Spectral index asymmetry

Both GCL and Liu & Pooley (1991) noted that the spectral index on the jet side was flatter than that on the counterjet side. Liu & Pooley claimed, in addition, to find the asymmetry for the lobe and hotspot separately. The lobe/hotspot separation is not straightforward, as the hotspot might not be well defined, and the lobe could be contaminated with emission from the jet. (GCL note that the flux asymmetry is confined to the hotspots whereas the lobe fluxes are symmetric.) These asymmetries need to be confirmed with data of higher resolution, and I will only consider the simple component spectra here, although I consider a possible asymmetry in the hotspot spectral indices in Section 5.

As noted by GCL, the spectral index asymmetry can be explained as a dilution effect. The hotspot on the jet side is brighter (Bridle & Perley 1984; Laing 1989) and flattens the overall spectrum more than the fainter hotspot on the counterjet side. The asymmetry in spectral index arises if the hotspot spectral indices are reasonably flat, perhaps with alpha=0.5 as expected for first-order Fermi acceleration (Bell 1978) and observed in at least some hotspots (Meisenheimer et al. 1989).

The natural explanation for the enhanced hotspot brightness on the jet side is Doppler boosting. If the hotspots are moving at velocities betac away from the nucleus of the radio source at an angle arccosmu=pi/2-theta_0 to our line of sight then the observed flux density is (Ryle & Longair 1967; Peacock 1987)

S = S_0 (1-betamu)^{-(3+alpha)}, (18)

where alpha is the radio spectral index and I have included the angle independent Lorentz factors in S_0. Wilson & Scheuer (1983) showed that this simplified description is valid in practice. For the jets the exponent is (2+alpha), but I here consider the hotspots as monolithic entities. The hotspot on the counterjet side has its flux density correspondingly reduced. The median ratio of peak intensities is reported by GCL to be 1.37; the median flux density ratio of 1.24 is smaller because the lobe emission is unaffected by orientation and dilutes the difference in the hotspots.

The jet is only a crude orientation indicator, simply indicating which side is which. The asymmetry in hotspot luminosity and spectral index is expected to be small for sources nearly in the plane of the sky, and large for sources close to the line of sight. Similar depolarization behaviour is expected, as sources close to the plane of the sky have similar depolarization on both sides, whereas sources close to the line of sight show a large depolarization asymmetry. In Fig. 4 I show the spectral index difference (counterjet spectral index less jet spectral index) plotted against the depolarization ratio (counterjet side depolarization divided by jet side depolarization) for the GCL and Liu & Pooley (1991) samples.

Fig 4. Spectral index difference plotted against depolarization ratio for the sources in the GCL sample (crosses) and the Liu & Pooley (1991) sample (filled squares). Lines represent an expected relation for beta=0.1, 0.25, and 0.4 (from bottom).

Using both Spearman's rank and Kendall's tau, the spectral index difference and depolarization ratio are anticorrelated at the 99% confidence level. This is exactly as expected---large differences in spectral index should be associated with small depolarization ratios. Furthermore, the zero point of the correlation is correct. As can be seen in Fig. 4, those sources not at small angles to the line of sight for which the depolarization ratio is near unity have spectral index differences which scatter around zero. The amplitude of the scatter is consistent with measurement errors in the spectral index of about 0.1 (GCL).

I have modelled the spectral index asymmetry as follows. The flux density on the jet side is

S_j = (nu/nu_0)^{-alpha_l} + f(nu_0) (nu/nu_0)^{-alpha_h} g(mu,beta), (19)

where f is the intrinsic flux density ratio of the hotspot to the lobe, alpha_l and alpha_h are the spectral indices of the lobe and hotspots respectively, and g(mu,beta) is the Doppler beaming factor (equation 18). The total spectral asymmetry can then be calculated as a function of orientation angle. I take alpha_h=0.6, alpha_l=1.1 and f=0.5 at 20 cm which can easily give the total flux asymmetry for beta=0.2 and maximises the spectral index asymmetry because for this choice the lobe and hotspot have comparable fluxes over the range of interest.

A simple model connecting depolarization ratio and angle is needed. At long wavelengths p proportional to 1 / (sigmalambda^2), and when both sides are depolarized the depolarization ratio is simply sigma_+/sigma_- which depends only on the geometry. I again take m=1, and in Fig. 4 show the expected relation between the two asymmetries in this model for beta=0.1, 0.25 and 0.4. The lines shown are not wholly unrealistic because the two endpoints are correct and the intermediate variation must be monotonic. Note that the curves cover the region occupied by the data, and that the scatter in the spectral index difference increases at small depolarization ratios consistent with a range of hotspot advance speeds (or of other source properties).

The model chosen clearly doesn't give the correct absolute spectral indices, but these can be obtained simply by adjusting both alpha_l and alpha_h together. Doing so has little effect on the asymmetry obtained. Changing the difference between alpha_l and alpha_h changes the resulting asymmetry, roughly in proportion. For these models, with a median angle of 50 degrees to the plane of the sky which is consistent with the source orientations derived earlier, the median spectral index asymmetry alpha_cj-alpha_j approx beta (alpha_l-alpha_h), although the asymmetry will be reduced for less optimal values of f. The major result is that reasonable choices of parameters can reproduce the observed asymmetry.

Liu & Pooley (1991) also considered Doppler effects. Their reasoning is incorrect as they neglect Doppler boosting entirely, and rely on red- and blue-shifting different portions of a curved spectrum to the observed wavelengths. As the spectral curvature is usually small, the required velocities are large.

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Peter Tribble, peter.tribble@gmail.com