The Equation of Time

The Equation of Time is a way of quantifying the variable part of the difference between time kept by an ordinary electrical or mechanical clock, keeping civil time, and the time kept by the Sun, such as what a sundial would read. Technically, the Equation of Time is the difference apparent solar time minus mean solar time, but to understand what that means, we need to understand what the two kinds of time represent.

### Introduction

First, it is important to realize that all of our modern timekeeping systems ultimately have been based on the apparent daily motion of celestial objects in the sky, with the Sun playing the leading role. That apparent motion is of course caused by the Earth's rotation. Objects in the sky appear to move over us from east to west, because the Earth is rotating from west to east. It is actually our horizon that is moving, not the sky, but we perceive it the other way.

There is an important detail to this familiar daily pattern: all solar system objects are moving in orbits around the Sun, so the planets have their own motions in the sky in addition to that caused by the Earth's rotation (the word planet is derived from a Greek word meaning wanderer). Furthermore, since the Earth is also in orbit around the Sun, we see all celestial objects from a slightly different vantage point each day. The resulting daily changes in the relative positions of objects in the sky are slow compared to the apparent daily motion of the whole sky due to the Earth's rotation, but over weeks and months they are significant nonetheless. As a result, all objects in the solar system appear to move across the sky at slightly different rates than do the background stars, and these rates are not constant. This is true even for the Sun, a fact that complicates our efforts to base our system of time on it. The Equation of Time is a reflection of that complication. In a sense, apparent solar time is a kind of raw measurement of where the Sun actually is in the sky, and mean solar time is an attempt to smooth out the variable part of the Sun's motion so that we can make mechanical or electric clocks that reflect the Sun's average motion in the sky.

### Apparent and Mean Solar Time

Two Definitions

Meridian: The plane containing the observer, the overhead point (zenith), and the north and south points on the horizon; it also passes through the center of the Earth. The meridian defines a line of constant longitude on the Earth and the north-south line in the sky.

Local Hour Angle: A measure of how far east or west of the observer's meridian a celestial object appears. Technically, it is the angle, measured at the celestial pole, between the observer's meridian and a line in the sky from the celestial pole to the object. The hour angle is zero for an object crossing the meridian; it is negative for objects east of the meridian (rising) and positive for objects that are west of the meridian (setting). Because of the Earth's rotation, hour angles of celestial objects are always increasing at a rate of approximately 15°/hour. Hour angles can be expressed in time units (e.g., hours) by dividing the angular measure (e.g., degrees) by 15.

Apparent solar time is what a perfectly constructed and calibrated sundial would read at a given location; it is based on the actual position of the Sun in the sky. In astronomical terms, it is determined by the local hour angle of the true Sun, which is a measure of the Sun's angular distance east or west of the local meridian (see box at right for definitions). Since each meridian is a line of constant longitude, apparent solar time is different for every longitude. We define apparent solar time at a specific location as 12h + the local hour angle (expressed in hours) of the apparent position of the Sun in the sky.

Apparent solar time is not a uniform time scale; the Sun crosses the sky at slightly different rates at different times of the year. This means the Sun runs "fast" part of the year and "slow" in others. Although the fractional change in the rate of the Sun's apparent daily motion is tiny (about 0.03%), the accumulated time difference can reach as much as 16 minutes. This effect is explained in detail below, but it is a function of where the Earth is in its orbit around the Sun.

In order to create a uniform time scale for practical use, we imagine a point in the sky called the fictitious mean sun, which moves at a constant rate across the sky (at the celestial equator), regardless of time of year. That is, the fictitious mean sun averages out the variations in the position and rate of motion of the true Sun over the course of an entire year. The fictitious mean sun is never more than about 4 degrees east or west of the actual Sun, although it is only an imaginary point. We can define mean solar time in the same way as apparent solar time:  mean solar time at a specific location is 12h + the local hour angle (expressed in hours) of the fictitious mean sun. Of course, the fictitious mean sun is not an observable point, so we need a mathematical expression to tell us where it is with respect to the true Sun; that is what the Equation of Time is.

