Improved Constraints on the Gravitational Lens Q0957+561. I. Weak
  Lensing
  Attempts to constrain the Hubble constant using the strong gravitational lens
system Q0957+561 are limited by systematic uncertainties in the mass model,
since the time delay is known very precisely. One important systematic effect
is the mass sheet degeneracy, which arises because strong lens modeling cannot
constrain the presence or absence of a uniform mass sheet $\kappa$, which
rescales $H_0$ by the factor $(1-\kappa)$. In this paper we present new
constraints on the mass sheet derived from a weak-lensing analysis of the
Hubble Space Telescope imaging of a 6 arcmin square region surrounding the
lensed quasar. The average mass sheet within a circular aperture (the strong
lens model region) is constrained by integrating the tangential weak
gravitational shear over the surrounding area. We find the average convergence
within a $30"$ radius around the lens galaxy to be $\kappa(<30") =
0.166\pm0.056$ ($1\sigma$ confidence level), normalized to the quasar redshift.
This includes contributions from both the lens galaxy and the surrounding
cluster. We also constrain a few other low-order terms in the lens potential by
applying a multipole aperture mass formalism to the gravitational shear in an
annulus around the strong lensing region. Implications for strong lens models
and the Hubble constant are discussed in an accompanying paper.
