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3. The radial velocity curve

 

3.1. The full data set

  The radial velocities measured from the SOFIN spectra (Table 1 (click here)) are supplemented by older measurements from the following sources (N is the number of RVs given):
Abt (1970), N=1
Bopp & Dempsey (1989), N=110
-> includes 19 values from Harper (1935)
Eker (1986), N=9
Harper (1914), N=38
Harper (1935), N=21
->includes 2 values from Harper (1914)
->includes 5 values from Moore (1928),
->includes of which are revised from Reese (1903)
->includes 4 values from Jones (1928),
->includes also listed by Lunt (1919).

An obvious misprint in one JD given by Harper (1935) has been corrected. Eker's RVs were not published before; he kindly made them available to us, and they are given in Table 2 (click here). The total number of RVs is thus 187 and the time interval covered by them ranges from 1902 to 1995.

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 0.9tex2html_wrap1865, 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 1.3tex2html_wrap1867, 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 0.5tex2html_wrap1869. We estimate the error of the SOFIN RVs to be 0.3tex2html_wrap1871 (see above).

3.2. The fit and error determination

 

The data consisting of triples tex2html_wrap_inline1873 with the weights tex2html_wrap_inline1875 are fitted (least-squares fit) by the function
 equation305
(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 tex2html_wrap_inline1877 with respect to the sun, and to the increasing light travel-time between periastron passages due to tex2html_wrap_inline1879). We use tex2html_wrap_inline1881 in the computation of tex2html_wrap_inline1883 and f(m).

There are two ways to determine standard deviations for the fit-parameters.

  1. The formal error of any fit-parameter p is given by
     equation318
    (Bevington 1969) with the derivatives taken at the minimum of tex2html_wrap_inline1889. With the fit-parameters and their formal errors known, the errors of the derived parameters can be obtained by the usual error propagation.
  2. The bootstrap errors (see e.g. Efron & Tibshirani 1993) are computed from the actual residuals of the measurements from the fit. B bootstrap samples are constructed by randomly selecting from the residuals of the N real measurements N residuals, with replacement, and adding them to the computed RVs using the best fit parameters. Since the residuals of measurements having low weight are on average larger than those of measurements having high weight, each residual picked is accompanied by its original weight. For each set of these new ``measurements'' a fit of the RV curve is performed, resulting in a new set of fit-parameters and derived parameters. For large B the parameter errors are given by the standard deviations of the bootstrap parameters (both fit-parameters and derived ones) from their values obtained from the set of real measurements.

3.3. The period search

 

 figure331
Figure 1:   Periodogram of the RV measurements of tex2html_wrap_inline1899 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 tex2html_wrap_inline1903 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 tex2html_wrap_inline1905 exceeding 10tex2html_wrap1911.

3.4. The radial velocity curve of tex2html_wrap_inline1913 Gem

 

 figure341
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 tex2html_wrap_inline2089 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:
 equation415
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 tex2html_wrap2119. 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 tex2html_wrap_inline2093 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 tex2html_wrap_inline2095 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.


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