### Where is the Sun?

Ecliptic plane [Wikimedia Commons].

The position of the Sun in the sky depends on where we are and on the time of day. Due to the Earth’s rotation, the Sun appears to cross the celestial sphere each day along a path called the ecliptic. The observer’s position on Earth is given by the geographic latitude and longitude. The path of the Sun depends on the latitude and the date, while the time when the Sun crosses the local meridian is determined by the longitude.

To find the Sun’s position at a given location and time, we must:

1. Calculate the Sun’s position in the ecliptic coordinate system
2. Convert the coordinates to the equatorial system
3. Convert to the observer’s horizontal coordinate system.

This process gives us the position of the Sun in terms of solar elevation and solar azimuth angle. The elevation is the angle of a line to the Sun above the horizontal plane. The azimuth is (essentially) the compass bearing of the Sun from the point of the observer. Plotting these angles on a polar diagram, we obtain a plot of the Sun as it moves across the sky.

An outline of several coordinate systems is given in the appendix below.

Fig. 1. Horizontal coordinates: the azimuth is the compass bearing and the elevation is the angle above the horizon [Image That’s Maths].

The path of the Sun can be calculated with great accuracy, but the computations are quite complicated. Moreover, because of the nature of the Earth’s orbit, solar time is not the same as (local) mean time. The difference is given by a formula called the Equation of Time.

We will denote the co-latitude (the angle relative to the North Pole axis) by ${\theta}$ and the geographic latitude by ${\varphi = 90^\circ - \theta}$. In Fig. 1 we show the path of the Sun as seen from a point at ${\varphi = 53.5^\circ}$N. The three curves are for the Summer solstice, the Equinoxes and the Winter solstice. The white area in the figure illustrates the region of the celestial dome where the Sun may be seen. It also gives the maximum (noonday) elevation and the azimuth at sunrise and sunset, when the elevation is zero.

Between the Winter solstice and Summer solstice, the declination of the sun, ${\delta}$, increases from ${-23.5^\circ}$ to ${+23.5^\circ}$. The solar elevation at noon, ${\theta+\delta = 90^\circ-\varphi+\delta}$, increases from ${13^\circ}$ to ${60^\circ}$. Between mid-summer and mid-winter, the midday elevation of the Sun decreases again, from ${60^\circ}$ to ${13^\circ}$.

The time may be estimated by assuming that solar time and local mean time agree. We know that this is not precisely true, but the error is not large. Thus, to a first approximation, the time in hours is given by ${h = 24 (\alpha/360^\circ)}$, where ${\alpha}$ is the azimuth. This implies that the Sun is due East at 6 am, due South at noon and due West at 6 pm. Then we can estimate the times of sunrise and sunset from Fig. 1. So, the earliest sunrise is at 3:20 and the latest at about 9:00 solar time.

Inverting the Problem

Suppose we are given the azimuth and elevation, ${(\alpha, \eta)}$. Can we deduce the time and date? Not unambiguously! For every point within the white region of the Fig. 1, there are two times each year when the azimuth and elevation have values corresponding to that point. For example, the values ${(\alpha,\eta) = (120^\circ, 30^\circ)}$ occur at 8 am on two dates, around May 10 and August 2, which are both separated by six weeks from the Summer solstice. It is not possible to determine which of these dates is relevant unless other evidence is available.

Appendix: Coordinate Systems

Several coordinate systems are useful in determining the position of the Sun. We do not give details here, but provide a brief sketch of each system. There are Wikipedia pages describing each system, with the equations required to transform from one to another.

Fig. 2. Horizontal coordinates, Azimuth and Altitude (or Elevation). Image Wikimedia Commons.

The Horizontal Coordinate System

The horizontal coordinate system uses the observer’s local horizon as the fundamental plane. Positions are given by two angles, azimuth ${\alpha}$ and altitude (or elevation) ${\eta}$. This topocentric system is used to specify positions relative to the location of the observer.

The Geographical Coordinate System

This is the usual latitude/longitude system, ${(\varphi,\lambda)}$, used to specify positions on the surface of the Earth. The origin is at the centre of Earth and the system rotates with the Earth. The fundamental plane is the plane of Earth’s equator, and the primary direction is through the equator and the Greenwich meridian.

The Equatorial Coordinate System

In this system, the origin is at the centre of Earth and the fundamental plane is the plane of Earth’s equator. The system does not rotate with the Earth, but remains fixed relative to the stars, with the reference direction pointing towards the vernal equinox. Positions of celestial objects are given by two angles, the right ascension ${a}$ and the declination ${\delta}$.

The Ecliptic Coordinate System

The ecliptic plane is the plane of the Earth’s orbit around the Sun. Most planets and smaller bodies have orbits lying close to this plane, so it is convenient to use it as the reference plane. This system is used for representing the positions and orbits of objects in the Solar System. The origin may be the taken either at the centre of Earth or of the Sun. The primary direction is towards the “first point of Aries”, corresponding to the vernal equinox.

Sources

Several Wikipedia articles were useful for the preparation of this note:

• Position of the Sun.
• Horizontal coordinate system
• Solar zenith angle
• Equatorial coordinate system
• Solar azimuth angle Declination