3.3. Velocity and Composition
The proper motion of features in the pc-scale jet has been measured directly implying a value 0.5c h-1 for the projected jet velocity. Whether this is simply a pattern speed or the true flow velocity is still unknown, as is the inclination angle with respect to the sky plane. Estimates of the velocity of the jet on kpc-scales are even more problematic, but there are a few indirect methods that have been used, which we review. In this section we assume h = 0.75, both for simplicity, and because the calculations are at best order-of-magnitude.
An estimate of the jet velocity can be obtained by considering the rate of bulk kinetic energy supplied by the jet, Lj, and the jet thrust, Tj. In the classic case the jet velocity is given by: vj = Lj/Tj for high Mach jets (see Dreher 1985, Bridle and Perley 1984, and Williams 1991 for the relativistic generalization). The standard assumption is that Lj = LR / where LR = lobe radio luminosity = 4.4 x 1044 erg sec-1, and is an efficiency factor for converting bulk kinetic energy into radio luminosity. The thrust can be estimated from: Tj = PHS x AHS = 4 x 1035 ergs cm-1, where PHS = minimum energy hotspot pressure = 3 x 10-9 dyn cm-2, AHS = hotspot area = 1 x 1044 cm2, leading to: vj = 0.04 c / .
Estimating is difficult. It has been suggested that jet shocks are very efficient at converting bulk kinetic energy into relativistic particles (Axford, Leer, and McKenzie 1982, Bell 1987). At the minimum the jet must also do work to expand the ambient medium: work PL x VL 1059 ergs, where PL = lobe pressure = 8 x 10-11 dyn cm-2, and VL = 7 x 1068 cm3. Dividing this by the source lifetime of ts 106.8 yrs (as estimated from synchrotron spectral ageing arguments; see below) yields 5 x 1044 ergs sec-1, thereby setting a very rough upper limit: 0.4. In reality, is likely to be considerably less than this (Dreher 1985, Leahy 1991), in particular if the lobe pressures are substantially different than dictated by minimum energy.
One can also consider the total mass flux in the jet and the upper limit to the mass in the radio lobes, ML, to derive a jet velocity: vj = ts x PHS x AHS / ML. Using a lobe density < 2 x 10-4 cm-3 derived from the lack of internal Faraday dispersion (Dreher et al. 1987) implies ML < 2.4 x 1041 gm. The lower limit to the jet velocity is then 0.01 c.
Both of the above calculations are invalid in the case of departure from minimum energy conditions, or a time-variable energy (or mass) supply in the jet. The mass estimate also depends on the assumptions of magnetic field strength and geometry inherent in the internal Faraday depolarization calculation.
Williams (1991) develops a simple hydrodynamic model relating observable parameters such as the ratio of jet width to lobe width and the ratio of hotspot pressure to lobe pressure to physical parameters such as the jet Mach number and ambient-to-jet density contrast. Applying this model to Cygnus A he finds Mj 8, and 2 x 10-4 for a Newtonian ideal fluid. He then balances jet ram pressure against minimum pressures in the hotspots and finds a required jet velocity somewhat greater than the speed of light. Williams uses this result to argue that the jet must be relativistic on kpc-scales, and must be composed primarily of a relativistic (pair) plasma. However, his calculations apply to axisymmetric flows in a constant density medium. Including jets which vary direction on short timescales, and/or density gradients in the ambient medium, would widen the lobes over the axisymmetric case, and invalidate the above arguments.
Muxlow, Pelletier, and Roland (1988) and Carilli et al. (1991a) have used the observed spectra of the Cygnus A radio hotspots to constrain various physical parameters of the jet. Their basic conclusion is that the various `features' in the hotspot spectra are consistent with a model involving particle acceleration at a strong shock in a Newtonian fluid with a mildly relativistic jet velocity ( 0.4 c).
Roland and Hermsen (1995) develop a two-component model for the jets in extragalactic radio sources involving a narrow `core' jet consisting of a pair-plasma, and moving at c, sheathed by a classical (proton-electron) plasma moving trans-relativistically ( 0.7c). The relativistic jet component gives rise to the large Lorentz factors, and ray emission, from superluminal radio jets, while the classical component dominates the dynamics, and gives rise to the large scale radio emitting structures (lobes, hotspots, etc...).
Overall, there are no data which necessitate a highly relativistic jet on kpc-scales or a jet consisting of only pair plasma (Begelman et al. 1984) in Cygnus A. On the other hand, neither of these is currently precluded, emphasizing our current ignorance of the real physical conditions in jets. Mildly relativistic flow would be consistent with the observed jet-to-counterjet surface brightness ratio, as discussed below.