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Astrometry is one of the main science case which drives the requirements ofthe next multiconjugate adaptive optics (MCAO) systems for future extremelylarge telescopes. The small diffraction limited point-spread function (PSF) andthe high Signal-to-Noise Ratio (SNR) of these instruments, promise astrometricprecision at the level of micro-arcseconds. However, optical distortions haveto be as low as possible to achieve the high demanding astrometry requirements.In addition to static distortions, the opto-mechanical instabilities causeastrometric errors that can be major contributors to the astrometry errorbudget. The present article describes the analysis, at design level, of theeffects of opto-mechanical instabilities when coupled with optical surfaceirregularities due to the manufacturing process. We analyse the notable exampleof the Multi-conjugate Adaptive Optics RelaY (MAORY) for the extremely largetelescope (ELT). Ray-tracing simulations combined with a Monte Carlo approachare used to estimate the geometrical structure and magnitude of fielddistortion resulting from the optical design. We consider the effects ofdistortion on the MCAO correction showing that it is possible achieve themicro-arcseconds astrometric precision once corresponding accuracy is obtainedby both optical design and manufacturing. We predict that for single-epochobservations, an astrometric error below 50$\mu$as can be achieved for exposuretimes up to 2 min, provided about 100 stars are available to remove fifth-orderdistortions. Such performance could be reproducible for multi-epochobservations despite the time-variable distortion induced by instrumentinstabilities.

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