# Math Chat: Today's Late Sunrise

Old challenge

Why do we have the latest sunrise now, around the first of the year, and the earliest sunset around Dec. 10, compared with the shortest day around Dec. 21? And how can I remember (from year to year) which comes first?

The period of daylight in the Northern Hemisphere is longest on the summer solstice, about June 21, when the North Pole tilts toward the sun, and shortest on the winter solstice, about Dec. 21, when the North Pole tilts away from the sun. So you would expect the latest sunrise and earliest sunset about Dec. 21.

The correction concerns the length of a 24-hour day. The earth rotates about its axis in 23 hours and 56 minutes. Meanwhile, the earth has traveled part of its annual journey around the sun and must rotate another four minutes to face the sun squarely again. This effect is bigger in summer and winter when the sun is over the shorter circles of latitude near the poles than in the spring and fall, when it is over the equator. Consequently, summer and winter days are a bit longer than 24 hours.

According to our 24-hour clock, sunrise and sunset come a bit later each day. So the earliest sunset comes before Dec. 21, and the latest sunrise comes after Dec. 21. Similarly in the summer, the earliest sunrise comes before June 21, and the latest sunset comes after June 21. It is easy to remember: the early daily event (sunrise or sunset) comes earlier in the month.

There is a secondary complication due to the fact that the earth's orbit is elliptical rather than circular, with fastest movement (longest "solar" day) early in January and slowest early in July. Therefore the above effect is greater in winter (when the latest sunrise comes some 10 days after Dec. 21) than in summer (when the latest sunset comes just about one week after June 21).

In his winning response, George Dillard cites an excellent article by Stan Wagon on "Why December 21 is the Longest Day of the Year" in the December 1990 Mathematics magazine.

Here is a riddle for your friends: What is the longest day of the year? (Answer: the 25-hour day in autumn when we turn the clocks back to go off daylight savings time.)

My daughter is in second grade and is having difficulty with math. She failed math in the first grade but they moved her into the second grade anyway. Our school system does not have a math resource to help her so I am on my own. Can you give me any advice?

- L.K.

Your concern is understandable and commendable. I'd recommend hiring a third-grade student as a tutor for your daughter. If you pay the tutor, you should be able to make sure that they meet at a regularly scheduled time and place (a library could be ideal), twice a week for a half-hour, and that the tutor helps her work problems (without doing them for her). Any third-grade teacher should be able to recommend a tutor.

New sequence challenge (Joe Shipman)

What is the next number in this sequence:

1 2 3 4 5 6 8 9 10 11 12 14 15 16 17 18 20 21 22 24 ... ?

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