On the Units of Time

Part I: From Seconds to Days

by Edward Hahn

February 7, 2001

What is a Day?

What could be simpler? A day is the time it takes the earth to rotate once on its axis. This is divided into 24 hours of equal length, which are then further subdivided into exactly 60 minutes of 60 seconds each.

Unfortunately, only one paragraph into this article I've already made one absolutely incorrect statement, and made several simplifications, some of which are only a couple of hundred years or so old.

The first problem is to figure out how long the day is. This requires us to set a point a starting point and ending point for our measurement. This is easy for us - since the 1884 International Meridian Conference of Washington DC, treaty has set it by convention at midnight. To our ancestors, however, there was no easily observable phenomenon by which to mark midnight, and even us modern folk are hard pressed to point to anything more substantial than the alignment of hands on a watch. So, midnight isn't the best of choices for the pre-technological society.

The next possibilities - sunrise and sunset - won't work either, because we all know that the times of sunrise and sunset are always changing (due to the tilt of the earth's axis), with the most amount of daylight at the Summer Solstice, and the least amount of daylight at Winter Solstice in the Northern Hemisphere. So, the only reasonably consistent observation we can make to nail down the length of the day is to mark the time at high noon, also known as the solar meridian passage. So, while we mark the beginning and end of the day at midnight, we can only make an accurate measurement of this phenomenon halfway through the day, at noon.

As a side note, the point at which the day begins and ends has varied widely among different society, and still varies today: for example in the Jewish nation of Israel and the Islamic Kingdom of Saudi Arabia, the day begins at sunset, which echoes the practice used in ancient Babylon. The ancient Egyptians, whose religion was obsessed with the daily death and rebirth of the sun god Ra, marked the beginning of the day at sunrise. Both the Romans and the Chinese foresaw the Meridian Conference and chose midnight. Finally, for practical reasons astronomers and sailors used noon as the beginning of the day; sailors began abandoning it in 1805, while astronomers held out until 1925.

A Day is 360 Degrees Plus a Little Extra...

OK, back to the length of the day. The absolute mistake I made was to say that the day is the time it takes the earth to spin once on its axis - that is, exactly 360 degrees of rotation as viewed from space, looking down directly over the North Pole. The problem is that the earth is also simultaneously moving around the sun, making a full revolution in about 365 days. This means that in the time it takes the earth to spin once, the earth has moved an extra amount, a little less than a degree, along its orbit around the sun. herefore, for the time that it takes between successive meridian passages of the sun must account for the extra time it takes to rotate that extra almost-degree. (Figure 1)

Figure 1: During the course of a day, the earth both rotates on its axis and moves from Point A to Point B. After one rotation, the earth has moved far along enough in its orbit to require an additional extra amount of rotation to get the sun directly overhead again relative to the same location. (Angles and distances have been exaggerated for clarity.)

Thus, we've run into the difference between the sidereal day, or the period of time it takes for the same star to appear in the same position in the sky, and the solar day - the time it takes for the sun to appear in the same position. Over the course of a year, there will be one extra sidereal day than solar day, because of the revolution around the sun. Since this is spread more or less evenly across the entire year, the difference in length between the solar day and the sidereal day is 24 hours divided by about about 365 days - or not quite 4 minutes per day.

Figure 2: This watch is calibrated to mark Sidereal Time rather than Solar Time. The watch would be useful to astronomers and navigators; astronomers could make a note of when a particular star was observable at a particular position, and be assured that the object would be at that same position at the same indicated time throughout the year at that location. Navigators can use the differences in time of stellar observation to assist with determination of longitude.

This four minutes per day can be observed by watching the slow drift of the visible constellations, say, at 10pm each evening, as the year goes by. While this concept is tricky for most people who aren't astronomers or navigators to grasp ("what do you mean, there are different kinds of days with different lengths?"), if this were the only detail affecting the length of the day, it would still be relatively easy to work with, since the length of the day would be still be constant.

