Theoretical limits on bright star astrometry with multi-conjugate adaptive optics using a diffractive pupil

We present a ground-based technique to detect or follow-up long-period exoplanets via precise relative astrometry of host stars using Multi-Conjugate Adaptive Optics (MCAO) on 8 meter telescopes equipped with diffractive masks. MCAO improves relative astrometry by sharpening PSFs, reducing the star centroiding error, and by providing a spatially stable, more easily modeled PSF. However, exoplanet mass determination requires multi-year reference grid stability of ∼10-100 uas or nanometer-level stability on the long-term average of out-of-pupil phase errors, which is difficult to achieve with MCAO. The diffractive pupil technique calibrates dynamic distortion via extended diffraction spikes generated by a dotted primary mirror, which are referenced against a grid of background stars. We calculate the astrometic performance of a diffractive 8-meter telescope with diffraction-limited MCAO in K using analytical techniques and a simplified MCAO simulation. Referencing the stellar grid to the diffraction spikes negates the cancellation of Differential Tip/Tilt Jitter normally achieved with MCAO. However, due to the substantial gains associated with sharper, more stable PSFs, diffractive 8-m MCAO reaches ∼ 4-6 μas relative astrometric error per coordinate in one hour on a bright target star (K - 7) in fields of moderate stellar density (∼10 stars arcmin^-2). Final relative astrometric precision with MCAO is limited by atmospheric differential tip/tilt jitter.