In model 1 the bow shock shape is simply assumed to be parabolic, i.e. that
the constant B in Eq. (1) in Dutrey et al. (1997) is zero. Dutrey
et al. (1997) assumed that the ambient material consists of a
low-velocity molecular outflow. The ratio of the ambient and shock
velocities () is described by the parameter
, which in the example spectrum is set to 0.2. This situation
represents the case where the ambient medium into which the jet is penetrating
moves at a relatively low velocity with respect to the exciting source.
Accordingly the peak emission is slightly shifted from the ambient cloud
velocity at small viewing angles. As in Dutrey et al. (1997) the
density distribution is supposed to be gaussian as a function of the
distance from the apex. This distribution is characterized by the density
dispersion
. This quantity as well as other quantities which have the
dimension of distance are expressed in the units of the parameter p of a
parabola (one half of the distance between the focus and the apex). The
observed line of sight velocity of a certain element on the surface of the
shock (in the units of the shock velocity) is
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(A1) |
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(A2) |
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(A3) |
In the model of Wilkin (1996) (model 2), analytical formulae for the shape of
the bow shock, tangential velocity and surface density are given as
functions of the angular distance from the apex (Eqs. 9-12). The
observed radial velocity for an element in the units of the central star
velocity is
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(A4) |
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||
(A5) | ||
Raga & Cabrit (1993) modeled turbulent wakes behind internal working
surfaces of jets and the associated bow shocks. Their purpose was to
investigate whether molecular outflows could be interpreted as gas entrained
in these wakes. The model is best understood by inspecting their Fig. 1.
The parameters of the model are the shape of the wake (determined by the
index and the length to width ratio x0/r0), and the ratio M1
between the jet velocity vj and the sound speed c1 in the mixing
layer. In the calculation of profiles we have used the set of parameters
depicted in Fig. 3b in Raga & Cabrit (1993), namely
, M1=20
and x0/r0 = 5. In this model, the systematic velocity toward the jet
axis (vr) is very small compared with the sound speed c1 in the mixing
layer, and is therefore neglected. Consequently, when the wake is seen from
the side (
) the the observed profile is determined by the
turbulent motion in the wake. Raga & Cabrit give the distributions of the
velocity along the jet axis (vx(x,r)), the density n(x,r) and
temperature T(x,r) in their Eqs. (16), (21) and (22), respectively. The
line of sight velocity distribution can be obtained by a multiplication by
the cosine of the viewing angle
. According to the assumption of
optically thin emission the spectrum has been calculated by adding all
volume elements weighted by a function of
density n and temperature T. The weighting function used is
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(A6) |
We thank the referee, Dr. J. Brand, for a careful reading of the manuscript and for very helpful comments and suggestions. We are grateful to him and Dr. J.G.A. Wouterloot for the program for calculating kinematic distances. We thank also Dr. C.M. Walmsley and Prof. K. Mattila for useful information and discussions. This work by J.H. and K.L. has been supported by the Finnish Academy through grant No. 1011055. The Swedish-ESO Submillimetre Telescope is operated jointly by ESO and the Swedish National Facility for Radio Astronomy, Onsala Space Observatory at Chalmers University of Technology.
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