The Equation of Time
Why Sundial time Differs From Clock Time Depending On Time Of Year
It is (at the time of initial writing) 12 December 2004, in Sheffield, United Kingdom. It is now getting dark quite early; but the time of sunset in the afternoon has just about reached its earliest for the season; and we should now gradually be seeing the period of daylight in the late afternoon start to increase again.
But, isn't the shortest period of daylight the winter solstice, on or about the 21st or 22nd of December?
Actually, it is; but what we've just been talking about is the period of daylight in the afternoon. The total period of daylight is that in the morning, before noon, plus that in the afternoon. And, as we shall see, the period of daylight before twelve occlock noon is not always the same length as the period after twelve occlock.
One reason for this, of course, is Daylight Saving Time, which in Britain, operates from the latter part of March to the latter part of October. During this period the clocks are put forward one hour from the actual astronomical-based time. During this period the sun reaches the meridian, when it is due south, at somewhere around 13.00 BST (which is, in fact, 12.00 GMT). Another factor is the fact that we (in Sheffield) are not exactly in the centre of the time zone, but are instead about 1.5 degrees west of the meridian on which the time zone is based. This has a slight effect on the times of sunrise, local noon (the time of the sun crossing the meridian, when it is highest in the sky, due south), and sunset.
The third factor, which is variable throughout the year, is what is known as the Equation of Time. In short, sundial time lags behind clock time at some parts of the year, and leads ahead of it at others. Early in November, sundial time is about 16 minutes ahead of mean clock time, which means that the mornings get half an hour more daylight than the afternoons. In February, sundial time can be 14 minutes behind mean clock time, which means that the afternoons get a longer period of daylight than the mornings. Sundial time and mean clock time coincide on or about April 15, June 14, September 2, and December 25 (provided we are on the exact meridian for which the time zone is set, and disregarding factors such as Daylight Saving Time). The length of daylight in the evenings seems to increase late in December not only because we have passed the winter solstice and the Sun is moving up north again, but also because sundial time, previously ahead of clock time, is now losing and falling behind clock time.
What is the Equation of Time, and to what is it due?
The Earth spins on its axis at an almost uniform rate, taking 23 hours, 56 minutes, and 4 seconds for each rotation. This is known as a siderial day, and is the time it takes for the distant ("fixed") stars to return to the same position in the sky.
The Sun, however, does not return to the same position in the sky in this time. During this time, the Earth has moved somewhat in its orbit round the Sun, and the Sun seems to have progressed further eastward relative to the distant stars, as we are looking at it from a different viewpoint. Thus it takes the Sun an average of 24 hours, from one meridian passage to the next. There are 366.25 siderial days in a year, but only 365.25 solar days, which is one less. (Because the true year is not a fixed number of days, the modern calendar uses a year of 365 days, except for one in four, which is leap year with an extra day, to enable the calendar to return to synchronism with the true year.)
If the Sun's movement eastwards relative to the distant stars was at a fixed rate, then the solar day would be constant, just as a siderial day is. However, the Sun's movement eastwards relative to the distant stars varies throughout the year, primarily for two reasons.
Reason One. The Earth's equator is not parallel to its orbit round the Sun, but is inclined to it by an angle of about 23.5 degrees. (This is what causes the seasons.) The Sun's apparent relative movement among the distant stars generally has an Easterly component (change in Right Ascension), and a North-South component (change in Declination). At the equinoxes (March and September), when the Sun is crossing the equator, the North-South component of its motion is at its greatest, and its Easterly component at its least. At this time, the Solar day will be less than 24 hours (but still greater than the siderial day). At the Solstices (June and December), the change in Declination is zero, but the change in Right Ascension is greater than average; at this time, the Solar day is more than 24 hours.
