Keck NGAO sky coverage modeling

This report details the sky coverage simulations carried out for the Keck Next Generation Adaptive Optics (NGAO) system. This work was carried out to meet the objectives stated in WBS Element Number: 3.1.2.2.10 Number and Type of Low Order WFS TS. As well as investigating the number of Natural Guide Star (NGS) Shack-Hartmann wavefront sensors (WFS), this report considers the number of modes to measure with the NGS WFS, the spectral band for the NGS WFS, the size of the NGS patrol field, and the amount of partial correction of the NGS. These sky coverage trade studies are completed for a selection of the science cases identified in Ref. 1. This report does not consider the type of low order wavefront sensor; Shack-Hartmann WFS(s) are assumed throughout, and a separate study will consider Shack-Hartmann vs pyramid WFSs. The sky coverage simulations presented here are calculated with the Sky Coverage Simulator developed for TMT and documented in Ref.s 2, 3. This code is a covariance based code, which allows fast simulation of large numbers of NGS constellations. The NGS constellations are generated using star statistics: for J,H, and K bands the Spagna4 model is used, and for V band the Bahcall-Soneira model5 model is used. The essence of this method is to transform the turbulence phase screens at each altitude, which are represented as a Zernike basis sum, to the aperture using geometric optics. This model accounts for the cone effect for the finite height of the laser guide stars LGSs, as well as the anisoplanatism caused by the GSs being off-axis with respect to the science object. The expected wavefront error is then calculated using a minimum variance estimator from these transformation matrices and the statistical properties of the atmosphere and the noise. The tip-tilt (TT) errors that are calculated are the sum of three terms: the servo lag, measurement noise and tilt anisoplanatism. The tilt anisoplanatism term includes the effect of the pairs of quadratic ”null modes” that