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The Geometry of Spring

A clear, bright sky greeted Tasha and me when we rounded the corner of the little courtyard just outside our door. The sun was just over the tops of the houses across the way. If the temperature hadnít been fifteen degrees, Iíd have thought it was spring. But the dog trotted down the driveway seeming oblivious to the cold air, her step light and her head high, eager to savor the smells of the neighborhood. I pulled the hood of my coat up over my head and followed her.

The sun makes such a difference in how the morning feels to me. Every day, my first task is to take Tasha for her morning walk. At our old home in Ann Arbor, we simply let her out the back door, where she could greet the day however she wished. A large fenced yard kept her safe and close. In our present co-op, the rules are that dogs may be outside only on a leash, and only when accompanied by a person. So Iíve experienced nearly every morning since our move by walking the dog. Whether itís rainy, snowy, cloudy or sunny, she and I are out there, making the most of the morning.

I used to be too busy to notice the weather much, unless it interfered with what I wanted to do. Farmers and others who have to be outside every day speak about how one gets more intimate with the weather, observing the subtle changes as they occur day to day. Recently, Iíve noticed that sunrise comes just a little earlier each day, so I began watching the Weather Channel on our television, where in every local weather broadcast the times of sunrise and sunset are listed, along with the phases of the moon. Last night it said that sunrise today would be two minutes earlier than yesterday, and sunset two minutes later. In a week, at that rate, weíll have almost a half-hour more daylight!

That much change, of course, happens only briefly this time of the year, and again in the fall. Curious, I found the website of the U.S. Naval Observatory, where one can type in a location anywhere in the world and get the sunrise and sunset times for a whole year. (The times do change a little bit, year to year.) Naturally, I downloaded two years of data (itís like finding a neat piece of lumber at the dumpóyou never know when you might have use for it!)

One interesting thing I found when I loaded the data into a spreadsheet and plotted it graphically: The time of year when the sunrise is the latest is not exactly the same as when the sunset is the earliest. In fact, there is almost a month between the two dates. So how do they determine the shortest (and longest) days of the year? Nominally, the winter solstice falls around December 21. But by then, sunsets have already begun to get later again! The earliest sunset in 2004 was on December 7ótwo weeks before the solstice. The latest sunrise was on January 3, so at the solstice, sunrises were continuing to get later every day. Not much, of course. If sunrise and sunset times were aligned, the result would be a day six minutes shorter. I did have to spend some time trying to see why that would happen.

Itís all geometry. The axis of rotation of the earth is tilted in relation to the sun, as you already know, and when the northern hemisphere is tilted away from the sun, the curve of the earthís surface obscures the sun for longer parts of the day. (If we lived on the Arctic Circle, the sun would never quite rise above the horizon.) But the tilt of the earthís axis isnít stationaryóit wobbles a bit, like a childís top when it begins to slow down. That wobble is what causes the sun to be obscured in the evening on an earlier date than when it is first revealed in the morning.

The opposite situation occurs around the summer solstice, except that there is only a twelve-day misalignment between earliest sunrise and latest sunset, and the difference in longest day between what is and what would be if they were aligned is only about two minutes.

Note: The above applies to latitudes close to mine, which is about 42 degrees north. Locations farther south will have less difference between summer and winter; those farther north will see more difference.

Unfortunately, my high-school math did not include spherical geometry, so I cannot provide the appropriate formulas.

My morning walks with Tasha are not always filled with such mental calculations. Usually, Iím just aware of the morning and the physical sensations of walking off my sleep mode. Lately, though, Iíve been looking forward to the coming spring. Even though the temperatures have not changed substantially in two months, the sun is rising earlier, and that cheers me up and gives me hope that warmer weather is, indeed, on its way.

Tasha, on the other hand, doesnít mind the cold. Her nose is awake long before I can detect spring in the air. Iíve seen her plunge her whole head into a foot of snow in order to smell the tracks of some small animal, laid down weeks before. The only difference spring will make to her is in the increased activity of those same animals after a long, cold hibernation. The increasingly earlier daybreaks of the next few months will mean that for her, our morning walks really should be starting sooner in the day.

 

March 11, 2005

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