Equation of time

The equation of time — above the axis the dial will appear fast, and below the dial will appear slow.

The equation of time is the difference between apparent solar time and mean solar time, both taken at a given place (or at another place with the same geographical longitude) at the same real instant of time.

Apparent (or true) solar time can be obtained for example by measurement of the current position (hour angle) of the Sun, or indicated (with limited accuracy) by a sundial. Mean solar time, for the same place, would be the time indicated by a steady clock set so that its differences over the year from apparent solar time average to zero (with zero net gain or loss over the year).

The equation of time varies over the course of a year, in a way that is almost exactly reproduced from one year to the next, except for a shift in phase of one day ahead in the calendar every 25 years as can be derived from ancient tables of the equation of time of the late 17th century (see e.g. Tho. Tompion 1683). Apparent time, and the sundial, can be ahead (fast) by as much as 16 min 33 s (around 3 November), or behind (slow) by as much as 14 min 6 s (around 12 February). The equation of time has zeros near 15 April, 13 June, 1 September and 25 December.

The equation of time results mainly from two different superposed astronomical causes (explained below), each causing a different non-uniformity in the apparent daily motion of the Sun relative to the stars, and contributing a part of the effect:

"A Table of the Equation of Days" — a table of clockmaker Tho. Tompion 1683, showing the difference in phase of the equation at the time.

• the obliquity of the ecliptic (the plane of the Earth's annual orbital motion around the Sun), which is inclined by about 23.44 degrees relative to the plane of the Earth's equator; and

• the eccentricity and elliptical form of the Earth's orbit around the Sun.

The equation of time is also the east or west component of the analemma, a curve representing the angular offset of the Sun from its mean position on the celestial sphere as viewed from Earth.

The equation of time was used historically to set clocks. Between the invention of accurate clocks in 1656 and the advent of commercial time distribution services around 1900, one of two common land-based ways to set clocks was by observing the passage of the sun across the local meridian at noon. The moment the sun passed overhead, the clock was set to noon, offset by the number of minutes given by the equation of time for that date. (The second method did not use the equation of time; instead, it used stellar observations to give sidereal time, in combination with the relation between sidereal time and solar time.) The equation of time values for each day of the year, compiled by astronomical observatories, were widely listed in almanacs and ephemerides.

Naturally, other planets will have an equation of time too. On Mars the difference between sundial time and clock time can be as much as 50 minutes, due to the considerably greater eccentricity of its orbit.

History

Ancient history — Babylon and Egypt

The irregular daily movement of the Sun was known by the Babylonians, and Book III of Ptolemy's Almagest is primarily concerned with the Sun's anomaly. Ptolemy discusses the correction needed to convert the meridian crossing of the Sun to mean solar time and takes into consideration the nonuniform motion of the Sun along the ecliptic and the meridian correction for the Sun's ecliptic longitude. He states the maximum correction is 8 1/3 time-degrees or 5/9 of an hour (Book III, chapter 9). However he did not consider the effect relevant for most calculations since it was negligible for the slow-moving luminaries and only applied it for the fastest-moving luminary, the Moon.

Medieval and Renaissance astronomy

Toomer uses the Medieval term equation, from the Latin term aequatio (equalization [adjustment]), for Ptolemy's difference between the mean solar time and the true solar time. Kepler's definition of the equation is "the difference between the number of degrees and minutes of the mean anomaly and the degrees and minutes of the corrected anomaly."

Apparent time versus mean time

Until the invention of the pendulum and the development of reliable clocks during the 17th century, the equation of time as defined by Ptolemy remained a curiosity, of importance only to astronomers. However, when mechanical clocks started to take over timekeeping from sundials, which had served humanity for centuries, the difference between clock time and solar time became an issue for everyday life. Apparent solar time (or true or real solar time) is the time indicated by the Sun on a sundial (or measured by its transit over the local meridian), while mean solar time is the average as indicated by well-regulated clocks. The first tables for the equation of time which accounted for its annual variations in an essentially correct way were published in 1665 by Christiaan Huygens. Huygens set his values for the equation of time so as to make all values positive throughout the year. This meant that a clock set by Huygens' tables would be consistently about 15 minutes slow on mean time.

Another set of tables was published in 1672/73 by John Flamsteed, who later became the first royal astronomer of the new Greenwich Observatory. These appear to have been the first essentially correct tables which also led to mean time without an offset. Flamsteed adopted the convention of tabulating and naming the correction in the sense that it was to be applied to the apparent time to give mean time.

