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.

Vernal Equinox plus Twelve

Finally a sunny weekend day for checking my solar calendar.*  (How quickly forgotten the drought that gave almost nothing but sunny days!)  The NOAA Solar Calculator page gave solar noon today for my address as 12:32:34 EDT.  I checked my watch against my "atomic"** bedside clock: my watch was 20 seconds slow. 

I figured that all this precision was a little over the top, as I got a chair to stand on and a Sharpie and ruler to mark the shadow of the gutter with.  Half-a-minute before the time, I lined up my ruler and began to draw, but found the shadow was moving visibly even as I did.  By the time the second had struck, I had to draw new lines.  I ran for alcohol to take off the original lines and made a discovery: alcohol merely faded and smudged it a little.  (With these marks  permanent, I'm glad I have a patient, understanding wife!)

If you make your own sundial/solar calendar, I wouldn't recommend a greater distance from gnomon to dial than maybe twenty feet--as you can see, the edge of the shadow is already pretty fuzzy and hard to pinpoint.  Greater distance would only make it fuzzier.  (The top of the school flagpole I once tried to use when I taught seventh grade science was so far away it didn't even cast a shadow!)

The corner of a gutter that serves as my gnomon is about twenty-two or twenty three feet away from the side of the house that is my "dial."  Clearly, that is more than distance enough to actually watch the shadow move.  You can see the change in the photos below, taken just eight minutes apart. 

 

I was also startled that the solar noon shadow today was nearly two feet higher on the siding than the one I made just eight days ago.  At that rate, will the solstice shadows fall on the house at all?  I have about six feet above and below my original mark.  That isn't much, but the shadow also moves farthest nearest the equinoxes, so it might be okay.  When I have time, maybe I'll try to work it out trigonomically.  Yeah, right.


*Another site will print solar noons for an entire year.
**The only thing "atomic" about atomic clocks (besides they're being made of atoms), is that they receive ultra-low-frequency radio signals from the atomic clock in Colorado that keeps the nation's time.  That IS pretty cool, though!

Make your own Solar Calendar

I've long been interested, in a general way, in the movements of the sun through the seasons.  The sun reaches its highest point each day at "solar noon."  And this high point gets higher and higher in the sky each winter and spring until it reaches its highest point at the summer solstice (usually June 21-22), then descends gradually towards its lowest point at the winter solstice around December 21-22. 

Since the sun's axis is tilted 23½ degrees, here in Massachusetts (latitude 42 degrees) the sun reaches a high of 71½ degrees above the horizon at the beginning of summer, but only 24½ degrees at the beginning of winter.  At the equinoxes (beginning of spring or fall), the noonday sun would be at 90-42=48 degrees. 
 
Along with this comes the lengthening and then shortening of the daylight hours.  The solstices, and the equinoxes midway between them, provide the defined beginnings of the four seasons; while the daylengths, together with the changing angle of the noonday sun, give our hemisphere more or less solar heat energy, providing us our seasonal weather.

 

There are lots of solar calendars in the world.  This is one of the better-known ones.

Ancient solar calendars like Stonehenge give these movements of the sun a mystical air that excites many people into spiritual experiences.  I used to impute to these New Age types a longing for intense spirituality that they never found in the staid worship of their parents, and which traditional worship they'd rebelled from anyway.  Such stuff.  At the same time, I felt condescension toward those ancients who worshiped thus.

But in the second episode of the new Cosmos,Neill Degrasse Tyson explained it in a way that made perfect sense to me.  Ancient people depended for their survival on knowing the time of year when prey animals migrated and food plants came into season, and, with the advent of agriculture, even more the dates when the killing frost would be safely past.  These calendars were intensely practical--even crucial to their survival.  And the idea that the stars governed their fates an entirely logical extension of the same practical experience.

For my part, my interest is also sometimes just as practical: my little vegetable garden is strategically placed for the longest full sun in late spring and early summer.   My tomatoes, cucumbers, zucchini, sage and rosemary enjoy over five hours of full sun at the summer solstice, but now at the autumnal equinox (September 23) receive less than four hours, since much of the day high trees block the lowering sun.  Lover of tomato sandwiches (on toast with mayo, salt & pepper, and a liberal sprinkling of dill weed) that I am, I check the times of sun and shadow whenever the sky is clear and I am home at the right moments.  (I count about two more tomato sandwich lunches for my wife and I from the last tomatoes that will ripen before the light fails.)

A few weeks ago I finally decided to act on the impulse to make a solar calendar of my own.  Such an enterprise takes a fair amount of thinking.  What to observe?  Several things change through the seasons: the greatest height of the sun, the times of sunrise and sunset, and the direction of sunrise and sunset.  (The sun rises directly east and sets directly west only on the equinoxes; toward summer it rises and sets northerly, and towards winter it rises and sets more southerly.)  Even the time of local noon varies through the year for a combination of reasons.  

I first thought to try to fix the direction of sunrise and sunset at the solstices and equinoxes, but the trees of my neighborhood make that impossible even from an upstairs window.  In fact, there is not a single broad, straight, east-west street I could use to even approach witnessing sunrise or sunset at the equinoxes.  Next I thought of a straight, vertical rod to made a gnomon for a sort of seasonal sundial.  I like the idea of having a sundial or calendar large enough that you can actually watch the motion of the shadow in real time.  The trouble is, the taller the gnomon, the less distinct its shadow.  (I once had my 7th grade science classes make a sundial using the school's flagpole for the gnomon; it was less than satisfactory, since the top of the flagpole made almost no shadow at all.)  I still think the idea of a moderate-sized gnomon a good one, but it would take a yard with a lot of open sky, and any simple pole would very easily go out of adjustment and become useless.

I also failed miserably at finding local noon experimentally: theoretically, it is the time that the sun is highest in the sky, and therefore casts the shortest shadow on a level surface, but in practice the shadow is too indistinct to measure and the differences in its length too slight.  If I had a clear horizon, and had a sextant and were good at using it, I could probably nail the time of solar noon to less than a minute (this is routine in traditional navigation aboard ship), but I have none of these.  Fortunately, NOAA's solar calculator comes to the rescue.

Last weekend I hit on the perfect plan for a practical solar calendar of my own: looking around outdoors near solar noon, I found a place where the edge of a gutter cast a pretty reasonable shadow on the side of the house a little distance away.  I waited with ruler and Sharpie in hand until it was exactly solar noon, and marked the fuzzy shadow as best I could on the siding.  If I make marks regularly--especially at the solstices and equinoxes--I will be able at a glance to know two things: first, whether it is before or after solar noon on that particular day; second, where we are in the march of seasons.  Since the shadow falls about six feet  up the wall, there will be plenty of room for the shadow to go higher (as the noonday sun drops towards the winter solstice), and lower (sun rising towards summer solstice).  It should work perfectly--at least as long as the gutters don't go out of alignment.  --and as long as my long-suffering wife doesn't object too strenuously to Sharpie on the white siding!

 

Gnomon: the corner of the gutter.  It's shadow falls at noon on the side of another part of the house.

  The red mark is labeled "AE +4," i.e. Autumnal Equinox +4 days.  (Not exactly Stonehenge, but it's mine!)