OGLE-2017-BLG-0482Lb: A Microlensing Super-Earth Orbiting a Low-mass Host Star

The observed light curve appears to be a typical case of a planetary lensing event in which the planetary signal is revealed as a short-term perturbation to the smooth light curve produced by the host of the planet (Mao & Paczyński 1991; Gould & Loeb 1992). For the basic description of the lensing light curve produced by a lens composed of two masses, one requires six parameters. Three of these parameters describe the geometry of the lens-source approach: the time of the closest lens-source separation, t₀, the lens-source separation at that time, u₀ (normalized to the angular Einstein radius ${\theta }_{{\rm{E}}}$), and the time for the source to cross the Einstein radius, t_E (Einstein timescale). The other three parameters describe the binarity of the lens: the separation between the primary (M₁) and the companion (M₂), s (also normalized to ${\theta }_{{\rm{E}}}$), their mass ratio, $q={M}_{2}/{M}_{1}$, and the angle between the source trajectory and the binary axis, α (source trajectory angle). If the source crosses the planet-induced caustic, the perturbation is affected by finite-source effects (Bennett & Rhie 1996). In such cases, one needs an additional parameter $\rho ={\theta }_{* }/{\theta }_{{\rm{E}}}$ (normalized source radius) to account for the light curve deviation caused by the finite-source effect. Here, θ_* represents the angular radius of the source star.

In cases for which planetary signals can be treated as perturbations, one can heuristically estimate the lensing parameters related to the planet, i.e., s and q, from the location and duration of the perturbation (Gould & Loeb 1992; Gaudi 2012). A planet induces two types of caustics: central and planetary caustics. The central caustic lies close to the primary of the lens, and thus the caustic produces perturbations near the peak of high magnification events. The planetary caustic, on the other hand, lies away from the primary lens with a separation

$$\begin{eqnarray}&&{u}_{{\rm{p}}}=s-\displaystyle \frac{1}{s},\end{eqnarray} \tag{ 1 }$$

and thus perturbations induced by the planetary caustic can appear anywhere along the lensing light curve. The perturbation of OGLE-2017-BLG-0482 lies away from the peak and thus it is produced by a planetary caustic. The perturbation occurred when the lensing magnification was $A={({u}^{2}+2)/[u({u}^{2}+4)}^{1/2}]\,\sim 7.5$, i.e., when the lens-source separation was u ∼ 0.135. By substituting u into u_p in Equation (1), one finds that the planet–host separation is either s ∼ 0.93 or 1.07, which are referred to as the close (s < 1) and wide-separation (s > 1) solutions, respectively. The caustic induced by a close planet usually produces a dip (negative deviation) in the light curve, while the caustic induced by a wide planet always produces a bump (positive deviation). The observed bump structure of the perturbation, therefore, suggests that the planet separation is greater than ${\theta }_{{\rm{E}}}$, i.e., s ∼ 1.07.

The duration of the planet-induced perturbation results from the combination of the sizes of the caustic, Δξ_p, and the source, ρ. If the source is bigger than the caustic, ρ > Δξ_p, the duration corresponds to the source crossing time, ${\rm{\Delta }}{t}_{{\rm{p}}}\sim 2\rho {t}_{{\rm{E}}}$, and thus is mostly determined by the source radius. If the source is smaller than the caustic, on the other hand, the duration is determined by the size of the caustic. In the case of OGLE-2017-BLG-0482, the source is a very faint main-sequence star and thus the duration is likely to depend on the caustic size. The size of the planetary caustic is related to the separation and the mass ratio between the planet and the host by

$$\begin{eqnarray}&&{\rm{\Delta }}{\xi }_{{\rm{p}}}=\displaystyle \frac{4{q}^{1/2}}{{s}^{2}}\left(1+\displaystyle \frac{1}{2{s}^{2}}\right)\end{eqnarray} \tag{ 2 }$$

Han (2006). As the duration of the planetary signal is ${\rm{\Delta }}{t}_{{\rm{p}}}\,\sim {\rm{\Delta }}{\xi }_{{\rm{p}}}{t}_{{\rm{E}}}$ , the mass ratio is expressed by

$$\begin{eqnarray}&&q={\left(\displaystyle \frac{{s}^{4}}{4{s}^{2}+2}\displaystyle \frac{{\rm{\Delta }}{t}_{{\rm{p}}}}{{t}_{{\rm{E}}}}\right)}^{2}.\end{eqnarray} \tag{ 3 }$$

With s ∼ 1.07, ${\rm{\Delta }}{t}_{{\rm{p}}}\sim 2$ days, and ${t}_{{\rm{E}}}\sim 40$ days, one finds that the mass ratio is ∼10⁻⁴.

