6.2. The Hotspot Spectra
The integrated spectra of the hotspots in Cygnus A at 4.5" resolution are shown in Muxlow et al. (1988) and Carilli et al. (1991a), while high resolution (0.4") spectral index images can be found in Carilli et al. (1989a) and Perley and Carilli (1996). The integrated spectra above 1 GHz are well fit by a continuous injection (CI) model spectrum, in which a power-law distribution of particles is continuously injected at the hotspot. The injection index is -0.5 ± 0.1 for both hotspots - typical for hotspots in high power radio galaxies (Meisenheimer et al. 1989). Diffusive shock acceleration theory predicts: in = 3/(2 - 2r), where r is the shock compression ratio (Bell 1978a, b, Blandford and Ostriker 1978). For a strong shock in a monatomic Newtonian fluid, r = 4, and hence in = -0.5. However, the effects on injection index assuming a relativistic jet velocity, or allowing for modification of shock structure due to up-streaming high energy particles, are minor, ± 0.1 or so in spectral index, and certainly cannot be ruled-out (Kirk and Schneider 1987, Drury and Völk 1981, Axford et al. 1982, Kirk 1989, Heavens 1989, Eilek and Hughes 1991).
The two dominant hotspots in Cygnus A (denoted A and D in Hargrave and Ryle 1974) have break frequencies around 10 GHz, implying spectral ages 105 yrs using minimum energy fields. Above the break the spectra are consistent with a power-law of index -1.0 out to at least 375 GHz. The hotspots have not been detected in the optical (Kronberg et al. 1977, Röser 1996). The optical upper limits fall two orders of magnitude below the extrapolation of the power-law from 375 GHz into the optical, suggesting a sharp cut-off in the spectra between 375 GHz and 1014 Hz (Harris et al. 1994a, Röser 1996).
The physical interpretation of spectral breaks in hotspot spectra is discussed at length in Muxlow et al. (1988), Roland et al. (1988), Meisenheimer et al. (1989), and Carilli et al. (1991a). The basic model involves relativistic particle injection at a `point', i.e., the terminal jet shock, and then convection away from the high field post-shock regions with the general outflow. The isotropic outflow velocity, vout, is given simply by the radius of the high surface brightness hotspot region (or the beam size, if it is smaller) divided by the spectral age derived from the hotspot spectrum. For Cygnus A the value for both hotspots is: vout 0.06c h-1. For a strong shock the inflow velocity is four times the outflow velocity (in the shock frame) implying an inflow velocity of 0.24c h-1. Carilli et al. (1991a) discuss the possible effects an anisotropic outflow may have on the above analysis.
The Cygnus A hotspot spectra continue to flatten below 1.5 GHz. Muxlow et al. (1988), Leahy et al. (1989), and Carilli et al. (1991a), have all argued that the low frequency flattening of the hotspot spectra in Cygnus A is due to a low energy cut-off in the relativistic electron population at Lorentz factors of 450 (assuming minimum energy magnetic fields). Such a cut-off was predicted by Bell (1978) in his original work on shock acceleration. Bell hypothesized that in order for a particle to be `injected into the acceleration process' it must have sufficient momentum to pass unperturbed through the potentials in the collisionless shock which act to stop the jet. In essence the relativistic electrons must have gyro-radii which are larger than the shock width, which in a collisionless shock is of order the gyro-radius of the thermal protons (see also Eilek and Hughes 1991). The alternative explanation of synchrotron self-absorption leads to fields strengths that are many orders of magnitude above equipartition values, while thermal absorption implies local densities inconsistent with redshifted H imaging of Cygnus A.