  We present extended modeling of the strong lens system RXJ1131-1231 with
archival data in two HST bands in combination with existing line-of-sight
contribution and velocity dispersion estimates. Our focus is on source size and
its influence on time-delay cosmography. We therefore examine the impact of
mass-sheet degeneracy and especially the degeneracy pointed out by Schneider &
Sluse (2013) using the source reconstruction scale. We also extend on previous
work by further exploring the effects of priors on the kinematics of the lens
and the external convergence in the environment of the lensing system. Our
results coming from RXJ1131-1231 are given in a simple analytic form so that
they can be easily combined with constraints coming from other cosmological
probes. We find that the choice of priors on lens model parameters and source
size are subdominant for the statistical errors for $H_0$ measurements of this
systems. The choice of prior for the source is sub-dominant at present (2%
uncertainty on $H_0$) but may be relevant for future studies. More importantly,
we find that the priors on the kinematic anisotropy of the lens galaxy have a
significant impact on our cosmological inference. When incorporating all the
above modeling uncertainties, we find $H_0 = 86.6^{+6.8}_{-6.9}$ km
s$^{-1}$Mpc$^{-1}$, when using kinematic priors similar to other studies. When
we use a different kinematic prior motivated by Barnab\`e et al. (2012) but
covering the same anisotropic range, we find $H_0 = 74.5^{+8.0}_{-7.8}$ km
s$^{-1}$Mpc$^{-1}$. This means that the choice of kinematic modeling and priors
have a significant impact on cosmographic inferences. The way forward is either
to get better velocity dispersion measures which would down weight the impact
of the priors or to construct physically motivated priors for the velocity
dispersion model.
