Dynamic orbit propagation in a gravitational field of an inhomogeneous attractive body using the Lagrange coefficients M.A. Sharifi a,1 , M.R. Seif b,⇑ a Department of Surveying and Geomatics Engineering, University College of Engineering, University of Tehran, Enghelab Ave., P.O. Box 11365-4563, Tehran, Iran b K.N. Toosi University of Technology, Faculty of Geodesy and Geomatics, No. 1346, Vali-Asr St., Mirdamad Cross, Tehran 1996715433, Iran Received 11 August 2010; received in revised form 19 April 2011; accepted 19 April 2011 Available online 27 April 2011 Abstract The analytical methods have nearly been replaced by the numerical methods due to their higher accuracy and accessibility of com- putation facilities. The semi-analytical Lagrange method of orbit propagation using f and g series is a competitive alternative to the numerical integration technique if the Lagrange coefficients are derived in a full gravitational field. In this paper, a generalization of the Lagrange method of orbit propagation is introduced. In other words, we introduce a complete form of the Lagrange coefficients in all force fields developed in the spherical harmonics for example full gravitational field of the Earth. The method is numerically com- pared with the numerical integration technique. In order to show the numerical performance of the method, it has been implemented for orbit propagation of a GPS-like MEO and CHAMP-like LEO satellites. Discrepancy at centimeter level for CHAMP-like and sub-mil- limeter accuracy for GPS-like satellites shows relatively high performance of the developed algorithm. Compared to integration method, the proposed Lagrange method is nearly faster by a factor two for small Nmax and four for large Nmax. Ó 2011 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Lagrange coefficients; f and g functions; Dynamic orbit; LEO satellite; GPS; CHAMP 1. Introduction Different numerical and analytical methods have been introduced for dynamic orbit propagation of the celestial bodies and artificial satellites. Dynamic orbit is the solution of the equation of motion of satellites or celestial bodies without using any observations (Seeber, 2003). Xu (2008) describes the analytic solutions of the equations of satellite motion perturbed by extraterrestrial and geopotential dis- turbances by means of discretization and approximated potential function. The method of Lagrange coefficients is a semi-analytical approach based on the position and velocity vectors in terms of the f and g series. In this method, we could be near into analytical solution by using highest-order series. The Lagrange method is based on expansion of the solution of Newton’s Law into Taylor ser- ies, and is thus a special kind of a numerical integration method using Taylor series (Beutler, 2005). The desired position and velocity vectors is propagated from initial position and velocity vectors with only one function evalu- ation without storing function values from previous steps and without extrapolation or interpolation. Like other Taylor series methods, the Lagrange method is combining all the advantages of classical single-step, e.g. Runge– Kutta methods with the additional freedom to choose the order in accord with runtime and accuracy requirements (Montenbruck, 1992). Then, it could be seen that Lagrange method has both advantages of the multi-step and single-step integrations. This method could represent the continues solution for equation of motion of satellite in a 0273-1177/$36.00 Ó 2011 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2011.04.021 ⇑ Corresponding author. Tel.: +98 21 88 78 62 13; fax: +98 21 88 78 62 12. E-mail addresses: sharifi@ut.ac.ir (M.A. Sharifi), seif.eng@gmail.com (M.R. Seif). 1 Tel.: +98 21 88 00 88 41; fax: +98 21 88 0088 37. www.elsevier.com/locate/asr Available online at www.sciencedirect.com Advances in Space Research 48 (2011) 904–913