**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.

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°.

The declination of the Sun δ is given by:

$sin\delta =sin\lambda sin\phi $

Where:

- λ is the true ecliptic longitude - the true anomaly plus the longitude of perihelion.
- φ = 23.4393° = 0.409093 radians, the angle between the Earth's axis and the plane of the ecliptic.

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

- 0 degrees: Center of Sun's disk touches a mathematical horizon
- -0.25°: Sun's upper limb touches a mathematical horizon
- -0.583°: Center of Sun's disk touches the horizon; atmospheric refraction accounted for
- -0.833°: Sun's upper limb touches the horizon; atmospheric refraction accounted for
- -6 °: Civil twilight (one can no longer read outside without artificial illumination)
- -12°: Nautical twilight (navigation using a sea horizon no longer possible)
- -15°: Amateur astronomical twilight (the sky is dark enough for most astronomical observations)
- -18°: Astronomical twilight (the sky is completely dark)

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

$cos\omega =\frac{sin\alpha -sin\Phi sin\delta}{cos\Phi cos\delta}$ (1)

Where:

- α = -0.833° is the altitude of the centre of the solar disc.
- Φ is the latitude of the observer.

First calculate the mean solar noon for the observer J^{*} = M +
l_{w}/2π where l_{w} 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 J_{transit} = J^{*} - EOT.

Sunrise is J_{rise} = J_{transit} - ω * 24/2π. Sunset is
J_{set} = J_{transit} + 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.