Australia Telescope National Facility

Source size measurement from observed quasar microlensing
Atsunori Yonehara
, PASA, 18 (2), in press.

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Method

First of all, I have to know about the magnification pattern to obtain ideal light curves of quasar microlensing. Generally, in the case of quasar microlensing, the magnification patterns are complicated and hugely varied (e.g., Wambsganss et al. 1990). In contrast, if the mass fraction of the objects that contribute to the microlensing events is small ( $\sim 10 \%$ ), there is a typical magnification pattern. The magnification pattern in the case of small mass fraction is almost identical to that of the ``Chang & Refsdal (Chang & Refsdal 1984)'' lens case. In Figure 1, an example of magnification patterns for quasar microlensing on the image C are presented.

**Figure 1:** Magnification patterns for quasar microlensing in the case of microlens mass fraction =0.9, 0.5, and 0.1 are displayed in left, middle, and right panel, respectively. These figures are calculated from the code developed by Wambsganss (1990).
$\begin{figure} \begin{center} \psfig{file=fig1.eps,height=6cm} \end{center} \end{figure}$

Then, I assume such a situation and apply the convergence ( $\kappa$ ) and shear ( $\gamma$ ) value on the image C which was obtained by Schmidt, Webster and Turner (1998), i.e., $\kappa = 0.69$ and $\gamma = 0.71$ . Even if I accept such simplification, it will be difficult to calculate the exact magnification factor in the case of finite-size sources. Thus, I adopt approximated formulae for the magnification factor in the vicinity of fold (Fluke & Webster 1999) and cusp caustics (Zakharov, 1995). Both of these two sorts of caustics seem to induce the observed, highly magnified microlensing event in image C. Furthermore, in the case of the Chang & Refsdal (1984) lens, two different types of cusp caustics and several fold caustics can be formed. Then, I consider two distinctive cusp caustics cases and four representative fold caustics cases for magnification, numerically integrate the magnification factor over the source, and obtain the effective magnification factor for the circular, finite-size source with a given radius (normalized by the Einstein-Ring radius, $r_{\rm E}$ ). Finally, to get the best-fit light curve and a corresponding set of parameters, I employ the standard method, i.e., the minimization of the $\chi ^2$ value between the observed light curve (number of data points = 83) and an ideal light curve for a given set of parameters. The set of parameters consists of source size, velocity relative to the caustic, impact parameter, the epoch of caustic crossing, constant magnification and its gradual change caused by other caustics. The total numbers of parameters to calculate the light curve are 6 and 7 for the fold and the cusp caustics cases, respectively. The adopted method to minimize the $\chi ^2$ value is a kind of downhill simplex method, so-called ``AMOEBA (Press et al. 1992)''. After obtaining the set of best-fit parameters, additionally, I have also estimated the confidence region for parameters by using a Monte-Carlo method as follows: (1) By supposing that the best-fit parameter is a real parameter, I calculate an ideal light curve without any errors. (2) By adding random errors to the magnitude corresponding to the observational error dispersion and by sampling this light curve at the times corresponding to the actually observed times, I obtain a mock light curve. (3) By using this mock light curve, I again perform a light-curve fitting and obtain a set of the best-fit parameters for the mock light curve. (4) By iterating procedures (2) and (3) many times, in this study 100 times, and summarizing the best-fit values for mock light curves, I can evaluate the confidence region. Details about these methods are shown in Yonehara (2001).

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