analemma

The Shadow Knows

by Pat Murphy & Paul Doherty

Gordon Van Gelder, the editor of this magazine, is a clever man. When he asked

us to write for Fantasy & Science Fiction, he wanted us to write four columns a

year. He cleverly set our deadlines on dates that we, as scientists who spend our

time observing the natural world, could not possibly forget. He said, "Let's

make your columns due on the solstices and equinoxes."

Very clever. We haven't forgotten a deadline yet.

In honor of that astute decision, we decided to write this column about about the

solstices and about the movement of the sun as perceived from the earth. We're

going to suggest that you spend some time watching shadows, a way of

indirectly observing the movement of the sun across the sky. These observations

can put you in touch with natural patterns that humans have been watching for

thousands of years--but that most of us modern folks have come to ignore.

Along the way, we'll talk about time--we just can't avoid it.

I'm not lazy; I'm a scientist

We'll start with an observation you can make on a sunny afternoon, while

lounging around in a hammock or kicking back in a poolside bar. Take a look at

the shadows around you. Find a place where a shadow, maybe the shadow of a

building or a fence, makes a straight line. Mark that line somehow--with a rock

if you're on the grass, a chalk line if you're on blacktop, or a swizzle stick if

you are sipping daquiris by the pool.

Guess where the shadow will be in 15 minutes and mark your guess in the same

way you marked the shadow. ("Hey, could we have some more swizzle sticks

over here? Oh, sure--put 'em in another round of daquiris.")

In 15 minutes, check your guess. You may be surprised at how quickly the

shadow moved. That is, you may be surprised at how quickly the sun moved.

Or, to be even more accurate, you may be surprised at how quickly the earth is

spinning--about 1000 miles per hour at the equator.

Right now, for simplicity's sake, we're going to talk about the movement of the

sun. We know and you know (and Galileo knew) that the sun isn't really moving

across the sky. But according to Paul, physicists have to be adept at jumping

from one point of view to another. According to Pat, so do writers. So we're

going to stay on earth (at that poolside bar, maybe) and tell our story from that

frame of reference for a while.

Next time you spend the day outside, pay attention to the movement of the

shadows as they move with the sun. The sun rises more or less in the east (more

on that "more or less" later) and sets more or less in the west. So the shadows

point more or less west in the morning and more or less east in the afternoon.

An interesting aside here: if you watch the movement of the shadow on a sundial

over the course of a day, you'll notice that it moves in a clockwise direction.

Coincidence? We don't think so! Early clockmakers designed clocks to mimic

the familiar sundial. Those readers who are fond of alternate history stories

might consider what might have happened if we used clocks based on sundials

that had been developed in the Southern Hemisphere where shadows move the

other way!

Shadows are at their longest at sunrise and sunset. When are they at their

shortest? Noon, you say? Well . . . more or less. You see, unlike sunrise and

sunset, the concept of noon relates to human time-keeping--and that gets a little

tricky.

Does Anybody Really Know What Time It Is?

If you are telling time by the sun, noon is defined as the time when the sun is at

its highest point in the sky. The important words in that sentence are "if you are

telling time by the sun." That is, if you are using solar time. Chances are, you

are telling time by that device strapped to your wrist. And the time on your

wrist watch isn't solar time; it's what's called standard time. You can blame that

on the railroads.

Back before 1883, people used solar time. Each community kept its own time,

basing that time on the sun's position in the sky. Since the sun is always moving

across the sky, noon where you are is at a slightly different time than noon at a

place a few miles to the east or west. Back before 1883, noon in one town would

be four minutes later than noon in a town fifty miles to the east.

In 1883, to regulate time for the sake of railroad schedules, the United States

adopted standard time, designating time zones and requiring all communities

within a time zone to keep the same time&emdash;even though that standard time

doesn't quite match solar time.

If you are smack dab in the middle of your time zone, the sun will be at its

highest point at noon. But if you are at one edge of your time zone, solar time

may differ from standard time by as much as 40 minutes.

