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.

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!)

Solstice Evening on the Pond

Early this morning came the summer solstice--the moment in Earth's orbit when the sun was directly above the Tropic of Cancer, so we in the northern hemisphere would experience the most direct rays and the longest day in the year.  That makes today the first day of summer; a surprise to those who think summer is defined by warm weather, instead of the the angle formed by the earth's orbit and its equator.  

This was a very significant moment for ancient peoples, reliant on such calendrical markers for agriculture and its attendant ceremonies.  Ceremony is still important to us today, of course.  (My wife tells me Stonehenge was mobbed.  To each his or her own, I suppose.)

I consciously acknowledge such days, though I have no ceremonies to mark them.  But I did think it lovely weather for a little paddle on a nearby pond, Nippenicket.  And appointments earlier in the day meant the opportunity would come later, so I resolved to be on the water at sunset.  (Sunrise would have been more appropriate, but I an NOT a morning person.)

I took my skin-on-frame boat, Musketaquid, and shoved-off at 7:30, paddled around a little island and back with many stops for photos, and landed just after the sun disappeared about 8:40.  Everything was growing.  The yellow water lilies just on the point of blooming, the white water lilies just behind them.   But the sunset alone made it worth the trip.

Winter Solstice--and the Reason for the Seasons

Winter solstice this year was on December 21.  (Depending on when leap years fall, it is sometimes the 22nd.)  The solstice (means "sun stops") occurs when, from our point of view, the noonday sun reaches its lowest point in the sky in the northern hemisphere--it stops getting lower, and will begin rising again after this date.  How low it goes depends on the observer's latitude: the higher your latitude, the lower the sun, until you reach the Arctic Circle inside which the sun will not rise at all on this date. 

The winter solstice is the boundary between fall and winter--all the seasons begin and end as astronomical events, rather than changes in the weather. 

Recall that all globes come with a built-in 23 1/2 degree tilt.  That tilt represents the degree to which earth's axis of rotation is out of perpendicular to its orbit around the sun--as it would be if you put a (small) model of the sun on the same table with the globe.  Model our seasons in the following way.  Put a powerful lamp in the middle of a large table.  Put the globe on the edge of that table with the north pole tilted as far as it can go toward the sun: it is June 21st, the summer solstice.  As you spin the globe, notice how high the sun would get in the imagined sky of North America; notice also that much more than half the northern hemisphere is illuminated.  (It's hard to see this with the usual shiny-surfaced globe; I dust it with chalk dust to make clearer how much is lit.)  Because of this, the northern hemisphere day will be longer than its night, and the sun will shine more directly (so more intensely) on the surface at noon.  Both of these factors mean more heating of the northern hemisphere, bringing on warmer weather as the heat builds up. 

Now slide the globe counter-clockwise around the edge of the table to make the weeks and months pass.  While you do, be careful to keep the axis of the globe point in the same direction all the time (keep it aimed always at the same side of the room).  When you reach the opposite side of the table, you will find the north pole now tilted away from the sun.   It is now Decenber 21 and the winter solstice.  Notice that the situation is reversed from six months ago: the sun will not rise nearly as high for North America, and since most of the hemisphere is in darkness, the daytime will be short.  The earth's surface in the northern hemisphere is receiving much less heat than six months ago, so it is getting colder.

Several lines on the globe are defined by the solstices.  The Tropic of Capricorn is the line at 23 1/2 degrees south latitude where the sun will be directly overhead on the winter solstice, while on that date nothing inside the antarctic circle will get any sun at all, while inside the arctic circle on that date the sun won't even set!  Similarly for the tropic of cancer and the summer solstice.  The "tropics" is therefore that band from 23 1/2 degrees north to 23 1/2 degrees south where the sun will pass directly overhead at least once each year.

This might put a few misconceptions to rest.  First, notice that our distance from the sun is not a big factor.   In fact, the earth is actually a bit closer to the sun right now than it will be in June!  Notice also that the seasons will be opposite in the two hemispheres: summer has just begun for my friends in New Zealand, who are enjoying their longest days right now.  Notice that the tilt of the earth'a axis is not changing--just where the sun is in relation to it.  [You might remember that the north pole always points (approximately) at Polaris, the "pole star."]  Finally, don't confuse orientation with distance: plenty of students do this very activity, then get mixed up explaining that when the north pole is tilted toward the sun that hemisphere is actually closer to the sun--true, sort of, but only by a miniscule fraction of a percent! 

Remember: the changing seasons are about the length of days and the directness of rays!

