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Astron. Astrophys. 350, L19-L22 (1999)

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3. Models

In this section, we will fit the OGLE V and I -band light curves simultaneously with theoretical models. We start with the simple standard model, and then consider a fit that takes into account both parallax and blending.

Most microlensing light curves are well described by the standard form (e.g., Paczynski 1986):

[EQUATION]

where [FORMULA] is the impact parameter (in units of the Einstein radius) and

[EQUATION]

with [FORMULA] being the time of the closest approach (maximum magnification), [FORMULA] the Einstein radius, v the lens transverse velocity relative to the observer-source line of sight, and [FORMULA] the Einstein radius crossing time. The Einstein radius is defined as

[EQUATION]

where M is the lens mass, [FORMULA] the distance to the source and [FORMULA] is the ratio of the distance to the lens and the distance to the source.

To fit both the I -band and V -band data with the standard model, we need five parameters, namely,

[EQUATION]

Best-fit parameters (and their errors) are found by minimizing the usual [FORMULA] using the MINUIT program in the CERN library and are tabulated in Table 1. The resulting [FORMULA] is 893.9 for 496 degrees of freedom. For convenience, we divide the data into one `unlensed' part and one `lensed' part; the former has [FORMULA] and the latter has [FORMULA]. For the standard fit, the lensed part has [FORMULA] for 161 data points, and the unlensed part has [FORMULA] for 340 data points; the somewhat high [FORMULA] for this part may be due to contaminations of nearby bright stars (as can be seen in the finding chart), particularly at poor seeing conditions. The [FORMULA] per degree of freedom for the `lensed' part (with 161 data points) is about 2.5, indicating that the fit is poor. This can also be seen from Fig. 1, where we show the model light curve together with the data points. As can be seen, the observed values are consistently brighter than the predicted ones for [FORMULA] in the I -band. Further, the prediction is fainter by about 0.05 magnitude at the peak in the V -band. We show next that both inconsistencies can be removed by incorporating parallax effect and blending.

[FIGURE] Fig. 1. The I -band (left) and V -band (right) light curves observed by the OGLE collaboration are shown. In each panel, the constant part of light curve is shown at the top with their amplitudes shifted by 1.5 magnitude and with their time intervals (from [FORMULA] to [FORMULA]) labelled at the top axis. The dotted line indicates the best-fit standard model (Eq. 1), while the solid line is for the best-fit model that takes into account both parallax and blending (Eqs. 5-9). Fit parameters are given in Table 1.


[TABLE]

Table 1. The best standard model (first row) and the best parallax model with blending (second row) for OGLE-1999-CAR-1.


To take into account the Earth motion around the Sun, we have to modify the expression for [FORMULA] in Eq. (1). This modification, to the first order of the Earth's orbital eccentricity ([FORMULA]), is given by Alcock et al. (1995) and Dominik (1998):

[EQUATION]

where [FORMULA] is the angle between [FORMULA] and the line formed by the north ecliptic axis projected onto the lens plane, [FORMULA] is now more appropriately the minimum distance between the lens and the Sun-source line. The expression of [FORMULA] and [FORMULA] are given by

[EQUATION]

and

[EQUATION]

where [FORMULA] is the time of perihelion, [FORMULA] is the transverse speed of the lens projected to the solar position, [FORMULA], and [FORMULA] is the longitude measured in the ecliptic plane from the perihelion toward the Earth's motion; this is given in the appendix of Dominik (1998),

[EQUATION]

where [FORMULA] is the longitude of the vernal equinox measured from the perihelion. [FORMULA] (rad), and the Julian day for Perihelion is [FORMULA]; the readers are referred to the The Astronomical Almanac (1999) for the relevant data. Note that the inclusion of the parallax effect introduces two more parameters, [FORMULA] and [FORMULA].

The two-color light curves show that the lensed object became bluer by [FORMULA] mag at the peak of magnification; such chromaticity is easily produced by blending. The additional source of light may be from the lens itself and/or it can come from another star which lies in the seeing disk of the lensed star by chance alignment. When blending is present, the observed magnification is given by

[EQUATION]

To model the blending in two colors, we need two further parameters - the fraction of light contributed by the unlensed component in I and V , [FORMULA] and [FORMULA], at the baseline. Therefore, a fit that takes into account both parallax and blending effects requires 9 parameters: [FORMULA], [FORMULA], [FORMULA], and [FORMULA].

The best-fit parameters for this model are given in Table 1. Compared with the standard fit, the [FORMULA] is reduced from 893.9 to 640.8. The reduction in the lensed part is dramatic: the [FORMULA] drops from 407.9 to 177.7 for 161 data points. The [FORMULA] for the unlensed part is 463.2 (as compared to 486.0 for the standard fit) for 340 data points. The [FORMULA] per degree of freedom is satisfactory. The predicted light curve (solid line in Fig. 1) matches the observed data both in the I -band and V -band. From Table 1, the blending fractions in the V and I bands are not well constrained, [FORMULA] and [FORMULA]. The differential blending, however, is reasonably constrained, [FORMULA] due to the observed differential magnification (0.05 mag) between the I -band and V -band. The projected lens velocity is well constrained while its direction has somewhat larger errors. For completeness, we mention that the best model that accounts for blending but not parallax has [FORMULA] while the best model that accounts for parallax but not blending has [FORMULA]. Hence the parallax effect reduces the [FORMULA] much more effectively than blending. This can be easily understood since the observed light curve is asymmetric, which cannot be produced by blending.

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© European Southern Observatory (ESO) 1999

Online publication: October 4, 1999
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