Equinoxes, Solstices, Apses, Sunrise and Sunset

by Phill Edwards

The Earth's axial tilt and the direction of the Vernal Equinox in this page are both subject to precession. The computation of the effects of precession are described on the Coordinates page.

Equinoxes, Solstices and Apses

The Earth's ecliptic longitude λ is the angle between the current position of the planet and the reference direction which is the March Equinox. Hence the date and time of the March Equinox is when λ = 0°. This can be determined from the VSOP theory by iteration to find the time of the zero. Consequently, the date and time of the October Equinox is when λ = 180°.

The June Solstice is when λ = 90°, the December Solstice is when λ = 270°.

The longitude of periapsis (perihelion) Π is the angle between the periapsis and the March Equinox, This is also determined by VSOP theory. Hence the date and time of periapsis is when λ = Π. Consequently, the date and time of the apapsis (aphelion) is when λ - Π = 180°.

Declination of the Sun

The declination of the Sun δ is given by:

sin δ = sin λ sin φ

Where:

Altitude of Solar Disc

The altitude of the solar disc is needed to define what is meant by sunrise and sunset. The values of α are:

The value of -0.833° is usually used for sunrise and sunset calculations.

Hour angle

The hour angle is used to calculate the offset for sunrise and sunset:

cos ω = sin α sin Φ sin δ cos Φ cos δ (1)

Where:

Sunrise and sunset

First calculate the mean solar noon for the observer J* = M + lw/2π where lw is the longitude West of the observer.

The solar noon, or solar transit is calculated by subtracting the Equation of Time value for the date Jtransit = J* - EOT.

Sunrise is Jrise = Jtransit - ω * 24/2π. Sunset is Jset = Jtransit + ω * 24/2π.

The calculations are quite good when the declination of the Sun is calculated for the mean solar noon of the day. The accuracy can be improved by interation. If the hour angle is recalculated using the Sun's declination at the calculated time of sunrise or sunset. One or two iterations give the same result to within a second.