ConstrainingH₀fromChandraObservations of Q0957+561

We report the detection of the lens cluster of the gravitational lens (GL) system Q0957+561 from a deep observation with the Advanced CCD Imaging Spectrometer on board the Chandra X-Ray Observatory. Intracluster X-ray emission is found to be centered 4".3+/-1"3 east and 3".5(-0".6)(+1".2) north of image B, nearer than previous estimates. Its spectrum can be modeled well with a thermal plasma model consistent with the emission's originating from a cluster at a redshift of 0.36. Our best-fit estimates of the cluster temperature, T-e = 2.09(-0.54)(+0.83) keV (90% confidence), and of the mass distribution of the cluster are used to derive the convergence parameter kappa, the ratio of the cluster surface mass density to the critical density required for lensing. We estimate the convergence parameter at the location of the lensed images A and B to be kappa(A) = 0.22(-0.07)(+0.14) and kappa(B) = 0.21(-0.07)(+0.12) respectively (90% confidence levels). The observed cluster center, mass distribution, and convergence parameter kappa provide additional constraints to lens models of this system. Our new results break a mass-sheet degeneracy in GL models of this system and provide better constraints of similar to29% (90% confidence levels) on the Hubble constant. We also present results from the detection of the most distant X-ray jet (z=1.41) detected to date. The jet extends approximately 8" northeast of image A, and three knots are resolved along the X-ray jet, with flux densities decreasing with distance from the core. The observed radio and optical flux densities of the knots are fitted well with a synchrotron model, and the X-ray emission is modeled well with inverse Compton scattering of cosmic microwave background photons by synchrotron-emitting electrons in the jet.