The impact of geometric distortions in multiconjugate adaptive optics astrometric observations with future extremely large telescopes

Astrometry is one of the main scientific fields driving the requirements for the next generation of multiconjugate adaptive optics (MCAO) systems for future extremely large telescopes. The small diffraction-limited point spread function (PSF) and the high signal-to-noise ratio (SNR) of these instruments promise astrometric precision at the level of micro-arcseconds. However, optical distortions have to be as low as possible to achieve the demanding requirements of astrometry. In addition to static distortions, opto-mechanical instabilities cause astrometric errors that can make major contributions to the astrometry error budget. This article describes the analysis, at design level, of the effects of opto-mechanical instabilities when coupled with optical surface irregularities resulting from the manufacturing process. We analyse the notable example of the Multi-conjugate Adaptive Optics RelaY (MAORY) for the Extremely Large Telescope (ELT). Ray-tracing simulations combined with a Monte Carlo approach are used to estimate the geometrical structure and magnitude of field distortion resulting from the optical design. We consider the effects of distortion on the MCAO correction, showing that it is possible to achieve micro-arcsecond astrometric precision once the corresponding accuracy is obtained by both optical design and manufacturing. We predict that for single-epoch observations, an astrometric error below 50 mu as can be achieved for exposure times up to 2 min, provided that about 100 stars are available to remove fifth-order distortions. Such a performance could be reproducible for multi-epoch observations despite the time-variable distortion induced by instrument instabilities.