The equation of time, correctly based on the two major components of the Sun's irregularity of apparent motion, i.e. the effect of the obliquity of the ecliptic and the effect of the Earth's orbital eccentricity, was not generally adopted until after Flamsteed's tables of 1672/3, published with the posthumous edition of the works of Jeremiah Horrocks.

Eighteenth and early nineteenth centuries

The corrections in Flamsteed's tables of 1672/3 and 1680 led to mean time computed essentially correctly and without an offset, i.e. in principle as we now know it. But the numerical values in tables of the equation of time have somewhat changed since then, owing to three kinds of factors:

• (1) general improvements in accuracy that came from refinements in astronomical measurement techniques,
• (2) slow intrinsic changes in the equation of time, occurring as a result of very slow long-term changes in the Earth's obliquity and eccentricity and the position of its perihelion (or, equivalently, of the Sun's perigee), and
• (3) the inclusion of small sources of additional variation in the apparent motion of the Sun, unknown in the 17th century, but discovered from the eighteenth century onwards, including the effects of the Moon, Venus and Jupiter.

Until 1833, the equation of time was tabulated in the sense 'mean minus apparent solar time' in the British Nautical Almanac and Astronomical Ephemeris published for the years 1767 onwards. Before the issue for 1834, all times in the almanac were in apparent solar time, because time aboard ship was most often determined by observing the Sun. In the unusual case that the mean solar time of an observation was needed, the extra step of adding the equation of time to apparent solar time was needed. In the Nautical Almanac issues for 1834 onwards, all times have been in mean solar time, because by then the time aboard ship was increasingly often determined by marine chronometers. In the unusual case that the apparent solar time of an observation was needed, the extra step of applying the equation of time to mean solar time was needed, requiring all differences in the equation of time to have the opposite sign than before.

As the apparent daily movement of the Sun is one revolution per day, that is 360° every 24 hours, and the Sun itself appears as a disc of about 0.5° in the sky, simple sundials can be read to a maximum accuracy of about one minute. Since the equation of time has a range of about 30 minutes, the difference between sundial time and clock time cannot be ignored. In addition to the equation of time, one also has to apply corrections due to one's distance from the local time zone meridian and summer time, if any.

The tiny increase of the mean solar day itself due to the slowing down of the Earth's rotation, by about 2 ms per day per century, which currently accumulates up to about 1 second every year, is not taken into account in traditional definitions of the equation of time, as it is imperceptible at the accuracy level of sundials.

Explanations for the major components of the equation of time

Eccentricity of the Earth's orbit

Graph showing the equation of time (red solid line) along with its two main components plotted separately, the part due to the obliquity of the ecliptic (mauve broken line) and the part due to the Sun's varying apparent speed along the ecliptic due to the eccentricity & ellipticity of the Earth's orbit (dark dash-dotted line)

The Earth revolves around the Sun. As such it appears that the Sun makes one rotation around the Earth through the background stars in one year. If the Earth orbited the Sun with a constant speed, in a circular orbit in a plane perpendicular to the Earth's axis, then the Sun would culminate every day at exactly the same time, and be a perfect time keeper (except for the very small effect of the slowing rotation of the Earth). But the orbit of the Earth is an ellipse, and its speed varies between 30.287 and 29.291 km/s, according to Kepler's laws of planetary motion, and its angular speed also varies, and thus the Sun appears to move faster at perihelion (currently around January 3) and slower at aphelion a half year later. At these extreme points, this effect increases (respectively, decreases) the real solar day by 7.9 seconds from its mean. This daily difference accumulates over a period. As a result, the eccentricity of the Earth's orbit contributes a sine wave variation with an amplitude of 7.66 minutes and a period of one year to the equation of time. The zero points are reached at perihelion (at the beginning of January) and aphelion (beginning of July) while the maximum values are in early April (negative) and early October (positive).

Obliquity of the ecliptic

Sun and planets at solar midday (Ecliptic in red, Sun and Mercury in yellow, Venus in white, Mars in red, Jupiter in yellow with red spot, Saturn in white with rings).