What About the Solstice?

If you spend some time watching shadows, you'll notice that the position and

length of a shadow depend not only on time of day--but also on the time of year.

That's because the sun's position at a certain time is different in different

seasons. And that, of course, brings us to the solstices.

What's the longest day of the year? Any good Druid could tell you the answer to

that one. The longest day is the summer solstice (June 21 or thereabouts) and the

shortest day is the winter solstice (December 21 or thereabouts).

As a knowledgeable fantasy reader, you probably even know of some of the

fantasy connections for these dates. The summer solstice is associated with

Midsummer's Night Eve, when witches and fairies and other supernatural forces

are in control. In The Hobbit, the keyhole that lets Bilbo, Thorin, and the

dwarves unlock the passage into the dragon's lair opens on Durin's Day, the first

day of the last moon of autumn on the threshold of winter.

You know the length of a day changes over the course of a year, but have you

ever really paid much attention to the position of the sun--other than squinting

when the summer sun comes in your window too early or complaining when the

days get too short? Well, here's your chance.

For those of you with a lot of patience, here's an activity that takes a year to

complete. You need a south-facing window, a pocket mirror, some small

Post-Its, and a lot of patience. Choose a time of day when you'll be home at least

once every couple of weeks for the next year. Put your little mirror on the

window sill and position it so that it reflects a spot of sunlight on the wall or the

ceiling. Cover most of the mirror with masking tape, leaving only a 1/4" square

exposed.

If you can, fasten the mirror in place so no one moves it by accident. Pat stuck

hers down with some stuff called "museum putty," that's sold in California

under the brand name Quake Hold™. You folks who are sensible enough to live

far away from the fault zone will have to come up with your own methods.

Note the time and date on a Post-It, and stick the Post-It to the wall or ceiling

where the spot of light reflecting from your mirror falls. A week later, at the

same time, do it again. And a week later, do it again. Repeat for an entire year.

(We warned you that you'd need patience.)

As you do this, you need to use standard time. If you move your clock forward

(or back) to adjust for daylight savings time, change the time that you make

your weekly mark by an hour.

Keep it up, and at the end of a year, the Post-Its will form a figure eight on

your wall. The marker from mid-December will be at one end of the eight and

the marker from mid-June will be at the other. This pattern is called the

analemma, which is Latin for "sundial." The analemma is a visual record of the

sun's changing position over the course of a year.

This is the same figure 8 you see on earth globes&emdash; usually in the middle of the

Pacific Ocean. It is also known as the equation of time. Each planet has its own

shape for the analemma. On Mars, the analemma is the shape of a teardrop.

Pat is in the middle of doing this activity. (On earth, not on Mars.) As we write

this column, she has Post-Its all over the ceiling of her sunporch. She started

back in December and we're writing this in March, so she's not even halfway

done yet.

If you (like most of us), prefer instant gratification, then you probably have

access to the World Wide Web. In that case, we suggest you visit

www.skypub.com/spc/staff/dic.html. On that web site, you find Dennis di

Cicco's award- winning, year-long photograph of the analemma made in the late

1970s. But to convince yourself that Dennis didn't cheat and do this in a

darkroom, you still might want to try the experiment with a mirror and

Post-Its. Depends on how trusting you are. (According to Paul, scientists must

be professional doubters. But he's not the one with Post-Its all over his ceiling,

so go figure.)

So why already?

You want to know why the analemma is a figure-eight, rather than a tear-drop

or an oval or a circle? You fool! Pat wanted to know why, once upon a time.

Days later, after much explanation with circles and arrows and too many

diagrams and too much math, she decided she didn't want to know the whole

story.

We're going to give you the short version of why the analemma is a

figure-eight. If you must understand every last detail (which Pat claims is

enough to make a person's head explode), we recommend you visit

www.analemma.com, a thoroughly detailed Web site with animations and full

discussion of why the sun does that.

