  Context. Adaptive optics (AO) systems greatly increase the resolution of
large telescopes, but produce complex point spread function (PSF) shapes,
varying in time and across the field of view. This PSF must be accurately known
since it provides crucial information about optical systems for design,
characterisation, diagnostics and image post processing. Aims. We develop here
a model of the AO long exposure PSF, adapted to various seeing conditions and
any AO system. This model is made to match accurately both the core of the PSF
and its turbulent halo. Methods. The PSF model we develop is based on a
parsimonious parameterization of the phase power spectral density with only
five parameters to describe circularly symmetric PSFs and seven parameters for
asymmetrical ones. Moreover, one of the parameters is directly the Fried
parameter r0 of the turbulence s strength. This physical parameter is an asset
in the PSF model since it can be correlated with external measurements of the
r0, such as phase slopes from the AO real time computer (RTC) or site seeing
monitoring. Results. We fit our model against endtoend simulated PSFs using
OOMAO tool, and against on sky PSFs from the SPHERE ZIMPOL imager and the MUSE
integral field spectrometer working in AO narrowfield mode. Our model matches
the shape of the AO PSF both in the core and the halo, with a sub 1 percent
relative error for simulated and experimental data. We also show that we
retrieve the r0 parameter with subcentimeter precision on simulated data. For
ZIMPOL data, we show a correlation of 97 percent between our r0 estimation and
the RTC estimation. Finally, MUSE allows us to test the spectral dependency of
the fitted r0 parameter. It follows the theoretical $\lambda^{6/5}$ evolution
with a standard deviation of 0.3 cm. Evolution of other PSF parameters, such as
residual phase variance or aliasing, is also discussed.
