## 3. ModelsIn this section, we will fit the OGLE V and Most microlensing light curves are well described by the standard form (e.g., Paczynski 1986): where is the impact parameter (in units of the Einstein radius) and with being the time of the
closest approach (maximum magnification),
the Einstein radius, where To fit both the Best-fit parameters (and their errors) are found by minimizing the
usual using the MINUIT program in
the CERN library and are tabulated in Table 1. The resulting
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 and
the latter has . For the standard
fit, the lensed part has for 161
data points, and the unlensed part has
for 340 data points; the somewhat
high 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
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
in the
To take into account the Earth motion around the Sun, we have to modify the expression for in Eq. (1). This modification, to the first order of the Earth's orbital eccentricity (), is given by Alcock et al. (1995) and Dominik (1998): where is the angle between and the line formed by the north ecliptic axis projected onto the lens plane, is now more appropriately the minimum distance between the lens and the Sun-source line. The expression of and are given by and where is the time of perihelion, is the transverse speed of the lens projected to the solar position, , and 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), where is the longitude of the
vernal equinox measured from the perihelion.
(rad), and the Julian day for
Perihelion is ; the readers are
referred to the The two-color light curves show that the lensed object became bluer by 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 To model the blending in two colors, we need two further parameters
- the fraction of light contributed by the unlensed component in
The best-fit parameters for this model are given in Table 1.
Compared with the standard fit, the
is reduced from 893.9 to 640.8. The reduction in the lensed part is
dramatic: the drops from 407.9 to
177.7 for 161 data points. The for
the unlensed part is 463.2 (as compared to 486.0 for the standard fit)
for 340 data points. The per degree
of freedom is satisfactory. The predicted light curve (solid line in
Fig. 1) matches the observed data both in the © European Southern Observatory (ESO) 1999 Online publication: October 4, 1999 |