Institute of Astronomy RAS, Moscow, Russia
The conclusion of the Encke's equations, in which the motion on an intermediate orbit of generalized problem of two fixed centres is taken for the reference motion is given. For c=0 and \sigma=0 the obtained equations turns into the classical Encke's equations, in which the reference motion is the Kepler's motion. The principle possibility of using the Encke's method to integrate the equations of motion of artificial Earth's satellite (AES) using the intermediate orbit of the problem of two fixed centres as reference motion is shown. This orbit allows to go round the difficulty which this problem faces when using the fixed Kepler's ellipse as reference orbit. Joint integration of the reference motion and the variational Encke's equations and its comparison with the integration of perturbated Kepler's motion prove the absolute accuracy of the found out equations. The perturbations of zonal geopotencial harmonics up to the 14-th order were used. The accuracy of separate integration of Encke's equations depends on the interpolation of the coordinates of the reference motion. The spline function of 3-th orders for interpolation was taken. When interpolating the reference orbit with 2-min knotes the co-ordinates differencies amount to a few mm for one revolution. The other versions (1 or 0.5 min knotes) gives similar results (smaller than 1 mm).
Separate integration was constructed using simultaneous computing of reference orbit with precise analytical formulaes of computer accuracy. Accuracy increasing is not obtained when integrating AES motion equations with a period of 2 hours for 30 days. Remarkable increasing of integration accuracy for Etalon-1 satellite is obtained. When integrating Encke's equation for 30 days with different parameters of Everhart's integrator 11 significant figures with an accuracy less than 1 mm for co-ordinates remained stable. When integrating Kepler motion equation only 9-10 significant figures remained.