  We present an analytical model of a single natural guide star astronomical
adaptive optics system, in closed loop mode. The model is used to simulate the
long exposure system point spread function, using the spatial frequency (or
Fourier) approach, and complement an initial open loop model. Applications
range from system design, science case analysis and AO data reduction. All the
classical phase errors have been included: deformable mirror fitting error,
wavefront sensor spatial aliasing, wavefront sensor noise, and the correlated
anisoplanatic and servo-lag error. The model includes the deformable mirror
spatial transfer function, and the actuator array geometry can be different
from the wavefront sensor lenslet array geometry. We also include the
dispersion between the sensing and the correction wavelengths. Illustrative
examples are given at the end of the paper.
