Fig 5. Intensity profiles for three hotspots with different _eff = 0.1, 0.25, 0.4, from left. The hotspots are intrinsically Gaussian with the emission 50% more extended than the velocity. The approaching hotspot is shown as a solid line, the receding hotspot as the lower dashed line. The upper dashed line is the receding hotspot amplified so that its shape can be compared with that of the approaching hotspot.
In fact, beaming alone can provide a trivial explanation of the compactness asymmetry. Describing the hotspot as a monolithic entity moving at a single speed is obviously incorrect. In reality, there will be some velocity gradient between the core of the hotspot and the stationary lobe.
To show this I consider a simple model with a monotonic velocity gradient from the central value of _0 to zero for the lobes which are assumed to be at rest. The important feature here is that for the approaching hotspot the centre is preferentially enhanced by beaming relative to the slower hotspot material, whereas for the receding hotspot the flux from the centre is preferentially reduced. The peakiness of the approaching hotspot is increased whereas the receding hotspot is flattened. In fact, it is possible to create an inverted receding hotspot with the observed flux decreasing toward the centre.
I take the emission I and the velocity to fall as Gaussians,
I(r) = I_0 exp(-r^2/r_I^2)
(r) = _0 exp(-r^2/r_^2). (20)
The results depend quite strongly on the ratio R=r_I/r_ of the sizes of these two distributions. The hotspot is lit up where the beam hits the working surface, so the emissivity process is connected with deceleration. This leads us to expect that R should be not too different from and possibly somewhat greater than unity. Some examples of hotspot profiles are shown in Fig. 5. I define the compactness ratio as the ratio of the full widths at half maximum of the approaching and receding hotspots. This is shown for different values of R and _eff=_0 in Fig. 6.
Fig 6. The compactness ratio as a function of _eff for Gaussian hotspots of different emission to velocity size ratios R.
An alternative model is to have material flowing through the hotspot at a velocity _0 and an associated backflow at velocity _1. Material is continually moving through the hotspot so that the pattern speed of the hotspot can be very much less than _0. This has the same qualitative features as the previous model with the exception that the outer emission might be brighter in the receding half of the source, boosted by the backflow approaching the observer.
The central value of _0 can be larger than either the value required to explain the hotspot luminosity and spectral index asymmetry or the upper limit on the advance speed from the lobe separation asymmetry (Longair & Riley 1979). This is because the hotspot flux and spectral index asymmetry depend on the emission weighted , and the separation asymmetry measures the pattern speed of the hotspot, which might both be substantially less than the flow speed of the emitting material.
If the hotspots are substantially beamed then they are also enfeebled by several powers of =(1-^2)^-1/2. This won't affect the asymmetry as both sides are equally affected. If, however, a secondary hotspot is present (Laing 1981; Valtaoja 1984; Lonsdale & Barthel 1984, 1986) then the relative properties of the primary and secondary hotspots will be affected. I assume that the primary jet loses a substantial fraction of its momentum at the primary hotspot, so that the reflected jet feeding the secondary hotspot has <<1 and Doppler boosting is negligible for the secondary. Therefore the two secondaries (assuming there to be one on each side) will appear rather similar. The approaching primary will be more compact, and the receding primary will be diffuse. If the approaching primary and secondary are of similar luminosity, then the receding primary will be much fainter than its corresponding secondary. Thus the receding side might only appear to contain one hotspot which is actually the secondary (see Lonsdale 1989). This picture is consistent with the hotspots of Cygnus~A (Carilli, Dreher & Perley 1989) where the primary hotspot on the counterjet side is rather weak compared to the secondary.
In Section 4 I showed how a spectral index asymmetry can be given by Doppler boosting of the hotspots. Exactly the same argument predicts a spectral index asymmetry for the hotspots alone. As the emitting plasma in the hotspots is decelerated, the relativistic electrons are losing energy and the emitted spectrum is steepening, giving a spectral index gradient across the hotspot. The approaching hotspot is dominated by the centre where the spectrum is flattest, so that if the hotspots are unresolved the total spectrum of the approaching hotspot will be flatter than that of the receding hotspot.