Why the earliest sunset, latest sunrise, and shortest day of the year occur on different dates
by John Holtz

If the first day of winter for us Northern Hemisphere dwellers is around December 21st, and this is the shortest day of the year (shortest amount of sunlight), then why does the earliest sunset occur around December 7th? Why does the latest sunrise occur around January 4th?

	To answer this difficult astronomical puzzle, try solving this everyday puzzle first.
	Puzzle: At 12 noon (12:00:00) and midnight, the hour and minute hands of a clock are perfectly aligned. (You remember those old fashion clocks with hands, correct?) At what time are the hands aligned again?
	Answer: 1:05:27.2727 (View this page for the solution)

Now imagine that the hour and minute hands are not connected together as they are in a real clock. Let the minute hand make one rotation in exactly 60 minutes, but let the time required for the hour hand to make one rotation vary by a few minutes. For a given period of the hour hand, you can calculate how long until the hands align again, just like you solved the real puzzle above. (You did solve the above puzzle, right?)

Surprise! The "hands of the clock" and the rotation periods that I was referring to in the clock puzzle can be replaced by the daily rotational period of the Earth (23 hours 56 minutes 4 seconds) and the revolution period of the Earth around the sun (365.25 days). If the Earth did not revolve around the Sun, then the day would be 23 hours 56 minutes long from high noon to high noon. But during the time it takes the Earth to rotate, it moves a small distance in its orbit around the Sun. Thus, the time from noon to noon is slightly longer than 23 hours 56 minutes (23:56).

Although the average time from noon to noon is 24 hours, the actual time varies from day to day because of two different effects. One effect that causes the variation is the Earth's orbit around the Sun — it is elliptical, not circular (eccentricity = 0.0167). Based on Kepler's Second Law of Planetary Motion, a planet travels fastest when closest to the Sun and slowest when farthest from the Sun. The Earth is closest to the Sun around January 4th, and therefore travels farther in one day than when the Earth is farthest from the Sun around July 3rd. So, the revolution speed of the Earth around the Sun varies from day to day. (The rotational period about the Earth's axis is fixed at 23:56:04.) In the clock puzzle, this would be like changing the speed of the hour hand throughout the year, leading to a different solution.

Earth's Orbit around the Sun and the Effect on the Length of Day

	Perihelion. The Earth is closest to the Sun around January 4th. Consequently, the orbital velocity is highest on this date. As a result, the length of the day, the period from noon to noon for example, is longer than the average of 24 hours. (The additional motion of the Earth in its revolution around the Sun during the 4 minutes between steps 2 and 3 is not shown.)
	Aphelion. The Earth is farthest from the Sun around July 3rd. Because the orbital speed is slowest at this time of year, the length of the day is shorter than the average of 24 hours. (The angle between the arrows in steps 2 and 3 is smaller in this figure than between steps 2 and 3 in the perihelion figure.)

The second effect that varies the length of time from noon to noon is due to the tilt of the Earth's axis (known as the obliquity) with respect to the Earth's orbit. You are well aware of this fact but perhaps did not realize the manifestation in the sky. The obliquity causes the sun to be highest in the sky on the first day of summer (around June 21st) and lowest in the sky on the first day of winter (around December 21st).

Earth's Tilt and the Effect on the Sun's Height

Summer. The north pole is tipped 23.5 degrees toward the Sun. The Sun appears high in the sky (straight overhead shown by red arrow).		Winter. The north pole is tipped 23.5 degrees away from the Sun. The Sun appears low in the sky (straight overhead shown by red arrow).

In reality, the Earth moves around the Sun. But for discussion purposes, it may help to imagine that the Earth is stationary while it rotates and that the Sun is moving around the Earth. To further simplify the discussion, imagine that the Sun appears to move at a uniform rate. The Sun moves along a path called the ecliptic, and as shown below, and it is tipped by the same 23.5 degrees to the Earth's axis. The consequence of the tilt is that the Sun appears to move at a non-uniform rate around the Earth's equator. As with the Earth's eccentricity, the varying speed results in the length of a day that is sometimes longer than 24 hours and sometimes shorter than 24 hours.

