3.6 Field ordering

Clues to the three-dimensional structure of the magnetic field come from the differences in polarization between the main and counter-jets, as summarized in Section 2.4. We initially tried the structure proposed by Laing 1993 in which a fast, transverse-field spine with equal radial and toroidal field components is surrounded by a slower longitudinal-field shear layer. This might naively be expected from a combination of expansion and interaction with the external medium. Such a structure produces a transverse apparent field on-axis and a longitudinal field at the edge, together with a transition from longitudinal to transverse apparent field on-axis, both as observed (Fig. 3). It could be rejected for 3C31, however, because it always requires the transition from longitudinal to transverse apparent field to occur closer to the nucleus in the main jet (where Doppler boosting makes the faster spine emission with its transverse apparent field relatively more prominent). The opposite is observed. The high degree of polarization observed in the outer counter-jet is also inconsistent with such a field configuration.

This suggested a model in which both the spine and the shear layer have toroidal and longitudinal field components (of roughly equal magnitude) but the radial component is everywhere very small (model B of Laing 1981). The field is then two-dimensional, in sheets wrapped around the jet axis. The apparent field is always longitudinal (with the theoretical maximum degree of polarization, p₀) at the edges of the jets, but can be either longitudinal or transverse on the axis, depending on the relative magnitudes of the two components and the angle to the line of sight. If this angle and the flow velocities are adjusted appropriately, then aberration can act so that the field sheets are seen face-on in their rest frames in the main jet (giving a low degree of polarization), but side-on in the counter-jet (leading to high polarization and a transverse apparent field). Models of this type produce much more realistic polarization distributions, especially when the ratio of toroidal to longitudinal field increases with distance from the nucleus, but still fail in two important respects. First, the field transition region in the main jet is too close to the nucleus and too short. Second, a high degree of polarization is predicted at the edge of the flaring region, where the observed values are quite low (Section 2.4). The solution to both problems is to allow a radial field component which increases from zero close to the axis to a finite value at the edge of the jet. This edge value must vary along the jet in such a way that the field is essentially isotropic at the boundary in parts of the flaring region, but the radial component vanishes at large distances from the nucleus. In contrast, we found no evidence for any transverse variation of the longitudinal/toroidal ratio in the shear layer.

The functional forms are again given in Tables 4 and 5. We use the ratios of rms field components $j(\rho, s)$ (radial/toroidal) and $k(\rho,s) = k_\rho(\rho)$ (longitudinal/toroidal), with no transverse variation in the spine for SSL models. We chose $j(\rho, s)= j_\rho(\rho)j_s(s)$ with $j_s(s) = s^p$ for the radial/toroidal ratio. If the functional forms given for the outer region are negative, the corresponding values of  j or k are set to zero.

Table 5:  Summary of the functional variations of velocity, emissivity and field ordering parameters across the model shear layers.

Quantity	Model	Functional variation
Velocity
$\beta_s(s)$ ($\rho > \rho_1$)	SSL	$1 + [\bar{v}(\rho)-1]s$
	Gaussian	$\exp [-s^2 \ln \bar{v}(\rho)]$
$\beta_s(s)$ ($\rho < \rho_1$)	SSL	$v_{\rm i}$
	Gaussian	1
Emissivity
$\epsilon_s(s)$	SSL	$1 + [\bar{e}(\rho)-1]s$
	Gaussian	$\exp [-s^2 \ln \bar{e}(\rho)]$
Radial/toroidal field ratio
$j_s(s)$		$s^p$

2002-06-13