The Varying Length of the Day

Kepler's Quest for Perfection Results in Something Quite Different

The first problem is that the earth's orbit is not a perfect circle, with the added result that the earth's speed in its orbit is not constant, either.

Historically, the problem of describing the motion of the sun in the sky has vexed astronomers for hundreds of years. The problem is that since the world was created by God (taken to be an absolute truth in those days), it was assumed that His creation would be perfect - meaning perfect circles for the motion of sun. However, the perfection had a glaring error in that the motion of the sun against the background of stars was not constant.

Figure 3: Johannes Kepler, German Astronomer (1571-1630)

Johannes Kepler, a German astronomer and mathematician (and deeply religious man) worked on this problem for years, trying to figure out ways in which the sun's and planets' motions could be shown to be perfect circles around the earth. What he discovered instead undermined forever the earth-centered perfect universe, and also provided one of the founding tenets of astronomy as a science independent of astrology - his three laws of planetary motion. (Fortunately for him, the Protestant reformation had made for a somewhat less contentious religious environment than for his contemporary Galileo in Roman Catholic Italy.)

The laws can be stated (in a simplified form) as follows:

The planets move about the sun in elliptical orbits, with the sun at one focus
The planets move faster when closer to the sun
The distance of a planet from the sun, and its orbital period are related mathematically.

What's remarkable about this achievement is that he came up with these laws by observational data alone - largely provided by Danish Astronomer Tycho Brahe - and without any knowledge of the gravitational force or calculus. What can be done in an hour lecture in a undergraduate physics class took Kepler many years to complete.

Anyway, it's the first two laws which have an effect on the varying length of the day. The implication of the first law is that the amount of extra rotation needed (as shown in the previous section) between meridian passages of the sun is going to be different when the earth is closer to the sun vs. when it is farther away.

Figure 4: The earth's orbit is an ellipse, not a circle. (Ellipticity of the orbit is exagerrated.)

...and the second law is going to make this effect even more pronounced:

Figure 5: The earth moves faster when it is closer to the sun

Some people in the Northern Hemisphere have the impression that the sun must be closest to the earth during the summertime - because the weather is hotter. This is incorrect, the seasons are caused by the tilt of the earth's axis, which accounts for why the seasons are reversed in the Southern Hemisphere. Since the earth's orbit is almost, but not quite circular, the effect of the distance from the sun is rather small: the earth's closest approach to the sun is currently happening around the second of January.

The net upshot of both effects is that the time between meridian passages are lengthening from January to June, while they are shortening from July to December. Note that this is not discussing the hours of daylight - but the actual duration of the entire day.

A Swiftly Tilting Planet

The second phenomenon which affects the length of the day is the axial tilt of the earth:

Figure 6: The axial tilt of the earth. This is the main cause of the seasons, as it alternately exposes the Northern and Southern hemispheres to more direct lighting at different times of the year.

At four times during the year - the summer and winter solstices, and the vernal and autumnal equinoxes - the effect of the tilt is zero, because the meridian is either aligned parallel or perpendicular to the tilt of the axis.

However, because of the axial tilt, the motion of the earth in its orbit has both components parallel and perpendicular to the meridian during all other times. The perpendicular component in particular causes the local noon deviates by up to a total of about ten minutes from what we might expect. (I haven't drawn a diagram which illustrates this effect - that's because I haven't figured out how to do it in a way that makes any sense. Hate to say it, but you're going to have to take it on faith. :-) )

Adding It Up Gives Us the "Equation"

What's amazing about all of this is that the inconstant length of day has been known since the time of Claudius Ptolemy (ca. 150 AD), although the reasons for them were a mystery. However, it was not really a big deal to society until much more recently. When timepieces were being developed which did not depend directly on a celestial phenomenon (e.g. a clock, and not a sundial), the variation in the length of the day was a problem. It's difficult enough to build a mechanical device which can measure a consistent interval of time; making one which can handle variations in the actual motion of the sun over the course of a year would be even harder.

