17.4.1 Absolute positions

To illustrate the potential of the IRAM array for astrometry we consider here observations of the SiO maser emission associated with evolved late-type stars. Strong maser line sources are excited in the v = 1, J = 2-1 transition of SiO at 86 GHz and easily observed with the sensitive IRAM array. Because of molecular energetic requirements (the vibrational state v = 1 lies some 2000 K above the ground-state) the SiO molecules must not be located too much above the stellar photosphere. In addition, we know that the inner layers of the shell expanding around the central star have sizes of order one arcsecond or less. Therefore, sub-arcsecond position accuracy is required to locate the SiO sources with respect to the underlying star whose apparent diameter is of order 20-50 milliarcseconds. For absolute position measurements one must primarily:

• select long baselines to synthesize small beamwidths,
• make a highly accurate baseline calibration observing several quasars selected for their small position errors,
• observe at regular intervals two or more quasars (phase calibrators) in the field of each program star in order to determine the instrumental phase and to correct for atmospheric phase fluctuations,
• observe the program star over a long hour angle interval, and use the best estimate of the stellar coordinates.

Our first accurate radio position measurements of SiO masers in stars and Orion were performed with the IRAM array in 1991/1992. We outline below some important features of these observations [Baudry et al 1994]. We used the longest E-W baseline available at that time, about 300 m, thus achieving beams of order 1.5 to 2 arcseconds. The RF and IF bandpass calibrations were made accurately using strong quasars only. To monitor the variable atmosphere above the array and to test the overall phase stability, we observed a minimum of 2 to 3 nearby phase calibrators. Prior to the source position analysis we determined accurate baseline components; for the longest baselines the r.m.s. uncertainties were in the range 0.1 to 0.3 mm. The positions were obtained from least-square fits to the imaginary part of the calibrated visibilities. (Note that the SiO sources being strong, working in the (u,v) or image planes is equivalent.)

The final position measurement accuracy must include all known sources of uncertainties. We begin with the formal errors related to the data noise. This is due to finite signal to noise ratio (depending of course on the source strength, the total observing time and the general quality of the data); poorly calibrated instrumental phases may also play a role. In our observations of 1991/1992 the formal errors were around 10 to 30 milliarcseconds. Secondly, phase errors arise in proportion with the baseline error $\delta{\vec B}$ and the offset between the unit vectors pointing toward the stellar source and the nearby phase calibrator. This phase error is $\delta(\phi -\phi_r) = (\delta{\vec B}.({\vec k}_{quasar} - {\vec k}_o)) 2\pi/\lambda$. Typical values are $\delta B \simeq 0.2$ mm and $\delta k \sim 10^\circ-20^\circ$ corresponding to phase errors of $3^\circ$ to $7^\circ$, that is to say less than the typical baseline residual phases. A third type of error is introduced by the position uncertainties of the calibrators. This is not important here because the accuracy of the quasar coordinates used during the observations were at the level of one milliarcsecond.

The quadratic addition of all known or measured errors is estimated to be around 0.07'' to 0.10''. In fact, to be conservative in our estimate of the position accuracy we measured the positions of nearby quasars using another quasar in the stellar field as the phase calibrator. The position offsets were around 0.1'' to 0.2'' depending on the observed stellar fields; we adopted 0.1'' to 0.2" as our final position accuracy of SiO sources. The SiO source coordinates are derived with respect to baseline vectors calibrated against distant quasars. They are thus determined in the quasi-inertial reference frame formed by these quasars.

Finally, it is interesting to remind a useful rule of thumb which one can use for astrometry-type projects with any connected-element array provided that the baselines are well calibrated and the instrumental phase is stable. The position accuracy we may expect from a radio interferometer is of the order of 1/10th of the synthesized beam (1/20th if we are optimistic). This applies to millimeter-wave arrays when the atmospheric fluctuations are well monitored and understood. With baseline lengths around 400 m the IRAM array cannot provide position uncertainties much better than about 0.05-0.1'' at 86 GHz. Extensions to one kilometer would be necessary to obtain a significant progress; the absolute position measurements could then be at the level of 50 milliarcseconds which is the accuracy reached by the best optical meridian circles.