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

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Abstract
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-rho(0), the central density; theta(c), the angular core radius; and eta, l, the radial index, which is defined such that mass increases as r(eta) 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 kappa and shear gamma. 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 chi(2) of the best-fit model is only 4.3, nevertheless we obtain a tight constraint on the radial index, 1.07 < eta < 1.18, at the 95% confidence level. Thus, the galaxy has mass increasing slightly more rapidly than isothermal (eta = 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 theta(c) < 0''.11 or linear core radius <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. We obtain H-0 = (60.5(-2.2)(+5.3))(1 - kappa)(Delta tau(BA)/1.5 yr)(-1) km s(-1) Mpc(-1), or = (82.5(-3.0)(+7.2))(1 - kappa)(Delta tau(BA)/1.1 yr)(-1) km s(-1) Mpc(-1). 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. In this case, we find that H-0 = (60.5(-4.1)(+6.4))(sigma/322 km s(-1))(2)(Delta tau(BA)/1.5 yr)(-1) km s(-1) Mpc(-1), or = (82.5(-5.6)(+8.7))(sigma/322 km s(-1))(2)(deltBA/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 dumpiness of the lensing cluster.

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