The astral body 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:
Altitude h: Angular distance above the horizon:
0 ≤ h ≤ 90°
Angular distance measured along the horizon, eastwards from the south point:
0 ≤ a < 360°
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 Z)×sin(ZS) =
- sin(a)×cos(h) = sin(τ)×cos(δ)
- 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 three 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 rising and setting time as well as at its zenith. Let us recall that the ecliptic plan, run by the Sun during its apparent move around the Earth, is titled by an angle of 23.5° with respect to the equator plan. It follows that the Sun declination varies from -23.5° to 23.5° (for more details, see the description of the celestial sphere).
Solstice Sunrise and Sunset azimuth
At Sunrise or Sunset, h = 0, and the last formula has the
- 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:
- - 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)
- - 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.
High Sun altitude
The Sun culminates in the sky when it goes through the meridian of the observation place (at noon, solar time), i.e. when a = τ = 0 or 180°.
Let us consider the second formula with τ = 0:
- sin(h) = sin(φ)×sin(δ) + cos(φ)×cos(δ)
- sin(h) = cos(φ-δ) = sin[90°±(φ - δ)]
- h = 90°±(φ - δ)
It follows that the Sun will only reach the zenith (h = 90°) if the latitude φ of the observation place equals the declination δ of the astral body. As mentioned before, the Sun declination is comprised between -23.5° et 23.5°. That means that the zenith will only be reached within the latitude band limited by the Cancer (φ = 23.5°) and Capricorn (φ = -23,5°) tropics.
The Sun reaches the zenith twice a year at most: a first time between the winter and summer solstice and a second one between the summer and winter solstice. This is notably the case of Mesoamerica regions, formerly inhabited by the Mayas. These two events coincide at the latitudes of the Cancer and Capricorn tropics. Then, the Sun reaches the zenith at the summer solstice.
When h < 90°, we obtain the following formula:
- h = 90°-φ+δ, if φ > δ
- h = 90°+φ-δ, if φ < δ
The formula gives, in particular, the Sun culmination for some remarkable declinations:
- - For δ = 23.5°
- h = 113.5°-φ, if φ > δ
- h = 66.5°+φ, if φ < δ
- - For δ = -23.5°
- h = 66.5°-φ, if φ > δ
- h = 113.5°+φ, if φ < δ
- - For δ = 0
- h = 90°-φ, if φ > δ
- h = 90°+φ, if φ < δ
The above picture gives the altitude results for Jerusalem located at latitude φ = 31.8°.
Let us notice that the Sun does not rise at all at winter solstice (h = 0) when the latitude equals (or exceeds) 66.5°, which corresponds to the Arctic Circle and, for instance, the Northern tip of Iceland.
A similar reasoning with τ = 180° would lead to comparable results.