Australia Telescope National Facility

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

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Results and discussions

**Figure 2:** Observational data (bars) and the best-fit light curves (solid lines) in the case of fold caustics (left 4 panels) and in the cause of cusp caustics (right 2 panels).
$\begin{figure} \begin{center} \psfig{file=fig2.eps,height=8cm} \end{center} \end{figure}$

The resultant, best-fit light curves are shown in Figure 2 and corresponding paths of the source relative to the caustic are also shown in Figure 3.

**Figure 3:** Considered caustics (solid curves) and paths to reproduce the best-fit light curves (dashed lines) are presented.
$\begin{figure} \begin{center} \psfig{file=fig3.eps,height=9cm} \end{center} \end{figure}$

Evidently, the best-fit parameters finely reproduce the observed light curves. The resultant source size and the reduced $\chi ^2$ ( $\bar{\chi} ^2$ ) are presented in Table 1. In this table the estimated confidence regions are also presented.

Table 1: The best-fit source sizes in the unit of the Einstein-Ring radius, 90 % confidence regions for the best-fit source sizes obtained by Monte-Carlo simulations (see text), and reduced $\chi ^2$ values are presented.

	fold-1	fold-2	fold-3	fold-4	cusp-1	cusp-2
size	0.18_-0.02^+0.02	0.24_-0.02^+0.02	0.85_-0.37^+0.13	0.76_-0.07^+0.09	0.21_-0.05^+0.14	0.10_-0.02^+0.08
$\bar{\chi} ^2$	1.46	1.49	1.48	1.47	1.33	1.30

In the case of an infinitely small-size source (a point source), the expected microlens light curves are quite different from case to case, e.g., light curves for the fold caustics case and those for the cusp caustics cases are clearly different. Moreover, in the case of a caustic crossing event, the light curve should show a spiky feature around its peak flux and, hence, such a case can be safely rejected (see Figure 4). Conversely, if the finite-size source effect is taken into account, such a spiky peak will be smeared out and I can manage to reproduce the observed light curve with an acceptable goodness of fit (

$\bar{\chi^2} \sim 1$ ) as you can see in table 1. For every case, if the source size is larger than the best-fit value, expected magnification will be suppressed too much, the light curve will become shallow, and the fit will not be as good. On the other hand, if the source is smaller than the best-fit value, the expected magnification will be large, the light curve will sharpen, and again the fit will not be as good. These are qualitative reasons why the resultant source sizes are limited in a somewhat small range as presented in Figure 4 clearly. In this figure, not only the total $\chi ^2$ values between the observed light curve and the best-fit light curve for given source sizes, but also the distributions of the best-fit source sizes for mock light curves obtained by Monte-Carlo simulations (the procedure is described before), are depicted. There are no clear differences in the goodness of fit between all the considered cases, and I can not unambiguously choose the best parameter. From these results, however, I can put a conservative limit on a source size in the units of $r_{\rm E}$ . This upper limit is given in the case of fold-3 and I can say that the source size of Q2237+0305 should be smaller than

$\sim 0.98 r_{\rm E}$ (more than 90% confidence level). Since the Einstein-Ring radius of quasar microlensing is typically

$\le 1 ~\mu{\rm as}$ , this result indicates the existence of a sub- $\mu{\rm as}$ source in a quasar! To convert the limit in the units of $r_{\rm E}$ obtained above into physical units I have to calculate $r_{\rm E}$ for relevant parameters. Fortunately, in the case of Q2237+0305, the Einstein-Ring radius weakly depends on cosmological parameters. Assuming the mass of the lens object and the Hubble's constant are equal to $1.0 M_{\odot}$ and

$70 {\rm ~km~s^{-1}~Mpc^{-1}}$ , respectively, the Einstein-Ring radius will correspond to

$\sim 10^{17} {\rm cm}$ and the source size should be smaller than

$\sim 10^{17} {\rm cm}$ . Moreover, for the

$\sim 0.1 M_{\odot}$ lens object case suggested by Wyithe, Webster and Turner (2000) as a mean lens mass, the upper limit for the source size will be reduced by a factor of $\sqrt{10}$ and become

$\sim 2 \times 10^3~{\rm AU}$ !

**Figure 4:** Solid lines show the total $\chi ^2$ values between observed light curve and the best-fit light curve for given source sizes. Histograms present the distributions of the best-fit source sizes for mock light curves obtained by Monte-Carlo simulations (see text). The unit of source size is the Einstein-Ring radius. Kinks of solid lines are caused by dramatic changes of path of the source to reproduce the best-fit light curve, e.g., from caustics crossing case to caustics gazing case.
$\begin{figure} \begin{center} \psfig{file=fig4.eps,height=8cm} \end{center} \end{figure}$

The result indicates that there is a luminous (

$\sim 10^{43}~{\rm erg~s^{-1}} \sim 10^{10}L_{\odot}$ without magnification due to the macrolens effect by foreground, lensing galaxy) but compact object in the central region of this quasar. Until now, at least, we have never discovered and/or recognized such a luminous but compact object except for an accretion disk. Therefore, this result strongly supports the existence of an accretion disk in a quasar.

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