Statistical Determination Of The Constants Of The Atmospheric Refraction Formula

In the present paper an efficient algorithm based on the least squares method was developed for the determination of the constants A&B of the atmospheric refraction formula. The principles of atmospheric refraction and the least squares method together with its error analysis were first developed and summarized together with its error analysis. The equations of condition for the problem were then established using N polar stars at their upper and lower culminations. Analytical formulae for the least squares solution of the equations of condition are given.  Analytical formulae of the errors estimate are also established  ,of these are: the  standard error of the fit , the  standard errors for the least squares solutions, the probable errors ,finally, the average squared distance between the exact solutions and the least squares solutions

Homotopy Continuation Method of Arbitrary Order of Convergence for Solving Differenced Hyperbolic Kepler's Equation

In this paper, an efficient iterative method of arbitrary integer order of >=2 will be established for the solution of differenced hyperbolic convergent Kepler's equation. The method is of dynamic nature in the sense that, on going from one iterative scheme to the subsequent one, only additional instruction is needed. Moreover, which is the most important, the method does not need any priori knowledge of the initial guess. Aproperty which avoids the critical situations between divergent to very slow convergent solutios, that may exist in other numberical methods which depend on initial guess. Computeational package for digital implementation of the method is given

Fourier Series Expansions of Powers of the Trigonometric Sine and Cosine Functions

In this paper, Fourier series expansions of powers of sine and cosine functions are established for any possible power real or complex or positive integer. Recurrence relations are established to facilities the computations of the coefficients of expansions formulae. Numerical applications for real and complex powers are also included , the accuracy of the computed values are at least of order . While the applications for positive integer powers are given as exact analytical expressions

Final state predictions for J2 gravity perturbed motion of the Earth’s artificial satellites using Bispherical coordinates

AbstractIn this paper, initial value problem for dynamical astronomy will be established using Bispherical coordinates. A computational algorithm is developed for the final state predictions for J2 gravity perturbed motion of the Earth’s artificial satellites. This algorithm is important in targeting, rendezvous maneuvers as well for scientific researches. The applications of the algorithm are illustrated by numerical examples of some test orbits of different eccentricities. The numerical results are extremely accurate and efficient

Universal symbolic expression for radial distance of conic motion

In the present paper, a universal symbolic expression for radial distance of conic motion in recursive power series form is developed. The importance of this analytical power series representation is that it is invariant under many operations because the result of addition, multiplication, exponentiation, integration, differentiation, etc. of a power series is also a power series. This is the fact that provides excellent flexibility in dealing with analytical, as well as computational developments of problems related to radial distance. For computational developments, a full recursive algorithm is developed for the series coefficients. An efficient method using the continued fraction theory is provided for series evolution, and two devices are proposed to secure the convergence when the time interval (t − t0) is large. In addition, the algorithm does not need the solution of Kepler’s equation and its variants for parabolic and hyperbolic orbits. Numerical applications of the algorithm are given for three orbits of different eccentricities; the results showed that it is accurate for any conic motion

Analytical formulations to the method of variation of parameters in terms of universal Y's functions

The method of variation of parameters still has a great interest and wide applications in mathematics, physics and astrodynamics. In this paper, universal functions (the Y's functions) based on Goodyear's time transformation formula were used to establish a variation of parameters method which is useful in slightly perturbed two-body initial value problem. Moreover due to its universality, the method avoids the switching among different conic orbits which are commonly occurring in space missions. The position and velocity vectors are written in terms of f and g series. The method is developed analytically and computationally. For the analytical developments, exact literal formulations for the differential system of variation of the epoch state vector are established. Symbolical series solution of the universal Kepler's equation was also established, and the literal analytical expressions of the coefficients of the series are listed in Horner form for efficient and stable evaluation. For computational developments of the method, an efficient algorithm was given using continued fraction theory. Finally, a short note on the method of solution was given just for the reader guidance

A mathematical model of star formation in the Galaxy

This paper is generally concerned with star formation in the Galaxy, especially blue stars. Blue stars are the most luminous, massive and the largest in radius. A simple mathematical model of the formation of the stars is established and put in computational algorithm. This algorithm enables us to know more about the formation of the star. Some real and artificial examples had been used to justify this model