(Right Ascension is measured Eastwards along the celestial equator from the vernal equinox. This is the point in the sky where the Sun crosses the celestial equator from south to north, and is known as the First Point of Aries. About twenty one centuries ago this was actually in the constellation of Aries; but now, due to Precession, it has moved back a constellation, so that it is now in Piscies. Declination is measured in degrees north or south of the celestial equator. The celestial equator is an imaginary line in the sky, which is in the plane of the Earth's equator. Right Ascension corresponds to longitude on the Earth's surface, and Declination to latitude.
Celestial Longitude is similar to Right Ascension, but is measured along the plane of the ecliptic, rather than the celestial equator. The ecliptic is the plane of the Earth's orbit round the Sun. Due to the inclination of the ecliptic to the equator, the Sun's celestial longitude is generally not the same as its Right Ascension, being the same only at the equinoxes (when they are both zero or 180 degrees), and at the solstices (when they are both 90 or 270 degrees). It is basically the difference between the Suns' Celestial Congitude and Right Ascension which is responsible for this inequality in the length of the solar day.)
Reason Two. The Earth's orbit around the Sun is not a perfect circle, but is instead an ellipse. In accordance with Kepler's Second Law, the Earth moves round the Sun faster when it is at its closest point (early in January), then when it is at its furthest point (early July). Thus in January the Sun's apparent Eastward movement relative to the distant background stars is greater in January than in July, with the consequence that the length of the solar day will be longer in January than in July.
Since official clock time is based on the mean length of the solar day as averaged through the year, when the true solar day is shorter than the mean solar day, the sun-dial time will very gradually gain on the clock time; whereas when the true solar day is longer than the mean solar day, the sun-dial time will gradually lose on clock time. In actual fact, the term GMT stands for Gweenwich Mean Time, which is the mean solar time at the meridian of Greenwich, in London, which is generally taken as the reference point for longtitude on the Earth, and thus, by definition, has zero longtitude. As stated above, Sheffield has a slight West longitude.
The Equation of Time is the amount by which true solar time differs from mean (clock) time. Apparently by general convention a positive value for the equation of time means that the sun-dial time is ahead of clock (mean) time, while a negative value means sun-dial time is behind clock time - viz, early November, the equation of time is +16 min, which means that the Sun is due south as seen from Greenwich, at 11.44 GMT; whereas in early February, the equation of time is -14 min, and the Sun is due south as seen from Greenwich, at 12.14 GMT.
It is interesting to note that if the Equation of Time were entirely due to Reason One above, then the Equation of Time would reach maximum positive twice a year, between the equinoxes and the solstices; maximum negative twice a year, between the solstices and the equinoxes; and zero four times a year. The graph of the Equation of Time due to this cause goes through TWO complete cycles per year.
If it were entirely due to Reason Two above, then the Equation of time would be zero twice a year, when the Earth were at perihelion (closest to the Sun) in January, and when at aphelion (farthest from the sun) in July. The Equation of time would be negative between January and July; and positive between July and the following January. The graph of the Equation of Time due to this cause goes through ONE complete cycle per year.
The dates when the Equation of Time is zero due to the two causes, do not coincide; but at some parts of the year the two effects reinforce each other, and at other times, they tend to counteract each other. On the whole, for planet Earth, Reason One (the inclination of the plane of the equator relative to the plane of the ecliptic) has a slightly greater effect, so we get two cycles per year, but they are not even nor of equal size.
Graph showing the Equation of time throughout the year can be found here, here, and here; there is a good table here, and another one here. (I think - at the time of writing - that one of the two tables may have the plus or minus sign for the equation of time, wrong for two, three or four dates in early September.) The two effects currently reinforce each other most when the Earth is near perihelion, in the northern hemisphere's winter months; and will sometimes tend to oppose each other around the northern hemisphere's summer months.
The Equation of Time for a planet with an orbit which is almost circular, but which has a comparitively high axial inclination, will be dominated by the effect referred to above as Reason One, and the planet's year will have two peaks (maximum positive equation of time), and two troughs (maximum negative equation of time) in each year; the two peaks and two troughs will be almost equal in magnitude. Such an example is Neptune, whose orbit is almost circular.