However, even if the Earth's orbit were circular, the motion of the Sun along the celestial equator would still not be uniform. This is a consequence of the tilt of the Earth's rotation with respect to its orbit, or equivalently, the tilt of the ecliptic (the path of the sun against the celestial sphere) with respect to the celestial equator. The projection of this motion onto the celestial equator, along which "clock time" is measured, is a maximum at the solstices, when the yearly movement of the Sun is parallel to the equator and appears as a change in right ascension, and is a minimum at the equinoxes, when the Sun moves in a sloping direction and appears mainly as a change in declination, leaving less for the component in right ascension, which is the only component that affects the duration of the solar day. As a consequence of that, the daily shift of the shadow cast by the Sun in a sundial, due to obliquity, is smaller close to the equinoxes and greater close to the solstices. At the equinoxes, the Sun is seen slowing down by up to 20.3 seconds every day and at the solstices speeding up by the same amount.

In the figure on the right, we can see the monthly variation of the apparent slope of the plane of the ecliptic at solar midday as seen from Earth. This variation is due to the apparent precession of the rotating Earth through the year, as seen from the Sun at solar midday.

In terms of the equation of time, the inclination of the ecliptic results in the contribution of another sine wave variation with an amplitude of 9.87 minutes and a period of a half year to the equation of time. The zero points of this sine wave are reached at the equinoxes and solstices, while the maxima are at the beginning of February and August (negative) and the beginning of May and November (positive).

Secular effects

The two above mentioned factors have different wavelengths, amplitudes and phases, so their combined contribution is an irregular wave. At epoch 2000 these are the values (in minutes and seconds with UT dates):

minimum	−14:15	11 February
zero	00:00	15 April
maximum	+03:41	14 May
zero	00:00	13 June
minimum	−06:30	26 July
zero	00:00	1 September
maximum	+16:25	3 November
zero	00:00	25 December

E.T. = apparent − mean. Positive means: Sun runs fast and culminates earlier, or the sundial is ahead of mean time. A slight yearly variation occurs due to presence of leap years, resetting itself every 4 years.

The exact shape of the equation of time curve and the associated analemma slowly changes over the centuries due to secular variations in both eccentricity and obliquity. At this moment both are slowly decreasing, but they increase and decrease over a timescale of hundreds of thousands of years. If/when the Earth's orbital eccentricity (now about 0.0167 and slowly decreasing) reaches 0.047, the eccentricity effect may in some circumstances overshadow the obliquity effect, leaving the equation of time curve with only one maximum and minimum per year, as is the case on Mars.

On shorter timescales (thousands of years) the shifts in the dates of equinox and perihelion will be more important. The former is caused by precession, and shifts the equinox backwards compared to the stars. But it can be ignored in the current discussion as our Gregorian calendar is constructed in such a way as to keep the vernal equinox date at 21 March (at least at sufficient accuracy for our aim here). The shift of the perihelion is forwards, about 1.7 days every century. For example in 1246 the perihelion occurred on 22 December, the day of the solstice. At that time the two contributing waves had common zero points, and the resulting equation of time curve was symmetrical. Before that time the February minimum was larger than the November maximum, and the May maximum larger than the July minimum. The secular change is evident when one compares a current graph of the equation of time (see below) with one from about 2000 years ago, for example, one constructed from the data of Ptolemy.

Practical use

Animation showing Equation of Time and Analemma path over one year.

If the gnomon (the shadow-casting object) is not an edge but a point (e.g., a hole in a plate), the shadow (or spot of light) will trace out a curve during the course of a day. If the shadow is cast on a plane surface, this curve will (usually) be the conic section of the hyperbola, since the circle of the Sun's motion together with the gnomon point define a cone. At the spring and fall equinoxes, the cone degenerates into a plane and the hyperbola into a line. With a different hyperbola for each day, hour marks can be put on each hyperbola which include any necessary corrections. Unfortunately, each hyperbola corresponds to two different days, one in each half of the year, and these two days will require different corrections. A convenient compromise is to draw the line for the "mean time" and add a curve showing the exact position of the shadow points at noon during the course of the year. This curve will take the form of a figure eight and is known as an "analemma". By comparing the analemma to the mean noon line, the amount of correction to be applied generally on that day can be determined.

