Abstract: The residual phase error after correction for turbulence-induced phase aberrations by a partially compensating adaptive optics system (with temporal lag and limited space resolution) is computed by using the phase expansion on the Zernike polynomials in this paper. The computation can be used to derive the maximum order of aberration to be corrected to get the most from the compensation. The partially compensating adaptive optics system with Rayleigh beacon or sodium beacon as reference is computed also in this paper. The results show, the system with sodium beacon requires higher space frequency than the system with Rayleigh beacon. And the off-axis anisoplanatism effect due to the angular separation between the object and the beacon or the system temporal lag is more serious for the syst