Re-Evaluation Solution Methods for Kepler's Equation of an Elliptical Orbit

Abstract

An evaluation was achieved by designing a matlab program to solve Kepler’s equation of an elliptical orbit for methods (Newton-Raphson, Danby, Halley and Mikkola). This involves calculating the Eccentric anomaly (E) from mean anomaly (M=0°-360°) for each step and for different values of eccentricities (e=0.1, 0.3, 0.5, 0.7 and 0.9). The results of E were demonstrated that Newton’s- Raphson Danby’s, Halley’s can be used for e between (0-1). Mikkola’s method can be used for e between (0-0.6).The term that added to Danby’s method to obtain the solution of Kepler’s equation is not influence too much on the value of E. The most appropriate initial Gauss value was also determined to be (En=M), this initial value gave a good result for (E) for these methods regardless the value of e to increasing the accuracy of E. After that the orbital elements converting into state vectors within one orbital period within time 50 second, the results demonstrated that all these four methods can be used in semi-circular orbit, but in case of elliptical orbit Danby’s and Halley’s method use only for e ≤ 0.7, Mikkola’s method for e ≤ 0.01 while Newton-Raphson uses for e < 1, which considers more applicable than others to use in semi-circular and elliptical orbit. The results gave a good agreement as compared with the state vectors of Cartosat-2B satellite that available on Two Line Element (TLE).