We'll start you off with an easy question: where does the sun rise? Did we hear

you say "east"? Sorry. It's an easy question, but the answer is tricky. We warned

you about that earlier, remember?

If you were to watch the sun rise each morning over the course of an entire

year, you'd see that the sun doesn't always rise in the same place. In the

summer, in the Northern Hemisphere, the sun rises a little bit north of due east.

The date on which it rises the farthest north of due east is June 21, the summer

solstice and the longest day of the year. In the winter, in the Northern

Hemisphere, the sun rises a little bit south of due east. The date on which it rises

the farthest to the south is December 21, the winter solstice and the shortest day

of the year.

Suppose you watched the path of the sun on the winter solstice and on the

summer solstice. On the summer solstice, the sun rises much higher above the

horizon at noon than it does on the winter solstice, taking a longer path across

the sky. On the winter solstice, the sun never gets as high in the sky.

Okay, now we're going to have to do one of those shifts in viewpoint that

physicists and writers like. Instead of staying on earth, we need to take a look at

the solar system from the outside, examining the earth's orbit.

The sun's path across the sky changes with the seasons partly because the earth's

axis (the imaginary line through the earth around which the planet spins) is

tilted with respect to the earth's orbit around the sun. As the earth orbits the

sun, the North Pole (the point where the axis intersects with the earth's

Northern Hemisphere) always points in the same direction, pointing near

Polaris, the North Star. (The direction of the Earth's axis does change over a

26,000 year cycle, which means that the analemma evolves with time. But we're

not going to get into that here.)

Because the earth's axis is tilted, during a portion of the earth's orbit, the earth's

Northern Hemisphere is tipped toward the sun. That's when it's summer in the

Northern Hemisphere. The North Pole is tipped toward the sun and the sun

shines on a greater area of the Northern Hemisphere. As the earth spins, places in

the Northern Hemisphere stay in the sunlit area longer, and the days are longer.

At the other extreme of the earth's orbit, the earth's Northern Hemisphere is

tipped away from the sun. That's when it's winter in the Northern Hemisphere.

The sun shines on a smaller area of the Northern Hemisphere, and the days are

shorter.

The earth's tilt affects the position of the sun in the sky&emdash;and so does the shape

of the earth's orbit around the sun. You might think that the earth always

traveled about the same speed on its way around the sun. That would be the case

if the earth's orbit were circular&emdash;but it's not. The earth's orbit is an elliptical,

which means that sometimes the sun is closer to the earth and sometimes it's

farther away. The difference in distance is only about 3% of the overall

distance. That may not seem like much, but it makes a difference to the speed of

the Earth.

Suppose you took the average speed of the earth --about 30 kilometers per

second. If you checked the planet's speed when it was closer to the sun (which

happens in January), you'd find it was a little faster than that average. When the

earth was farthest from the sun, in July, you'd find that it was moving a little

slower than average.

Meanwhile, Back on Earth

That's what all this looks like from outside the solar system. How does all this

affect what you see on the planet Earth?

Paul says that the analemma would be easier to understand if there were no

atmosphere on the earth. Without the atmosphere to scatter the sun's light, we

could see the stars during the daytime, and we'd be able to see the sun's

movement against the background stars. (Of course, if there were no

atmosphere we couldn't breathe. But we'd understand the analemma. Pat says

that seems like a small consolation.) Anyway, if we could see the sun moving

against a background of stars, we'd see that the sun moves on a regular path

through the stars, a path called the ecliptic.

Suppose you could see the stars when the sun is out. Suppose you're watching

the stars at around noon in mid-May. The sun is in the constellation of Scorpio,

perhaps near the star called Antares. Just before noon the next day, 23 hours and

56 minutes later, you check on the position of Antares. Antares will be back in

the same place in the sky. The sun, however, won't yet have reached its highest

point in the sky. That will take about four more minutes. From your point of

view on earth, the sun is lagging four minutes behind the stars.