### Is Civil Time the Same as Mean Solar Time?

Mean solar time is closely related to civil time, which is what our clocks read (if they are set accurately). The worldwide system of civil time has historically been based on mean solar time, but in the modern system of timekeeping, there are some subtle differences.

U.S. Time Zones as of 2010, see govpulse for any updates (National Atlas of the United States, Jan. 27, 2011)

Civil time is based on a worldwide system of 1-hour time zones, which are spaced 15 degrees of longitude apart. (The time zone boundaries are usually irregular over land, and the system has broad variations; local time within a country is the prerogative of that country's government.) All places within a time zone, regardless of their longitudes, will have the same civil time, and when we travel over a time zone boundary, we encounter a 1-hour shift in civil time. The time zones are set up so that each is an integral number of hours from a time scale called Coordinated Universal Time (UTC). Time laboratories all over the world contribute to UTC, which is really an average of the readings of many very precise atomic clocks. UTC is accurately distributed by GPS, the Internet, and radio time signals. So the minute and second "ticks" of civil time all over the world are synchronized and counted the same; it is only the hour count that is different. (There are a few odd time zones that are a ¼ or ½ hour offset from neighboring zones; for example, Newfoundland is ½ hour ahead of Labrador and Nova Scotia. The minute count is obviously different in these places.) See our page on U.S. Time Zones for more information.

UTC is required by international convention to remain within 0.9 seconds of another kind of Universal Time called UT1. UT1 is determined by astronomical observations, and it can be considered the modern equivalent of Greenwich Mean Time, that is, mean solar time at longitude zero. However, the position of the Sun in the sky is no longer used for determining astronomical time. Instead, quasars — very distant galaxies that emit natural radio noise — are continually monitored, as the Earth rotates, by a worldwide system of radio telescopes. The International Earth Rotation and Reference System Service (IERS) combines data from many observatories and publishes the results. The Sun no longer has to be observed daily to determine time astronomically; an equation suffices to relate the Earth's observed rotation angle, with respect to the quasars, to UT1. Because the Earth's rotation undergoes small variations, and is gradually slowing, UT1 and UTC tend to drift apart, and an extra second must be inserted into UTC every year or so (a "leap second") to keep the difference between the two kinds of time within the allowed range. See our page on Universal Time for more information.

To summarize: our civil time is based on UTC, which is always within 0.9 seconds of UT1, which is the modern equivalent of mean solar time at Greenwich. But suppose we are not at Greenwich? Mean solar time is different at every longitude. However, civil time is the same for all locations within a time zone. Therefore, civil time can be considered a good approximation to local mean solar time only along the defining meridian of longitude in each time zone, and only when daylight time (summer time) is not in effect. The longitudes of the defining meridians of the time zones, expressed in degrees, are evenly divisible by 15, and are usually near the center of their zone. For other places in the same time zone, mean solar time is 4 minutes earlier (later) than civil time for each degree of longitude east (west) of the defining meridian. For example, Memphis, Tennessee, is at longitude W 90°, on the defining meridian of the U.S. central time zone, so there, mean solar time is essentially the same as central standard time. However, Omaha, Nebraska, is at longitude W 96° and would measure mean solar time that is always 24 minutes later than central standard time. This constant offset, different for each location, actually applies to both kinds of solar time, although apparent solar time also is subject to the Equation of Time variations.

### The Equation of Time

The graph on the right shows the Equation of Time — the difference apparent solar time minus mean solar time — in minutes as a function of date (click on it for a larger version).

The plot is computed for the Greenwich meridian and year 2014, but the plot is essentially the same from year to year and for all longitudes. Basically, the plot shows whether the true Sun is ahead of (+) or behind (–) the fictitious mean sun in the sky; equivalently, whether a sundial would show times "fast" or "slow" with respect to local mean solar time. For example, at the end of July, the Equation of Time value is about –7 minutes, so a sundial would show local apparent solar time that is about 7 minutes behind local mean solar time. (As explained above, local mean solar time is the same as civil time, to within 0.9 second, if the sundial is located on the central meridian of a time zone. Otherwise there is an additional constant time offset, depending on longitude.)

