Position of the Sun

To derive the position of the Sun in the sky we need to take three things into account:

1. The movement of the Earth in its orbit around the Sun, which does not proceed at a constant rate because of the eccentricity of the orbit of the Earth.

1. The tilt of the Earth's axis of rotation with respect to the plane of its orbit around the Sun. This tilt causes the seasons.

1. The position of the observer on the Earth, which (together with time) determines the direction of the horizon and the zenith.

Below, formulae are derived that take all of these effects into account. To make the formulae easy, they are given for 1997, small variations from year to year are ignored, the formulae are all stated in terms of the day number in the current year, and small effects are ignored. This means that things may be off by at most about half a day. All angles are measured in degrees. Remember that you can add or subtract multiples of 360 degrees to any angle without changing its direction.

The Earth's orbit around the Sun: the ecliptic coordinates

Because we watch the Sun from the Earth, we see the movement of the Earth around the Sun reflected in the apparent movement of the Sun relative to the stars along the ecliptic. If the orbit of the Earth around the Sun were a perfect circle, then the Sun would traverse the ecliptic at a constant speed, and it would be easy to calculate its position at any time. This position is called the mean anomaly, and denoted M. The mean anomaly is measured from the perigee of the orbit. The mean anomaly of the Sun is approximately equal to

(Eq 1) M = -3.18 + 0.98560 d degrees,

where d is the day number in the current year (1 January = 1, 2 January = 2, 1 February = 32, etcetera).

To find the position of the Sun along the ecliptic, relative to the vernal equinox, we need to know the position of the perihelion relative to the vernal equinox. This is called the longitude of the perihelion, here denoted omega:

(Eq 2) omega = -77.11 degrees

The position the Sun would have relative to the vernal equinox if the orbit of the Earth were a perfect circle is called the mean longitude, denoted L:

(Eq 3) L = M + omega

The Earth's orbit is not a perfect circle, but rather an ellipse with an eccentricity of about 0.0167. Because of the eccentricity of the orbit, the speed of the Sun along the ecliptic varies through the seasons. The difference between the true position of the Sun along the ecliptic (the true anomaly) and the easy-to-calculate position it would have in a circular orbit (the mean anomaly) is called the Equation of Center. As explained elsewhere, the Equation of Center of the Earth's orbit (and of the Sun along the ecliptic) is approximately equal to

(Eq 4) C = 1.915 sin M + 0.020 sin 2M degrees.

The true anomaly nu of the Sun, i.e., its true angular distance to the perihelion, is equal to

(Eq 5) nu = M + C

and the (true, ecliptic) longitude lambda of the Sun is equal to

(Eq 6) lambda = L + C = M + omega + C = nu + omega

The ecliptic latitude of the Sun, i.e., its distance to the ecliptic, is always very small, and is taken to be zero in the rest of this derivation. The coordinates relative to the ecliptic (the ecliptic longitude and latitude) are the ecliptic coordinates.

Tilt of the Earth's axis: equatorial coordinates

The axis of the Earth is tilted relative to its orbit, i.e., relative to the ecliptic. The equator of the Earth makes an angle with the ecliptic which is called the obliquity of the ecliptic, denoted epsilon:

(Eq 7) epsilon = 23.4397 degrees

The coordinate system linked to the Earth's rotation axis is called the equatorial coordinate system. The coordinate that corresponds to the ecliptic longitude is called the right ascension, denoted alpha. The coordinate that corresponds to the ecliptic latitude is called the declination, denoted delta. The declination of an object indicates from which parts of the Earth the object is visible. The right ascension determines (together with other things) when the object is visible.

The following formulae transform from the ecliptic to the equatorial coordinates:

(Eq 8) tan alpha = (sin lambda cos epsilon - tan beta sin epsilon) / cos lambda

which is derived from

(Eq 8') sin alpha cos delta = sin lambda cos beta cos epsilon - sin beta sin epsilon

and

(Eq 8'') cos alpha cos delta = cos lambda cos beta.

Also

(Eq 9) sin delta = sin beta cos epsilon + cos beta sin epsilon sin lambda

where beta is the ecliptic latitude. For the Sun, we set beta equal to zero and find

(Eq 10) tan alpha = tan lambda cos epsilon

(Eq 11) sin delta = sin epsilon sin lambda

Observer's position: horizontal coordinates

Where an object is in your sky depends on your geographical coordinates, the position of the object among the stars (its equatorial coordinates), and the rotation angle of the Earth relative to the stars. The latter quantity is expressed as the sidereal time, denoted theta:

(Eq 12) theta = 99.698 + 0.98562 d + 15.041 (t + t_z) - L_g degrees

where t is the local time in hours at the observer's position, in the 24-hour system, t_z is the time in hours that you have to add to the local time to get UT (Greenwich Mean Time), and L_g is the observer's geographical longitude (counted positive West of Greenwich, negative East of Greenwich). The quantity t_z is equal to -2 for CEDT, -1 for CET, 0 for GMT, +4 for EDT, +5 for EST and CDT, +6 for CST and MDT, +7 for MST and PDT, and +8 for PST. It is usually close to L_g/15, or one less during daylight savings time.

The sidereal time indicates which stars are out at midnight local time and is therefore equal to the mean longitude L of the Sun + omega + 180 degrees.

