17.4.2 Relative Positions and Self-calibration Techniques

We have measured with the IRAM array the absolute position of the SiO emission sources associated with each spectral channel across the entire SiO emission profile. Any spatial structure related to the profile implies different position offsets in the direction of the star. Such a structure with total extent of about 50 milliarcseconds is observed in several late-type stars. This is confirmed by recent VLBI observations of SiO emission in a few stars. VLBI offers very high spatial resolution but poor absolute position measurements in line observations.

The best way to map the relative spatial structure of the SiO emission is to use the phase of one reference feature to map all other features. This spectral self-calibration technique is accurate because all frequency-independent terms are cancelled out. The terms related to the baseline or instrumental phase uncertainties as well as uncalibrated atmospheric effects are similar for all spectral channels and cancel out in channel to channel phase differences. By making the difference

$$(\phi(\nu) - \phi(\nu_{\mathrm{ref}})) (\lambda/2\pi) = ({\vec B}.\delta{\vec k}(\nu) - {\vec B}.\delta{\vec k}(\nu_{ref})$$ (17.19)

where the SiO reference channel is at frequency $\nu_{ref}$ we obtain a phase difference equation whose solution gives the coordinate offsets $\Delta\alpha(\nu)$ and $\Delta\delta(\nu)$ relative to channel $\nu_{ref}$. The main limitation in such self-calibration techniques comes from the thermal noise and the achieved signal to noise ratio SNR. The angular uncertainty $\Delta \theta$ can then be estimated with the simple equation $\Delta\theta = 0.5 (\lambda / B) / \mathrm{SNR}$. Common practice with connected-element arrays shows that selection of a reference channel is not critical; it must be strong in general. Self-calibration proved to be successful with the IRAM array in several stars and in Orion where we have obtained very detailed relative maps of SiO emission. Detailed relative maps were also obtained for the rare isotope ²⁹SiO; this emission is nearly 2 orders of magnitude weaker than that of the main isotope.

The relative spot maps obtained with connected-element arrays do not give the detailed spatial extent of each individual channel. This would require a spatial resolution of about one milliarcsecond which can only be achieved with VLBI techniques. Note however that VLBI is sensitive to strong emission features while the IRAM array allows detection of very weak emission; thus the two techniques appear to be complementary.

With SiO spatial extents of about 50 milliarcseconds and absolute positions at the level of 0.1 arcsecond it is still difficult to locate the underlying star. We have thus attempted to obtain simultaneously the position of one strong SiO feature relative to the stellar photosphere and the relative positions of the SiO sources using the 1 and 3 mm receivers of the IRAM array. This new dual frequency self-calibration technique is still experimental but seems promising.