The next graph on the right shows the rate of change of the Equation of Time, in minutes per day, as as function of date.

This graph indicates whether the true Sun is moving faster (+) or slower (–) across the sky, in its daily east–west motion, than the fictitious mean sun, which moves at a constant rate. (To obtain the fractional difference in the rate of motion, divide the value shown on the plot by 1440 minutes/day.) Another way of thinking about this graph is that it shows the length of a solar day, measured by two successive passages of the true Sun cross the local meridian (the time between two successive solar noons), compared to a 24-hour clock day. For example, the graph shows that at the end of March, the Equation of Time values are changing by about +0.3 minutes/day, which means the Sun is moving slightly faster than average across the sky (by about 0.02%) and each solar day during this time is about 0.3 minutes shorter than 24 hours.

It is this second graph that is relevant to the issue of when the earliest sunset or latest sunrise occurs near the winter solstice, or when the latest sunset or earliest sunrise occurs near the summer solstice. In particular, the graph shows that the Sun reaches its slowest rate of motion across the sky in late December, with each solar day almost 0.5 minute longer than 24 hours. That circumstance, taken alone, tends to move both sunrises and sunsets later with respect to the clock each day from early November until mid February. However, this effect is in competition with a purely geometric phenomenon: the length of the Sun's track across the sky (as we perceive it) is also changing, and that determines the amount of time each day that the Sun is above the horizon. In the Northern Hemisphere, the winter solstice, around December 21, is the day with the least amount of daylight. If we did not know about the Equation of Time, we would expect the earliest sunset and the latest sunrise to occur at the solstice. But the Equation of Time in effect pushes both sunrises and sunsets later than when they would occur purely due to geometry. Because of this, sunsets start to shift later each day at an earlier date, and sunrises don't start shifting earlier each day until a later date. That is, the Equation of Time moves the date of earliest sunset to before the solstice and the date of latest sunrise to after the solstice. A similar situation, although less extreme, occurs in June near the summer solstice. This issue is explained more thoroughly in our page on Sunrise and Sunset Times Near the Solstices.

Sometimes a graph of the Equation of Time is combined with a graph of the Sun's angular height above the horizon at noon, since both are a function of the day of year and both repeat (with small differences) from one year to the next. If the Equation of Time is plotted as the x-axis and the Sun's noon height is plotted as the y-axis, the result is a figure-8 curve for the year called the analemma. (Most often, the Sun's height is measured in degrees, + or –, relative to the celestial equator, a quantity called the Sun's declination.) The analemma is the pattern of Sun images that you would obtain if, using a fixed camera and a multiple exposure setting, you recorded the Sun at noon mean solar time on every day of the year that it was clear; see, for example this picture.

A table of Equation of Time values, at one-day intervals, is printed in the annual booklet Astronomical Phenomena. You can get the same information online, for a range of specific dates, from our page on the Geocentric Positions of Solar System Objects (be sure to select Apparent Geocentric Right Ascension and Declination for the Sun). A simple formula for the Equation of Time is given on our page about computing Approximate Solar Coordinates.

### What Causes the Equation of Time?

Where does the difference between apparent and mean solar time — the Equation of Time — come from? Why does the Sun not cross the sky at a uniform rate, regardless of time of year?

