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

Abstract

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, $⟨n⟩\propto v_{eff}/\sqrt{⟨m⟩}$, 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 $v_{eff}\gtrsim 240\sqrt{⟨m⟩/0.17M_{\odot }}\mathrm{km}{\mathrm{s}}^{-1}$ at 68\% of confidence. From this value and the average $FWHM$ of 4 high magnification events we obtain a lower limit of $r_s\gtrsim 1.4\sqrt{⟨m⟩/0.17M_{\odot }}$ 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 $v_{eff}\simeq (493\pm 246)\sqrt{⟨m⟩/0.17M_{\odot }}\mathrm{km}{\mathrm{s}}^{-1}$ for the effective transverse velocity and of $r_s\simeq (2.7\pm 1.3)\sqrt{⟨m⟩/0.17M_{\odot }}$ light-days for the source size. The estimated transverse peculiar velocity of the galaxy is $v_t\simeq (429\pm 246)\sqrt{⟨m⟩/0.17M_{\odot }}\mathrm{km}{\mathrm{s}}^{-1}$.

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