Equation of time

Apparent time versus mean time

The rotation of the Earth makes a good clock because it is, for all practical purposes, constant. Of course, scientists are not practical and care about the fact that the length of the day increases by one second every 40 000yrs. For the rest of us, it's just a matter of finding a convenient way to determine which way the Earth is pointing. Stars would be good, but they are too dim (and too many) at night and go away during the day. A useful aid is the Sun, which is out and about when we are and hard to overlook. Unfortunately, the apparent position of the sun is determined not just by the rotation of the Earth about its axis, but also by the revolution of the Earth around the Sun. The following will explain exactly how this complication works, and what you can do about it.

The diameter of the Sun as seen from the Earth is 1/2 degree, so it moves by its own radius every minute.

That means it will be hard to read a sundial to better than the nearest minute, but then, we don't bother to set our clocks much more accurately than that either. Unfortunately, if we define the second to be constant (say, the fraction 1/31 556 925.974 7 of the year 1900, the "ephemeris second"), then we find that some days (from high noon to high noon) have more than 86,400 seconds, and some have less. The solar Christmas day, for example, is 86,430 seconds long. The discrepancy between "apparent time" and "mean time" can add up to +/- 15min. How does it come about?

The inclination of the ecliptic

First note that the Earth rotates on its axis not once in 24hrs but once in 23hrs 56min 4sec. It's just that in the course of a 365dy year, the Earth must turn an extra time to make up for its orbit about the sun.

The trouble comes in because this 3min 56sec is only an average value. Think of an observer sitting at the north pole on a platform which rotates once every 23 hrs 56 min 4 sec. She will see the stars as stationary and the sun as moving in a circle. The plane of this circle is called the "ecliptic" and is tilted by 23.45 deg relative to the equatorial plane. The observer will see the sun move from the horizon, up to 23.45 deg, then back down to the horizon. The sun will move at a constant speed (I'm lying, but wait till later) along its circle, but the shadow cast by the North Pole (the one with the red and white candy stripes) will not move at a constant rate. When the sun is near the horizon, it must climb at a 23.45 deg angle, so that it has to move 1.09 deg before the shadow moves 1 deg.

On the other hand, in the middle of summer, the sun is high in the sky taking a short cut, so it must move only 1 deg along its circle to cause the shadow to move 1.09 deg. This effect generalizes to more temperate climates, so that in spring and fall the 3 min 56 sec is reduced by the factor 1.09 to 3 min 37 sec, whereas in summer and winter it is correspondingly increased to 4 min 17 sec. Thus a sundial can gain or lose up to 20 sec/dy due to the inclination of the ecliptic, depending on the time of year. If it is accurate on one day, six weeks later it will have accumulated the maximum error of 10 min.

The seasonal correction is known as the "equation of time" and must obviously be taken into account if we want our sundial to be exact to the minute. If the gnomon (the shadow casting object) is not an edge but a point (e.g., a hole in a plate), the shadow (or spot of light) will trace out a curve during the course of a day. If the shadow is cast on a plane surface, this curve will (usually) be a hyperbola, since the circle of the sun's motion together with the gnomon point define a cone, and a plane intersects a cone in a conic section (hyperbola, parabola, ellipse, or circle). At the spring and fall equinox, the cone degenerates to a plane and the hyperbola to a line. With a different hyperbola for each day, hour marks can be put on each hyperbola which include any necessary corrections. Unfortunately, each hyperbola corresponds to two different days, one in the first half and one in the second half of the year, and these two days will require different corrections. A convenient compromise is to draw the line for the "mean time" and add a curve showing the exact position of the shadow points at noon during the course of the year. This curve will take the form of a figure eight and is known as an "analemma". By comparing the analemma to the mean noon line, the amount of correction to be applied generally on that day can be determined. At the equinox, we found that the solar day is closer to the sidereal day than average, that is, it is shorter, so the sundial is running fast. That means in fall and spring the correct time will be earlier than the shadow indicates, by an amount given by the curve. In summer and winter the correct time will be later than indicated.

The eccentricity of the Earth's orbit

If you look at such a figure eight calculated correctly, you will see that the fall and winter loop is actually somewhat larger than the spring and summer loop. This is due to the lie I told above. The Earth does not actually orbit at a constant speed around the sun. On January 2, the Earth is 1.7% closer to the Sun than average and thus the angular velocity is 3.4% larger (conservation of angular momentum). This make the solar day longer than the sidereal day by about 8 sec more than average,

and in the course of 3 months a sundial accumulates an error of 8 min due to the eccentricity of the Earth's orbit.

Thus the correct time will be later than the shadow indicates at the spring equinox and earlier at the fall equinox. This shifts the dates at which the sundial is exactly right from the equinoxes into the summer, making the summer loop of the figure eight smaller. The 20 sec/dy error due to the inclination of the ecliptic and the 8 sec/dy error due to the eccentricity work in the same direction around Christmas time and add up exactly (well, almost) to the 30 sec/dy mentioned earlier. The accumulated errors of 10min and 8min due to these two effects don't add up quite so neatly, so the maximum accumulated error turns out to be somewhat less than 18 min. If you calculate everything correctly, you find that during the course of a year a sundial will be up to 16 min 23 sec fast (on November 3) and up to 14 min 20 sec slow (on February 12). Suppose in October you start a 15 min coffee break at 10:45 by the wall clock. If you believe the sundial outside, without accounting for the equation of time. you will already be late for the 11:00 session as soon as you step out the door.

More details

The Equation of time can be approximated by

day number, January 1 = day 1

There is also a table giving the Equation of Time and the declination of the sun for every day of the year. See also the page from Sundials on the Internet. But the best site I have seen on the Equation of Time is doubtless this one.

Referenced By