1 Introduction

Long-baseline optical interferometry do have a potential for sub-mas precision in astrometric measurements as first suggested by Shao & Staelin (1977). Furthermore, with the dual-field technique and simultaneous observation of two nearby sources, long time averaging of differential path length fluctuations makes it achievable precision at a $10-20~\mu$ as level (Shao & Colavita 1992). This is the range of precision required for the detection of Jupiter-like extrasolar planets through the reflex motion of their central star, projected onto the celestial sphere; e.g. the amplitude of the reflex motion of the sun at 10 parsec is 0.5 mas.

Astrometric interferometers for wide-angle measurements, MkIII (Shao et al. 1988) and Navy Prototype Optical Interferometer or NPOI (Armstrong et al. 1998), have their main delay lines located in evacuated pipes. Zeroing the residual phase of the white-fringe holds the scales between an outside geometric delay depending on the source position in the sky, and the compensating vacuum delay within the interferometer arms. With non-evacuated delay lines, longitudinal dispersion has to be considered: the relation between residual phase and wavenumber is no longer linear, and there is no longer any white-fringe.

After the first implementation of a dual-field interferometer, the Palomar Testbed Interferometer or PTI (Colavita et al. 1994; Colavita et al. 1999), unique astrometric results appear to be on the reach. The Very Large Telescope Interferometer (VLTI) at Paranal will have all its optical paths along non-evacuated pipes (von der Lühe et al. 1994; Mariotti et al. 1998). Although residual air turbulence is not expected to be a limiting factor in its operation (Koehler et al. 1997), longitudinal dispersion effects have to be considered (Lévêque 1997). Although this instrument is not supposed to work as an astrometric interferometer for wide-angle measurements, a dual-field instrument for narrow-angle astrometry is being studied (Quirrenbach et al. 1998) and some wide-angle astrometry is needed, at least for baseline calibration.

With an astrometric interferometer, fringe tracking is not just trying to keep the residual delay small enough for the visibility remaining near its maximum value. The residual delay has to be accurately measured, as well as the optical path and/or group delay introduced by the delay line. With non-evacuated pipes, these last quantities are wavelength dependent, and the metrology has to be scaled to some effective wavelength really observed. In this paper, we shall examine the operation of an astrometric interferometer in the near-IR range and draw some inferences on the design of a fringe sensor unit, in terms of its sensitivity and dependence on ambient parameters. The scheme proposed for coherence tracking is the simplest form of group-delay tracking or GDT (Lawson 1995), with the observation of two wavebands only. It has not to be confused with the two-color technique developed in the visible range and used with the MkIII interferometer (Colavita et al. 1987). Indeed, the air dispersion factor is more than 10 times smaller in the near-IR band (between H and K bands), as compared to the visible, so that highly precise phase measurements would be required for it to be useful. The two-color technique was implemented with the MkIII interferometer, for partial correction of the atmospheric path length fluctuations ahead. A two-bands group-delay tracking is here investigated to get rid of longitudinal dispersion along the non-evacuated delay lines of an IR astrometric interferometer.

A dual-field interferometer is a kind of a two-stages apparatus: the uncertainty on small angle measurements is the product of the wide-angle, or single-field astrometric accuracy, by the separation angle to be measured, expressed in radian. The single-field accuracy is about the same as the relative uncertainty on the interferometer baseline. For angles as small as a few arcminutes, or 10^-3, and error on relative baseline of about 10^-7, differential astrometry can be performed with a 10^-10 uncertainty, or $20~\mu$as. These are the figures to be kept in mind for performance estimates.

We first point out the difference between optical path and group delay within interferometer arms with non-evacuated lines. A simple and operating model is proposed for longitudinal dispersion. Group-delay tracking and the effects of stellar emission spectra (or color temperature) on astrometric measurements are developed in Sect. 3. Sensitivity and performances of a proposed concept for fringe tracking are further analyzed.