Why doesn't the length of each day change much around the solstices?
According to the sunset sunrise tables, the length of daylight hours remains fairly constant at both solstices for a period of about ten days. Can you explain why this occurs?
The amount by which the number of daylight hours changes from day to day has to do with the direction that the sun appears to move each day with respect to the background stars. This motion, which is due to the earth's orbit around the sun, can be measured by taking a picture of the sun at the same time every day for an entire year. If you do this, you will find that the sun traces out a shape on the sky called the analemma (seen in this image from Astronomy Picture of the Day):
As you can see, the analemma has a somewhat complicated shape (the reasons for which are explained in detail at this site), but for our purposes the important point is to look at the direction the sun is moving from day to day. If the sun is moving mostly north or south (away from or towards the horizon) then the length of daylight hours will change rapidly from day to day, whereas if the sun is moving mostly east or west (parallel to the horizon), then the length of daylight hours will not change much from day to day, since the sun's maximum height will be roughly the same one day as it was the day before.
So to answer your question, all you have to do is look at the shape of the analemma during the solstices (the upper left and lower right points in the picture, when the sun is either farthest from or closest to the horizon). As you can see, the curve flattens out at this point, and the sun will be moving in a roughly east-west direction during the days near the solstice (note that the spacing between "suns" in the picture is much larger than the distance the sun moves each day - if you actually took a picture every day instead of only once in a while then the analemma would be filled in with 365 suns!). Since the sun is moving mostly east-west, the length of the day will not be changing very much.
If you think about it, you can see that the solstices almost have to have this property, regardless of the exact shape of the analemma. By definition, the solstice is the point at which the sun is as far north or south as it can possibly be, so if it were still moving north or south when it reaches that point then it wouldn't really be the solstice! The only way you could avoid having the day-to-day change in the number of daylight hours slow down near the solstice is if the analemma had a sharp "spike" in it at that point. But if you think about the physical situation (the earth's motion around the sun), you can see that there is no reason to expect a "spike". The earth moves in a smooth orbit that doesn't have any violent, rapid changes, so we would expect the analemma to be smooth too.
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