Add together the time it takes for Antares to return to its original position and

the four minutes that it takes to get the sun back to its original position. You get

24 hours or one average day. Very tidy, isn't it?

Each day, the sun lags behind the stars. Over the course of months, this

accumulating difference means that different stars rise at different times. In the

Northern Hemisphere, for example, Scorpio is a summer constellation--you

don't see it in the night sky during the winter. The difference between the sun's

movement and the stars' movement has shifted the rising time for the stars that

make up Scorpio so that the constellation is up during daylight hours.

Why do the sun and Antares move across the sky at slightly different rates? Ah,

that takes us back to outer space. The earth is spinning, and that's what brings

Antares back to its starting position. But the earth is also orbiting the sun. That's

why it takes an extra four minutes (or so) for the sun to get back into position.

No doubt you caught that weasely little "or so" in the previous sentence. It

doesn't always take the sun exactly four minutes to get back into place. After all,

the earth isn't always orbiting the sun at the same speed. From your point of

view on earth, that means that the time it takes the sun to return to a particular

place in the sky isn't always the same. In early January, when the earth is nearest

to the sun, the sun moves farther from one day to the next. It takes longer for

the earth to overtake the sun and return it to the same place in the sky. It can

take 8 seconds longer each day. These 8 seconds add up from day to day, and the

sun begins to lag behind. In June, when the sun is farthest from the earth, the

sun takes less than four minutes to return to its original position.

But that's not all. Remember the earth's tilt? Over the course of the year, the

position of the noontime sun moves up and down in the sky, because of the

earth's tilt. Consider the position of the sun at noon. Maybe you've been told

that the sun is overhead at noon. That's not necessarily so. (Sorry. Someone's

been telling you fibs.) In fact, if you are in North America, the sun is never

directly overhead. For the sun to be directly overhead, you have to be in the

tropics, the belt around the earth between the Tropic of Cancer at 23.5 degrees

north latitude and the Tropic of Capricorn at 23.5 degrees south latitude.

On the summer solstice, when the North Pole is tilted 23º 21' toward the sun,

the sun is directly overhead on the Tropic of Cancer. Six months later, on the

winter solstice, the South Pole is tilted toward the sun and the sun is directly

overhead on the Tropic of Capricorn.

Putting It All Together

From our earth-based viewpoint, the movement of the sun changes in two ways

over the course of the year. The daily path of the sun moves up and down in the

sky, and the time it takes the sun to reach its noontime position changes, with the

average time being four minutes.

Those of you who are familiar with electronics may have seen Lissajous figures,

very cool patterns that appear on an oscilloscope screen when you have two

signals out of phase with each other. Paul says of the analemma: "It's a Lissajous

figure with the sun moving up and down in the sky once a year and ahead of and

behind the rotation of the earth twice a year." Pat agrees that makes a certain

amount of sense: after all, you have two movements that are out of phase and

that could certainly create a figure 8. But she says that we have already caused

the heads of our audience to explode and we should stop now.

But Wait! There's More!

For those of you who are still with us, here is one more question. You know

that the winter solstice is the shortest day of the year. On what day of the year

does the sun rise latest? Or, for those of us who prefer not to be up at dawn, on

what day of the year does the sun set earliest?

Did you say the winter solstice? Not a bad guess, but wrong, nevertheless.

Though the winter solstice is shortest day, but it's not the day when the sun rises

latest or sets earliest. The exact date of the latest sunset depends on your exact

latitude, but around here the earliest sunset is around about December 7 or so.

The latest sunrise is around about January 4. And the winter solstice is

December 21, somewhere between the two.

Weird. To understand why this happens, you need to apply the concept of

analemma rise and analemma set. And that's something that Paul says makes his

head hurt. So we'll stop here, with Pat cheerily putting Post-Its on her ceiling

and Paul puzzling over analemma rise. Then maybe we'll have another round of

drinks by the pool and watch a few shadows move. After all, it's science.

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