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. We represent the galaxy as a softened power-law sphere (SPLS), a generalization of the singular isothermal sphere with three parameters-ρ₀_, the central density; θc_, the angular core radius; and η, the radial index, which is defined such that mass increases as reta^ at large radius. As in previous studies, we approximate the galaxy cluster surrounding the lensing galaxy by means of a quadratic potential described by its convergence κ and shear γ. A feature of the model is that it does not require a large central compact mass. We fit the model to a recent high-resolution VLBI map of the two images of 0957+561. The data provide a number of independent constraints, and the model fit has 6 degrees of freedom, which is a significant improvement over previous models. Although the reduced χ²^ of the best-fit model is only 4. 3, nevertheless we obtain a tight constraint on the radial index, 1. 07 < η < 1. 18, at the 95% confidence level. Thus, the galaxy has mass increasing slightly more rapidly than isothermal (η = 1) out to at least 15 h^-1^ kpc. Since the light from the galaxy follows a de Vaucouleurs profile, we deduce that the mass-to-light ratio of the galaxy increases rapidly with increasing radius. We also obtain an upper limit on the core radius, namely θc_ < 0. 11" or linear core radius < 330 h^-1^ pc. We use the model to calculate the Hubble constant H₀_ as a function of the time delay ATBA DELTAtBA_ between the two images. We obtain H₀_ = (60. 5^+5. 3^_-2. 2_) (1 - κ) (DELTAtBA_/1. 5 yr) ^-1^ km s^-1^ Mpc^-1^, or = (82. 5^+7. 2^_-3. 0_) (1 - κ) (DELTAtBA_/1. 1 yr) ^-1^ km s^-1^ Mpc^- 1^ Once DELTAtBA_ is measured, this will provide an upper bound on H₀_ 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. In this case, we find that H₀_ = (60. 5^+6. 4^_-4. 1_) (σ/322 km s^-1^) ²^ (DELTAtBA_/1. 5 yr) ^- 1^ km s^-1^ Mpc^-1^, or = (82. 5^+8. 7^_-5. 6_) (σ/322 km s^-1^) ²^ (DELTAtBA_/1. 1 yr) ^-1^ km s^-1^ Mpc^-1^. We find that these results are virtually unchanged if we include the ellipticity of the lensing galaxy or clumpiness of the lensing cluster.