For this reason, the actual movement of the sun in the sky began to be abandoned in urban areas, and a convenient simplification, mean solar time was adopted in several European cities around 1800. This creates an average or "mean" day - which is essentially the average length of all the days over the course of a year - and provides a consistent time interval for clocks and watches to measure.

But the motion of the sun couldn't just be ignored with the invention of mean solar time - the sun was still the most convenient observable event by which to set one's clock or watch. Thus, some way to relate mean solar time to the sun was needed.

Taking the two effects - the elliptical shape of earth's orbit, combined with the axial tilt - it is possible to construct the famous "Equation of Time". This is not really an equation, rather a table or chart which gives the difference between when the sun actually crosses the meridian compared with mean solar time:

Figure 7: The Equation of Time. This gives the difference between the observed meridian passage and the "average" or mean solar time.

Using the Equation of Time is fairly straightforward: the local noon is observed by noting when the shadow of a vertical stick or other object aligned with the north-south meridian line. The Equation of Time value for the particular calendar day is then added to 12:00, and the clock is set to that time.

For example, on May 1, the Equation of Time shows the sun as 2.84 minutes (or 2 mins 50.4 secs) fast; therefore, when the local meridian passage is observed ("apparent noon"), the clock is set to 11:57:09.6 am.

The Equation of Time has been used extensively for several hundred years; E.G. Richards documents the following stanza from the 1752 Wing's Sheet Almanac:

April the Fourth, and June the Sixth remember,
August the Twentieth, and Twenty-fourth December;
On these Four Days and none else in the year,
The Sun and the Watch both the same Time declare.

Nowadays, however, there is an additional complication - standard timezones, adopted worldwide in 1884. A reference meridian is used to define the time for an entire geographic region, and local time is no longer used. Any use of the sundial and Equation of Time to set your watch must also add in an additional compensation of up to 30 minutes or more - or risk being early or late for your favorite television program.

As should be obvious by now, the Equation of Time complication as recently implemented on a watch by Breguet, Audemars Piguet, and Patek Phillipe, is mainly to show off watchmaking design skills, and is not very useful - unless you happen to set your watch by sundial.

Finally, the shape of the curve which describes the Equation of Time changes as time goes by, since the point of closest approach of the earth is slowly advancing through the calendar at a rate of one day every 57 years. The full cycle of the point of closest approach against the calendar is about 21,000 years. Write Phillipe Stern now to insist that your descendants 5,000 years from now have a properly working Star Caliber 2000.

Figure 8a: The Audemars Piguet Equation du Temps. Audemars Piguet displays the numerical value of the equation of time as a separate indication, using the bezel as the scale. Note the correction for Geneva local time vs. Central European Standard Time engraved on the bezel.

Figure 8b: The Patek Phillipe Star Calibre 2000. In contrast to the AP, Patek Phillipe has implemented a "running" equation of time - the sun hand acts like a minute hand which is tied to the local apparent sun. At local apparent noon, the hour hand will be close to 12 o'clock, and the sun hand will be at the 60 minute mark. Note that the watchmaker must place the sun hand on the watch with the correct geographic correction for standard vs. local time.

The Mummy Invents the Hour

The Egyptians are generally credited with inventing the 24 hour day. However, their hour was not constant in length - there were exactly twelve hours of daylight and twelve hours of night in every single day. Because the latitude of many of the great cities of ancient Egypt are similar to those of the Southern US (e.g. ancient Memphis and the Pyramids of Giza are at the same latitude as New Orleans), there was significant variation in the length of an hour throughout the year, and except for the equinoxes, the length of an hour during the day was different from the length of an hour at night.

The reasons why they insisted on 12 hours for daylight and night was due to their religion. In the Egyptian Book of the Underworld (or "Duat"), the story of the journey of the sun god Ra through the underworld beginning at sunset is described. Ra must pass through 12 stages or "hours" of night before being reborn in the eastern sky. The Egyptian Book of the Underworld is found reproduced in many tombs of pharaohs in Egypt - since the pharaoh was considered the living incarnation of any number of gods, it was often painted or even carved onto the stones of the tombs, to assist the spirit ("ka") of the pharaoh on his way through the underworld.

