Calculation of Solar Trajectory in the Sky and Solar Analemma as Observed from the Earth

Abstract

In a previous companion paper “On the elliptical orbit of the Earth and the position of the Sun in the sky: an engineering approach,” published in The NUCLEUS, we presented various computational methodologies for the position/trajectory of the Sun in the sky of an observer at Earth [1]. In this paper, the methodology for calculation of solar analemma (as observed from the earth surface) has been presented, along with an elaboration of the “Equation of Time,” as called in literature [2,3]. The computational methodologies presented in the earlier paper included: 1) an analytical approach; 2) a numerical algorithm; and 3) a Solar Position Algorithm commonly abbreviated as PSA from the Spanish name of its developer Plataforma Solar de Almería [4]. In the numerical approach, Earth’s momentum equation written in a polar coordinate system (r, θ) was numerically solved. It was also demonstrated that if the Earth’s momentum equation was transformed to eliminate the time dependence, it could be solved analytically. In this paper, a Cartesian coordinate system is used to calculate the coordinates of the pole star (Polaris) and its declination angle. The position vector of an observer that rotates with the Earth is calculated using a new Cartesian coordinate system, whose origin is located at the center of the Earth. The solar elevation angle and the solar azimuth angle are obtained by performing a set of rotations of this new coordinate system. Towards the end, the Equation of Time (EOT) is explained and used for calculating the solar analemma.