The following is a slightly edited version of an article of mine that appeared in the Kalamazoo Astronomical Society (KAS) newsletter in January 2001.
Sunrise, Sunset, and the Solstice
A KAS member recently sent me the following question in an email, slightly paraphrased. I thought it might be a question that a lot of KAS members have,  especially about this time of the year. Here it is: "I have a question for you. Why does the shortest day of year in the Northern Hemisphere (Winter Solstice) not correspond to the the date of the earliest sunset? I have been keeping track of sunrise and sunset times (minutes of daylight) and I have noticed (by looking ahead) that the Sun will actually set later on the 22nd of December than it does right about now. There must be something that I am overlooking in this picture."

Yes, that's right, the Winter Solstice doesn't correspond to the earliest sunset or the latest sunrise. The earliest sunset occurs around 2 weeks before the solstice (about the 7th), depending upon your latitude. The latest sunrise occurs some two weeks after the solstice (about January 4th), again depending upon your latitude. These differences also occur for the summer solstice. How come? First, a definition: mean solar day. The Earth rotates once upon its axis in 23 hours, 56 minutes, 4 seconds, and some fractions of a second (and even this is variable, but that's another story). If the Earth were otherwise unmoving, then all observers on Earth would note that the Sun would take just exactly that long to return to the meridian (the line connecting the points due north and due south along your horizon with the zenith overhead), completing one circuit in the sky. However, the Earth is involved in another motion: it orbits the Sun. Since it takes approximately 365.25 mean solar days to orbit the Sun once (360 degrees), the Sun appears to move eastward along the ecliptic by just under 1 degree per day. This means that we have to wait for the Earth to rotate just a bit more, corresponding to an extra 3 minutes and 56 seconds of time, on average, to bring the Sun back to the meridian. Adding 3 minutes 56 seconds to Earth's  rotation period of 23 hours, 56 minutes, 4 seconds brings us to a 24 hour mean solar day. So all else being equal the Sun should return to the meridian one full mean solar day later. Now we are about to find out why it is called a "mean" solar day, as well as the answer to the original question.

Ok, here we go. There are two reasons, and they are the same reasons that give us the analemma (that figure-8 thingy on a globe), and so explain why sundials need correcting to match mean solar time = 24.00 hours: (1) the Earth is tilted on its rotation axis, and (2) the Earth's orbit is elliptical, so that its orbital speed varies. The Earth makes its closest approach to the Sun on about January 4, and it is therefore moving fastest in its orbit on that day (northern hemisphere winters are 3 days shorter than our summers!). This affects how rapidly the Sun moves in our sky along the ecliptic, due to the Earth's orbit (i.e., the eastward motion from west to east of about 1 degree per day). In early January that motion in the sky would be near maximal, if all else were equal. But all else is not equal, because at this time the Sun's position on the celestial sphere is near its most southerly declination (-23.44 degrees), due to the the 23.44 degree tilt of Earth's rotation axis and its position around the Sun. An object on the celestial equator moves 15 degrees in 1 hour's time (1 degree every 4 minutes) due to the Earth's rotation, and just under 1 degree per day due to the Earth's revolution around the Sun. But this angular motion in the sky depends upon the cosine of the object's declination (i.e., the  number of degrees north or south of the celestial equator), and this factor is about 0.92 (= cosine(-23.44)) for our Sun in December, compared to an object lying on the celestial equator. This means that the Earth's rotation and revolution do not "move" the object in as large an angle on the sky as it would if the object were on the celestial equator. To help understand this, think of the north celestial pole (near Polaris, the North Star) which is stationary in our sky despite the Earth's rotation and revolution (declination 90 degrees, cosine(90) = 0, so no angular motion). The bottom line is that during the time that the Earth rotates once, the Sun's declination in the sky has changed and so has the Earth's revolution speed, and thus so has the rate of the Sun's apparent motion in the sky. The Sun can "run fast" by up to +16.55 minutes (around 3 November) and "run slow" by up to -14.1 minutes (around 12 February) through the course of the year. I always  tell my students that stuff's always moving around up there. Now if the Sun "runs fast" it reaches noon (crosses the meridian) earlier than expected (so that it must have breached the eastern horizon earlier than expected and will breach the western horizon earlier than expected), and if it runs slow it reaches noon later than expected (so that it must have breached the eastern horizon later than expected and will breach the western horizon later than expected). "Expected" means based upon the 24.00 hour mean solar day. Keeping this in mind....

These two effects (Earth's rotation axis tilt and its elliptical orbit around the Sun) duke it out, with the net effect accumulating to differing values over the course of the year, to give us the  variation in solar time that is called the equation of time (or the analemma). In early December as we approach the solstice, the Sun is still "running fast," though by an amount that is quickly declining toward zero, and so the earliest sunset comes earlier than the winter solstice. A couple of days after the solstice, the equation of time goes to zero and thereafter runs the other way, and the Sun increasingly "runs slow.'' This results in a date in early January that gives us our latest sunrise. The last bit of complication has to do with the observer's latitude, since this affects the overall length of the day. The separation between the two dates (earliest sunset, latest sunrise) will depend upon your latitude. They are separated by the greatest number of days for observers on the Earth's equator (who actually have two sets of these per year), but are nearly simultaneous (just to either side of the solstice) for high latitudes. It is a bit complex, and I probably haven't quite made it transparent, but I hope this has helped some. For an explanation with some helpful pictures, go to http://www.astronomynotes.com/nakedeye/s9.htm. You might also try visiting this web page on the analemma: http://www.analemma.com/Pages/framesPage.html

May your winter skies sparkle.
 

Kirk T. Korista
Professor of Astronomy
Department of Physics
Western Michigan University
Kalamazoo, MI 49008-5252