  We investigate the ellipticity of the point-spread function (PSF) produced by
imaging an unresolved source with a telescope, subject to the effects of
atmospheric turbulence. It is important to quantify these effects in order to
understand the errors in shape measurements of astronomical objects, such as
those used to study weak gravitational lensing of field galaxies. The PSF
modeling involves either a Fourier transform of the phase information in the
pupil plane or a ray-tracing approach, which has the advantage of requiring
fewer computations than the Fourier transform. Using a standard method,
involving the Gaussian weighted second moments of intensity, we then calculate
the ellipticity of the PSF patterns. We find significant ellipticity for the
instantaneous patterns (up to more than 10%). Longer exposures, which we
approximate by combining multiple (N) images from uncorrelated atmospheric
realizations, yield progressively lower ellipticity (as 1 / sqrt(N)). We also
verify that the measured ellipticity does not depend on the sampling interval
in the pupil plane using the Fourier method. However, we find that the results
using the ray-tracing technique do depend on the pupil sampling interval,
representing a gradual breakdown of the geometric approximation at high spatial
frequencies. Therefore, ray tracing is generally not an accurate method of
modeling PSF ellipticity induced by atmospheric turbulence unless some
additional procedure is implemented to correctly account for the effects of
high spatial frequency aberrations. The Fourier method, however, can be used
directly to accurately model PSF ellipticity, which can give insights into
errors in the statistics of field galaxy shapes used in studies of weak
gravitational lensing.
