Gravitational lens magnification and the mass of Abell 1689

We present the first application of lens magnification to measure the absolute mass of a galaxy cluster: Abell 1689. The absolute mass of a galaxy cluster can be measured by the gravitational lens magnification of a background galaxy population by the cluster gravitational potential. The lensing signal is complicated by the intrinsic variation in number counts resulting from galaxy clustering and shot noise and by additional uncertainties in relating magnification to mass in the strong lensing regime. Clustering and shot noise can be dealt with using maximum likelihood methods. Local approximations can then be used to estimate the mass from magnification. Alternatively, if the lens is axially symmetric we show that the amplification equation can be solved nonlocally for the surface mass density and the tangential shear. In this paper we present the first maps of the total mass distribution in Abell 1689, measured from the deficit of lensed red galaxies behind the cluster. Although noisier, these reproduce the main features of mass maps made using the shear distortion of background galaxies, but have the correct normalization, finally breaking the "sheet-mass" degeneracy that has plagued lensing methods based on shear. Averaging over annular bins centered on the peak of the light distribution, we derive the cluster mass profile in the inner 4' (0.48 h(-1) Mpc). These show a profile with a near-isothermal surface mass density kappa approximate to (0.5 +/- 0.1)(theta/1')(-1) out to a radius of 2.'4 (0.28 h(-1) Mpc), followed by a sudden drop into noise. We find that the projected mass interior to 0.24 h(-1) Mpc is M(<0.24 h(-1) Mpc)= (0.50 +/- 0.09) x 10(15) h(-1) M,. We compare our results to masses estimated from X-ray temperatures and line-of-sight velocity dispersions, as well as to weak shear and lensing arclets. We find that the masses inferred from X-ray, line-of-sight velocity dispersions, arclets, and weak shear are all in fair agreement for Abell 1689.