The

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 [9], 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

Using the modified time, the vector equation of motion becomes

Power series

Representations of

Using power series (1) and (2), to compute for a body of mass

The coordinates of positions vector (_{0}, _{0}, _{0}) and also the velocities components vector _{0}, _{0}, _{0}) and _{0}.

To compute the distance

_{0}, (_{0}, _{0}, _{0}),

End

Finally, we would like to mention that: It enough to find only ten coefficients from the functions _{j}_{j}_{0}) must be small. But in the case of the interval of time is large, we will follow the strategy of repeating decrements of time interval many times over, thus the desired time interval must be very small in this procedures. At end, it should remark that the application of the aforementioned algorithm of _{0} and _{0} at the initial time _{0} is effective for some factors), such as the algorithm are satisfied the following properties and advantages:

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 _{0} = 3/8/1985,

_{0} = +0.169342, −0.355991, −0.207717 AU,

The out put of the application of the above algorithm is listed in Table 1

The components of position and velocity vectors

t | x | y | z | |||
---|---|---|---|---|---|---|

3/8/1985 | 0.169342 | –0.355991 | 0.207717 | 1.183760 | 0.669777 | 0.234931 |

5/8/1985 | 0.208826 | –0.330584 | –0.198243 | 1.108300 | 0.806612 | 0.315848 |

7/8/1985 | 0.245372 | –0.300538 | –0.185986 | 1.012790 | 0.939173 | 0.396564 |

9/8/1985 | 0.278269 | –0.266032 | –0.170969 | 0.895868 | 1.065510 | 0.476178 |

11/8/1985 | 0.306754 | –0.227321 | –0.153247 | 0.756154 | 1.183160 | 0.553512 |

13/8/1985 | 0.330022 | –0.184756 | –0.132926 | 0.592352 | 1.288970 | 0.627024 |

15/8/1985 | 0.347225 | –0.138810 | –0.110169 | 0.403457 | 1.379000 | 0.694714 |

17/8/1985 | 0.357491 | –0.090106 | –0.085220 | 0.189079 | 1.448410 | 0.754027 |

19/8/1985 | 0.359950 | –0.039451 | –0.058419 | –0.050101 | 1.491330 | 0.801771 |

21/8/1985 | 0.353785 | 0.0121299 | –0.0302278 | –0.311690 | 1.501050 | 0.834100 |

23/8/1985 | 0.338301 | 0.0633692 | –0.0012525 | –0.590801 | 1.470260 | 0.846615 |

25/8/1985 | 0.313025 | 0.1127500 | 0.0277459 | –0.879227 | 1.391890 | 0.834681 |

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

Date | Computed | Reference | Residuals |
---|---|---|---|

13/8/1985 | 0.4008980 | 0.4008978 | –0.0000002 |

23/8/1985 | 0.3441870 | 0.3441872 | +0.0000002 |

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

Date | Computed | Reference | Residuals |
---|---|---|---|

13/8/1985 | 0.4008980 | 0.4008978 | –0.0000002 |

23/8/1985 | 0.3441870 | 0.3441872 | +0.0000002 |

#### The components of position and velocity vectors r and r˙

t | x | y | z | |||
---|---|---|---|---|---|---|

3/8/1985 | 0.169342 | –0.355991 | 0.207717 | 1.183760 | 0.669777 | 0.234931 |

5/8/1985 | 0.208826 | –0.330584 | –0.198243 | 1.108300 | 0.806612 | 0.315848 |

7/8/1985 | 0.245372 | –0.300538 | –0.185986 | 1.012790 | 0.939173 | 0.396564 |

9/8/1985 | 0.278269 | –0.266032 | –0.170969 | 0.895868 | 1.065510 | 0.476178 |

11/8/1985 | 0.306754 | –0.227321 | –0.153247 | 0.756154 | 1.183160 | 0.553512 |

13/8/1985 | 0.330022 | –0.184756 | –0.132926 | 0.592352 | 1.288970 | 0.627024 |

15/8/1985 | 0.347225 | –0.138810 | –0.110169 | 0.403457 | 1.379000 | 0.694714 |

17/8/1985 | 0.357491 | –0.090106 | –0.085220 | 0.189079 | 1.448410 | 0.754027 |

19/8/1985 | 0.359950 | –0.039451 | –0.058419 | –0.050101 | 1.491330 | 0.801771 |

21/8/1985 | 0.353785 | 0.0121299 | –0.0302278 | –0.311690 | 1.501050 | 0.834100 |

23/8/1985 | 0.338301 | 0.0633692 | –0.0012525 | –0.590801 | 1.470260 | 0.846615 |

25/8/1985 | 0.313025 | 0.1127500 | 0.0277459 | –0.879227 | 1.391890 | 0.834681 |

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