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Astron. Astrophys. 342, 717-735 (1999)

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Appendix: calculation of the recombination curves

In the absence of energetic radiative sources and assuming that local collisional processes are too weak to produce substantial hydrogen ionization the equation governing the decoupled time-dependent recombination of the gas can be easily found combining the standard continuity equations for the total and electron densities (see, e.g., Osterbrock 1989) through [FORMULA]. The final equation for the ionization fraction in this simple case reads:

[EQUATION]

where [FORMULA] is the substantial time derivative, with [FORMULA] the flow speed, and [FORMULA] is the recombination coefficient valid in Case B (Hummer & Storey 1987).

We assume that in the recollimated portion of the jet the gas flows on average along a set of axisymmetric nested flow surfaces, which in a cylindrical coordinate system and in the proper frame of the source are described by the equations [FORMULA], where [FORMULA] is the axial coordinate, and [FORMULA] is the local radius of the jet channel. The parameter [FORMULA] labels the different surfaces, the axis being identified by [FORMULA] and the external boundary of the jet, by [FORMULA]. We neglect azimuthal motions, and take a constant value of the axial velocity [FORMULA] all over the flow. Variations of the quantities with time can therefore be described as variations along the spatial coordinate [FORMULA]. It is easily shown that in these conditions the total density of a fluid particle behaves as: [FORMULA], where [FORMULA] and [FORMULA] are the hydrogen density and the channel radius at the arbitrarily chosen initial point of the integration [FORMULA]. Here we will assume for simplicity that the flow surfaces are shaped as nested cones, so that [FORMULA], where [FORMULA] is the average opening angle of the jet (which is positive for diverging jets and negative for converging ones). Coherently with the spectral read-out we adopted, we shall limit ourselves to the study of the fluid particles located on the axis of the flow, assigning to each of them the average physical properties of the jet section the particle belongs to.

Under these assumptions, and expressing variables and coordinates through quantities directly related to observations, the solution of equation (2) can be written as:

[EQUATION]

where z is the axial distance from the source projected on the plane of the sky and expressed in arcseconds, [FORMULA] is the starting point of the integration, that in practice is the first position along the beam for which a determination of the ionization fraction is available, or the position of an ionization jump (see e.g. Figs. 8, 9, 10). In expression (3) the quantities f and g are defined as follows:

[EQUATION]

[EQUATION]

[EQUATION]

where i is the inclination angle of the beam with respect to the plane of the sky, [FORMULA] is the jet radius expressed in arcseconds, D is the distance to the object in parsecs, and [FORMULA] the observed line of sight velocity of the emitting gas in the jet. The free parameters are the opening angle [FORMULA], which is varied until the best fit is obtained, and the inclination angle i of the jet to the plane of the sky, if not known independently. It should be kept in mind that in the framework of this simple model the jet is assumed to diverge or converge monotonically over its entire length or over separate portions of the flow: the jet radius to which we refer is, however, that corresponding to the zones of maximum emission, which in turn are the zones of maximum compression. The curves, therefore, refer to average properties of the flow, weighted more by the brightest regions.

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© European Southern Observatory (ESO) 1999

Online publication: February 23, 1999
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