20.4 Accurate Position Measurements with the IRAM Interferometer

Let us start with two general and simple remarks. First, the phase equation in Section 20.2 or the least-square analysis of the $uv$ data in Section 20.3 show that higher position accuracy is achieved for smaller values of the fringe spacing $\lambda/{\cal B}$. Thus, for astrometry it is desirable to use long baselines and/or to go to short wavelengths. However, the latter case implies that the phases are more difficult to calibrate especially at mm wavelengths where the atmospheric phase fluctuations increase with long baselines. Second, sensitivity is always important in radio astrometry. For a point-like or compact source the sensitivity of the array varies directly as $D^2 (n(n- 1))^{0.5}$ where $D$ is the antenna diameter and $n$ the number of antennas. Thus, the detection speed varies as $D^4 n(n-1)$ and big antennas are clearly advantageous [Baudry 1996].

Comparison of the IRAM 5-element array with one of its competitors, the Owens Valley Radio Observatory array (OVRO) with 6 $\times10.4$m, gives a ratio of detection speed of 1 over 0.36 at 3mm and 1 over 0.65 at 1.3mm in favour of the Plateau de Bure array (see Table 1 below where the two entries correspond to 3mm and 1mm; system temperatures have been adopted according to advertised array specifications [June 2000]; sensitivity and speed are defined in Table 1). (Note also that the sixth antenna in the Bure array will increase its detection speed by 50%.) For comparison we include in Table 1 the BIMA array located in California and the Nobeyama array in Japan (NMA). In addition, it is interesting to note that the large dishes of the IRAM array are well adapted to quick baseline and phase calibrations; this is another clear advantage of the IRAM interferometer in astrometric observations.

Table 1. Comparison of Sensitivity and Speed of mm-wave Interferometers

	BIMA		IRAM		NMA		OVRO
Antennas	9		5		6		6
Baseline (m)	2000		400		400		480
Sensitivity	0.31	0.26	1.00	1.00	0.42	0.06	0.36	0.65
Speed	0.10	0.07	1.00	1.00	0.18	--	0.13	0.42

Sensitivity $=\frac{\eta_AD^2\sqrt{n(n-1)}}{T_{\rm sys}}$, Speed $=[\frac{\eta_AD^2\sqrt{n(n-1)}}{T_{\rm sys}}]^2$

20.4.1 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:

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select long baselines to synthesize small beamwidths,

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make a highly accurate baseline calibration observing several quasars selected for their small position errors,

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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,

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observe the program star over a long hour angle interval, and use the best estimate of the stellar coordinates (corrected for proper motion).

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 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 $\delta{\vec {\cal B}}$ and the offset between the unit vectors pointing toward the stellar source and the nearby phase calibrator. This phase error is $\delta(\phi -\phi_r) = (\delta{\vec {\cal B}}.({\vec k}_{\rm quasar} - {\vec k}_0)) 2\pi/\lambda$. Typical values are $\delta {\cal B} \simeq 0.2$ 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.

20.4.2 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

$$(\phi(\nu) - \phi(\nu_{\mathrm{ref}})) (\lambda/2\pi) = {\vec {\cal B}}.\delta{\vec k}(\nu) - {\vec {\cal B}}.\delta{\vec k}(\nu_{ref})$$ (20.20)

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. In this case [Reid et al. 1988] showed that the one sigma position uncertainty or angular uncertainty $\Delta\theta$ is approximately given by the equation

$$\Delta\theta = 0.5 (\lambda /{\cal B}) / \mathrm{SNR}$$ (20.21)

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 Orion where we have obtained accurate relative maps of SiO emission. Detailed and accurate relative maps were also obtained for the rare isotope $^{29}$SiO emission which is nearly 2 orders of magnitude weaker than that of the main isotope [Baudry et al. 1998]. A relative position accuracy of 2 to 5 milliarcsec was obtained in the Orion spot map of $^{28}$SiO emission (Fig. 20.1).

Figure: Spot map of $^{28}$SiO $v=1, J=2 \rightarrow 1$ emission observed on August 1995 in the direction of Orion IRc2 [Baudry et al. 1998]. The right ascension and declination offsets are in arcsec. Each small open square marks the center of an individual channel. The diameter of each circle, given every 3 channels, is proportional to the line intensity. The two main ridge of $^{28}$SiO emission cover $-1 \rightarrow -10$ (southern ridge) and $12 \rightarrow 20$ kms$^{-1}$.

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.