For the accurate determinations of the lensing parameters, we conduct numerical modeling of the observed light curve. We search for the solution of the lensing parameters in two steps. In the first step, we conduct a dense grid search over (log s, log q) plane. At each point on this plane, we hold the two grid parameters fixed while allowing the remaining five parameters $({t}_{0},{u}_{0},{t}_{{\rm{E}}},\rho ,\alpha )$ to vary in six Markov Chain Monte Carlo (MCMC) χ² minimizations that are equally spaced around the circle in their seed values of α. The seed values of $({t}_{0},{u}_{0},{t}_{{\rm{E}}})$ are taken from the point-lens fit, and we seed ρ at ρ = 1.0 × 10⁻³. From this preliminary search, we identify local χ² minima on the $(\mathrm{log}s,\mathrm{log}q)$ plane. In the second step, we refine the individual local minima by allowing all parameters to vary. Figure 2 displays the Δχ² map over the $(\mathrm{log}s,\mathrm{log}q)$ plane obtained from the grid search. It shows that there exists a unique planetary solution with the planet separation slightly greater than unity and a mass ratio of q = (1–3) × 10⁻⁴, which roughly matches the prediction of the heuristic analysis.

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When a source passes over or approaches very close to caustics, the planetary anomaly is affected by finite-source effects and the analysis of the deviation enables one to measure the normalized source radius ρ. In the case of OGLE-2017-BLG-0482, it is found that ρ cannot be measured due to the lack of finite-source effects. Measuring ρ is important because the angular Einstein radius, which is needed to determine the lens mass, is estimated from ρ by ${\theta }_{{\rm{E}}}={\theta }_{* }/\rho$. The difficulty of the ρ measurement is caused by the fact that the source did not cross the caustic.

Nevertheless, the source approached close to a strong cusp of the caustic around which the gradient of lensing magnification is high. In this case, one can set an upper limit on the source size. To test this possibility, we draw light curves expected from various values of ρ (see Figure 3). From this, combined with the MCMC chain obtained from modeling, we find that the 3σ upper limit of the normalized source size is ρ_max ∼ 0.006. However, a lower limit cannot be set because the best-fit model cannot be distinguished from a point-source model within 2σ.

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It is known that a subset of binary-source events can produce short-term anomalies similar to planetary perturbations and thus masquerade as planetary events (Gaudi 1998; Hwang et al. 2013, 2017). We, therefore, check the possibility of the interpretation in which the perturbation is produced by a source companion. The lensing magnification of a binary-source event corresponds to the flux-weighted mean of the magnifications associated with the individual source stars, A₁ and A₂, i.e.,

$$\begin{eqnarray}&&A=\displaystyle \frac{{A}_{1}{F}_{1}+{A}_{2}{F}_{2}}{{F}_{1}+{F}_{2}}=\displaystyle \frac{{A}_{1}+{q}_{F}{A}_{2}}{1+{q}_{F}}.\end{eqnarray} \tag{ 4 }$$

Here, ${q}_{F}={F}_{2}/{F}_{1}$ represents the ratio between the unmagnified fluxes of the individual source stars (Han & Jeong 1998). We conduct modeling of the observed data with the binary-source interpretation. For a binary-source event mimicking a planetary event, the flux ratio is usually very small, and the faint source approaches close to the lens. In this case, lensing magnifications during the perturbation can be affected by finite-source effects. We, therefore, consider finite-source effects in the modeling.

In Figure 4, we compare the fits of the planetary and binary-source models in the neighborhood of the anomaly. The best-fit binary-source model yields a flux ratio ${q}_{F,I}\sim 0.005$. One finds that the binary-source model yields an unsatisfactory description of the region before the major anomaly, i.e., 7878.5 ≲HJD' ≲ 7879.7. Numerically, we find that the binary-source model is worse than the planetary model by Δχ² = 175.2. We, therefore, exclude the binary-source interpretation.

