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Multi-conjugate adaptive optics (MCAO) will assist a new era of ground-basedastronomical observations with the extremely large telescopes and the VeryLarge Telescope. High precision relative astrometry is among the main sciencedrivers of these systems and challenging requirements have been set for theastrometric measurements. A clear understanding of the astrometric error budgetis needed and the impact of the MCAO correction has to be taken into account.In this context, we propose an analytical formulation to estimate the residualphase produced by an MCAO correction in any direction of the scientific fieldof view. The residual phase, computed in the temporal frequency domain, allowsto consider the temporal filtering of the turbulent phase from the MCAO loopand to extract the temporal spectrum of the residuals, as well as to includeother temporal effects such as the scientific integration time. The formulationis kept general and allows to consider specific frameworks by setting thetelescope diameter, the turbulence profile, the guide stars constellation, thedeformable mirrors configuration, the modes sensed and corrected and thetomographic reconstruction algorithm. The formalism is presented for both aclosed loop and a pseudo-open loop control. We use our results to investigatethe effect of tip-tilt residuals on MCAO-assisted astrometric observations. Wederive an expression for the differential tilt jitter power spectrum that alsoincludes the dependence on the scientific exposure time. Finally, weinvestigate the contribution of the differential tilt jitter error on thefuture astrometric observations with MAVIS and MAORY.

Temporal spectrum of multi-conjugate adaptive optics residuals and impact of tip-tilt anisoplanatism on astrometric observations