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. The PSF must be accurately known since it provides crucial information about optical systems for design, characterization, 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 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 end-to-end simulated PSFs using the OOMAO tool, and against on-sky PSFs from the SPHERE/ZIMPOL imager and the MUSE integral field spectrometer working in AO narrow-field mode. Our model matches the shape of the AO PSF both in the core and the halo, with a relative error smaller than 1% for simulated and experimental data. We also show that we retrieve the r₀ parameter with sub-centimeter precision on simulated data. For ZIMPOL data, we show a correlation of 97% 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 λ^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.