A New Model of the Gravitational Lens 0957+561 and a Limit on the Hubble Constant

Abstract

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, $η$, defined such that mass increases as $r^η$ at large radius. We approximate the galaxy cluster surrounding the lensing galaxy with a quadratic potential described by its convergence $κ$ and shear $γ$. 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, $1.07 < η<1.18$, 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 $Δτ_{BA}$ between the two images: $H_0 = \left({ 82.5^{+5.9} _{-3.0} }\right) (1 - κ) \left({ Δτ_{BA}/1.1 \,{\rm yr} }\right)^{-1}$ km/s/Mpc. Once $Δτ_{BA}$ is measured, this will provide an upper bound on $H_0$ since $κ$ cannot be negative. In addition, the model degeneracy due to $κ$ can be eliminated if the one-dimensional velocity dispersion $σ$ of the lensing galaxy is measured. We then have $H_0 = \left({ 82.5^{+8.7} _{-5.6} }\right) (σ/ 322\,{\rm km/s})^2 \left({ Δτ_{BA} /1.1 \,{\rm yr} }\right)^{-1}$ 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.