Ecliptic coordinates

These equations, from the Astronomical Almanac,^[4][5] 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. These equations are coded into a Fortran 90 routine in Ref.^[3] 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

${\displaystyle n=\mathrm {JD} -2451545.0}$

The mean longitude of the Sun, corrected for the aberration of light, is:

${\displaystyle L=280.460^{\circ }+0.9856474^{\circ }n}$

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:

${\displaystyle g=357.528^{\circ }+0.9856003^{\circ }n}$

Put ${\displaystyle L}$ and ${\displaystyle g}$ in the range 0° to 360° by adding or subtracting multiples of 360° as needed.

Finally, the ecliptic longitude of the Sun is:

${\displaystyle \lambda =L+1.915^{\circ }\sin g+0.020^{\circ }\sin 2g}$

The ecliptic latitude of the Sun is nearly:

${\displaystyle \beta =0}$,

as the ecliptic latitude of the Sun never exceeds 0.00033°,^[6]

and the distance of the Sun from the Earth, in astronomical units, is:

${\displaystyle R=1.00014-0.01671\cos g-0.00014\cos 2g}$.

Obliquity of the ecliptic

Where the obliquity of the ecliptic is not obtained elsewhere, it can be approximated:

${\displaystyle \epsilon =23.439^{\circ }-0.0000004^{\circ }n}$

Equatorial coordinates

${\displaystyle \lambda }$, ${\displaystyle \beta }$ and ${\displaystyle R}$ 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, ${\displaystyle \epsilon }$, and continuing:

Right ascension,

${\displaystyle \alpha =\arctan(\cos \epsilon \tan \lambda )}$, where ${\displaystyle \alpha }$ is in the same quadrant as ${\displaystyle \lambda }$,

To get RA at the right quadrant on computer programs use double argument Arctan function such as ATAN2(y,x)

${\displaystyle \alpha =\arctan 2(\cos \epsilon \sin \lambda ,\cos \lambda )}$

and declination,

${\displaystyle \delta =\arcsin(\sin \epsilon \sin \lambda )}$.

Rectangular equatorial coordinates

Right-handed rectangular equatorial coordinates in astronomical units are:

${\displaystyle X=R\cos \lambda }$

${\displaystyle Y=R\cos \epsilon \sin \lambda }$

${\displaystyle Z=R\sin \epsilon \sin \lambda }$

Where ${\displaystyle X}$ axis is in the direction of the March equinox, the ${\displaystyle Y}$ axis towards June Solstice, and the ${\displaystyle Z}$ axis towards the North celestial pole.^[7]

Horizontal coordinates

Calculations

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°26' 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°26'. Therefore, δ_☉ = +23°26' at the northern summer solstice and δ_☉ = −23°26' 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:

${\displaystyle \delta _{\odot }=\arcsin \left[\sin \left(-23.44^{\circ }\right)\cdot \sin \left(EL\right)\right]}$

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:

${\displaystyle \delta _{\odot }=-23.44^{\circ }\cdot \cos \left[{\frac {360^{\circ }}{365}}\cdot \left(N+10\right)\right]}$

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.^[2] The 1971 Spencer formula^[10] (based on a Fourier series) is also discouraged for having an error of up to 0.28°.^[11] 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:^[12]

${\displaystyle \delta _{\odot }=\arcsin \left[\sin \left(-23.44^{\circ }\right)\cdot \cos \left({\frac {360^{\circ }}{365.24}}\left(N+10\right)+{\frac {360^{\circ }}{\pi }}\cdot 0.0167\sin \left({\frac {360^{\circ }}{365.24}}\left(N-2\right)\right)\right)\right]}$

which can be simplified by evaluating constants to:

${\displaystyle \delta _{\odot }=-\arcsin \left[0.39779\cos \left(0.98565^{\circ }\left(N+10\right)+1.914^{\circ }\sin \left(0.98565^{\circ }\left(N-2\right)\right)\right)\right]}$

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^[13][14] which are based on the 1999 Jean Meeus algorithm that is accurate to within 0.01°.^[15]

(The above formula is related to a reasonably simple and accurate calculation of the Equation of Time, which is described here.)

More complicated algorithms^[16][17] 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

The declination calculations described above do not include the effects of the refraction of light in the atmosphere, which causes the apparent angle of elevation of the Sun as seen by an observer to be higher than the actual angle of elevation, especially at low Sun elevations.^[2] For example, when the Sun is at an elevation of 10°, it appears to be at 10.1°. The Sun's declination can be used, along with its right ascension, to calculate its azimuth and also its true elevation, which can then be corrected for refraction to give its apparent position.^[2][14][18]