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

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

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:

  1. The planets move about the sun in elliptical orbits,
    with the sun at one focus
  2. The planets move faster when closer to the sun
  3. 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

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

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.

here to continue to Part II

Image credits – all images are copyright © 2001 by Edward
Hahn, except as follows:

  • 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
  • Image of Patek Phillipe Star Caliber 2000 by Patek
  • Image of Egyptian Book of the Underworld by Egyptian
    Museum, Cairo

**Note: Thanks to Jay Lewis for catching an error.

Copyright Edward Hahn © 2001