4.2 Model parameters and confidence limits

The optimization problem is complicated, and estimates of some of the parameters are strongly correlated. In addition, we do not know the statistics (or even the rms level) of the ``noise'' a priori and we have imposed additional constraints by our choice of fitting functions. The $\chi ^2$ statistic is effective in optimizing the fit, but assessing confidence limits (e.g. by a Bayesian analysis or using bootstrap techniques) would be far from straightforward. We have instead adopted a simple ad hoc procedure, by which we vary a single parameter until the fractional increase in the $\chi ^2$ values for I or Q and U in one of the inner, flaring or outer regions corresponds to the formal 99% confidence limit for independent Gaussian errors and the appropriate number of degrees of freedom. Most parameters affect the fit significantly only for part of the jet, or for a subset of the Stokes parameters, so this approach is superior to one based on the total $\chi ^2$. The estimates are qualitatively reasonable, in the sense that varying a parameter by its assigned error leads to a visibly unacceptable fit, and we believe that they give a good general impression of the range of allowed models. The numerical confidence levels should not be taken too seriously, however.

 

Table 7: Fitted parameters and error estimates.

Quantity	SSL			Gauss
	opt	min1	max2
Angle to line of sight $\theta$ (degrees)	52.4	48.9	54.1	51.4
Jet half-opening angles (degrees)3
inner region $\xi_{\rm i}$	6.7	$-$	$-$	6.6
outer region $\xi_{\rm o}$	13.2	$-$	$-$	13.0
Boundary positions (kpc)
inner $r_1$	1.1	$-$	$-$	1.1
outer $r_0$	3.5	$-$	$-$	3.6
arbitrary fiducial $r_{\rm f}$	9.6	$-$	$-$	9.8
Spine half opening angles (degrees)
inner region $\zeta_{\rm i}$	4.06	3.1	6.5	$-$
outer region $\zeta_{\rm o}$	2.79	0.7	4.5	$-$
On-axis velocities / $c$
inner jet $\beta_{\rm i}$	0.87	0.83	0.93	0.20
inner boundary $\beta_1$	0.77	0.68	0.83	0.76
outer boundary $\beta_0$	0.55	0.45	0.63	0.54
outer fiducial $\beta_{\rm f}$	0.28	0.25	0.33	0.27
velocity exponent $H$	9.5	3.6	$>$	8.8
Fractional velocity at edge of jet4
inner jet $v_{\rm i}$	0.06	0.0	1.15	$-$
inner boundary $v_1$	0.74	0.4	1.30	0.97
outer boundary $v_0$	0.67	0.51	0.87	0.63
On-axis emissivity exponents
inner spine $E_{\rm i}$	1.96	$<$	2.3	$-$
flaring spine $E_{\rm f}$	2.52	1.9	2.9	$-$
outer spine $E_{\rm o}$	2.14	1.4	3.8	$-$
inner shear layer $E_{\rm i}$	1.33	$<$	2.2	0.75
flaring shear layer $E_{\rm f}$	3.10	2.9	3.4	3.08
outer shear layer $E_{\rm o}$	1.42	1.33	1.54	1.44
Fractional emissivity at edge of jet
inner boundary $e_1$	0.27	0.05	0.52	0.37
outer boundary $e_0$	0.20	0.09	0.28	0.26
Shear layer / spine emissivity	2.11	1.5	3.1	$-$
ratio at inner boundary
Emissivity ratio at inner boundary
(inner / flaring region)
spine $g$	0.37	0.13	0.53	$-$
shear layer $g$	0.04	0.003	0.08	0.05

¹ The symbol $<$ means that any value smaller than the quoted maximum is allowed.

² The symbol $>$ means that any value larger than the quoted minimum is allowed.

³ Opening angles and boundary locations are given in the jet coordinate system. The jet opening angles and the boundary locations are determined by the outer isophotes once the angle to the line of sight is specified , so no errors are quoted.

⁴ The upper limits on the fractional velocity at the edge of the jet in the inner region and at the inner boundary are set not by the $\chi ^2$ constraint but rather by the condition that the velocity must be $<c$.

Table 7: Fitted parameters and error estimates (continued).

Quantity	SSL			Gauss
	opt	min	max
RMS field ratios (shear layer)5
radial/toroidal
inner jet centre $j_{\rm i}$	0.37	0.0	$>$	0.38
inner jet edge	0.0			0.0
inner boundary centre $j_1$	0.93	0.3	1.4	0.78
inner boundary edge $\bar{j}$	0.0			0.00
outer boundary centre $j_0$	1.00	0.52	1.38	0.92
outer boundary edge	0.0			0.00
fiducial distance centre $j_{\rm f}$	0.0	0.0	0.62	0.24
fiducial distance edge	0.0			0.00
index $p$	0.53	0.3	1.5	0.41
longitudinal/toroidal
inner jet $k_{\rm i}$	1.23	0.2	2.3	1.43
inner boundary $k_1$	1.16	1.05	1.35	1.17
outer boundary $k_0$	0.73	0.63	0.80	0.82
fiducial distance $k_{\rm f}$	0.50	0.41	0.58	0.54
RMS field ratios (spine)
radial/toroidal6
inner jet $j_{\rm i}$	0.0	0.0	1.5	$-$
inner boundary $j_1$	0.0	0.0	1.3	$-$
outer boundary $j_0$	0.0	0.0	1.9	$-$
fiducial distance $j_{\rm f}$	0.0	0.0	10.0	$-$
longitudinal/toroidal
inner jet $k_{\rm i}$	1.75	1.1	2.4	$-$
inner boundary $k_1$	1.06	0.7	1.8	$-$
outer boundary $k_0$	1.40	0.8	4.0	$-$
fiducial distance $k_{\rm f}$	0.84	0.0	8.0	$-$

⁵ The radial/toroidal ratios always vary from 0 at the spine/shear-layer interface (SSL) or axis (Gaussian) to a maximum value at the edge of the jet (Table 5).The values quoted are for the edge and centre of the shear layer.

⁶ radial/toroidal field ratios for the spine in the SSL models are consistent with 0 but poorly constrained everywhere (to the extent of having negligible influence on the $\chi ^2$ values). The relevant parameters were fixed at 0 throughout the optimization process.

Table 7 gives the fitted parameters and error estimates for the spine/shear layer and Gaussian models. The parameters are the angle to the line of sight, the spine opening angles and those defined in Tables 4 - 5. The columns are:

1. A description of the parameter. The symbols are those used in Tables 4 -  5.
2. The best fit for the SSL model
3. The minimum value allowed by the $\chi ^2$ recipe given earlier, again for the SSL model.
4. The corresponding maximum value.
5. The best fit for the Gaussian model (the allowed ranges are very similar to those for the SSL model).

In general, the parameters for the Gaussian and SSL models are very similar and always agree within the quoted errors - the contribution of the spine component to the emission (and therefore to the $\chi ^2$ value) is quite small.

2002-06-13