In this paper, relation was developed for Hyades stars between a function of the right ascensions and the angular distances from the vertex. The precision criteria of this relation are very satisfactory and a correlation coefficient value of ≃ 1 was found which proves that the attributes are completely related linearly. The importance of this relation was illustrated through its usages as:

In this paper, an efficient iterative method of arbitrary integer order of convergence ≥ 2 has been established for solving the hyperbolic form of Kepler’s equation. The method is of a dynamic nature in the sense that, moving from one iterative scheme to the subsequent one, only additional instruction is needed. Most importantly, the method does not need any prior knowledge of the initial guess. A property which avoids the critical situations between divergent and very slow convergent solutions that may exist in other numerical methods which depend on initial guess. Computational Package for digital implementation of the method is given and is applied to many case studies.

In this paper, we consider a statistical method for distance determination of stellar groups. The method depends on the assumption that the members of the group scatter around a mean absolute magnitude in Gaussian distribution. The mean apparent magnitude of the members is then expressed by frequency function, so as to correct for observational incompleteness at the faint end. The problem reduces to the solution of a highly transcendental equation for a given magnitude parameter 𝛼. For the computational developments of the problem, continued fraction by the Top–Down algorithm was developed and applied for the evaluation of the error function erf(𝑧). The distance equation 𝛬(𝑦) = 0 was solved by an iterative method of second order of convergence using homotopy continuation technique. This technique does not need any prior knowledge of the initial guess, a property which avoids the critical situations between divergent and very slow convergent solutions, that may exist in the applications of other iterative methods depending on initial guess.

Finally, we apply the method for the nearby main sequence late type stars assuming that the stars of each group of the same spectral type scatter around a mean absolute magnitude in a Gaussian distribution. The accuracies of the numerical results are satisfactory, in that, the percentage errors between 𝑟 and the mean values are respectively: (2.4%, 1.6%, 0.72%, 0.66%, 3.5%, 2.4%, 2%, 2.5%, 0.9%) for the stars of spectral types: (F5V, F6V, F7V, F8V, F9V, G0V, G2V, G5V, G8V).

In this paper, four parameters analytical solution will be established for the stellar density function in globular clusters. The solution could be used for any arbitrary order of outward decrease of the cluster’s density.

In this paper, an efficient algorithm is established for computing the maximum (minimum) angular separation ρ_max(ρ_min), the corresponding apparent position angles (𝜃|_ρmax , 𝜃|_ρmin) and the individual masses of visual binary systems. The algorithm uses Reed’s formulae (1984) for the masses, and a technique of one-dimensional unconstrained minimization, together with the solution of Kepler’s equation for (ρ_max, 𝜃 |_ρmax) and (ρ_min, 𝜃 |_ρmin). Iterative schemes of quadratic coverage up to any positive integer order are developed for the solution of Kepler’s equation. A sample of 110 systems is selected from the Sixth Catalog of Orbits (Hartkopf et al. 2001). Numerical studies are included and some important results are as follows: