# The course of the sun over an observer on the ground

At a glance

The sun is so far from the Earth that it appears as a sphere of small diameter; its apparent diameter is about 32' ± 0.5' of arc, or about half a degree. Therefore, the rays arriving at the top of the atmosphere can be considered to be parallel.

The solar zenithal angle is the angle formed by the direction of the sun and the local vertical. The azimuth of the sun, or solar azimuthal angle, is the angle formed by the projection of the direction of the sun on the horizontal plane and the north. The azimuth increases clockwise.

The direction of the sun perceived at the top of the atmosphere is very close to that perceived by an observer on the ground; the refraction of solar rays by the atmosphere induces a very slight variation that can be neglected. Consequently, the solar angles are nearly identical whether on the ground or at the top of the atmosphere, or more generally, at any altitude.

The rotation of the Earth on its axis determines the duration of day and night. To avoid confusion with the fact that the day actually lasts 24h, daytime, or day- length, is defined here as the period during which the sun is above the horizon, i.e., between sunrise and sunset. Night is the period when the sun is below the horizon. These definitions are taken in the astronomical sense, that is to say, without obstacle in the line of sight. Otherwise, as, for example, in a city or in an area with a relief, the effective daytime may be different from the astronomical one.

The notion of effective solar angles is a means of associating unique angles with each measurement that has been taken over a certain time interval.

At any time and in any place, the solar radiation received on the ground depends on two main elements: the terrestrial atmosphere and the radiation from the sun arriving at the top of the atmosphere, also called extraterrestrial radiation. The atmosphere depletes the solar radiation as the latter makes its way downward to the ground. Extraterrestrial radiation is therefore the driving quantity because radiation at ground level cannot exceed it, except in rare conditions. The extraterrestrial radiation received by a surface at any time and in any place is a function of the day in the year, that is to say, of the position of the Earth on its orbit and of the solar declination, of the time in the day, of the latitude, and of the inclination of the surface relative to the horizontal. The direction of the sun determines the angle of incidence of the solar rays onto a surface at the top of the atmosphere. The previous chapter has shown the importance of this angle in the transfer of energy by radiation between the incident radiative energy and the receiving plane. The greater the angle of incidence, the lower the energy received by the plane.

This chapter deals with the calculation of the direction of the sun and more generally with the course of the sun over an observer at ground. The relative geometry between the direction of the sun and an observer on the ground, in other words the apparent course of the sun in the sky above an observer, determines the angle of incidence. Therefore, the direction of the sun as seen by the observer must be known accurately. It is described by the solar zenithal angle, or equivalently the angular elevation above the horizon, and the solar azimuth. These angles are defined in this chapter.

Several examples are given. A series of simple but fairly accurate equations is provided to calculate the various elements discussed. They are those of the European Solar Radiation Atlas (ESRA). The calculation errors of the solar angles are less than 0.2° for the solar zenithal angle and 0.4° for the azimuth. The accuracy in time is of the order of a few minutes, and these equations are suitable if one is interested in radiation values on timescales of hour or larger. For better accuracy, more accurate equations should be preferred.^{[1]}

The mathematical expression of the course of the sun in the sky makes it possible to calculate the solar angles at sunrise and sunset. From these results, hours of sunrise and sunset can be calculated, and therefore the length of the day, called daytime or daylength, and that of night.

Since the radiation measurements are made over a certain period of time, during which the solar angles change, what are the solar angles to be assigned to each measurement? This chapter answers this problem by presenting the concept of effective solar angles which is a means of associating unique angles with each measurement that has been taken over a time interval greater than 1 min, w'hile the solar angles are only defined for very short moments, less than 1 min. Solutions for associating the solar angles with each measurement are discussed.

Given the enormous size of the sun, is there a single direction of the sun seen from Earth or more than one? In fact, the sun is so far from the Earth that despite its size, it appears as a very small sphere to an observer on the ground or at the top of the atmosphere. For many practical purposes, the sun can be thought of as a point, and there is only one direction of the sun. One consequence is that the solar rays arriving at the top of the atmosphere can be considered as parallel. Mathematically, it is a little different because the solid angle under which the sun appears is not zero and is equal to 0.68 10^{-4} sr. In other words, the apparent diameter of the sun is about 32' ± 0.5' of arc, or about half a degree (0.53°±0.008° of arc). The solar rays arriving at the same point at the top of the atmosphere from all points included in this solid angle are not perfectly parallel. Nevertheless, assuming that the direction of the sun is that of its center and that the solar rays are parallel, this results in very small errors on the solar radiation which are perfectly acceptable in practice.

- [1] Software libraries are available in C and Matlab at rredc.nrel.gov/solar/codesandalgorithms/spa(SPA algorithm) and http://www.oie.mines-paristech.fr/Valorisation/Outils/Solar-Geometry/ (SG2algorithm), and also in Python at pysolar.org (PySolar) or https://rhodesmill.org/skyfield/ (SkyField).