A new model of the gravitational lens 0957+561 and a limit on the Hubble constant

We present a simple mass model for the lensing galaxy in the gravitationally lensed quasar 0957+561. The model is a generalization of the singular isothermal sphere and includes a core radius, r_c, and a power-law index, \eta, defined such that mass increases as r^\eta at large radius. We approximate the galaxy cluster surrounding the lensing galaxy with a quadratic potential described by its convergence \kappa and shear \gamma. We fit the model to a recent high resolution VLBI map of the two images of 0957+561. We obtain a tight constraint on the radial index, 1.07 < \eta <1.18, which means that the lens galaxy is nearly isothermal with increasing mass-to-light ratio out to 15 h^{-1} kpc. We also obtain an upper limit on the core radius, r_c < 330 h^{-1} pc. We use the model to calculate the Hubble constant H_0 as a function of the time delay \Delta\tau_{BA} between the two images: H_0 = \left({ 82.5^{+5.9} _{-3.0} }\right) (1 - \kappa) \left({ \Delta\tau_{BA}/1.1 \,{\rm yr} }\right)^{-1} km/s/Mpc. Once \Delta \tau_{BA} is measured, this will provide an upper bound on H_0 since \kappa cannot be negative. In addition, the model degeneracy due to \kappa can be eliminated if the one-dimensional velocity dispersion \sigma of the lensing galaxy is measured. We then have H_0 = \left({ 82.5^{+8.7} _{-5.6} }\right) (\sigma / 322\,{\rm km/s})^2 \left({ \Delta \tau_{BA} /1.1 \,{\rm yr} }\right)^{-1} km/s/Mpc. We investigate the effects of ellipticity in the lensing galaxy and clumpiness in the lensing cluster and find that these cause little change in our results