Declination of the Sun (simplified)

Description

The position of the Sun in the sky is a function of both time and the geographic coordinates of the observer on the surface of the Earth. As the Earth moves around the Sun during the course of the year, the Sun appears to move with respect to the fixed stars on the celestial sphere, along a path called the “ecliptic”. The Earth’s rotation about its axis causes the fixed stars to move in the sky in a way that depends on the observer’s geographic latitude. The time when a given fixed star crosses the observer’s meridian depends on the geographic longitude. To find the Sun’s position for a given observer at a given time, one may therefore proceed in three steps:
calculate the Sun’s position in the ecliptic coordinate system,
convert to the equatorial coordinate system, and
convert to the horizontal coordinate system, for the observer’s local time and position.
This calculation is useful in astronomy, navigation, surveying, meteorology, climatology, solar energy, and for designing sundials.

Declination of the Sun as seen from Earth

The Sun appears to move northward during the northern spring. Its declination reaches a maximum equal to the angle of the Earth’s axial tilt (23.44 degrees) on the June solstice, then decreases until the December solstice, when its value is the opposite of (-1 times) the axial tilt. This variation produces the seasons.
A graph of solar declination during a year looks like a sine wave with an amplitude of 23.44 degrees, but one lobe of the “sine wave” is several days longer than the other, among other differences.
Imagine that the Earth is spherical, in a circular orbit around the Sun, and that its axis is tilted 90 degrees, so that the axis itself is in the plane of the orbit (similar to Uranus). At one date in the year the Sun would be vertically overhead at the North Pole, so its declination would be +90 degrees. For the next few months, the sub-solar point would move toward the South Pole at constant speed, crossing the lines of latitude at a constant rate, so that the solar declination would decrease linearly with time. Eventually the Sun would be over the South Pole, with a declination of -90 degrees. Then it would start to move northward at a constant speed. Thus the graph of the Sun’s declination, as seen from this highly tilted Earth, would not resemble a sine wave — it would be a sawtooth, zig-zagging between plus and minus 90 degrees, with linear segments between the maxima and minima.
Now suppose that the axial tilt decreases. The absolute maximum and minimum values of the declination would decrease, to equal the axial tilt. Also, the shapes of the maxima and minima on the graph would become less acute (“pointy”), being curved to resemble the maxima and minima of a sine wave. However, even when the axial tilt equals that of the real Earth, the maxima and minima remain more acute than those of a sine wave.
The real Earth’s orbit is elliptical. The Earth moves more rapidly around the Sun near perihelion, in early January, than near aphelion, in early July. This makes processes like the variation of the solar declination happen faster in January than July. On the graph, this makes the minima more acute than the maxima. Also, since perihelion and aphelion do not happen on exactly the same dates as the solstices, the maxima and minima are slightly asymmetrical. The rates of change before and after are not quite equal.

The declination of the Sun, δ☉, is the angle between the rays of the Sun and the plane of the Earth’s equator. The Earth’s axial tilt (called the obliquity of the ecliptic by astronomers) is the angle between the Earth’s axis and a line perpendicular to the Earth’s orbit. The Earth’s axial tilt changes slowly over thousands of years but its current value of about ε = 23°26’ is nearly constant, so the change in solar declination during one year is nearly the same as during the next year.
At the solstices, the angle between the rays of the Sun and the plane of the Earth’s equator reaches its maximum value of 23°26’. Therefore δ☉ = +23°26’ at the northern summer solstice and δ☉ = −23°26’ at the southern summer solstice.
At the moment of each equinox, the center of the Sun appears to pass through the celestial equator, and δ☉ is 0°.

The declination can be calculated using the parameters of the Earth’s orbit to more accurately estimate EL, as shown here.

Related formulas

Variables

δSdeclination of the Sun as seen from Earth (dimensionless)
πpi
N day of the year beginning with N=0 at midnight Coordinated Universal Time as January 1 begins (i.e. the days part of the ordinal date -1). The number 10, in (N+10), is the approximate number of days after the December solstice to January 1 (dimensionless)