The position of the Sun or the direction of the Sun in the sky is a function of both the time and the geographic location of observation on Earth's surface. As Earth orbits the Sun over the course of a year, the Sun appears to move with respect to the fixed stars on the celestial sphere, along a circular path called the ecliptic.
Earth's rotation about its axis causes diurnal motion, so that the Sun appears to move across the sky in a Sun path that depends on the observer's latitude. The time when the Sun transits the observer's meridian depends on the longitude.
To find the Sun's position for a given geographic location at a given local time, one may proceed in three steps: calculate the Sun's position in the ecliptic coordinate system, convert to the equatorial coordinate system, and convert to the horizontal coordinate system. The resulting Sun angles are the solar zenith angle and solar azimuth angle, which can be used to depict the Sun path.
This calculation is useful in astronomy, navigation, surveying, meteorology, climatology, solar energy, and sundial design.
Approximate position
Ecliptic coordinates These equations, from the Astronomical Almanac,Much the same set of equations, covering the years 1800 to 2200, can be found at , at the U.S. Naval Observatory website . Graphs of the error of these equations, compared to an accurate ephemeris, can also be viewed. can be used to calculate the apparent coordinates of the Sun, mean equinox and ecliptic of date, to a precision of about 0°.01 (36″), for dates between 1950 and 2050. Similar equations are coded into a Fortran 90 routine in Ref.Zhang, T., Stackhouse, P.W., Macpherson, B., and Mikovitz, J.C., 2021. A solar azimuth formula that renders circumstantial treatment unnecessary without compromising mathematical rigor: Mathematical setup, application and extension of a formula based on the subsolar point and atan2 function. Renewable Energy, 172, 1333-1340. DOI: https://doi.org/10.1016/j.renene.2021.03.047 and are used to calculate the solar zenith angle and solar azimuth angle as observed from the surface of the Earth.
Start by calculating n, the number of days (positive or negative, including fractional days) since Greenwich noon, Terrestrial Time, on 1 January 2000 (J2000.0). If the Julian date for the desired time is known, then
The mean longitude of the Sun, corrected for the aberration of light, is:
The mean anomaly of the Sun (actually, of the Earth in its orbit around the Sun, but it is convenient to pretend the Sun orbits the Earth), is:
Put and in the range 0° to 360° by adding or subtracting multiples of 360° as neededwhich is to say, and are really to be evaluated (mod 360).
Finally, the ecliptic longitude of the Sun is:
The ecliptic latitude of the Sun is nearly:
, as the ecliptic latitude of the Sun never exceeds 0.00033° (a little over 1″),Meeus (1991), p. 152 and the distance of the Sun from the Earth, in astronomical units, is:
.
Obliquity of the ecliptic Where the obliquity of the ecliptic is not obtained elsewhere, it can be approximated:
Equatorial coordinates , and form a complete position of the Sun in the ecliptic coordinate system. This can be converted to the equatorial coordinate system by calculating the obliquity of the ecliptic, , and continuing:
Right ascension,
, where is in the same quadrant as ,
To get RA at the right quadrant on computer programs use double argument Arctan function such as ATAN2(y,x)
and declination,
.
Rectangular equatorial coordinates Right-handed rectangular equatorial coordinates in astronomical units are:
Where axis is in the direction of the March equinox, the axis towards June Solstice, and the axis towards the North celestial pole.
Horizontal coordinates
Declination of the Sun as seen from Earth
The Sun appears to move northward during the northern spring, crossing the celestial equator on the March equinox. Its declination reaches a maximum equal to the angle of Earth's axial tilt (23.44° or 23°26') on the June solstice, then decreases until reaching its minimum (−23.44° or -23°26') on the December solstice, when its value is the negative of the axial tilt. This variation produces the seasons.
A line graph of the Sun's declination during a year resembles a sine wave with an amplitude of 23.44°, but one lobe of the wave is several days longer than the other, among other differences.
The following phenomena would occur if Earth were a perfect sphere, in a circular orbit around the Sun, and if its axis were tilted 90°, so that the axis itself is on the orbital plane (similar to Uranus). At one date in the year, the Sun would be directly overhead at the North Pole, so its declination would be +90°. For the next few months, the subsolar point would move toward the South Pole at constant speed, crossing the circles of latitude at a constant rate, so that the solar declination would decrease linearly with time. Eventually, the Sun would be directly above the South Pole, with a declination of −90°; then it would start to move northward at a constant speed. Thus, the graph of solar declination, as seen from this highly tilted Earth, would resemble a triangle wave rather than a sine wave, zigzagging between plus and minus 90°, with linear segments between the maxima and minima.
If the 90° axial tilt is decreased, then the absolute maximum and minimum values of the declination would decrease, to equal the axial tilt. Also, the shapes of the maxima and minima on the graph would become less acute, being curved to resemble the maxima and minima of a sine wave. However, even when the axial tilt equals that of the actual Earth, the maxima and minima remain more acute than those of a sine wave.
