Greg Dobler edited Microlensing.tex  over 10 years ago

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Microlensing has long been acknowledged as a significant source of potential error when estimating time delays from optical monitoring data \citep[see e.g.][and references therein]{TewesEtal2013a}. Except for caustic crossing events, the microlensing signal tends to be smoothly varying over long ($\sim$decade) timescales. While strongest at bluer wavelengths, where the accretion disk size is smaller, microlensing variability is seen in the optical bands where the image quality is highest, and is hence unavoidable in a synoptic optical survey. The solution is to model the microlensing in each quasar image individually, at the same time as inferring the time delay \citep[e.g.]{Kochanek2004,TewesEtal2013b}.  We include realistic microlensing signals in each quasar image lightcurve by tracking the position of the source as it crosses a static starfield with the appropriate stellar density, convergence and shear, with the appropriate velocity, and computing and applying the corresponding microlensing magnification. Our approach is similar to that of \citet{XXX} and \citet{YYY}. For each lens in the OM10 catalog (for which the gravitational lens shear and convergence, $\gamma$ and $\kappa$, are known), we estimate the fraction of mass in stars, $f^*$, at each image position, given a simple model for the stellar mass distribution: we estimate the i-band luminosity given the lens velocity dispersion from the Faber-Jackson relation of \citet{XXX}, and then convert this to a stellar mass assuming a mass to light ratio of XXX \citep{YYY}. We then distribute that stellar mass in a \citet{deVaucoleurs1948} density profile centred on the lens with the given ellipticity and orientation; the effective radius we draw from the Fundamental Plane relation of \citet{}. Given values of $f*$, $\gamma$, $\kappa$ we generate a starfield and compute its magnification map on the source plane following the method of \citet{Wambsganss}.\footnote{The microlensing code used in this work, {\sc MULE} MULES}  is freely available at \texttt{https://github.com/gdobler/mule}.} \texttt{https://github.com/gdobler/mules}.}  The effect of the source having non-zero size is to smooth out the magnification map; this can be an important effect \citep{DoblerAndKeeton}, not least because it enables the measurement of accretion disk sizes \citep{Kochanek2004}. We draw a source size from a distribution implied by recent disk size estimates, and assume a suitable brightness profile before convolving this profile with the magnification map. We then draw a random 10-year path across this smoothed magnification map, with path length given by the source's velocity relative to the lens, $v_{\rm rel}$, and read off the magnification values. To obtain a plausible value of $v_{\rm rel}$, we draw two objects at their given redshifts from the Millennium Simulation \citep{MS}, and compute their relative velocity \citepp[as in][]{relativevelocity}. The resulting microlensing magnification curve is then multiplied by that quasar image's flux values to give a microlensed lightcurve.