The Equation of Time for a planet with a relatively eccentric orbit, but negligible axial inclination, will be dominated by the effect referred to above as Reason Two, and there will likely be just one peak (maximum positive equation of time), and one trough (maximum negative equation of time), in the planet's year. Such examples are Mars (which has an axial tilt simlar to Earth's but much higher orbital eccentricity), for which sundial time can be up to 50 minutes different from mean time; and Jupiter (which has neglible axial tilt). It should be noted that Mars' rotation period is only about 40 minutes longer than Earth's, so the day is similar in length. Jupiter's axial tilt (3.5 degrees) is so slight that it probably has no significant effect on the equation of time, which is thus dominated entirely by the ellipicity of the orbit, and thus has only one period in the planet's year.
The Analemma. At noon, according to mean clock time (GMT if on the Gweenwich zero meridian), the sun will be almost due south. Its altitude will vary substantial throughout the year, as its declination varies from -23.5 to +23.5 degrees. It will also generally vary slightly in position, to the east or west of due south, due to the mean clock time not being the same as sundial time. If one could support a camera stationary throughout the year, pointing due south, and upwards at an angle of about 40 degrees to the horizontal; and expose the film very briefly at exactly noon (mean time) every few days, whenever the sky was clear, throughout the year, then the multiple images of the sun on the resultant multi-exposure photograph would form a very narrow, elongated, figure-of-eight, with the top (northern) lobe smaller than the bottom (southern) part. Such a figure is known as an analemma. In actual fact, if one could travel westwards along the equator at a uniform speed such that we took exactly 24 hours to arrive back where we started (more than 1000 miles per hour), then the Sun would not appear to move noticeably during the course of the day. If however, we continued for a whole year, the Sun would appear to trace out this slender figure-of-eight we call the analemma.
If instead, one points the camera in a different direction, and takes the exposures at a different time of day, e.g. late afternoon, one will get a similar effect, but the figure-of-eight will not be vertical, as here. An analemma for Mars can be seen here, from a multiple exposure photograph taken in late afternoons on Mars by the Pathfinder Project. This, instead of being a figure-of-eight, is shaped like a pear. As stated above, due to the eccentricity of its orbit, the Equation of Time on Mars just follows one cycle per Martian year, and the analemma does not cross over itself.
There is an encyclopedia article about the analemma here, which shows the Earth's and Mars' analemmas.
The fact that the northern lobe of the Earth's analemma is smaller than the southern reflects the fact that the crossover point in the figure-of-eight is north of the celestial equator; and although the Equation of Time is currently zero close to the solstices, the other two date of zero Equation of Time are not at the equinoxes, but are about three weeks later than the vernal equinox and three weeks earlier than the autumnal equinox (for the northern hemisphere). This is due to the effects of the axial tilt and the orbital eccentricity tending to reinforce each other around the time of perihelion, and opposing each other around the time of aphelion.
If one greatly stretches the image of the analemma horizontally, one might notice that it is slopes slightly, and is not completely symmetrical. This is because the times of perihelion and aphelion passage do not quite coincide with the solstices. Due to precession of the equinoxes, the seasons gradually shift round with respect to the perihelion and aphelion of the Earth's orbit, so over thousands of years, the graph of the Equation of Time, and the analemma, alter their shapes.
