3.4 Velocity field

We have chosen to model the velocity field as a separable function $\beta(\rho, s) = \beta_\rho(\rho) \beta_s(s)$ with $\beta_s(0) = 1$. The inner region is faint and poorly resolved so, in the absence of evidence to the contrary, we assume that the on-axis velocity is constant there. To generate the sideness profile of the rest of the jet (Fig. 2), the on-axis velocity must remain fairly constant throughout much of the flaring region, drop rapidly just before the outer boundary and then fall smoothly and uniformly. We chose simple functional forms for $\beta_\rho(\rho)$ to satisfy these requirements (see Table 4). The constants b₀ - b₂  , c₀ and c₁ are chosen to match specified velocities at the inner and outer boundaries, and at an arbitrary fiducial point in the outer region. In addition, we require continuity of velocity and acceleration across the outer boundary so the constants are uniquely determined. These continuity conditions are not strictly necessary for the calculation described here, but are physically reasonable and essential for the adiabatic models that we discuss elsewhere.

In the flaring and outer region, $\beta_s = 1$ in the spine for SSL models, dropping linearly with s from 1 at the spine/shear layer interface to a minimum value at the edge of the jet. For Gaussian models, $\beta_s$ is a truncated Gaussian function. The fact that the sidedness ratio at the edge of the jets exceeds unity over most of the modelled area (Fig. 2) means that the fractional velocity at the edge is significantly greater than zero. In both classes of model, this minimum fractional velocity, $\bar{v}(\rho)$, is allowed to vary along the jet (Table 4).

In the inner region, we found that we could not obtain satisfactory fits with linear or Gaussian transverse velocity profiles. The data required a mixture of fast and slow material, without much at intermediate velocity. Given the poor transverse resolution, we took the simple approach of assigning a single fractional velocity $\beta_s = v_{\rm i}$ to material in the ``shear layer'' (there is actually no shear). This means that the velocities of the spine and shear layer in the SSL models are decoupled, as required. For the Gaussian model, there is no separate spine component, so the inner region has a constant velocity $\beta = \beta_i$ everywhere (i.e. $\beta_s = 1$). An unphysical acceleration is required in the shear layer at the flaring point in both classes of model: this is an inevitable consequence of the increase in sidedness ratio. We discuss this problem and a possible solution in Section 5.1.

2002-06-13