Figure 9: Papyrus Version of the Egyptian Book of the Underworld (Duat). Each hour is represented by a vertical section of the scroll, and depicts the hazards of the underworld that the sun god Ra must overcome during that hour. At dawn, Ra is reborn at the eastern horizon in the form of the rising sun.
This method of having 24 hours or parts to a day was then picked up by the Greeks and Romans, and were handed down to us (by the way, horology uses the same root as hour: Latin for time, "hora".) Despite having lengths of hours which were constantly shifting, this was really not that big a deal prior to the invention of clocks.

However, once clocks were invented around the fourteenth century, and made very accurate with the development of a pendulum escapement in the 1600s, it became imperative to have equal hours for a clock to work with. This is why the hours measured by a clock are "of the clock" or "o'clock" - to distinguish them from hours measured by the sky.

Finally, minutes and seconds were developed by adapting the Babylonian principle of dividing things into 60 parts. The first and second divisions of the hour (Latin "pars minuta prima" (first small part) and "partes minutae secundae" (second small part - as in first, second, third, etc.)) have given us our names for these units of time (and angular divisions of degrees).

Now That We Have That Straightened Out, Let's Throw It Out the Window

So, we now know what a day is in detail: the period which it takes the earth to rotate through successive solar meridian passages, as made consistent by averaging over the course of the entire year to cancel out the effects of an elliptical orbit and the axial tilt of the earth. The problem is that we are 44 years too late.

As of 1956, the length of a second has been freed from the vagaries of the earth's motion, and is now defined by the Systemé International d'Unités equal to "the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom in zero magnetic field." This means that the values for units of time are no longer tied to the motion of the earth, and instead are tied to innate properties of matter. In fact, the minute, hour, day, and even the average "tropical" year are defined as exactly 60; 3600; 86,400; and 31,556,925.9747 seconds respectively.

The arbitrary reference year of 1900 AD was chosen to fix the value of the second and tropical year. Unfortunately, the earth has slowed down slightly over this time, such that the mean solar day in the year 2000 is about 1.7 milliseconds longer than it was in 1900, and is getting longer. This is what necessitates the addition of a "leap second" every 588 days or so - pausing the march of the clock and calendar so that we can keep it aligned with the motion of the sun.

Unfortunately for us, the fact that the mean solar day is getting longer means that these leap seconds will need to be added more and more frequently. Such is the fate of nailing our units of time to things independent of the motion of the earth.

Bibliography and Suggested Reading

Primary scientific references for writing this article has been provided largely by two excellent works which have been written for the millennium: Mapping Time: the Calendar and Its History by E.G. Richards (Oxford University Press, 1998, ISBN 0-19-286205-7), and Marking Time: the Epic Quest to Invent the Perfect Calendar by Duncan Steel (John Wiley and Sons, 2000, ISBN 0-471-29827-1). These two works are written primarily for the layman, although they both contain a good dose of mathematics and astronomy.

Of the two, Steel's book appears to be more definitive, as he has written this book in response to incorrect statements about the calendar made by such august authorities as the US Naval Observatory and the Royal Greenwich Observatory - in addition to those in more "popular" accounts of the calendar.

Richards' book may appeal to both the sociologists and the mathematicians out there, as he investigates several ancient and modern non-western calendars, and includes a section on algorithms to convert between these calendars.

Of particular utility in understanding how the stars and planets move is a freeware program, called Home Planet, which has a host of astronomical functions available. This program can be found for free at www.fourmilab.ch.

There are some interesting Equation of Time calculators out on the internet; SunAngle has an interesting Java applet which uses a very accurate algorithm.

Click here to continue to Part II

Image of Sidereal Time Watch is reproduced by kind permission of Pieces of Time of London

Image of Johannes Kepler is in the public domain

Image of Audemars Piguet Equation of Time by Christian Schörg

Image of Patek Phillipe Star Caliber 2000 by Patek Phillipe

Image of Egyptian Book of the Underworld by Egyptian Museum, Cairo