Graphs of various kinds, for some of the other planets, can be found here. For some reason, graphs of analemmas are plotted in various ways, and some people seem to place them horizontal when one might expect them to be vertical. The one for Jupiter, like that of Mars, does not cross over itself (because of Jupiter's extremely small axial tilt). The one for Saturn seems to cross over, but only just - one lobe is extremely minute compared to the other; Saturn spends most of its 29-Earth-Year year in the larger lobe. The two lobes of Neptune's, unlike those of the Earth, will be pretty similar in size, because Neptune's axial tilt is slightly greater than Earth's but its orbit has only a very slight eccentricity. Whether the analemma for a particular planet at a particular time will cross over itself depends not only on the axial tilt and orbital eccentricity, but also on the relative position of its equinoxes relative to the major axis of its orbital ellipse. At present, the earth's perihelion passage almost, but not quite, coincides with the northern hemisphere's winter soltice. If instead it coincided with one of the equinoxes (as it will in a some thousands of years), then the figure-of-eight would be greatly distorted, and at a slant. The analemmas for Mars would slant, and possibly cross over itself, if the perihelion passage happened to coincide with an equinox or be very near it.
In actual fact, for the Earth, no solar day ever differs from 24 hours by more than a matter of seconds. The reason clock time can be a quarter of an hour ahead or behind sun time is because of the cumulative effect of a slightly fast or slow day, over periods of weeks or months.
So, the date of earliest sunset is earlier than the winter solstice, and the date of latest sunrise is after the winter solstice. In Sheffield, the earliest sunset and latest sunrise are about nine or so days before and after the winter solstice; but closer to the equator, the difference between these various dates will be greater. Also, the date of date of earliest sunrise is before the summer solstice and the latest sunset after the summer solstice. (Bear in mind that Britain, and many other countries, have Daylight Saving Time in force during the summer months.)
Times of sunrise and sunset can be found in some diaries. There is a web-based program here for calculating times of sunrise and sunset at a particular date, at various places throughout the world. This is java-based, and may only be fully functional on certain compatible browsers, which should have java enabled in the browser's preferences.
This site here contains a link to a C program which can plot analemmas under various criteria. It seems to use formulae which are more accurate than those some other sites use, which may only give tolerably accurate results if the axial tilt and orbital eccentricity are low. I have myself downloaded and compiled this program, and have plotted analemmas for real and hypothetical planets with various axial tils and eccentricities, as well as various angles between the equinox and the major axis of the orbital ellipse.
The left hand picture below shows the appoximate shape of the Earth's analemma about eight to nine hundred years ago, when the orbital line of apsides coincided with the solstices (perihelion coincided with the northern winter solstice). The centre one shows the analemma of the present epoche, with the perihelion early in January, whilst the northern winter solstice is December 21-22. The right picture shows the shape of the analemma when the line of apsides coincide with the equinoxes, as will probably occur in around five thousand years time.
In summary, then:
Sun-dial time does not always coincide with mean clock time, but can be up to about a quarter of an hour ahead or behind clock time depending on time of year (taking into allowance adjustments for Daylight Saving Time and longitude differences).
In Sheffield (and similar latitudes) the earliest sunset occurs about nine days before the winter solstice, and the latest sunrise a similar amount of time after the winter solstice. The difference in time between the date of eariest sunset and latest sunrise is greatest in the lower latitudes as one approaches the tropics, and least as one approaches the arctic or antarctic circles.
The difference between sun-dial time and local mean time is known as the Equation of Time (being by convention positive when the sun-dial time is ahead of mean time, and negative when behind).
Multiple photographic exposures of the Sun, taken with a stationary mounted camera, at intervals of 24 hours (or exact multiples thereof), throughout the year, would show the Sun trace a path like a long slender figure-of-eight, known as an analemma.
The corresponding analemma for Mars is pear-shaped. (The intervals between exposures here must be multiples of approximately 24 hours 40 minutes, the duration of the Martian mean solar day.) In addition, due to Mars' orbit being more eccentric than the Earth's, sun-dial time on Mars can be up to about 50 minutes behind or 40 minutes ahead of local mean time.
Analemmas for other planets are of varying shapes, and may or may not cross over themselves (figure-of-eight style), depending on the relative importance of the planet's axial tilt and the eccentricity of its orbit, in determining the Equation of Time for the planet.
Last updated 28 December 2005