  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.
