Sunny Stuff

The following are a few equations that you may find interesting or useful. They come from the British Sundial Society Sundial Glossary. The definitions come from the glossary as well. Some of the equations are best fit approximations of the parameter being defined. There are more accurate methods of determining these but they are not shown here.

Hour Angle (h, HA): the angle corresponding to the sun's position around its daily (apparent) orbit. Measured westward from local noon, it increases at a rate of 15° per hour. Thus 3 pm (Local Apparent Time) is 45° and 9 am is -45°.

h = (T₂₄ - 12) x 15°

where T₂₄ is the time in 24-hour clock notation (hours after midnight) in decimal hours.

Declination (of the sun) (δ, DELTA, DEC): the angular distance of the Sun above or below the celestial equator. Its value follows an annual sine wave like curve varying between 0° at the equinoxes and ±23.4° (approx.) at the solstices. It has positive values when the Sun is above the celestial equator (summer in the Northern hemisphere) and negative when below.

The Fourier transform approximation (best fit equation) below yields a maximum error of 0.0006 radians (less than 3 arcminutes) or, if the final two terms are omitted, 0.0035 radians (12 arcminutes).

δ = 0.006918 - 0.399912 cos w + 0.070257 sin w - 0.006758 cos 2w + 0.000907 sin 2w - 0.002697 cos 3w + 0.001480 sin 3w

where δ is in radians and w is calculated from day number n_d ( ranging from 1 on 1 January to 365 on 31 December) by:

w = 2π n_d / 365

1 radian = 57° 17' 44" .80625 = 57.2957795131 degrees and 360° = 2π radians.

Azimuth (of the sun) (A, AZ): the angle of the sun, measured in the horizontal plane and from true south. Angles to the west are positive, those to the east, negative. Thus due west is 90°, north is ±180°, east -90°.

A = arctan { sin (h) / [ sin ø cos (h) - cos ø tan δ ] }

Altitude (of the sun) (a, ALT): the angular distance of the centre of the sun's disk above the observer's horizon (negative numbers indicate that the sun is below the horizon). It is measured along the principal plane to the sun's centre, and is the complement of the zenith distance. It is part of the horizontal co-ordinate system.

a = arcsin { sin ø sin δ + cos ø cos δ cos (h)}

Sunrise, Sunset: the fist (last) appearance of the sun above the horizon each day. This occurs when the sun's altitude reaches -0° 50'. Note that some astronomers define the rising of an object as an altitude of 0°. The difference is due to the combined effects of the sun's mean semi-diameter (16 arcmin) and atmospheric refraction (34 arcmin).

The time (hour angle) of sunrise/sunset is given by:

h_sr,ss = ± arccos (-tan ø tan δ)

The azimuth of the rising/setting sun is given by

A_sr,ss = ± arccos (-sin δ / cos ø)

Note that these times are for astronomical sunrise/sunset, i.e. when the centre of the sun is on the true horizon, neglecting atmospheric refraction.

Equation of Time (E, EoT): the time difference between Local Apparent Time (apparent solar time) and Mean Solar Time at the same location. Its value varies between extremes of about +14 minutes in February and -16 minutes in October. It arises because of the elliptical orbit of the earth, and the tilt of the earth's axis to the ecliptic. The preferred usage by diallists is:

Mean Solar Time = Apparent Solar Time + EoT

but this convention is by no means universal and the opposite sign is used in modern almanacs. Irrespective of the sign convention adopted, the sundial will appear slow compared to the mean time in February, and fast in October/November.

EoT varies continuously, but is usually tabulated for noon each day at a particular location.

A full calculation of the EoT for any time in any epoch is complex and the reader is referred to Meuss, an Astronomical Almanac, or the NASS Dialist's Companion computer program. For many practical purposes, the Fourier transform approximation (best fit equation) given below, which has a worst-case error of 0.0025 radians (35 seconds of time), will be sufficient.

E_a = -0.0000075 - 0.001868 cos w + 0.032077 sin w + 0.014615 cos 2w + 0.040849 sin 2w

where E_a is in radians at 12:00 UT and w is as defined for the declination above.

To convert to the EoT in seconds (of time), multiply E_a by 43,200/π.