An Improved Algorithm For The Solution of Kepler‘s Equation For An Elliptical Orbit

In this paper, a root finding method due to iterative method is used first to thesolution of Kepler's equation for an elliptical orbit. Then the extrapolationtechnique in the form of Aitken D2 - acceleration is applied to improve theconvergence of the iterative method.In addition, by making use a new improvement to Aitken's method enables oneto obtain efficiently the numerical solution of the Kepler's equation. The speed ofthe proposed algorithms is compared using different values of eccentricity( e ) inthe range eÎ( 0,1 ) and for given mean anomaly (M).

section_content: The iterative on the solution of the Kepler problem is extensive. There is a large number of researchers were devoted to its solution in numerical form [1, 2, 45, 7, 10]. 

Nowadays, papers on Kepler's equation and its solution are still of scientific interest (see e. g. [3, 6, 8] ) and it shall be in the future. 

In this paper, we will discuss the solution of Kepler's equation in the conventional form,

and for elliptic orbits. 

Since the elliptical orbit is periodic, then solution of eq. ( 1 ) is considered with π 2 0 ≤ ≤ M for which the solution will be similarly bounded, ) ( e is taken to be ) 1, 0 (.
section_name: Introduction

section_content: It is well known that Kepler's equation can be solved by means of an iterative method defined, in a natural way, from the equation itself. This method yields to the unique solution if the eccentricity is in the range of the elliptic orbits. In [11], Newton method was used to solve Kepler's equation while Thomas [9] used Gauss method to solve Kepler's equation which depend on Picard type iteration. 

A root -finding method due to Laguerre [4] was applied to the solution of the kepler problem. In addition, Danby [13] and Serafin [11] described numerical solution for solving Kepler's equation. 

In this paper, Aitken's acceleration technique with an improvement is applied to find the solution of kepler equation.
section_name: Solutions of Kepler's Equation

section_content: The solution proceeds with the choice of a successive approximation algorithm and an appropriate starting value for the iteration, which we will call o E. 

The iterative method uses the following scheme: writing Kepler 's equation eq. ( 1 ) in the form [2] :

Then precede further the procedure as indicated below:

Remark: Aitken -∆ 2 Acceleration [12] will be applied in this paper to find an approximate solution for Kepler's equation as well as an improvement to Aitken -∆ 2 Acceleration is considered here which allowed some improvement at each step.
section_name: The Iterative Method

section_name: Aitken

section_content: Aitken's method is a way of solving KE quickly and accurately enough that astronomers might make repeated calculations for determining orbits. Given

as the solution of KE. In general, when an iteration ) ( 5 ) Equivalently eq. ( 5 ) for acceleration the convergence can be written as

or in terms of the forward difference operator ∆, as and hence the name -∆ 2 acceleration.
section_name: -∆ 2 Acceleration

section_content: In this section, the improved Aitken's method is proposed. It will show the difference in the number of iterations required to converge to a given accuracy. 

The improved Aitken's method uses the following scheme:

Given e and M, taking o E from eq. ( 3 ), then calculating

where o E is determining using eq. ( 3 )

E are determining from eq. ( 4 ). 

Then define the improve value E

In general, the improved sequence {E } can be defined by:
section_name: The Improved Aitken's Method

section_content: Newton's method for iterative solution of equations is a standard technique. 

The procedure of Newton's method provides correct values of the true anomaly, but can take a large number of iterations, especially for large values of eccentricity. 

The method becomes unstable for certain values of mean anomaly and eccentricity. 

For certain kinds of function, the method either does not converge, or generates very large values before converging. 

The efficient the algorithms described in section two in solving Kepler's equation.
section_name: Results and Conclusions

section_content: . Wait, R., "The Numerical Solution of Algebric Equations", John Wiley & Sons, Ltd., 1979. [13]"Celestial Computing with matlab by Sciences software", 2008.