In reality, Earth's orbit is elliptical. Earth moves more rapidly around the Sun near perihelion, in early January, than near aphelion, in early July. This makes processes like the variation of the solar declination happen faster in January than in July. On the graph, this makes the minima more acute than the maxima. Also, since perihelion and aphelion do not happen on the exact dates as the solstices, the maxima and minima are slightly asymmetrical. The rates of change before and after are not quite equal. Furthermore, Earth's subsolar point only occurs within the tropics.
The graph of apparent solar declination is therefore different in several ways from a sine wave. Calculating it accurately involves some complexity, as shown below.
Calculations Simple Formula of finding altitude of Sun on various latitudinal positions on Earth:
Equinoxes-> (90°- X°)
Summer and Winter solstices-> 90°-|(X° - 23½°)| Conditions-
(i) Where X is the angle of latitude of any place on Earth, expressed in degrees(excluding minutes and seconds part).
(ii) X is positive in Northern Hemisphere and negative in Southern Hemisphere, on Summer Solstice.
(iii) X is negative in Northern Hemisphere and positive in Southern Hemisphere, on Winter Solstice.
Examples:
For Summer Solstice-
Altitude of Sun at the Tropic of Cancer is:- 90° - |(23½° - 23½°)|= 90° at 23½°N
Altitude of Sun at the Tropic of Capricorn is:- 90° - |(-23½° - 23½°)|= 90°-47°=43° at 23½°S
For Winter Solstice-
Altitude of Sun at the Tropic of Cancer is:- 90° - |(-23½° - 23½°)|= 90°-47°=43° at 23½°N
Altitude of Sun at the Tropic of Capricorn is:- 90° - |(23½° - 23½°)|= 90° at 23½°S
The declination of the Sun, δ☉, is the angle between the rays of the Sun and the plane of the Earth's equator. The Earth's axial tilt (called the obliquity of the ecliptic by astronomers) is the angle between the Earth's axis and a line perpendicular to the Earth's orbit. The Earth's axial tilt changes slowly over thousands of years but its current value of about ε = 23.44° is nearly constant, so the change in solar declination during one year is nearly the same as during the next year.
At the solstices, the angle between the rays of the Sun and the plane of the Earth's equator reaches its maximum value of 23.44°. Therefore, δ☉ = +23.44° at the northern summer solstice and δ☉ = −23.44° at the southern summer solstice.
At the moment of each equinox, the center of the Sun appears to pass through the celestial equator, and δ☉ is 0°.
The Sun's declination at any given moment is calculated by:
where EL is the ecliptic longitude (essentially, the Earth's position in its orbit). Since the Earth's orbital eccentricity is small, its orbit can be approximated as a circle which causes up to 1° of error. The circle approximation means the EL would be 90° ahead of the solstices in Earth's orbit (at the equinoxes), so that sin(EL) can be written as sin(90+NDS)=cos(NDS) where NDS is the number of days after the December solstice. By also using the approximation that arcsin[sin(d)·cos(NDS)] is close to d·cos(NDS), the following frequently used formula is obtained:
where N is the day of the year beginning with N=0 at midnight Universal Time (UT) as January 1 begins (i.e. the days part of the ordinal date −1). The number 10, in (N+10), is the approximate number of days after the December solstice to January 1. This equation overestimates the declination near the September equinox by up to +1.5°. The sine function approximation by itself leads to an error of up to 0.26° and has been discouraged for use in solar energy applications. The 1971 Spencer formula (based on a Fourier series) is also discouraged for having an error of up to 0.28°. An additional error of up to 0.5° can occur in all equations around the equinoxes if not using a decimal place when selecting N to adjust for the time after UT midnight for the beginning of that day. So the above equation can have up to 2.0° of error, about four times the Sun's angular width, depending on how it is used.
The declination can be more accurately calculated by not making the two approximations, using the parameters of the Earth's orbit to more accurately estimate EL:
which can be simplified by evaluating constants to:
N is the number of days since midnight UT as January 1 begins (i.e. the days part of the ordinal date −1) and can include decimals to adjust for local times later or earlier in the day. The number 2, in (N-2), is the approximate number of days after January 1 to the Earth's perihelion. The number 0.0167 is the current value of the eccentricity of the Earth's orbit. The eccentricity varies very slowly over time, but for dates fairly close to the present, it can be considered to be constant. The largest errors in this equation are less than ± 0.2°, but are less than ± 0.03° for a given year if the number 10 is adjusted up or down in fractional days as determined by how far the previous year's December solstice occurred before or after noon on December 22. These accuracies are compared to NOAA's advanced calculations which are based on the 1999 Jean Meeus algorithm that is accurate to within 0.01°.
(The above formula is related to a reasonably simple and accurate calculation of the Equation of Time, which is described here.)
More complicated algorithms correct for changes to the ecliptic longitude by using terms in addition to the 1st-order eccentricity correction above. They also correct the 23.44° obliquity which changes very slightly with time. Corrections may also include the effects of the moon in offsetting the Earth's position from the center of the pair's orbit around the Sun. After obtaining the declination relative to the center of the Earth, a further correction for parallax is applied, which depends on the observer's distance away from the center of the Earth. This correction is less than 0.0025°. The error in calculating the position of the center of the Sun can be less than 0.00015°. For comparison, the Sun's width is about 0.5°.
Atmospheric refraction