Equation of Time

by Phill Edwards

Clock time versus solar time

The question is, why are sunrise and sunset not symmetrical about noon? This effect is most apparent around the December Solstice when, in the Northern Hemisphere, sunrise is about the same time for several weeks, but sunset gets later every day. The answer lies in the Equation of Time.

Modern clocks assume a 24 hour day. The Sun doesn't culminate at exactly noon clock time every day due to the fact that the solar day varies in length due to orbital effects. The difference between clock time and solar time is described by the equation of time. The equation of time is usually displayed as a graph. It shows the cumulative difference between clock time and solar time. The Equation of Time has two major components due to the Earth's elliptical orbit and the Earth's axial tilt, plus some smaller effects due to the gravity of other planets and relativistic effects.

Eccentricity of orbit

Clock time assumes that every day is of equal length. This assumes that the Earth sweeps out a fixed angle of `360/24 = 15°` in its orbit every day. The Earth's orbit is elliptical as described by Kepler's first law. This means that every solar day has a slightly different length. This is due to Kepler's second law which states that an orbiting body sweeps an equal area in each given time interval. This means that the angle swept out in a day is different from day to day. This means that the times between noon varies from day to day.

Figure 1 Kepler's Laws

Earth parameters

There are a number of parameters which define the Earth's orbit. These are not actually constants as they vary over long periods of time. These parameters values were determined for the J2000 Epoch which was 1 January 2000 at 1200 (noon not midnight!) UTC. These values form a starting point for quite accurate predictions for a few decades before and after J2000.

• Anomalistic year ty = 365.2596358 day, the period between consecutive periapses (perihelia)
• Eccentricity of the orbit ε = 0.01671123
• Mean Anomaly at epoch MD = 6.24004027
• Axial tilt φ = 23.4393° = 0.409093 radians, the angle between the Earth's axis and the plane of the ecliptic
• The angle from the Vernal Equinox to the periapsis λp = 4.9358
• The factor f = 24 * 60 / 2π converts radian day angles into minutes

Mean amomaly

Clock time is based on days of fixed length of 24 hours. The term anomaly is used to describe the angle in radians from the periapsis which the Earth has swept out in a period of time since the periapsis. The mean anomaly M is the angle swept out of the ellipse's containing circle after from periapsis as shown in figure 1. As time t is measured from the start of the year, the mean anomaly is calculated by subtracting the time of the periapsis from the start of the year tp.

M = 2π(t - tp)/ty

The value tp is difficult to calculate as the perihelion date advances each year due to the anomalistic year being longer than the calendar year and other effects such as precession. A better way of calculating M is to use the value of M = MD at the J2000 epoch as a starting point. Then M can be calculated from the number of days since the epoch tD. Suitable multiples of 2π need to be added or subtracted to get M in the tange [0,2π].

M = MD + 2πtD/ty (1)

Eccentric anomaly

The eccentric anomaly E from, figure 1, was introduced by Kepler to find the relationship between the mean anomaly and the true anomaly. The mean anomaly and the eccentric anomaly are related by Kelper's equation.

M = E - ε sin E (2)

This equation can't be solved analytically, but it can be easily solved using Newton's method using M as the initial estimate for E which is a good choice as ε is small.

Real anomaly

The real anomaly ν (θ in figure 1) is the actual angle swept out by the Earth. Kepler's laws state that the area swept out by the mean anomaly = area swept out by the real anomaly. This means that the area of the circular sector ZPEm equals the area of the eliptic sector SPEr. By trigonometry.

$\mathrm{tan}\frac{\nu }{2}=\sqrt{\frac{1 + \varepsilon }{1 - \varepsilon }}\mathrm{tan}\frac{E}{2}$ (3)

Equation of Time Component

The eccentricity component of the Equation of Time can be calculated by solving equations 1, 2 and 3 for each day of the year's value of M. The offset between the noon times of the clock time and solar time is given by Δt = f(M - ν). This is plotted as the orange curve in figure 3 below. This is a sine curve with a period of 1 anomalistic year, zeros at the apses and an amplitude of 7.66 minutes. The amplitude is the value of Δt when M = π/2.

Figure 2 Eccentricity Equation of Time Component

Obliquity of the ecliptic

Clock time assumes that every day is of equal length. The Earth's axis of rotation is tilted and is therefore not normal to the plane of the ecliptic (the plane of the Earth's orbit around the Sun). This means that the position of the Sun as observed from Earth is the projection of the position of the Sun from the ecliptic plane onto the equatorial plane. This introduces a shift in the observed position of the Sun's longitude which changes the length of the solar day.

To take the simplest case. Consider an observer on the equator. The Sun is only directly overhead at noon on the equinoxes. At the solstices, the ecliptic is at 23.5° to the equatorial plane. This means that the projection of the Sun at noon has a longitudinal and latitudinal offset. The longitudinal component is only zero at the equinoxes and solstices. At other times the longitudinal component causes solar noon and clock noon to differ by the longitude component expressed in minutes and seconds.

Figure 3 Ecliptic and Equatorial planes

The Earth's axis of rotation is inclined at an angle φ to the Ecliptic. This means that the Equatorial plane is inclined at an angle φ to the Ecliptic plane. These two planes are only parallel at the Equinoxes.

The observed position of the Sun is actually the projection of the Sun's position on the Ecliptic plane onto the Equatorial plane. This results in a rotation of the Sun's position which changes the length of the solar day.

The eccentricity calculations were based on angles measured from periapsis. The obliquity calculations are based on angles from an Equinox. The Vernal Equinox is chosen to be the zero angle. Thus the angle between the Vernal Equinox and the periapsis λp needs to be added to the angles. The true anomaly becomes λ = θ + λp. The angle α which is the projection of the Sun's position onto the Equatorial plane is shown in figure 2:

tan α = x2/y = x1 cos φ/y = cos φ tan λ (4)

The difference between the mean and solar days becomes:

Δt = f(M + λp - α) (5)

Equation of Time Component

The obliquity component of the Equation of time is calculated by solving equation 5 for each day of the year's value of M, assuming a circular orbit where ε = 0. Then θ = E = M. The green curve in figure 3 is the obliquity component of the Equation of Time. This is a sine curve with period of half a year. It has zeros at the solstices and the equinoxes and an aplitude of 9.86 minutes. The amplitude is the value of Δt when M = π/4.

Figure 4 Obliquity Equation of Time Component

Equation of Time

The Equation of time is calculated by solving equation 5 for each day of the year's value of M using the Earth's eccentricity value to solve equations 1-4. The blue curve in figure 3 shows the Equation of Time. Positive values mean that the Sun culminates that many minutes before noon clock time. It has zeros on 15 April, 13 June, 1 September and 25 December. The maxima are on 14 May value 3.69 and 3 November value 16.41. The minima are on 11 February value -14.26 and 26 July value -6.48.

Now we can explain why sunrise stays about the same time while sunset gets progressively later in the Northern hemisphere im the weeks following the Winter Solstice. The Equation of Time shows a steep descent from +16 minutes in November to -14 minutes in February. During this period the Sun's culmination is getting later each day. The culmination occurs at noon at the zero of the Equation of Time on 25 December. After the solstice, the daylight hours are getting progressively longer, the time between sunrise and sunset is increasing.

On 1 January 2013, sunrise was at 0806 and sunset was at 1602. The midpoint between these two times is 1204. The Equation of Time for that day has a value of -4, which puts culmination at 1204.
On 4 January 2013, sunrise was at 0806 and sunset was at 1606. The midpoint between these two times is 1206. The Equation of Time for that day has a value of -6, which puts culmination at 1206.