Simultaneous approximation of a function and its derivatives by Sobolev polynomials: Applications in satellite geodesy and precise orbit determination for LEO CubeSats

The objectives of this paper are twofold: (1) to present a new method of approximation such that the function and some of its derivatives are simultaneously approximated; and (2) to investigate the potential applications of this new method in the field of satellite geodesy by deriving a numerical solver of ordinary differential equations. To fulfill both the objectives, Sobolev polynomials are used. Explicit formulas for these polynomials are presented. A quadrature rule is derived based on the explicit forms of these polynomials. Solution of first order ordinary differential equations is presented based on the numerical integration method by the new quadrature rule. To present the applications of the results, three problems are investigated in the field of satellite geodesy. In the first application, orbit propagated by the numerical solution of orbital equations is compared with the Keplerian motion. It is shown that the new method for solving ordinary differential equations can be used alongside any other method, including the adaptive Runge-Kutta and Adams-Bashforth-Moulton integration methods. The comparison of this new method with the adaptive Runge-Kutta, Adams-Bashforth-Moulton, and the powerful and newly established adaptive Gaussian numerical integration methods reveals that it is at least 580, 666667, and 72 times more accurate than the mentioned methods, respectively, in any given absolute tolerance and time increment. The method is fast, stable, consistent, and capable of handling high accuracy, even with large time increment and low absolute tolerance, which is shown in this application. In the second application, the Gravity Recovery and Climate Experiment (GRACE) satellites' orbits are propagated using both the Ensemble Kalman Filter (EnKF) with stochastic updates and smoothing for the position and the Sobolev numerical integration. The orbits are then compared with the observed positions of the satellites. It is shown that the Sobolev polynomials work better in this problem, being more than 2 times more accurate. In the third application, the precision orbit determination problem for the Low Earth Orbit (LEO) CubeSats is investigated in the reduced dynamic form. A case study is presented for a pico-nano CubeSat in China. It is shown that the reduced dynamic orbit is approximately 32 percent more accurate than either of the kinematic or static Precise Point Positioning (PPP),representing a maximum error of 3.4 cm, with respect to the observation of the ground tracking stations.