REANALYSIS OF THE GRAVITATIONAL MICROLENSING EVENT MACHO-97-BLG-41 BASED ON COMBINED DATA

To describe the observed light curve of the event, we begin with a standard binary-lens model. In this model, the light curve is described by seven lensing parameters. Among them, three parameters describe the geometry of the lens–source approach: the time of the closest lens–source approach, t₀, the separation between the lens and the source at the moment of the closest approach, u₀ (impact parameter), and the duration required for the source to cross the radius of the Einstein ring, t_E (Einstein timescale). The Einstein ring denotes the image of a point source when the lens and source are exactly aligned and its radius, θ_E, is commonly used as a length scale in describing lensing phenomenon. The impact parameter is expressed in θ_E. Another three parameters are needed to describe the binarity of the lens: the projected separation s and the mass ratio q between the binary-lens components, and the angle between the source trajectory and the binary axis α (source trajectory angle). The binary separation s is also expressed in units of θ_E. The two peaks of the light curve of MACHO-97-BLG-41 are likely to be features involved with caustic crossings or approaches during which finite-source effect becomes important (Dominik 1995; Gaudi & Gould 1999; Gaudi & Petters 2002). To account for this effect, an additional parameter, the normalized source radius ρ_* = θ_*/θ_E is needed, where θ_* is the angular source radius. In our standard binary-lens modeling, we additionally consider the limb-darkening variation of the source star surface brightness by introducing linear-limb-darkening coefficients, Γ_λ. The surface brightness profile is modeled as S_λ∝1 − Γ_λ(1 − 3cos ϕ/2), where λ denotes the observed passband and ϕ is the angle between the normal to the surface of the source and the line of sight toward the source (Albrow et al. 1999). Based on the source type (subgiant) determined by the spectroscopic observation conducted by Lennon et al. (1997), we adopt coefficients from Claret (2000). The adopted values are Γ_B = 0.793, Γ_V = 0.666, Γ_R = 0.575, and Γ_I = 0.479 for data sets acquired with a standard filter system. For the MSO 1.3 m data, which used a non-standard filter system, we adopt (Γ_B + Γ_V)/2 for the B-band data and (Γ_R + Γ_I)/2 for the R-band data.

Although the two sets of solutions presented by Bennett et al. (1999) and Albrow et al. (2000) already exist, we separately search for solutions in the vast parameter space encompassing wide ranges of binary separations and mass ratios in order to check the possible existence of other solutions. From this, we find a unique solution with s ∼ 0.49 and q ∼ 0.48. See Table 1 for the complete solution of the model. We note that the solution is basically consistent with the results of Bennett et al. (1999) and Albrow et al. (2000) in the interpretation of the main part of the light curve including the peak at HJD' ∼ 654. At the bottom panel of Figure 1, we present the residual of the standard binary model. It is found that there exist some significant residuals for the standard model. This is also consistent with the previous analyses that a basic binary model is not adequate to precisely describe the light curve.

The existence of residuals in the fit of the standard binary-lens model suggests the need for considering higher-order effects. We consider the following effects.

First, we consider the effect of the motion of an observer caused by the orbital motion of the Earth around the Sun. This "parallax" effect causes the source trajectory to deviate from rectilinear, resulting in long-term deviations in lensing light curves (Gould 1992). The event MACHO 97-BLG-41 lasted ∼100 days, which is an important portion of the Earth's orbital period, i.e., 1 yr, and thus the parallax effect can be important. To describe the parallax effect, we add two more lensing parameters: π_{E, N} and π_{E, E}. They denote the two components of the lens parallax vector ${\boldsymbol {\pi }}_{\rm E}$ projected onto the sky along the north and east equatorial coordinates, respectively. The direction of the lens parallax vector corresponds to the relative lens–source proper motion seen from the Earth at a certain time and its magnitude is the ratio of the relative lens–source parallax, $\pi _{\rm rel}={\rm AU}(D_{\rm L}^{-1}-D_{\rm S}^{-1})$, to the angular Einstein radius, i.e., π_E = π_rel/θ_E (Gould 2004). In our modeling, we use t₀ for the reference time of the lens parallax measurement (Gould 2004).

Second, the orbital motion of a binary lens can also cause the source trajectory to deviate from rectilinear (Dominik 1998). The orbital motion causes further deviations in lensing light curves by deforming the caustic over the course of the event. The "lens orbital" effect can be important for long timescale events produced by close binary-lens events for which the event duration comprises an important portion of the orbital period of the lens system (Shin et al. 2013). For modest orbital effects, the lens orbital motion can be described by two parameters: ds/dt and dα/dt. These parameters represent the rate of change of the binary separation and the source trajectory angle, respectively (Albrow et al. 2000).

Third, we also check the possible existence of a third body in the lens system. Introducing an additional lens component requires three additional lensing parameters including the normalized projected separation, s₂, and the mass ratio, q₂, between the primary and the third body, and the position angle of the third body with respect to the line connecting the primary and secondary of the lens, ψ.

We test models considering various combinations of the higher-order effects. In Table 2, we list the goodness of the fits and the best-fit parameters for the individual tested models. From the comparison of the models, we find the following results.

First, we confirm the result of Albrow et al. (2000) that the consideration of the orbital effect substantially improves the fit. We find that the improvement is Δχ² ∼ 441 compared to the static binary model. When we additionally consider the parallax effect, the improvement of the fit, Δχ² ∼ 2.7, is very meager. This implies that between the two effects, the orbital motion of the lens is the dominant higher-order effect in explaining the residual from the standard model. In Figure 1, we present the model light curve on the top of the observed light curve and the residual of the model. In Figure 2, we also present the geometry of the lens system.

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Second, we find that the existence of a third body does not provide a fully acceptable fit. Our best-fit solution of three-body lens modeling is consistent with the solution of Bennett et al. (1999) in the sense that the third body is a circumbinary planet with a small mass ratio. See Table 2 for the best-fit parameters and Figure 2 for the lens system geometry. With the introduction of a planetary third body, the fit does improve from the standard binary model with Δχ² ∼ 270. The additional consideration of the parallax effect further improves the fit with Δχ² ∼ 9, but it is still substantially poorer than the orbiting binary-lens model with Δχ² ∼ 166. From the comparison of the residual (see Figure 1), it is found that the triple lens model cannot precisely describe the light curve during and before the first peak, 595 ≲ HJD' ≲ 625.

The key PLANET data that exclude the triple lens are from SAAO. Their smooth "parabolic" decline signals a caustic exit along the axis of cusp. This is compatible with the triangular caustic generated by the close binary, but not with the quadrilateral caustic induced by the putative planet. In particular, the near-symmetric shape of the quadrilateral caustic implies that a cusp-axis trajectory would have a near-symmetric excess flux in the approach to the first peak as after its exit, which is not seen in the pre-peak MSO data.