Density gradients in Galactic planetary nebulae

Certain hydrodynamic models of planetary nebulae (PNe) suggest that their shells possess appreciable radial density gradients. However, the observational evidence for such gradients is far from clear. On the one hand, Taylor et al. claim to find evidence for radio spectral indices  0.6 < α < 1.8  , a trend which is taken to imply a variation   n _e∝ r ⁻²  in most of their sample of PNe. On the other hand, Siódmiak & Tylenda find no evidence for any such variations in density; shell inhomogeneities, where they occur, are primarily attributable to 'blobs or condensations'.
It will be suggested that both of these analyses are unreliable, and should be treated with a considerable degree of caution. A new analysis within the  log( F (5 GHz)/ F (1.4 GHz))–log( T _B(5 GHz))  plane will be used to show that at least 10–20 per cent of PNe are associated with strong density gradients. We shall also show that the ratio   F (5 GHz)/ F (1.4 GHz)  varies with nebular radius; an evolution that can be interpreted in terms of varying shell masses, and declining electron densities.