On the elliptical orbit of the earth and position of the sun in the sky: an engineering approach

The position of the Sun as seen by an observer on the Earth's surface and the position and velocity vectors of the Earth revolving in an elliptical orbit around the Sun can be calculated using several computational approaches. These approaches include (but are not limited to) the use of an analytical approach; a numerical approach, and the use of a Solar Position Algorithm (PSA). In the analytical methodology, the Earth's momentum equation is transformed to eliminate its time dependence, and the equation is solved analytically. Whereas, using the numerical approach, the dimensionless momentum equation of the revolving Earth is written in the polar coordinate system (r, θ) and solved numerically. The solar position algorithm known as PSA (Plataforma Solar de Almeria, abbreviated from its Spanish origin: https://www.psa.es), is a numerical algorithm that uses several empirical relations to calculate the solar declination and the ecliptic longitude angles, etc. The algorithm uses Cartesian coordinate system to calculate the dimensionless coordinates of the pole star (Polaris) and its declination angle to calculate the position vector of an observer that rotates with the Earth. This coordinate system is referred to as a new Cartesian coordinate system whose origin is located at the center of the Earth. The solar elevation angle and azimuth angle are obtained by performing a set of rotations of this new Cartesian coordinate system. In this article, we have used basic physical principles (analytical approach) to obtain the main parameters of the Sun's trajectory and position, at certain time in the sky. The methodology presented here can easily be used by professionals and engineers working in the area of solar/alternate energy, as well as for the design of intelligent/green buildings/cities for a sustainable environment. (author)