3. Relative data calibration

Great care on the calibration has to be taken in a project like this one, to be able to compare measurements made at very distant epochs. We adopted several observational constraints during the entire period in order to improve the relative calibration of all the measurements as much as possible. These special procedures are presented in this section.

1.

Uncertainties in the detected signal due to pointing errors were specially treated for each individual observation in order to reduce them as much as possible. We made a small map of the beam response by pointing the telescope at the star's coordinates and at four other points located $\pm$30'' off in azimuth or elevation. Assuming the beam to be Gaussian, we calculated the pointing errors from the intensities measured at the five different points (using the values of the peak or integrated area intensities as input), and corrected the flux for each observation. This procedure is useless when the signal-to-noise ratio of the individual points is low. In such cases we took as final spectrum the average of those obtained at the different positions, after correcting the intensities assuming that there were no pointing errors. Note that this five-point map procedure is less efficient than a standard observation, and therefore somewhat longer integration times are needed to attain a certain signal-to-noise ratio.

2.

In order to minimize the effect of possible changes in the efficiency of the telescope with the elevation, not properly modeled, each source has been always observed at similar elevations (see Table 2). For each source, the rms of the average elevation of the individual observations is $\sim$ 3$^{\circ}$ (also see additional reasons for this observational constraint in point 3 of this section).

3.

It was also taken into account the possible influence of changes in the observed polarization on the recorded intensity variations. It has been shown that circumstellar SiO masers can present a relatively high degree of polarization. For example Clark et al. (1985) have studied the degree of linear polarization of the v=1, J=2-1 SiO transition in several strong SiO maser emitters like o Cet, W Hya, and R Leo (these three sources are included in our monitoring). They concluded that in this transition the different maser emission peaks have a well defined angle of polarization. The degree of linear polarization could reach up to 80%, with typical values of 10 - 30%. McIntosh et al. (1994) published measurements of the circular and linear polarization of the J=2-1 and J=1-0, v=1 SiO maser lines in the red supergiant VY CMa. Their results show up to 10% of linear polarization at some velocities, the circular polarization being always less than 5%. Barvainis et al. (1987) also found evidence of circular polarization in five circumstellar SiO masers, but at a degree smaller than 10% in all cases. We have also made polarization measurements of 43 GHz SiO masers with the CAY-13.7 m telescope (Martínez et al. 1988). Since only one receiver polarization, horizontal linear, was available, we had to measure the linear polarization of SiO masers by observing the objects at very different parallactic angles, with the drawback that the observations are not simultaneous neither performed at the same elevation. Due to these limitations, this method is unable to reliably detect linear polarization degrees smaller than 20%. However, it is useful to check the presence of highly linearly polarized SiO maser emission, which could influence our monitoring data. In almost all cases, the linear polarization degree was found to be less than 30%, mainly for the peak flux and the velocity integrated intensity (the two parameters considered for our SiO light curves, see Sect. 4).

According to the previous discussion, we decided to observe each individual star always at the same range of sidereal time, in order to maintain the same averaged parallactic angle (and also the same elevation angle, see point 2 in this section). This method ensures that the projection of the receiver linear polarization on the object is always the same.

Regarding to the polarization state, the response of the system has been as follows. Until December 1988 the receiver polarization was horizontal linear. Then, it was changed to vertical linear until May 1989, when a $\lambda$/4 plate was placed in front of the receiver, setting the effective polarization sensitivity of the system to left-hand circular for the rest of the monitoring.

The observational method followed and the small degree of both linear and circular polarization of SiO masers ensures that we can compare the spectra corresponding to each individual object, at least for the periods mentioned above, and that the variability curves obtained correspond to changes in the intensity or polarization state of the sources.

Figure 1: Examples of individual observations of the v=1 J=1-0 maser line obtained during the monitoring for five objects: VY CMa, o Cet, IRC +10011, IK Tau, and VX Sgr

4.

The calibration for the different atmospheric conditions of each observational run is probably the most important source of uncertainty when comparisons of different individual observations have to be done. In order to reduce this uncertainty as much as possible the observations were always performed under clear sky conditions, or partially cloudy with only high stratiform clouds (that do not affect the transmission of electromagnetic waves at 7 mm). At 43 GHz, the atmospheric effects are usually dominated by O₂ absorption and, at a lower degree, by H₂O absorption ( $\tau_{\rm H_{2}O}\simeq \tau_{\rm O_{2}}/6$ for 6 mm of integrated water vapor above the telescope; Pardo 1996). At the location of the Yebes Observatory, the atmospheric transmission at 43 GHz under good weather conditions is in any case high, $\sim93\%$ at the zenith (Cernicharo 1988; Pardo 1996). To first approximation, variations in the ambient and atmosphere temperatures are accounted for by the calibration method. From our experience at Yebes, observing molecular lines of constant intensity (thermally excited lines in molecular clouds) under different weather conditions, we conclude that the relative calibration errors in the monitoring due to atmospheric effects are smaller than 10%. This factor can be taken as an uncertainty upper limit for our data set.

5.

In addition to all the cautions explained above, a thermal continuum source, the HII region W 51, has been observed twice (one at each frequency) every observing run. This was done to detect long-term changes in the whole calibration procedure during the entire observational program. The measured intensity of W 51 over the entire program shows an rms $\sim$ 10%, in agreement with the estimates previously given. Neither systematic nor seasonal changes were found, and therefore no corrections have been applied to the data based on the results from the observations of W 51.