line
Simulations were carried out to observe the behaviour of
and 
as a
function of the choice of the two segments in order to find their optimal
position and length.
Synthetic spectra were computed at 6000 K and 7500 K with a logg of 4 and solar metallicity. They were then convoluted by instrumental and rotational profiles. We considered projected rotational velocities of 50 and 100 kms-1. Finally, a random noise was added, corresponding to a signal-to-noise ratio of 150.
Two types of simulations were performed:
and
was then computed for each position of
these segments.
and were
again computed.
Figure 12: Variation of and
as a function of the limits for a star of
6000 K with a logg of 4 and a solar metallicity. The graph at the left
simulates variations for a 
of 50 kms-1 and at the right of
100 kms-1. See text for comments
Figure 13: Same as Fig. 11 (click here), but with 
The results are given in Figs. 12 (click here) and 13 (click here), the upper part
of them showing the variation of and the
lower part the variation of as a
function of the limits for the 4 spectra. Each curve corresponds to
different initial conditions, i.e. to different fixed limits. The first type
of simulation is represented by dotted lines and the second type by dots and
dashes. The dispersion and the mean central wavelength are given as a
function of the moving limit. To make these graphics easier to read, we have
represented the right wing of the 
line with the scale in
normalised flux (right axis of the graphics).
First, let us discuss the behaviour of .
When the internal and external limits are the same on each wing, the
dispersion is evidently null, increasing when the internal and external
limits are separated. This increase depends on each profile as seen in
Figs. 12 (click here) and 13 (click here). Nevertheless, we can make general
remarks. First, the nearer the limits to the centre of the line, the
stronger the dispersion. This is easily understood, because near the centre
only a few points are taken into account to make the fit and a weak change
of limits leads to important changes of the three parameters of
. These parameters are not well defined either when we choose
external limits too far from the centre, because have a
lorentz profile only near the centre. Thus limits that are neither too
close nor too far from the centre have to be chosen. Generally speaking,
these subjective criteria are satisfactory when the segments lie on the
linear part of the profile.
Increased and 
make the determination of the
central wavelength more difficult because the line is wider. The dispersion
therefore increases with and . When the segments
are on the linear part of the profile, the following dispersions result: at
6000 K, varies from 0.003 to 0.007 Å for
equal to 50 and 100 kms-1 respectively and at 7500 K, its
values are 0.004 to 0.007 Å for the same . Thus the dispersion
mainly depends on the rotational velocity and barely on the effective
temperature.
On the lower graphics, one sees that
varies much as a function of the position and lengths of the segments.
Nevertheless, if we consider only the linear portion, the variability of
this parameter is not so important and corresponds to values given above.
We observe that the shape of these fluctuations depends essentially on the
: for a of 100 kms-1,
regularly decreases when the limits
approach the centre of the line; for a of 50 kms-1 we
observe an opposite trend.
The results of these simulations give only a qualitative idea about the
behaviour of and
as a function of the chosen segments. We
have shown that these parameters are very sensitive to the limits, which
have to be put on the linear parts of the line profile to alleviate this
problem. For a given star, we always use the same limits in order
to have a good internal coherence. Nevertheless, we may have systematic errors,
but this is not a severe problem because we are interested in variations of
radial velocity rather than in absolute values.