Mark the solstice--literally

I've given up--for now, at least--the idea of making a shadow calendar that would mark the passage of the seasons as a shadow moves from one solar noon to the next.  But simply marking solstices or  equinoxes is simplicity itself.  Try it yourself, tomorrow!

Theoretically, you could find the position of the sun at solar noon on any particular day, choose an object to cast the shadow, and measure the angle to see where that shadow would fall.  But it's far, far easier to simply wait for the event in question, find a convenient shadow, and mark where it falls.  

You could do such a mark at sunrise, sunset, or solar noon.  Sunsets and sunrises change direction as the seasons pass (really!*), while solar noon marks the greatest height the sun reaches on a given day.  (All three of these things also vary in time.)  Since trees and buildings block our view of sunrises and sunsets around here, solar noon makes the best target, and it's what I marked. 

Your first job is to find out the time of solar noon tomorrow--the day of the summer solstice.  Go to NOAA's solar calculator.  Enter enter your latitude and longitude, OR) move the nearest orange pin on the map to your location.  I like to put the pin right on my house, but for practical purposes that's overkill: a few miles either way won't matter.  (The sun moves [okay, the earth turns] one degree of longitude every four minutes.  At my latitude of 42 degrees north, a degree of longitude is forty-five or so miles.)  Change the date to June 21.  Click the box for daylight savings time, then click Save.  You now know the times of sunrise and sunset, and the time of solar noon.  (Other check boxes will make lines on the map showing directions of sunrise and set, and the current position of the sun.)  WRITE DOWN THE TIME OF SOLAR NOON.  

As I write, the NOAA solar calculator page keeps timing out--probably swamped with interest so close to the solstice.  Keep trying!

Notice that, not only is it not at noon, it's also not at one o'clock, as you'd expect with daylight savings time.  Here, solar noon comes at 12:45pm, EDT right now.  Part of the reason is that time zones are about fifteen degrees of longitude (one hour of time) wide.  Only those in the east-west center of the time zone could reasonably expect noon to be on the hour.  (Since we in eastern Massachusetts are east of the center of the eastern time zone, solar noon always falls before clock noon.)  The rest of the difference is more subtle, and also causes the time of solar noon to vary by a minute or more from day to day, and tens of minutes from season to season. 

On Sunday go outdoors half-an-hour or so before the time of solar noon.  You need to scope out a good shadow.  The shadow ought best be pointed, and should be cast by a something permanent at some distance away (say, ten or twenty feet away).  I used a corner of the roof, whose gutter casts an angled shadow.  As the sun continues to move westward, that shadow will move eastward, so you need to make sure it will still fall in a convenient place at solar noon.  (In the event it doesn't, you may want to find a back-up shadow.)  

Marked yesterday, when it was sunny.  (Two days before the solstice is close enough

in my book.)  The shadow of the roof corner will just cover this rock tomorrow.

Added 6/22: A cloud blocked the sun right at solar noon on 6/22, but a minute 

later the shadow appeared.  In that minute, the shadow had already advanced

an inch to the right.  (The roof casting the shadow is about 20 feet away.)

That's pretty much it.  Just figure out how you can mark the location of this shadow when solar noon arrives.  This shadow falls on the ground right now, which I marked by pushing a little stone into the ground at the point of the shadow with my foot.  At the equinoxes, on the other hand, that shadow falls on the side of the house, and I marked it in indelible marker.  

Henceforward, every summer solstice (usually about June 21st) that shadow will appear in exactly the same place.  If you're lucky enough that that same shadow will fall on the same surface at the equinoxes and winter solstice (a very tall order), you will be able to mark those the same way.  Then you could watch the sun's noontime progress between those marks, as the earth makes its stately, tilted way around the sun season year after year.  

This was a few days after the fall equinox.  (The shadow has passed, 

since solar noon was ten or fifteen minutes before.

To summarize:
1. Enter your location in the NOAA solar calculator site (not forgetting daylight savings) and find the time of solar noon for June 21.
2. Go out safely before solar noon on the 21st and find a convenient shadow.
3. At solar noon, mark the location of that shadow. If it's cloudy, do it tomorrow!  (Close enough!)


* You can see from the figure below that the sun rises and sets south of east and west in the fall and winter, but north of east and west in the spring and summer.  Curiously, the paths of the sun across the sky are always parallel to each other.  The figure is approximately correct for mid-northern latitudes like those here in New England.

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.