Statistics of Microlensing Caustic Crossings in Q~2237+0305: Peculiar Velocity of the Lens Galaxy and Accretion Disk Size

We use the statistics of caustic crossings induced by microlensing in the lens system Q~2237+0305 to study the lens galaxy peculiar velocity. We calculate the caustic crossing rates for a comprehensive family of stellar mass functions and find a dependence of the average number of caustic crossings with the effective transverse velocity and the average mass, $\langle n \rangle \propto {v_{eff} / \sqrt{\langle m \rangle}}$, equivalent to the theoretical prediction for the case of microlenses with identical masses. We explore the possibilities of the method to measure $v_{eff}$ using the $\sim$12 years of OGLE monitoring of the four images of Q 2237+0305. To determine a lower limit for $v_{eff}$ we count, conservatively, a single caustic crossing for each one of the 4 high magnification events identified in the literature (plus one additional proposed by us) obtaining $v_{eff} \gtrsim 240\sqrt{\langle m \rangle/0.17M_\odot}\rm\,km\, s^{-1}$ at 68\% of confidence. From this value and the average $FWHM$ of 4 high magnification events we obtain a lower limit of $r_s \gtrsim 1.4 \sqrt{\langle m \rangle/0.17M_\odot}$ light-days for the radius of the source ($r_s=FWHM/2.35$). Tentative identification of 3 additional caustic crossing events leads to estimates of $v_{eff}\simeq (493\pm 246)\sqrt{\langle m \rangle/0.17M_\odot}\rm\,km\, s^{-1}$ for the effective transverse velocity and of $r_s \simeq (2.7\pm 1.3)\sqrt{\langle m \rangle/0.17M_\odot}$ light-days for the source size. The estimated transverse peculiar velocity of the galaxy is $v_t \simeq(429\pm246)\sqrt{\langle m \rangle/0.17M_\odot}\rm\,km\, s^{-1}$.