Mathematical details

The mathematical expression of the equation of time can be written as

$\Delta t={\frac {(M-\nu )+(\lambda -\alpha )}{\omega _{E}}}$

Δt = t_s - t is the time difference between solar time t_s (essentially time measured by a sundial) and mean solar time t (essentially time measured by a mechanical clock). The first term in parentheses is the time difference due to the eccentricity of the Earth's orbit, and the second term is the time difference due to the inclination of the Earth's rotational axis with respect to its orbital plane. This formulation ignores perturbations due to the moon and other planets, as well as the very small variations in the orbital and rotational parameters with time. If the right ascension of the Sun is RA_S and that of a mean Sun moving uniformly along the celestial equator is RA_MS then the difference between the two is also the equation of time,

Δt = RA_MS - RA_S.

The various quantities appearing on the right are the following:

• M is the Sun's mean anomaly,
• ν the Sun's true anomaly,
• λ - the angle from the vernal equinox to the Sun in the plane of the ecliptic.,
• α is the Sun's right ascension - the angle from the vernal equinox to the Sun in the plane of the equator.
• ω_E = 1 rev/day = 0.0013889π min⁻¹ is the Earth's axial rotation rate (day is a mean solar day) — most astronomers do not include this quantity, they write instead Δt = (M-ν) + (λ-α), because they regard these angles and time as instances of the same dimensional quantity that are related by conversion factors such as; 2π radian = 360° = 1 day = 24 hour.

In order to calculate Δt as a function of M, two additional angles are required; they are

• E the Sun's eccentric anomaly,
• λ_p = 1.794207 = 102.8° is the angle from the vernal equinox to the periapsis in the plane of the ecliptic,

The celestial sphere and the Sun's elliptical orbit as seen by a geocentric observer looking normal to the ecliptic showing the 6 angles (M, λ_p, α, ν, λ, E) needed for the calculation of the equation of time

All these angles are shown in the figure on the right, which shows the celestial sphere and the Sun's elliptical orbit seen from the Earth (the same as the Earth's orbit seen from the Sun). In this figure ε = 0.40910 = 23.44° is the obliquity, while the eccentricity of the ellipse is e = [1 − (b/a)²]^1/2 = 0.01671.

Now given a value of 0≤M≤2π, one can calculate α(M) by means of the following procedure:

First, knowing M, calculate E from Kepler's equation

$M=E-e\sin E$

A numerical value can be obtained by infinite series, or numerical methods such as iteration.

Next, knowing E, calculate the true anomaly ν

$\nu =2\tan ^{-1}\left[{\sqrt {\frac {1+e}{1-e}}}\tan {\frac {E}{2}}\right]$

The correct branch of the multiple valued function tan⁻¹x to use is the one that makes ν a continuous function of E(M) starting from ν(E=0) = 0. Thus for 0≤ E < π use tan⁻¹x = Tan⁻¹x, and for π < E ≤ 2π use tan⁻¹x = Tan⁻¹x + π. Here Tan⁻¹x is the principal branch, |Tan⁻¹x| < π/2; the function that is returned by calculators and computer applications. The time difference due to the Earth's orbital eccentricity may now be calculated.

λ is related to ν as:

$\lambda =\nu +\lambda _{p}$

The value of λ varies non-linearly with M because the orbit is elliptical. Note that if the orbit were circular (e = 0) one would find λ = M + λ_p.

Next, knowing λ, from spherical trigonometry for the right triangle on the celestial sphere shown above, the relationship between α and λ is found to be:

$\tan \alpha =\cos \varepsilon \,\tan \lambda$

with a formal solution for α:

$\alpha =\tan ^{-1}[\cos \varepsilon \,\tan \lambda ]$

The solution for α is not unique since the arctangent function is multivalued, however it is required that the solution for α be continuous over the angles of interest. The correct branch of tan⁻¹x to use makes α a continuous function of λ(M) starting from α(λ=0)=0. Since λ_p≤λ≤λ_p+2π when M varies from 0 to 2π, we have: for λ_p≤λ<5π/2 use tan⁻¹x = Tan⁻¹x + 2π, for 5π/2<λ<7π/2, use tan⁻¹x = Tan⁻¹x + 3π, and for 7π/2<λ≤λ_p+2π use tan⁻¹x = Tan⁻¹x + 4π.

An alternative method is to note,

$\tan \alpha ={\frac {\sin \alpha }{\cos \alpha }}={\frac {\cos \varepsilon \sin \lambda }{\cos \lambda }}$

and that α is the angle of a right triangle with base cosλ, height cosε sinλ and hypotenuse √cos²λ + cos²ε sin²λ. One can then avoid zeros in the denominator for tanα and the signs of the sides allow one to determine the quadrant of α. The time difference due to the inclination of the Earth's rotational axis with respect to its orbital plane may now be determined.