One question might still occur to you: why isn't the winter sostice the coldest day, and the summer solstice the warmest?  It's strange to think that, as the days lengthen into January, the weather is still cooling off!  The secret is to think in terms of the balance between heat gain and loss: the summer solstice is when the northern hemisphere gains heat fastest, but it takes time for that temperature to rise; similarly, the decreased heating that reaches its lowest ebb on Dec 21 will take time to have its full effect.  So it really does make sense to begin winter with the winter solstice: though the fastest cooling is past, the chill will still deepen further.

 

If the shortest day is Dec 21st, why is the earliest sunset weeks before?

A recent post at EarthSky explains a confusing fact: the earliest sunset of the year is NOT on the shortest day of the year--December 21st--but weeks earlier.

The days shorten until December 21st or so, due to the earth's revolving around the sun to the point at which the earth's tilt has the northern hemisphere tilted as far as possible away from the sun.  It would be reasonable to expect that the days would shorten equally at both "ends"--so later sunrises and earlier sunsets--but that turns out to be wrong.

On the right is the situation we're approaching: notice that

much of the northern hemisphere in darkness, so that our daylight hours are few.

I first discovered this years ago as a junior high school science teacher.  I liked to get out of the book sometimes, and do big outdoor things.  One favorite was to make the school yard into a giant sundial using the flagpole as a gnomon.  (My hope was that such a tall pointer would make a shadow you could watch move just standing there for a few minutes, but the shadow turned out to be too indistinct to work well.)  I wanted to use the shadow to establish the exact direction of south by looking at the shadow at "local noon"--the time when the sun is directly over your meridian, that is, your longitude line.  I figured to find that time as the half-way point between sunrise and sunset.  (That didn't work as planned, owing to the definitions of sunrise and sunset!)  It was in studying the newspaper almanac day after day that I became confused by the seeming lack of a pattern in the times. 

Note: I've discovered that the explanation below (as well as the EarthSky post referenced above) has errors. I will fix these in a post later in January. (Edited 1/3/14)

EarthSky describes the reason for the mismatch this way: "The time difference is due to the fact that the December solstice occurs when Earth is near its perihelion – or closest point to the sun* – around which time we’re moving fastest in orbit. Meanwhile, the June solstice occurs when Earth is near aphelion – our farthest point from the sun – around which time we’re moving at our slowest in orbit."

That explanation is good, but has gaps I think need filling.

First, why should the speed of the earth in its orbit around the sun affect sunrise and sunset times?  For convenience, let's count a day as being from one solar noon to the next.  (The Royal Navy used to do this.)  We think of the 24-hour day as the result of the earth's rotation on its axis, but in fact the sun is also moving about one degree of its 360 degree annual trip around the sun in that same day.  That means that the earth not only has to rotate one degree MORE than 360 degrees on its axis in order for our location to return to pointing at the sun--that is, back to solar noon.  As long as the earth kept a steady speed in it's orbit, all would be well.  But instead the earth speeds up a bit as winter approaches, and begins slowing again after early January.  Because it is going more than one degree around the sun in December, the earth must rotate farther to bring it around to the same solar noon, making everything (all things being equal) a bit later.  The opposite occurs when the earth is at its slowest, in June and July.  This effect combines with the shortening days of fall to create the odd timing of sunrises and sunsets.

Here's an animation that makes the below clearer.

Imagine you are standing on earth where the left-pointing arrow begins. From "Day 1" to "Day 2" your location has rotated 360 degrees PLUS an additional amount to point back toward the sun, since the earth has moved a bit further around the sun.

At least one more question occurs: why does the earth change speed in the first place?  That has to do with the elliptical shape of earth's orbit, and a law first discovered by Johannes Kepler centuries ago (soon after Copernicus and Kepler established that the earth revolved around the sun, instead of the reverse).  Earth's orbit is an ellipse (oval) that is not quite circular, with the sun a bit nearer to one end of the oval.  Kepler discovered that planets move slowest when they are farthest from their parent body, and fastest when closest.  To be precise, any orbiting body sweeps out equal areas of its orbit in equal times.  (You can think of these areas as pie-slices of the whole orbit: it will take a wider pie slice to cover the same area when the wedge is shorter.) 

Enough for now.  More about the "reasons for the seasons" soon!

*This surprises a lot of people, who assume that summer is warm because the earth is closer to the sun, while winter is colder because it is farther--but the opposite is true.  It turns out that the difference in earth-sun distance from summer to winter is not very great, and the reason for the seasons lies elsewhere, as I'll explain at the solstice in a couple of weeks.