Earth's Obliquity and the Effect on the Length of Day

Even if the Sun were to follow the ecliptic (black arc) at a uniform rate, the time from noon to noon, when the Sun is on the meridian, would vary. Due to the tilt of the Earth's axis with respect to the ecliptic, the Sun's apparent speed along the celestial equator (green arc) is non-uniform; that is, the right ascension of the sun changes at a non-uniform speed. The speed along the equator depends on where the Sun is relative to the equinoxes because the position changes the angle a between the speed along the ecliptic and the speed along the equator (see figure at right). The speed along the equator is slowest near the equinoxes (when a is the largest) and fastest halfway between: at the solstices (a is near 0).

These two factors — the Earth's elliptical orbit around the Sun and tilt of the Earth's axis — cause the Sun to be at a slightly different position in the sky from day to day when viewed at a particular time of the day. If you were to note the Sun's position, say at 12 noon, for an entire year, you'd discover that it traces a figure-eight. This pattern is the analemma, and an approximate representation is shown at right. (Some globes plot the analemma, usually in the Pacific Ocean.) The Earth's elliptical orbit and tilt of the axis causes the Sun to be at a different position horizontally each day, while the tilt of the Earth's axis causes the Sun to be at a different position vertically. (To explore the relative importance of each effect on the horizontal width of the analemma, view this page.)

Okay, were are now ready to answer the original problem: why the earliest sunset, latest sunrise, and shortest day of the year occur on different dates. Imagine the figure-eight analemma rising, crossing the sky, and setting. This simply shows where the sun is at any given time of day, on any day of the year. From northern latitudes, the analemma slants upward and to the left at "sunrise" and upward and to the right at "sunset". (The analemma is vertical at local noon.) The shortest day of the year (first day of winter) occurs when the sun is at the bottom of the analemma; thus, the sun spends the least amount of time above the horizon. The longest day of the year (first day of summer) occurs when the sun is at the top of the analemma so that the sun spends the greatest amount of time above the horizon.

However, if you've drawn your figure-eight with the correct slant, you'll notice that the latest sunrise occurs when the last part of the analemma rises. This does not occur with the sun "at the bottom" of the analemma (December 21st) but a few days later (January 4th). Likewise, the earliest sunset occurs when the first part of the analemma sets below the western horizon. Again, this occurs a few days (December 7th) before the sun reaches "the bottom" of the analemma (December 21st).

Highly Magnified View of Analemma Rising and Setting
for 79° 48.8' W, 40° 37.6' N

Rising. The analemma at 7:43 am EST for every day of the year. If the date is January 4th, then the Sun is at the "lowest" position of the analemma, and sunrise is just occurring. The Sun has already risen at any other position on the analemma; that is, on any other date. Thus, January 4th is the date of the latest sunrise.	Setting. The analemma at 4:52 pm EST for every day of the year. If the date is December 7th, then the Sun is at the "lowest" position of the analemma, and sunset is just occurring. The Sun has not yet set at any other position on the analemma; that is, on any other date. Thus, December 7th is the date of the earliest sunset.

The same process can be used to see why the dates differ in the summer. However, the two effects causing the width of the analemma — the Earth's elliptical orbit and tilt — combine in the winter and partially cancel each other during the summer. Thus, the top of the analemma is very narrow. So, the effect is not as drastic in the summer as in the winter..

Many thanks to Larry Denenberg for pointing out that both the eccentricity and obliquity have a significant effect on the horizontal width of the analemma. For an interesting article on the other planets, refer to "The Analemmas of the Planets" in the March 1982 Sky & Telescope, pages 237-239.

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Last modified 2003 Dec 21. added effects of obliquity