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To precisely describe lensing light curves, it is often necessary to consider higher-order effects. In the case of OGLE-2017-BLG-0482, the event duration of ${\rm{\Delta }}t\,=2{(1-{u}_{0}^{2})}^{1/2}{t}_{{\rm{E}}}\sim 80$ days, as measured by the time during which the source is within the Einstein ring, comprises an important portion of Earth's orbital period, i.e., 1 year. In this case, the light curve can deviate from the one expected from a rectilinear lens-source relative motion due to the orbital motion of Earth: "microlens-parallax" effect (Gould 1992). Similarly, the orbital motion of the lens can also induce deviation in the lensing light curve: "lens-orbital" effect (Albrow et al. 2000).

Consideration of the higher-order effects requires additional parameters in lensing modeling. To account for the microlens-parallax effect, one needs two parameters: ${\pi }_{{\rm{E}},N}$ and ${\pi }_{{\rm{E}},E}$. These parameters denote the north and east components of the microlens-parallax vector, ${{\boldsymbol{\pi }}}_{{\rm{E}}}$, projected onto the sky along the north and east equatorial coordinates, respectively. The magnitude of the microlens-parallax vector is

$$\begin{eqnarray}&&{\pi }_{{\rm{E}}}={({\pi }_{{\rm{E}},N}^{2}+{\pi }_{{\rm{E}},E}^{2})}^{1/2}=\displaystyle \frac{{\pi }_{\mathrm{rel}}}{{\theta }_{{\rm{E}}}},\end{eqnarray} \tag{ 5 }$$

where ${\pi }_{\mathrm{rel}}=\mathrm{au}({D}_{{\rm{L}}}^{-1}-{D}_{{\rm{S}}}^{-1})$ is the lens-source relative parallax, and ${D}_{{\rm{L}}}$ and D_S represent the distances to the lens and source, respectively. The direction of ${{\boldsymbol{\pi }}}_{{\rm{E}}}$ corresponds to the relative lens-source motion, ${\boldsymbol{\mu }}$. To the first-order approximation that the change rates of the binary separation and source trajectory angle are constant, the lens-orbital effect is described by two parameters, ds/dt and $d\alpha /{dt}$, which represent the change rates of the binary separation and the source trajectory angle, respectively. The measurement of the microlens parallax is important to determine the physical parameters of the lens mass and distance because ${\pi }_{{\rm{E}}}$ is related to these parameters by

$$\begin{eqnarray}&&M=\displaystyle \frac{{\theta }_{{\rm{E}}}}{\kappa {\pi }_{{\rm{E}}}},\end{eqnarray} \tag{ 6 }$$

and

$$\begin{eqnarray}&&{D}_{{\rm{L}}}={\displaystyle \frac{\mathrm{au}}{{\pi }_{{\rm{E}}}{\theta }_{{\rm{E}}}+\pi }}_{{\rm{S}}},\end{eqnarray} \tag{ 7 }$$

where $\kappa =4G/({c}^{2}\mathrm{au})$ and ${\pi }_{{\rm{S}}}=\mathrm{au}/{D}_{{\rm{S}}}$ is the parallax of the source.

In order to check the higher-order effects, we conduct additional modeling of the observed data. From this modeling, we find that the fit substantially improves (by ${\rm{\Delta }}{\chi }^{2}\sim 110$) with the consideration of the higher-order effects. However, it is found find that the signal of the higher-order effects varies depending on the data sets. From the inspection of the cumulative Δχ² distributions as a function of time for the individual data sets, we find that most of the signal comes from the KMTC (by Δχ² ∼ 90) and the KMTS (by Δχ² ∼ 20) data sets, while the signal from the other data sets is minor. The inconsistency of the signal among different data sets is of concern because signals of the higher-order effects are subtle long-term deviations from the standard model, and thus they can be affected by the stability of photometry data. We additionally check the reality of the signal by replacing the KMTC and KMTS photometry data with new ones processed using a different software, pySIS (Albrow et al. 2000). From this, we find that the inconsistency between the KMTC+KMTS and the other data sets still persists. These results suggest the possibility that the KMTC and KMTS data are not stable enough to securely measure the higher-order parameters.

