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We present a new method of extracting kinematic information from multi-epoch VLBI observations that, subject to a simplifying assumption, enables us to reconstruct the full three dimensional trajectory of individual components. Analyzing multiple observations of the quasar 3C\,345 made by ourselves and by the Caltech group, we find that components C2 and C3 move along a common path and at the same speed. This trajectory makes an angle $\theta \simeq 2\deg$ with the line of sight and is gently curving away from it. The Lorentz factor of the pattern velocity is at least $11.5\,h^{-1}$($H_0 = 100\,h$ km s$^{-1}$ Mpc$^{-1}$).
We then combine these kinematic constraints with the polarization information for component C3 and the unshocked jet obtained with VLBI polarimetry (Brown, Roberts, \& Wardle 1994). If the component is interpreted as a shock in a fluid jet with an ultra-relativistic equation of state, then the shock parameters can be explored in some detail. We determine the ranges of shock strength, thickness of the shocked region, upstream and downstream fluid velocities, and degree of order of the magnetic field in the unshocked jet that are consistent with the observations. We find that the shock speed is greater than that of the underlying jet (i.e., the fluid velocity is towards the nucleus in the frame of the shock), and that the shock must be rather weak. There can be little particle acceleration in the shock apart from that due to adiabatic compression.
This work was supported by grants from the National Science Foundation to DHR and JFCW.
