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In the present paper, an efficient algorithm based on the continued fractions theory was established for the universal Y’s functions of space dynamics. The algorithm is valid for any conic motion (elliptic, parabolic or hyperbolic).

Today, one of the well-known facts of space dynamics is the desperate needs of the universal formulations of orbital motion. This is because, in complete interplanetary transfer, all types of the two body motion (elliptic, parabolic, or hyperbolic) appear, moreover, the given type of an orbit is occasionally changed by perturbing forces acting during finite interval of time. Thus far, we have been obliged to use different functional representations for motion depending upon the energy state (elliptic, parabolic, or hyperbolic) and a simulation code must then contain branching to handle a switch from one state to another. In cases where this switching is not smooth, branching can occur many times during a single integration time-step causing some numerical “chatter”. Consequently, through the use of the universal formulations, orbit predictions will be free of the troubles, since a single functional representation suffices to describe all possible states.

Recently Sharaf and Saad [

The universal Y’s functions are given by:

where

What concerns us among the properties of the Y’s functions given in Paper I are:

In fact, continued fraction expansions are generally far more efficient tools for evaluating the classical functions

than the more familiar infinite power series. Their convergence is typically faster and more extensive than the series.

Top-Down Continued Fraction EvaluationThere are several methods available for the evaluation of continued fraction. Traditionally, either the fraction was computed from the bottom up, or the numerator and denominator of the nth convergent were accumulated separately with three-term recurrence formulae. The drawback of the first method is obviously, having to decide far down the fraction to being in order to ensure convergence. The drawback to the second method is that the numerator and denominator rapidly overflow numerically even though their ratio tends to a well-defined limit. Thus, it is clear that an algorithm that works from top down while avoiding numerical difficulties would be ideal from a programming standpoint.

Gautschi [

then initialize the following parameters

and iterate

In the limit, the c sequence converges to the value of the continued fraction. Continued fraction method was used in many problems in astrophysics [

In the following, we shall consider the evaluations of the four functions

From the expressions of tanx and

where

Input:

Output

Computational sequence

1-Compute a’s from

2-Compute u from the continued fraction

by using Gautschi’s algorithm of Subsection 3.1

3-

4-

5-

6-

7-

8-The algorithm is completed.

The applications of the above algorithm for the numerical values of

No | a | c_{ } | Y_{0 } | Y_{1 } | Y_{2 } | Y_{3 } |
---|---|---|---|---|---|---|

1 | −3 | −3.14159 | 115.384 | −66.6147 | 38.1282 | −21.1577 |

2 | −2 | −2.14159 | 10.359 | −7.29074 | 4.67952 | −2.57457 |

3 | −1 | −1.14159 | 1.72553 | −1.40622 | 0.725531 | −0.264628 |

4 | 0 | −0.141593 | 1.00000 | −0.141593 | 0.0100242 | −0.0004731 |

5 | 1 | 0.858407 | 0.653644 | 0.756802 | 0.346356 | 0.1016050 |

6 | 2 | 1.85841 | −0.871076 | 0.347294 | 0.935538 | 0.7555560 |

7 | 3 | 2.85841 | 0.236263 | −0.561005 | 0.254579 | 1.198000 |

The more accurate calculation of

In concluding the present paper, an efficient algorithm based on the continued fractions theory was established for the recent universal Y’s functions of space dynamics. The algorithm is valid for any conic motion (elliptic, parabolic or hyperbolic).