{\chi }^{2}$ of the binary-source model with respect to the binary-lens model as a function of time. This shows that the major improvements occur during the ascending part of the strong anomaly near $7498\lt \mathrm{HJD}^{\prime} \lt 7501$ and the descending part of the light curve near $7514\lt \mathrm{HJD}^{\prime} \lt 7520$, where the KMTNet survey densely and continuously covers the anomaly. The fit improvement from KMTNet data is ${\rm{\Delta }}{\chi }^{2}\sim 484$. This demonstrates the importance of the high-cadence observation for the unambiguous characterization of the lensing event.

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Knowing that OGLE-2016-BLG-0733 is better explained by the binary-source interpretation, we characterize the individual source stars by estimating the source type from the de-reddened color ${(V-I)}_{0}$ and brightness I₀. For this, we use four KMTNet pyDIA reductions ([CTIO+SAAO]×[BLG02+BLG42]) to construct the instrumental color–magnitude diagrams (CMDs). Based on the method of Yoo et al. (2004), we first estimate the offsets in color ${\rm{\Delta }}(V-I)$ and magnitude ${\rm{\Delta }}I$ between individual sources and the centroid of the giant clump (GC). Figure 7 shows the instrumental CMD of one of four KMTNet fields (CTIO BLG02), where the locations of individual sources and GC are shown. Assuming that the source and GC suffer the same extinction combined with the known de-reddened color ${(V-I)}_{0,\mathrm{GC}}=1.06$ and magnitude ${I}_{0,\mathrm{GC}}=14.43$ of GC (Bensby et al. 2011; Nataf et al. 2013), we then estimate the de-reddened color and magnitude of the individual sources. By applying this procedure to each KMTNet field, we find that the mean de-reddened color and magnitude of the individual sources are ${(V-I,I)}_{\mathrm{0,1}}=(0.79\pm 0.10,19.32\pm 0.05)$ for the primary and ${(V-I,I)}_{\mathrm{0,2}}=(0.68\pm 0.07,18.63\pm 0.03)$ for the secondary. These correspond to a late and an early G-type main-sequence star for the primary and the secondary, respectively.

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Note that we also characterize the source star based on the binary-lens interpretation solely for the purpose of incorporating the limb-darkening coefficients into the binary-lens model. By following the same procedure, we determine the mean de-reddened color and magnitude of the source as ${(V-I,I)}_{0}\,=(0.76\pm 0.08,19.60\pm 0.05)$ corresponding to a G-type main-sequence star.