The data are weighted, with the weight of an individual RV being the inverse
of its variance. Since no individual errors for the older RVs are known, for
each set of data a common mean standard deviation is adopted: Bopp & Dempsey
(1989)
estimate their RV-errors to be 0.9, and they assign to the data given
by Harper (1914, 1935) a weight of 0.5 with respect to their own
measurements. This translates to a standard deviation of 1.3
, which is
adopted here and also used for the RV given by Abt (1970). Eker
(1986, see Table 2 (click here)) gives an error of 0.5
. We estimate
the error of the SOFIN RVs to be 0.3
(see above).
The data consisting of triples with the weights
are
fitted (least-squares fit) by the function
(see e.g. Heintz 1978
for the derivation and the meaning of the symbols).
The parameters are determined with a non-linear least-squares fit using the simplex-algorithm (e.g. Caceci & Cacheris 1984; Press et al. 1994).
As an additional parameter derived from the fit-parameters, we compute the
period in the rest frame of the system (corrected for the time dilation in
the inertial frame moving with the velocity with respect to the sun,
and to the
increasing light travel-time between periastron passages due to
).
We use
in the computation of
and f(m).
There are two ways to determine standard deviations for the fit-parameters.
Figure 1: Periodogram of the RV measurements of
Gem. The only significant period is at 19.604471 days
Assuming the orbit to be circular, i.e. fixing e=0 in the fits, and removing the last 3 measurements from Table 1 (click here) (see Sect. 3.4 (click here)), the best orbital period was searched in the interval 18.0 to 21.0 days.
Figure 1 (click here) shows the periodogram obtained by varying the
start value of the period in fits of
RV curves to the data set. The plot
gives the standard deviation of an individual measurement of mean
weight from the fit vs. the period.
It is obvious that
there is only one period close to the value given by Bopp & Dempsey
(1989); all other minima have
exceeding 10
.
Figure 2: The SOFIN measurements (squares) and Eker's
measurements (crosses) with the radial velocity curve of
Solution 4
Table 3: The results of 4 different fits of radial velocity
curves to the data. Solution 1 includes all data; the errors given
in parentheses are the formal fit errors, the others bootstrap
errors from B=1000 bootstrap runs. For Solutions 2-4, only the
bootstrap errors from B=1000 runs are given. Solution 2 includes
only the SOFIN data supplemented by Eker's RVs
(Table 1 (click here) and Table 2 (click here)). Solution 3 includes
the same data
as Solution 2 except the last 3 SOFIN measurements. Solution 4
is a circular fit to the data used for Solution 3.
For Solutions 2-4 the
period was kept fixed at 19.604471 days. For comparison, the
parameters given by Bopp & Dempsey (1989) are given in the
column BD89
The results of four different fits of radial-velocity curves are given in
Table 3 (click here). Except for Solution 4, all allow for a non-zero
eccentricity of the orbit. The formal parameter errors are only given for
Solution 1: one can see that in most cases the formal and the bootstrap errors
are very close to each other. The main exception is which has a very
large bootstrap error. The bootstrap error is certainly more realistic due to
the very small eccentricity. This relation between the formal and the bootstrap
errors is the same for all fits; only the (more reliable) bootstrap errors are
therefore given in the following.
A forced circular fit to all measurements excluding the last 3 SOFIN
measurements (see below) yields a slightly different period than that given
in Solution 1:
which is adopted as the final period. This period is kept fixed in fits leading
to Solutions 2-4, because the limited time-coverage of the data sets used
for them does not allow a
reliable determination of the period.
The eccentricity of the orbital solution 1 is only half of what Bopp & Dempsey
(1989) got. Solutions 2 and 3 are obtained with the aim to establish the
significance (or otherwise) of this eccentricity. Only the data sets having the
smallest velocity errors, i.e. the SOFIN and Eker's data sets
(Table 1 (click here) and Table 2 (click here)), are used. Solution 2 uses all
of these data. The eccentricity has slightly decreased
further. When computing the O-C from the orbit, the last 3 SOFIN measurements
in Table 1 (click here) show systematical deviations from the fit. They range up
to 2.6 . Since these measurements are at a sensitive phase (close to the
maximum RV,
where there are also not many other measurements) they might be responsible for
a large part of the eccentricity. For Solution 3, they are therefore removed
from
the data set. The eccentricity dropped again by a factor 2, now being only
a
result and extremely small. We thus conclude (as was already
suspected by
Bopp & Dempsey 1989) that the orbit is in fact circular.
Solution 4 is accordingly a forced circular fit to the data set used for
Solution 3 (SOFIN + Eker,
excluding the last 3 measurements in Table 1 (click here)) and we adopt it as
the final solution. The insignificant changes in the orbit parameters and in
from Solution 3 to Solution 4 support our conclusion of
circularity. Note, that the period of the final orbital solution is still
based on the whole data
set covering the time interval 1902-1995; only all other parameters are
derived from the limited, but most accurate data.
As can be seen from Table 3 (click here) the parameters are very close to
those given by Bopp & Dempsey (1989).
Figure 2 (click here) shows the measurements of SOFIN and Eker (1986) together with the fit of Solution 4. The systematic deviation of the last 3 SOFIN measurements close to phase 0 is clearly seen. These and the significant deviations of a few other data points are probably caused by strong distortions of the line profiles due to spot features.