As stated in the Introduction, the Sun's daily motion in the sky is the sum of two components: one due to the rotation of the Earth and the other due to the Earth's orbital motion about the Sun. The component due to the rotation of the Earth is fast (15.04°/hour from east to west) and constant to within milliseconds per day. The component due to the Earth's orbital motion is much slower (1°/day = 0.04°/hour from west to east) and can vary over the course of a year by as much as ±12%. It is the variations in this latter component that cause the Equation of Time. The motion of Earth in its orbit affects the Sun's angular speed across the sky in two ways:

• Seasonal variations due to the 23.4° tilt, or obliquity, of the Earth's axis. Because of this, from our vantage point on the Earth, we see the Sun traverse the daytime sky in a much higher path in summer than in winter; spring and fall are transition times. In the early part of the year, as we change seasons (from winter to summer in the Northern Hemisphere), the Sun runs along a more northerly track in the sky every day; around the time of the March equinox, the Sun's path is about 0.4° higher each day. What is actually happening then is that the Sun's apparent daily motion across the sky is not purely east-to-west — there is a small component toward the north. As a result, at this time of year, the direction of sunset along the horizon is always a bit more northerly than the direction of the previous sunrise. In most of the second half of the year, the Sun has a component of motion toward the south and the opposite is true. The geometry of the sky is such that the Sun's east-west rate is slower when the north-south rate is large and faster when the north-south rate is small (in such a way that the combined rate from the Earth's orbital motion is constant, considering this effect alone). For timekeeping purposes, it is only the east-west rate that matters. The cumulative effect of the east-west rate variations is that the true Sun falls behind then moves ahead of the fictitious mean sun in several cycles over the course of a year. The part of the Equation of Time due just to this effect is zero at the equinoxes and solstices, and can reach ±10 minutes at other times of the year. If the Earth's axis were perpendicular to its orbit, this effect would disappear (as would our seasons).
• The non-circularity, or eccentricity of the Earth's orbit around the Sun. The Earth's orbit is almost a perfect circle, but not quite; it is an ellipse that is flattened by 0.014% (the orbital eccentricity is 0.0167). The Sun is at one focus of the ellipse. The Earth is closest to the Sun, that is, at perihelion, in early January and is farthest from the Sun, at aphelion, in early July. By Kepler's third law, the Earth's speed in its orbit is maximum at perihelion and minimum at aphelion, in such a way that a line from the Sun to the Earth sweeps out equal areas in equal times. Viewed from the Sun, then, the Earth's angular rate of advance in its orbit would be faster near the beginning and end of each year (by almost 7%) than in the middle. For us on the Earth, we see this variation reflected in that part of the Sun's apparent motion across the sky that is due to the Earth's orbital motion. This means that the Sun's total rate of motion across the sky (the sum of the components due to the Earth's rotation and orbital motion) seems slowest in December and January and fastest in June and July. The Equation of Time reflects the cumulative effect of these changes in orbital speed, with the largest offsets in early April (–7.5 minutes) and early October (+7.5 minutes). If the Earth's orbit were a perfect circle, its orbital speed would be constant and this effect would disappear.

The graph at the right shows the contributions of these two components to the Equation of Time: the effect of the obliquity of the Earth's axis is the green line and the effect of the eccentricity of the Earth's orbit is the orange line.

The blue line is the resulting Equation of Time from the two effects together, the same as the blue line in the first graph above. If you click on this graph to obtain a larger version, you may be able to notice that the values represented by the green and orange lines at any date do not quite sum up to the value represented by the blue line on the same date (although close, within about 0.5 minute). That is because the two effects are not strictly additive; the obliquity effect modulates the eccentricity effect.

As can be seen from the graph, the two effects have similar magnitudes, so that it is not correct to say that either the obliquity effect or the eccentricity effect dominates or is the "main cause" of the Equation of Time. However, there have been times in the Earth's past (and will be in the future) when both the obliquity and the eccentricity were quite different than now, and the Equation of Time graph would look quite different. Of course, there were no humans with clocks then to notice. We can also imagine establishing a mean solar time for future astronauts on Mars. Mars has a length of day and obliquity that are quite similar to those of the Earth (24.6 hours and 25.2°, respectively), but its orbit is five times more eccentric than that of the Earth, so in a Martian Equation of Time, the eccentricity effect would have a more prominent role than it has for us.