Saturday, January 3, 2015

Solar Calendar, again

I will call it a shadow calendar from now on, since solar calendar really means something like this. 
My first attempt at a shadow clock ended in disappointment, when it became clear that the corner of a roof gutter (my gnomon) was too far from the wall the shadow fell on.  Because the noonday sun moves (seems to, from our point of view) a full 47 degrees from solstice to solstice, my wall was not nearly tall enough to contain the whole range of the shadow.  (A little simple math would have saved me wasted time, had I not been overconfident.)  In addition, the fact that the wall was not really east-west meant that the shadow did not reach its highest point at solar noon, as would be expected.*  This is more an aesthetic than a practical point, since it’s hard to stand there long enough to confirm anyway.

My next, more modest attempt used the shadow cast by the house eave on the wall of the house, only a few feet away.  One fortuitous advantage of this “calendar” was that the shadow of the downspout falls “just so,” telling you when it is really solar noon without need for a clock.  But in this case, the distance was too short to show much shadow movement from season to season.  (It amounted to less than a full clapboard in height over several months.)  I did not even get the satisfaction of seeing the shadow on Dec 21, since it was cloudy.

The math involves trigonometry of a right triangle, and relies on having the shadow falling on a vertical wall (opposite side) a known horizontal distance (adjacent side) from the gnomon.  Then you need the angle above the horizon of the noonday sun at each solstice (angle theta).  These angles can be found from your latitude: summer solstice theta = 90 degrees – latitude + 23.4 degrees, while winter solstice theta is the same, but minus those 23.4 degrees.  (These angles, by the way, are the height of the noonday sun above the horizon on those days.)  Opposite side = Tan(theta) X distance.

Then minimum height of wall needed can be found by working out the opposite sides, and subtracting them.  IF the height of your wall is no smaller than this difference, AND IF the noonday shadow falls at the bottom of this wall on the summer solstice, THEN the winter solstice shadow will fall at (or short of) the top of the wall.  (Phew!)

To save you the trouble, here at 42 degrees north latitude, the noonday sun is at 24.6 degrees at the winter solstice, 48 degrees at the equinoxes, and 71.4 degrees at the summer solstice.  My gnomon was about 20 feet from the wall, so the opposite side would be about 59 feet and 9 feet: a difference of 50 feet!  That result surprises me even now.  Needless to say, my house isn’t that tall.  (So great a height is needed partly because of the downward slant of the rays: near the north or south pole, with the rays shining nearly horizontally, the height needed would only be about 20 feet.  –while at the equator the sun couldn’t shine on the same wall at both solstices at all—it would hop to the other side, shining in the southern sky at the end of December, but the northern sky at the end of June.)

In the meantime, I realized that even a flat, vertical, and perfectly east-west wall would distort: the sun’s path from the gnomon would be changing continuously through the day, and also be different lengths at different times of year—that means the position of the shadow would not change in even increments week by week.  In fact, the ONLY way to give the shadow a steady march would be to project it on a semicircle whose radius was the length of the shadow.  (Got anything like that outside of YOUR house?)  So much for my plan of having a “found” shadow calendar!

On the other hand, a length of heavy aluminum bar would be pretty easy to bend into the necessary quarter-circle.  The main problem would be adjusting it and holding it solidly in place.  And I wonder how expansion and contraction with temperature would affect it?  Hmmm…

Here, since I missed posting for so long due to Life, computer death, etc, is Everything You Need to Know About the Winter Solstice.

*An email reply from the folks at NOAA made it clear to me: the shadow moves left to right on the wall, but since the sun’s rays slant downward, and the wall is tilted so the path of those rays gets longer, the shadow continues to move downward a long while even after the sun has passed its high point for the day.

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