3.2 Geometry

We define $\theta$ to be the angle between the jet axis and the line of sight. $z$ is a coordinate along the jet axis with its origin at the nucleus, $x$ is measured perpendicular to the axis, $r = (x^2+z^2)^{1/2}$ is the distance from the nucleus and $\phi$ is an angle measured from the jet axis ($x = z\tan\phi$). The first step in our procedure is to define functional forms for the outer surfaces of the jets and for the flow streamlines. The latter inevitably involves some guesswork, to be justified post hoc by the quality of the model fit. Inspection of the outer isophotes shows that the jets can be divided into three regions:

1. Inner (0 - 2.5 arcsec): a cone, centred on the nucleus, with a half-opening angle of 8.5 degrees.
2. Flaring (2.5 - 8.3 arcsec): a region in which the jet initially expands much more rapidly and then recollimates.
3. Outer (8.3 - 28.3 arcsec): a second region of conical expansion, also centred on the nucleus, but with a half-opening angle of 16.75 degrees.

All dimensions given above are as observed, i.e. projected on the plane of the sky. This pattern of an initially narrow base and a rapid expansion followed by recollimation is general in FRI jets (Bridle & Perley , ). In what follows, we use subscripts i, f and o to refer to quantities associated with the inner, flaring and outer regions. We refer to the inner and outer boundaries separating the regions by subscripts $1$ and $0$. The inner boundary is the flaring point defined by Parma et al. (1987) and Laing et al. (1999), and we also use this term.

Guided by the shape of the outer isophotes, we assume that the flow in the inner and outer regions is along straight lines passing through the nucleus. Our general approach is to devise simple analytical functions to describe the flow in these regions, and then to interpolate across the more complex geometry of the flaring region in such a way as to preserve continuity. Families of streamlines are parameterized by the streamline index $s$, which varies from 0 at the inner edge of a component (spine or shear layer) to 1 at the outside edge. In the inner and outer regions, the streamlines make constant angles $\phi_{\rm i}$ and $\phi_{\rm o}$ with the jet axis. We define $\xi_{\rm i}$ and $\xi_{\rm o}$ to be the half-opening angles of the jet in the inner and outer regions, and $\zeta_{\rm i}$, $\zeta_{\rm o}$ to be the corresponding angles for the spine. s is defined in terms of these angles in Table 3.

Table 3: Definitions of streamline indices for inner and outer regions.

Model	Inner (conical)	Outer (conical)
SSL spine	$\phi_{\rm i}= \zeta_{\rm i} s$	$\phi_{\rm o}= \zeta_{\rm o} s$
SSL shear layer	$\phi_{\rm i}= \zeta_{\rm i} + (\xi_{\rm i}-\zeta_{\rm i})s$	$\phi_{\rm o}= \zeta_{\rm o} + (\xi_{\rm o}-\zeta_{\rm o})s$
Gaussian	$\phi_{\rm i}= \xi_{\rm i} s$	$\phi_{\rm o} = \xi_{\rm o} s$

We require continuity of the streamlines and their first derivatives with respect to $z$ across the flaring region. The simplest functional form that satisfies these constraints and fits the outer isophote shape for $s = 1$ in the shear layer is:

$$\begin{eqnarray*} x & = & a_0(s) + a_1(s) z + a_2(s) z^2 + a_3(s) z^3 \\ \end{eqnarray*}$$

For each streamline, the values of $a_0(s)$ - $a_3(s)$ are determined uniquely and in analytic form by the continuity conditions. The natural boundaries between regions are then spherical, centred on the nucleus at distances $r_1$ and $r_0$ and therefore perpendicular to the streamlines. Fig. 4 shows sketches of the assumed geometry for the SSL model (the equivalent for the Gaussian model is essential identical, but with the spine removed).

In order to describe variations along a streamline, we use a coordinate $\rho$, defined as:

$$\begin{eqnarray*} \rho & = & r \makebox{~~(inner region)} \\ \rho & = & r_1 + (... ...flaring region)} \\ \rho & = & r \makebox{~~(outer region)} \\ \end{eqnarray*}$$

$\rho$ is monotonic along any streamline and varies smoothly from $r_1$ to $r_0$ through the flaring region ($\rho = r = z$ on the axis). This allows us to match on to simple functional forms which depend only on $r$.

The functions defining the edge of the jet are constrained to match the observed outer isophotes and are fixed in a coordinate system projected on the sky. Their values in the jet coordinate system then depend only on the angle to the line of sight. The outer edge of the spine in SSL models is not constrained in this way, and the relevant parameters may be varied in order to obtain a good fit to the data.

In what follows we will refer to streamline coordinates defined by longitudinal (along a streamline), radial (outwards from the axis) and toroidal orthonormal vectors.

Figure 4: Geometry of the spine/shear-layer model, showing the inner, flaring and outer regions in the plane containing the jet axis. The thick full curves represent the edge of the jet, the boundaries between regions are represented by thin full curves and the $s = 0.5$ streamlines for the spine and shear layer are drawn as dashed curves. (a) The entire modelled region; (b) the base of the jet on a larger scale, showing the boundary surfaces at distances of $r_1$ and $r_0$ from the nucleus. The Gaussian model is essentially the same, but with the spine component removed.

2002-06-13