Knowing the possibility of systematics in the KMTC and KMTS data, we mainly use the OGLE data set to measure the higher-order parameters. The KMTNet survey started in 2015 season and thus the system is still under development. On the other hand, the OGLE system is very stable from its 25-year operation since 1992. While the higher-order effect parameters are determined based on the overall shape of the lensing light curve, the planet parameters are determined by the planetary anomaly. The overall light curve was well covered by the OGLE data, but the OGLE coverage around the planetary anomaly is poor. In the analysis, we therefore use the combination of data sets with the whole OGLE data set plus partial data sets from the other data sets. The KMTNet+MOA data used in the analysis cover the anomaly region during $7875\lt \mathrm{HJD}^{\prime} \lt 7887$. We exclude MOA data outside this region as well because of the instability in the baseline.

In Figure 5, we present the Δχ² map of MCMC chains in the $({\pi }_{{\rm{E}},E},{\pi }_{{\rm{E}},N})$ plane obtained using the the restricted data set. In the model considering the microlens-parallax effect, it is known that there exist a pair of degenerate solutions with u₀ > 0 and u₀ < 0 due to the mirror symmetry of the source trajectories between the two degenerate solutions (Smith et al. 2003; Skowron et al. 2011), and thus we check this so-called ecliptic degeneracy. We note that the lensing parameters of the two solutions resulting from the ecliptic degeneracy are approximately in the relation $({u}_{0},\alpha ,{\pi }_{{\rm{E}},N},d\alpha /{dt})\,\leftrightarrow -({u}_{0},\alpha ,{\pi }_{{\rm{E}},N},d\alpha /{dt})$. From the distributions of MCMC points, we find that both u₀ > 0 and u₀ < 0 solutions result in similar values of ${\pi }_{{\rm{E}}}$. The measured microlens-parallax parameters are (π_E,N, π_E,E) = (0.19 ± 0.75, −0.06 ± 0.12) for the u₀ > 0 model and (−0.17 ± 0.73, −0.12 ± 0.11) for the u₀ < 0 model. It is found that the east component of the microlens-parallax vector, ${\pi }_{{\rm{E}},E}$, is well constrained but the uncertainty of the north component, ${\pi }_{{\rm{E}},N}$, is considerable.

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In Table 1, we list the lensing parameters of the best-fit solutions. Because the ecliptic degeneracy is severe with Δχ² < 1, we present both solutions with u₀ > 0 and u₀ < 0. Also presented are the flux from the source F_s and the blended light F_b measured from the OGLE data. We find that the event was produced by a planetary system with s ∼ 1.07 and q ∼ 1.4 × 10⁻⁴.

Table 1.  Best-fit Lensing Parameters

Parameter	Value
	${u}_{0}\gt 0$	${u}_{0}\lt 0$
${\chi }^{2}$	1548.4	1548.6
t0 (HJD')	7873.948 ± 0.013	7873.952 ± 0.013
u0	0.059 ± 0.002	−0.058 ± 0.002
${t}_{{\rm{E}}}$ (days)	40.01 ± 1.45	40.39 ± 1.21
s	1.07 ± 0.01	1.07 ± 0.01
q (10−4)	1.35 ± 0.20	1.41 ± 0.30
α (rad)	0.369 ± 0.023	−0.365 ± 0.026
${\pi }_{{\rm{E}},N}$	0.19 ± 0.75	−0.17 ± 0.73
${\pi }_{{\rm{E}},E}$	−0.06 ± 0.12	−0.12 ± 0.11
ds/dt (yr−1)	−1.94 ± 0.82	−1.96 ± 0.96
$d\alpha /{dt}$(yr−1)	−0.29 ± 0.27	0.87 ± 0.50
${F}_{s,\mathrm{OGLE}}$	0.124 ± 0.006	0.122 ± 0.004
${F}_{b,\mathrm{OGLE}}$	0.029 ± 0.005	0.031 ± 0.004

Note. $\mathrm{HJD}^{\prime} =\mathrm{HJD}-2450000$.

