We describe new Chandra spectroscopy data of the cluster which harbors the prototypical head tail radio galaxy 3C 129 and the weaker radio galaxy 3C 129.1. We combined the Chandra data with Very Large Array (VLA) radio data taken at 0.33, 5, and 8 GHz (archival data) and 1.4 GHz (new data). We also obtained new HI observations at the Dominion Radio Astrophysical Observatory (DRAO) to measure the neutral Hydrogen column density in the direction of the cluster with arcminute angular resolution. The Chandra observation reveals extended X-ray emission from the radio galaxy 3C 129.1 with a total luminosity of 1.5E+41 erg/s. The X-ray excess is resolved into an extended central source of ~2 arcsec (1 kpc) diameter and several point sources with an individual luminosity up to 2.1E+40 erg/s. In the case of the radio galaxy 3C 129, the Chandra observation shows, in addition to core and jet X-ray emission reported in an earlier paper, some evidence for extended, diffuse X-ray emission from a region east of the radio core. The 12 arcsec x 36 arcsec (6 kpc x 17 kpc) region lies in front of the radio core, in the same direction into which the radio galaxy is moving. We use the radio and X-ray data to study in detail the pressure balance between the non-thermal radio plasma and the thermal Intra Cluster Medium (ICM) along the tail of 3C 129 which extends over 15 arcmin (427 kpc). Depending on the assumed lower energy cutoff of the electron energy spectrum, the minimum pressure of the radio plasma lies a factor of between 10 and 40 below the ICM pressure for a large part of the tail. We discuss several possibilities to explain the apparent pressure mismatch.
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