The height in degrees of an object above the horizon is called the altitude, denoted h. The distance in degrees along the horizon of the direction of an object from the South point is called the azimuth, denoted A. Here, the azimuth is measured positive from South in the direction of the West (i.e., West has azimuth +90 degrees, South has 0 degrees, East has -90 degrees, and North has 180 degrees). These coordinates are called the horizontal coordinates. The following formulae transform from equatorial to horizontal coordinates:

(Eq 13) tan A = sin H / (cos H sin phi - tan delta cos phi)

(Eq 14) sin h = sin phi sin delta + cos phi cos delta cos H

(Eq 15) H = theta - alpha,

where phi is the observer's geographical latitude, positive North of the equator, and negative South of the equator, and H is the hour angle. If the hour angle is zero, then the object is due South of the celestial pole.

Sunset and Sunrise

Sunrise is the moment when the top of the solar disk touches the horizon in the morning. Sunset is the similar moment in the evening. Besides the things described above, there are two more effects that must be taken into account when calculating the times of sunset and sunrise:

1. The disk of the Sun has a radius of about 0.2666 degrees, so when the center of the Sun sets (h = 0), then half of the Sun is still above the horizon. The size of the Sun must be taken into account.

1. The Earth's atmosphere bends light downward so that an object appears a bit higher in the sky than it would be without the atmosphere. This effect is called refraction and is largest near the horizon, where the light has to travel through the most atmosphere to get to you. At the horizon, the effect amounts to some 0.57 degrees on average, i.e., an object that would already be 0.57 degrees below the horizon if there were no atmosphere is just at the horizon with the atmosphere.

To compensate for these two effects, h must be set to h0 = -0.8333 degrees in the formulae given above to find the times of sunset and sunrise.

[LS 17 November 1996 - 27 December 1997]

Equation of Time

The time of true noon, when the Sun is due south, usually differs from the time of mean noon (according to a clock). The main reasons are:

1. Because the obliquity of the ecliptic is not equal to zero, so that the fraction of the Sun's motion that is in the east-west direction (i.e., that affects the time of true noon) varies throughout the year.

1. Because the eccentricity of the Earth's orbit is not zero, so that the apparent speed of the Sun between the stars varies throughout the year.

1. Because most places use the standard time of another place rather than the local time for their own geographic longitude.

The third reason yields a constant offset and is ignored below. The remaining influence on the time of true noon is called the Equation of Time.

From Eqs. 4, 6, and 10 follows, to first order in eccentricity e and second order in obliquity epsilon of the ecliptic,

(Eq 16) alpha = L + 2 e sin M - (1/4) epsilon^2 sin 2L + higher-order terms.

where, following Eq. 3, L = M + omega.

The Equation of Time is measured by L - alpha. The mean longitude L of the Sun advances at a constant rate (disregarding changes on time scales of many years because of the changes in the length of the year), and if e and epsilon were equal to zero, then we'd have alpha = L and the Sun's right ascension alpha would advance at a constant rate, too, leading to an Equation of Time equal to zero. In general, then, the Equation of Time (here denoted E) is equal to

(Eq 17) E = -2 e sin M + (1/4) epsilon^2 sin 2L + higher-order terms.

Using the previously defined values for e and epsilon, we find

(Eq 18) E = -7.7 sin M + 9.6 sin 2L minutes.

The contribution of the dominant term because of eccentricity to the Equation of Time amounts to about 8 minutes, and the contribution of the dominant term because of obliquity of the ecliptic to about 10 minutes. The 2L term (due to the obliquity of the ecliptic) accounts for the secondary minima and maxima in the Equation of Time.

Times of Earliest and Latest Sunrise and Sunset

With Eq. 14 we find for sunrise and sunset

(Eq 19) H = (pi/2) + epsilon tan phi sin L + higher-order terms.

This approximation breaks down near and beyond the latitudes of the arctic circles, i.e., when phi gets near or exceeds 90 degrees - epsilon. The obliquity of the ecliptic accounts for the seasons and the change in the amount of daylight throughout the year. For the time of sunrise, we find

(Eq 20) t_rise = 6 - 2 e sin M + (1/4) epsilon^2 sin 2L - epsilon tan phi sin L hours + higher-order terms

where e and epsilon must now be expressed in hours (with 15 degrees = 1 hour). An extreme in the time of sunrise is reached when its derivative with respect to time is equal to zero. This is when

(Eq 21) (1/2) epsilon^2 cos 2L - 2 e cos M - epsilon tan phi cos L = 0.

The third of the three terms is the dominant one for moderate latitudes on the Earth. If the other two terms were absent, then Eq. 21 would be satisfied for L = ± pi/2, i.e., at the solstices. Because the other two terms are in fact not equal to zero, the condition is satisfied for slightly different values of L. If we set L = ± pi/2 + dL (with the - for the December solstice and the + for the June solstice), expand to first order in dL, and solve the equation, then we find

(Eq 22) dL = - (2 e sin omega ± (1/2) epsilon^2)/(epsilon tan phi - 2 e cos omega)

With the current numerical values for the various parameters we find

(Eq 23) dL = -(-4.63 ± 11.89)/(tan phi - 0.018) days

so the date of the latest sunrise differs from the date of the shortest day by an amount that increases as one goes nearer to the equator, and the effect of the obliquity of the ecliptic is about 2.6 times bigger than that of the eccentricity of the Earth's orbit. For the date of earliest sunset the result is similar, but with an extra minus sign in front of the tan phi term.

[LS 27 December 1997]