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In Figure 6, we present the lens system geometry (for the u₀ > 0 solution), in which the source trajectory with respect to the lens and caustic are shown. Due to the proximity of the normalized planet–host separation to unity, i.e., s ∼ 1, the caustics form a single closed curve for which the central and planetary caustics are merged into a single caustic, i.e., resonant caustic. The planetary perturbation was produced when the source approached the strong planet-side cusp of the caustic located on the planet–host axis. As mentioned in Section 3.2, the source trajectory did not pass over the caustic and thus finite-source effects cannot be measured. We note that the uncertainties of the lens-orbital parameters ds/dt and $d\alpha /{dt}$ are big, indicating that the lens-orbital motion is poorly constrained. Nevertheless, it is important to fit for these parameters because they can be correlated with the parallax parameters (Batista et al. 2011; Skowron et al. 2011).

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In the analysis of lensing events, characterizing the source star is important because the angular Einstein radius, which is needed to determine the lens mass and distance by Equations (6) and (7), is derived from the angular source radius by ${\theta }_{{\rm{E}}}={\theta }_{* }/\rho$. In the case of OGLE-2017-BLG-0482, the normalized source radius ρ and thus the angular Einstein radius ${\theta }_{{\rm{E}}}$ cannot be measured. However, the upper limit on ρ leads to a lower limit on ${\theta }_{{\rm{E}}}$, and this lower limit may help to constrain the physical lens parameters. Thus, determination of θ_* could be important depending on what limit is obtained.

We characterize the source based on its dereddened color ${(V-I)}_{0}$ and brightness I₀. To determine ${(V-I)}_{0}$ and I₀ from the instrumental color V − I and brightness I, we apply the method of Yoo et al. (2004) using the centroid of red giant clump (RGC) as a reference. Figure 7 shows the locations of the source and the RGC centroid in the instrumental color–magnitude diagram. With the offsets in color and brightness ${\rm{\Delta }}(V-I,I)\,=(-0.69,3.56)$ with respect to the RGC centroid and the known dereddened color and brightness of the RGC centroid of ${(V-I,I)}_{0,\mathrm{RGC}}=(1.06,14.46)$ (Bensby et al. 2013; Nataf et al. 2013), it is estimated that the dereddened color and brightness of the source are ${(V-I,I)}_{0}$ = ${(V-I,I)}_{0,\mathrm{RGC}}$ +${\rm{\Delta }}(V-I,I)=(0.37,18.02)$. We note that the relatively blue color indicates that the source is a main-sequence star located in the disk. Considering the faintness, the star is likely to lie behind the obscuring dust.

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Once the dereddened V − I color is measured, we then use the VIK color–color relations of Bessell & Brett (1988) to convert from V − I to V − K and apply the color/surface-brightness relation of Kervella et al. (2004) to obtain ${\theta }_{* }=0.56\,\mu$as. Combined with the 3σ upper limit we have derived on ρ and the measured Einstein timescale, this implies

$$\begin{eqnarray}&&{\theta }_{{\rm{E}}}=\displaystyle \frac{{\theta }_{* }}{\rho }\gt 0.093\,\mathrm{mas}\end{eqnarray} \tag{ 8 }$$

and

$$\begin{eqnarray}&&\qquad \mu =\displaystyle \frac{{\theta }_{{\rm{E}}}}{{t}_{{\rm{E}}}}\gt 0.86\,\mathrm{mas}\,{\mathrm{yr}}^{-1}.\end{eqnarray} \tag{ 9 }$$

We note that the lower limits of ${\theta }_{{\rm{E}}}$ and μ can be subject to additional uncertainties because of the uncertain ratio between the extinction values toward the source and the RGC centroid. The method using a color–magnitude diagram to derive ${\theta }_{* }$ assumes that the amount of extinction toward the source and RGC stars is same. For OGLE-2017-BLG-0482, the source is likely to be in the disk, and thus the hypothesis of the same extinction may not be valid. If the source suffers less extinction than RGC stars, it would be actually fainter and redder than if it were at the same distance as the RGC stars. Therefore, the derived value of ${\theta }_{* }=0.56\,\mu \mathrm{as}$ is an upper limit and the additional uncertainty would propagate into ${\theta }_{{\rm{E}},\min }$ and ${\mu }_{\min }$.