Key symbols: position on the celestial sphere

The two coordinate systems

The following article recalls basic calculations to determinate the astral object position on the celestial sphere. These elements allow particularly to evaluate the solstitial rectangle dimensions, a rectangle joining sunrise and sunset positions at solstices and playing a key role within traditions facing East such as Celtic, Biblical, Roman and Christian ones.

According to the description of the celestial sphere, the observer position on earth is determined by the latitude φ (phi). He may identify the position of stars and other astral objects on the celestial sphere by two coordinate systems, related either to the horizon or to the equator:

Horizontal coordinates

Altitude h: Angular distance above the horizon:

0 ≤ h ≤ 90°

Azimuth a: Angular distance measured along the horizon, eastwards from the south point:

0 ≤ a < 360°

Equatorial coordinates

Declination δ (delta): Angular distance measured along the star meridian, north or south the equator:

0 ≤ δ ≤ 90°, north the equator;

- 90° ≤ δ < 0, south the equator.

Hour angle τ (tau): Angular distance measured along the equator, from the observer to the star meridian:

0 ≤ τ < 360°

The change from one system to the other

The conversion of horizontal coordinates into equatorial coordinates and conversely can be derived from the properties of the spherical triangle ZNS delimited by the Zenith, the North Pole and the Star.

The angles and sides of that spherical triangle, measured respectively on the surface and from the centre of the celestial sphere, are the following:

ang S= ?; ZN = 90°-φ

ang Z= 180°-a; NS = 90°-δ

ang N = τ; ZS = 90°-h

From spherical geometry, we get the three following main formulas:

sin(ang S)/sin(NS) = sin(ang N)/sin(ZS)

sin(180°-a)/sin(90°-δ)=sin(τ)/sin(90°-h)

sin(a)/cos(δ) = sin(τ)/cos(h)

cos(ZS) = cos(ZN)×cos(NS)+sin(ZN)×sin(NS)×cos(ang N)

cos(90°-h) = cos(90°φ)×cos(90°-δ)+sin(90°-φ)×sin(90°-δ)×cos(τ)

sin(h) = sin(φ)×sin(δ)+cos(φ)×cos(δ)×cos(τ)

cos(NS) = cos(ZN)×cos(ZS)+sin(ZN)×sin(ZS)×cos(ang Z)

cos(90°- δ) = cos(90°-φ)×cos(90°-h)+sin(90°-φ)×sin(90°-h)×cos(180°-a)

sin(δ) = sin(φ)×sin(h)-cos(φ)×cos(h)×cos(a)

These formulas allow us to express a and h in terms of δ and τ and conversely. They will be useful to determinate the apparent sun position at noon as well as at rising and setting time.

Solstice sunrise and sunset azimuth

At sunrise or sunset, h = 0, and the last formula has the simple form:

sin(δ) = - cos(φ)×cos(a)

cos(a) = - sin(δ)/cos(φ)

Considering particularly the solstice and equinox sun positions, we get the following results:

Solstice sunrise or sunset:

- At Summer solstice,

δ = 23.5° and cos(a) = - sin(23.5°)/cos(φ)

- At Winter solstice,

δ = - 23.5° and cos(a) = sin(23.5°)/cos(φ)

Equinox sunrise and sunset:

- At Spring equinox,

δ = 0 and a = 90°

- At Autumn equinox,.

δ = 0 and a = 270°

The above picture depicts the “solstitial rectangle” for Jerusalem located at latitude φ = 31.8°.

The “solstitial rectangle” is a square if a = 45° and cos(φ) = sin(23.5°)/cos(45°), that means at the latitude φ = 55.7° corresponding, for instance, to the Northern tip of Ireland.

Noon sun altitude

At noon sun, a = 0 and τ = 0, and the second formula gives:

sin(h) = sin(φ)×sin(δ)+cos(φ)×cos(δ) = cos(φ-δ)

h = 90°- φ + δ

Simple result which can be set up directly.

Considering particularly the solstice and equinox sun positions, we get:

Solstice noon sun:

- At Summer solstice:

δ = 23.5° and h = 113.5° - φ

- At Winter solstice:

δ = - 23.5° and h = 66.5° - φ

Equinox noon sun:

δ = 0 and h = 90° - φ

The opposite picture gives the altitude results for Jerusalem located at latitude φ = 31.8°.

The sun does not rise at all at winter solstice (h = 0) when the latitude equals (or exceeds) 66.5° corresponding to the Arctic Circle and, for instance, the Northern tip of Iceland.