The study of the relationship between the coordinates of the sun and the moon with the crescent visibility factors has not been previously treated in a detailed and accurate way in research and previous studies, despite its religious importance. Accordingly, this paper aims to study the relationship between the crescent visibility factors (age, lag time, elongation (ARCL), arc of vision or relative altitude (ARCV), relative azimuth (DAZ), and crescent width (W), with coordinates of the sun and the moon), and how it varies during the day of the crescent's observation. In this paper, Matlab programs were designed to calculate the ecliptic sun and moon coordinates (λ, β) and in the presence of all perturbation impacts (planets), then convert these coordinates to the equatorial (α, δ) and horizontal coordinates (A, a). The results were compared with other programs in this field and references such as the ephemeris, accurate times, and astronomical programs. The variation of the sun and moon coordinates (ecliptic, equatorial, and horizontal) was studied with time, and the relationship of those coordinates was explained with visibility crescent factors during the day of the crescent's observation. Finally, the relationship of those factors with each other was studied. The results indicated that there was a relationship between the sun and the moon coordinates with some factors of crescent visibility in critical and standard observations, where the critical observations of the moon (age, lag time, elongation) were increased when the sun and the moon coordinates approach from two equinox points (spring and autumn), while the other factors, such as the relative azimuth and relative altitude, were independent on the change of the coordinates through the year. Also, a relationship was found between the visibility factors with each other, this led to a direct relationship between increasing the elongation with the crescent width. The values of the relative altitude were also increased with the increase in the lag time. Lastly, there was a direct relationship between the increase in the relative azimuth and the relative altitude.
