So, it has many applications in scientific researches, where
Furthermore, there are some considerable contributions to find the periodic solutions of perturbed two–body problem using some perturbation methods are developed in , the authors have presented an analytical study to find periodic solutions of the perturbed two–body problem using the classic perturbation theory, the Lindstedt–Poincaré technique, and the Krylov–Bogoliubov–Mitropolsky method, as well as, a numerical integration by Runge–Kutta algorithm is applied to compare the obtained results by the aforemention methods. Also, some work to find also periodic solutions by others powerful methods, such as multiple scales technique, the averaging theory and KB averaging method [10–12]
It is exceeding important to estimate the space bodies positions and velocities vectors at any time. The equations of motion for the
In this paper, a universal computational algorithm will be constructed to find the solution of two–body problem using
In this section, the practical form of the equations of motion will be presented. The gravitational parameter μ could be written as
The Earth at the centre of the system. The length is measured in terms of the Earth’s radius (gr). The mass is measured in terms of the Earth’s mass (gm). The time is measured in minutes (min). The value of k in this system equals 07436680 (gr)3/2/min. The modified time Τ is defined by the equation
The Earth at the centre of the system.
The length is measured in terms of the Earth’s radius (gr).
The mass is measured in terms of the Earth’s mass (gm).
The time is measured in minutes (min).
The value of k in this system equals 07436680 (gr)3/2/min.
The modified time Τ is defined by the equation
Using the modified time, the vector equation of motion becomes
Using power series (1) and (2), to compute for a body of mass The coordinates of positions vector ( To compute the distance
The coordinates of positions vector (
To compute the distance
Finally, we would like to mention that: It enough to find only ten coefficients from the functions Its simple reparation provide us with powerful method to find any required number from the coefficients for precise predictions of There is no neediness to find the Kepler’s equation solution and its variants for parabolic and hyperbolic orbits to evaluate the position vector The algorithm is considered a global procedure, where it can be applied for any of conic orbit. The algorithm can be applied whatever the length of interval time
Its simple reparation provide us with powerful method to find any required number from the coefficients for precise predictions of
There is no neediness to find the Kepler’s equation solution and its variants for parabolic and hyperbolic orbits to evaluate the position vector
The algorithm is considered a global procedure, where it can be applied for any of conic orbit.
The algorithm can be applied whatever the length of interval time
In this section, we will take as an example the heliocentric motion of the planet Mercury. We will calculate its position for 20 days. The numerical values are
Sun’s mass = 1MU, Mass of Mercury in terms of Sun’s mass = 0.000000166, κ = 0.01720209895, The initial time
Sun’s mass = 1MU,
Mass of Mercury in terms of Sun’s mass = 0.000000166,
κ = 0.01720209895,
The initial time
The out put of the application of the above algorithm is listed in Table 1
The components of position and velocity vectors
In Table 2, the comparison between the computed values from the above algorithm with the corresponding values of
The comparison between the computed values and the given in Astronomical Almanac
It can be seen from the residuals that the two–body motion aggress. Finally, we demonstrate that a universal computational algorithm is constructed by using
The comparison between the computed values and the given in Astronomical Almanac
The components of position and velocity vectors r and r˙
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