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 $15h^{-1}$ kpc. We also obtain an upper limit on the core radius, $r_c < 330 h^{-1}$pc\mathrm{.}WeusethemodeltocalculatetheHubbleconstant$H_0$ as a function of the time delay $\mathrm{\Delta }{\tau }_{BA}$ 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\mathrm{/}s\mathrm{/}Mpc\mathrm{.}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.