August 2006 Feature: The Equation of Time

For simplicity, let's say you live on the meridian that governs your standard time zone. Let's suppose, also, that you have a clock and a sundial that are telling you the "correct" time. On August 1, you'd notice that the clock runs 6 minutes ahead of the sundial. When the sundial reads 10 a.m., the clock reads 10:06 a.m. Or when the sundial reads 12:00 noon, the clock says it's 12:06 p.m.
Let's fast forward to September 1. On this date, the sundial reads 10 a.m. when the clock reads 10 a.m. At 12:00 noon by the sundial, the clock reads 12:00 noon as well. On or near September 1, the sundial and the clock agree -- very convenient! Let's now jump to September 30 -- on which date the sundial reads 12:00 noon at the same time the clock says it's 11:50 a.m.
What in the world is going on? Assuming that the clock is constant, does the Earth's daily period of rotation vary throughout the year? Well, it's said that the Earth's rotation does vary and that overall, it is slowing down. But this subtle change is hard to detect and measure, especially in the course of just one year. For our purposes, we'll assume a uniform rotation for our planet Earth.

Earth's Rotation

It's often said that the Earth rotates once in 24 hours. What is really meant is that in a mean period of about 24 hours, the Earth rotates once on its axis relative to the Sun. One 360-degree rotation of Earth happens in 23 hours 56 minutes and 4 seconds. But during this rotational period, the Earth has orbited nearly one degree in its journey around the sun. So the Earth has to rotate another degree or so (for a total of about 361 degrees), before it can rotate once in reference to the sun. The Earth takes about 4 minutes more to rotate this extra degree, so the mean time period from solar noon to solar noon represents 24 hours (23 hours 56 minutes + 4 minutes = 24 hours).

Solar or Sundial Noon

I suppose an explanation of solar noon (sometimes called local apparent noon) is in order. Solar noon is pretty much synonymous with midday. Halfway between sunrise and sunset, the Sun reaches its highest elevation for the day and casts the day's shortest shadow (in the tropics, however, it's inevitable that some days display no noonday shadow at all). At true solar noon, the Sun is in one of three places: at zenith (straight overhead), due south of zenith or due north of zenith. In our northern temperate latitudes, true solar (sundial) noon always finds the Sun shining due south, with the day's shortest shadow pointing due north.
At true solar noon, the Sun is said to be on the meridian or at upper transit. If you wish to know the time of the Sun's transit in your sky, check out this US Naval Observatory web page. Remember that if you use daylight savings time, you are pretending that it is one hour later than it actually is. In other words, you have to subtract one hour to convert to standard time.
Leave it to our harried species to decree by law that it's one hour later than what it is! I suspect modern day humanity could cure itself of this neurotic obsession to hurry through life if it'd settle down to enjoy the moment and to live more in rhythm with natural cycles.

Equinoxes & Solstices

The period of time lapsing from true solar noon to true solar noon varies, depending on the time of year. If you're mindful of what season it is, it's fairly easy to know whether the days are longer or shorter than 24 hours. For several weeks preceeding and following the March equinox, and for several weeks before and after the September equinox, the days as measured from noon to noon are shorter than 24 hours long. On the other hand, the several weeks preceeding and following the June solstice and the December solstice give us days that are longer than 24 hours.
The shortest days of the year always happen near the equinoxes, whereas the longest days always happen near the solstices. Halfway between an equinox and a solstice, or halfway between a solstice and an equinox, the days as measured from noon to noon are almost exactly 24 hours long.

Geometry

If you wish to understand the geometry involved, there's no substitute for a globe that shows the ecliptic. The ecliptic maps out the Sun's position relative to the equator for every day of the year. First, use a caliper or circle compass to measure 15 degrees of longitude along the Earth's equator. Now put one end of the caliper at one of the two equinox points (where the equator intersects the ecliptic), and place the other end of the caliper on the ecliptic. You'll note that 15 degrees along the ecliptic from either equinox point amounts to less than 15 degrees of longitude -- which tells you the days as measured from noon to noon are shorter than 24 hours in length.
Now find either the northern or southern solstice point (where the ecliptic intersects the Tropic of Cancer or the Tropic of Capricorn). If you place one end of the caliper at the solstice point and measure 15 degrees along the ecliptic, you'll see that it amounts to more than 15 degrees of longitude -- hence, the days are longer than 24 hours.
By the way, 15 degrees along the ecliptic represents the Sun's apparent change of position in about one-half calendar month. Click here for an illustration of the celestial coordinate system.

Enter Aphelion & Perihelion

In the first week of July, the Sun is farthest from Earth for the year, an event that is called aphelion; and during the first week of January, the Sun is closest for the year, an event that is called perihelion.
Around aphelion, the Earth travels most slowly in its orbit. Therefore, as the Earth rotates 360 degrees on its axis, the Earth covers the least amount of distance in its orbit. Thus, the Earth needs to rotate a little less angular distance to bring the Sun back to the meridian or its noontime position. In other words, aphelion acts to shorten the day. However, as aphelion takes place rather close to the June solstice, the solstice trumps aphelion. The days are still over 24 hours long at this time of year, though they'd be longer still if it weren't for aphelion partially canceling out the solstice effect.
Days in June are about 13 to 14 seconds longer than 24 hours, but this time difference accumulates throughout the month. For example, around mid June, the sundial and the clock show little if any discrepancy -- noon by the clock and noon by the sundial happen at nearly the same moment. But each day therafter, sundial noon keeps coming some 13 to 14 seconds later according to the clock. By the first of July, this time difference accumulates: the sundial reads noon while the clock reads 12:04 p.m.
Around perihelion, the Earth moves most swiftly in its orbit, so the Earth travels a greater distance daily than it does at other times of the year. Since perihelion coincides closely with the December solstice, the combination of the two adds up to the longest days of the year. The days near the December solstice are a whopping 30 seconds longer than 24 hours. Around Christmas Day, the sundial and the clock agree. But in deep, dark December, the Sun returns to its noontime position 30 seconds later daily according to the clock. Therefore, four days after Christmas, sundial noon comes at 12:02 p.m. local clock time.

Equation of Time

The equation of time is a simple graph that enables you to convert sundial time to clock time, and vice versa. Looking at the graph below, you can see that sundial noon and clock noon are one and the same around September 1. Since we are approaching the equinox at this time of year, the days are shorter than 24 hours -- by about 20 seconds. Since these 20 seconds accumulate daily, this adds up to 10 minutes by the end of the month (20 seconds x 30 days = 600 seconds = 10 minutes). Thus, around September 30 or October 1, 12 noon by the sundial coincides with 11:50 a.m. by the clock.
The clock time on this equation of time graph refers to your local clock time. If you live on the meridian that regulates your time zone, then local time and standard time are one and the same. Otherwise, for every degree that you live east of your time zone meridian, true noon comes 4 minutes earlier by standard time. On the other hand, for every degree that you live to the west of your time zone meridian, then true solar noon comes 4 minutes later than is indicated on the graph.

Equation of Time Graph