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Measure Heights With Shadows

Skywatch

By Dennis Mammana- Though much of the Northern Hemisphere remains quite cold and snowy, spring is definitely on its way. In fact, you’ve probably noticed that the sun climbs higher in our midday sky and our daytime hours are already becoming significantly longer.

Keep an eye out and you’ll find that the sun will continue its daily climb until the first day of summer, when day will be longest, and will stand quite high above the southern horizon at noon.

From that day forward, the sun will begin its six-month descent, reaching its lowest elevation of the year once again on the first day of winter.

Astronomers know each of these dates as a solstice, a word that comes from Latin and means “sun stands still.” And it’s true that on those days, the sun does seem to stand still — at least in its oscillation in height above our southern horizon.

The length of shadows cast by the sunlight also appears to change during the year. Around the summer solstice — when the sun appears high in the sky — midday shadows appear quite short, while around the winter solstice — when the sun is rather low — they appear long.

Anyone familiar with basic trigonometry knows that we can calculate the height of any object simply by knowing the angle at which sunlight falls on it and the length of the shadow it casts. Now, if the very thought of trigonometry makes your eyes roll back into your head, don’t worry, because Mother Nature has it all worked out for us.

At this time of year, you can do the same thing without knowing a sine from a sign … at least for a few days.

The reason is twice during the sun’s annual oscillation in the midday sky, it reaches a point 45 degrees above the southern horizon. At those times — and for a few days before and after — an interesting phenomenon takes place: The length of all midday shadows equals the height of the objects that cast them.

The dates on which this happens depend on your location. Sky watchers in Southern California will see this occur during mid-February. In the New York area, it will happen at the end of the first week of March, and in the Midwest around mid-March.

This physical circumstance makes it possible to, without knowing trigonometry, learn the heights of objects too tall to measure directly. For example, suppose you’d like to monitor the growth of a tree in your backyard. Simply measure the length of its noontime shadow along the flat ground on these days each year, and you’ll be able to keep track of its progress.

You can look up the maximum midday height the sun reaches at your location by visiting the United States Naval Observatory website at aa.usno.navy.mil/data/docs/mrst.php. Simply enter your town and state, along with a range of dates, and you will get a table that shows when the sun transits the midday sky and its height at that moment.

When the sun appears 45 degrees above the horizon at its midday transit, that’s your time to do your measurements.

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