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’sequation of an elliptical orbit for methods (Newton-Raphson, Danby, Halley andMikkola). 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 ebetween (0-0.6).The term that added to Danby’s method to obtain the solutionof Kepler’s equation is not influence too much on the value of E. The mostappropriate initial Gauss value was also determined to be (En=M), this initial valuegave a good result for (E) for these methods regardless the value of e to increasingthe accuracy of E. After that the orbital elements converting into state vectors withinone orbital period within time 50 second, the results demonstrated that all these fourmethods can be used in semi-circular orbit, but in case of elliptical orbit Danby’sand Halley’s method use only for e ≤ 0.7, Mikkola’s method for e ≤ 0.01 whileNewton-Raphson uses for e < 1, which considers more applicable than others to usein semi-circular and elliptical orbit. The results gave a good agreement as comparedwith the state vectors of Cartosat-2B satellite that available on Two Line Element(TLE).