Notice that if the Earth's rotation axis was perpendicular to the ecliptic (ε = 0) one would find α = λ (if simultaneously e = 0, one would have α = M + λ_p so that Δt = 0, in which case there would be no difference between solar time and clock time). Notice also, that mechanism which causes the time difference due to the inclination of the Earth's axis is the same as that which causes a variation in the velocity of the driven axle in a universal joint.

Finally, Δt can be calculated using the starting value of M and the calculated α(M). The result is usually given as either a set of tabular values, or a graph of Δt as a function of the number of days past periapsis, n, where 0≤n≤ 365.242 (365.242 is the number of days in a tropical year); so that

$M={\frac {2\pi \,n}{365.242}}$

Each of the equations (for E, ν, and α) in the algorithm described above can be approximated based on the small values of obliquity and eccentricity so that Δt itself can be written as a simple explicit expression, which is designated Δt_a because it is only an approximation. Expressions for Δt_a have been obtained in many forms; the earliest one of them can be written as

$\Delta t_{a}={\frac {-2e\sin M+\tan ^{2}{\frac {\varepsilon }{2}}\,\sin(2M+2\lambda _{p})}{\omega _{E}}}=[-7.655\sin M+9.873\sin(2M+3.588)]{\mbox{min}}$

The equation of time as calculated by the exact procedure for Δt described in the text and the asymptotic expression for Δt_a given there.

A comparative plot of the two calculations is shown in the figure on the right. The approximate calculation is seen to be close to the exact one, the absolute error, Err = |(Δt− Δt_a)|, is less than 45 seconds throughout the year; its largest value is 44.8 sec and occurs on day 273. More accurate approximations can be obtained by retaining higher order terms, but they are necessarily more time consuming to evaluate. At some point it is simpler to just evaluate Δt, but Δt_a as written above is easy to evaluate, even with a calculator, and has a nice physical explanation as the sum of two terms, one due to obliquity and the other to eccentricity.

When evaluating the expression for Δt_a as given above, a calculator must be in radian mode to obtain correct values. Note also that the date and time of periapsis (perihelion of the Earth orbit) varies from year to year; a table giving the connection can be found in perihelion.

See also

• azimuth
• declination

References

• Burington R S 1949 Handbook of Mathematical Tables and Formulas (Sandusky, Ohio: Handbook Publishers)
• Duffett-Smith P 1988 Practical Astronomy with your Calculator Third Edition (Cambridge: Cambridge University Press)
• Hinch E J 1991 Perturbation Methods, (Cambridge: Cambridge University Press)
• Meeus, J 1997 Mathematical Astronomy Morsels, (Richmond, Virginia: Willman-Bell)
• Milne R M 1921, "Note on the Equation of Time", The Mathematical Gazette 10 (The Mathematical Association) pp 372–375.
• Moulton F R 1970 An Introduction to Celestial Mechanics, Second Revised Edition, (New York: Dover).
• Muller M 1995, "Equation of Time - Problem in Astronomy", Acta Phys Pol A 88 Supplement, S-49.
• United States Naval Observatory April 2010, Multiyear Computer Interactive Almanac (version 2.2.1), Richmond VA: Willmann-Bell.
• Whitman A M 2007, "A Simple Expression for the Equation of Time", Journal Of the North American Sundial Society 14 pp 29–33.
• A Table of the Equation of Time. Printed for Tho. Tompion, Clockmaker 1683.

External links

• Graphical Visualisation of Equation of Time - Constantly updated
• Table giving the Equation of Time and the declination of the sun for every day of the year
• The equation of time described on the Royal Greenwich Observatory website
• An analemma site with many illustrations
• The Equation of Time and the Analemma, by Kieron Taylor
• An article by Brian Tung containing a link to a C program using a more accurate formula than most (particularly at high inclinations and eccentricities). The program can calculate solar declination, Equation of Time, or Analemma.
• Doing calculations using Ptolemy's geocentric planetary models with a discussion of his E.T. graph
• The equation of time correction-table A page describing how to correct a clock to a sundial.
• Cosmology and the Equation of Time by Jack Forster. Watches that include the equation of time.
• Equation of Time Longcase Clock by John Topping C.1720
• Solar tempometer - Calculate your solar time including the equation of time.