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

Astrometry is one of the main science case which drives the requirements of the next 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 high demanding astrometry requirements. In addition to static distortions, the opto-mechanical instabilities cause astrometric errors that can be major contributors to the astrometry error budget. The present article describes the analysis, at design level, of the effects of opto-mechanical instabilities when coupled with optical surface irregularities due to 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 achieve the micro-arcseconds astrometric precision once 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 about 100 stars are available to remove fifth-order distortions. Such performance could be reproducible for multi-epoch observations despite the time-variable distortion induced by instrument instabilities.