We study the kinematics of ridge lines on the pc-scale jet of the active galactic nucleus BL Lac. We show that the ridge lines display transverse patterns that move superluminally downstream, and that the moving patterns are analogous to waves on a w
hip. Their apparent speeds $beta_mathrm{app}$ (units of $c$) range from 3.9 to 13.5, corresponding to $beta_mathrm{wave}^mathrm{gal}= 0.981 - 0.998$ in the galaxy frame. We show that the magnetic field in the jet is well-ordered with a strong transverse component, and assume that it is helical and that the transverse patterns are Alfven waves propagating downstream on the longitudinal component of the magnetic field. The wave-induced transverse speed of the jet is non-relativistic ($beta_mathrm{tr}^mathrm{gal} lesssim 0.09$). In 2010 the wave activity subsided and the jet then displayed a mild wiggle that had a complex oscillatory behaviour. The Alfven waves appear to be excited by changes in the position angle of the recollimation shock, in analogy to exciting a wave on a whip by shaking the handle. A simple model of the system with plasma sound speed $beta_mathrm{s}=0.3$ and apparent speed of a slow MHD wave $beta_mathrm{app,S}=4$ yields Lorentz factor of the beam $Gamma_mathrm{beam} sim 4.5$, pitch angle of the helix (in the beam frame) $alphasim 67^circ$, Alfven speed $beta_mathrm{A}sim 0.64$, and magnetosonic Mach number $M_mathrm{ms}sim 4.7$. This describes a plasma in which the magnetic field is dominant and in a rather tight helix, and Alfven waves are responsible for the moving transverse patterns.
Parsec-scale VLBA images of BL Lac at 15 GHz show that the jet contains a permanent quasi-stationary emission feature 0.26 mas (0.34 pc projected) from the core, along with numerous moving features. In projection, the tracks of the moving features cl
uster around an axis at position angle -166.6 deg that connects the core with the standing feature. The moving features appear to emanate from the standing feature in a manner strikingly similar to the results of numerical 2-D relativistic magneto-hydrodynamic (RMHD) simulations in which moving shocks are generated at a recollimation shock. Because of this, and the close analogy to the jet feature HST-1 in M87, we identify the standing feature in BL Lac as a recollimation shock. We assume that the magnetic field dominates the dynamics in the jet, and that the field is predominantly toroidal. From this we suggest that the moving features are compressions established by slow and fast mode magneto-acoustic MHD waves. We illustrate the situation with a simple model in which the slowest moving feature is a slow-mode wave, and the fastest feature is a fast-mode wave. In the model the beam has Lorentz factor about 3.5 in the frame of the host galaxy, and the fast mode wave has Lorentz factor about 1.6 in the frame of the beam. This gives a maximum apparent speed for the moving features 10c. In this model the Lorentz factor of the pattern in the galaxy frame is approximately 3 times larger than that of the beam itself.
We discuss the jet kinematics of a complete flux-density-limited sample of 135 radio-loud active galactic nuclei (AGN) resulting from a 13 year program to investigate the structure and evolution of parsec-scale jet phenomena. Our analysis is based on
new 2 cm Very Long Baseline Array (VLBA) images obtained between 2002 and 2007, but includes our previously published observations made at the same wavelength, and is supplemented by VLBA archive data. In all, we have used 2424 images spanning the years 1994-2007 to study and determine the motions of 526 separate jet features in 127 jets. The data quality and temporal coverage (a median of 15 epochs per source) of this complete AGN jet sample represents a significant advance over previous kinematics surveys. In all but five AGNs, the jets appear one-sided, most likely the result of differential Doppler boosting. In general the observed motions are directed along the jet ridge line, outward from the optically thick core feature. We directly observe changes in speed and/or direction in one third of the well-sampled jet components in our survey. While there is some spread in the apparent speeds of separate features within an individual jet, the dispersion is about three times smaller than the overall dispersion of speeds among all jets. This supports the idea that there is a characteristic flow that describes each jet, which we have characterized by the fastest observed component speed. The observed maximum speed distribution is peaked at ~10c, with a tail that extends out to ~50c. This requires a distribution of intrinsic Lorentz factors in the parent population that range up to ~50. We also note the presence of some rare low-pattern speeds or even stationary features in otherwise rapidly flowing jets... (abridged)
