Thursday, 18 February 2010
Where Did the Analemma Go?
(Thanks to Lia C. for the pic)
The word is drawn from the Greek for a "lofty structure" or "upraised portion" of something, (a reference to the gnomen of a sundial) but the more common use of the term is to describe that figure eight that is, as I recently told me students, "on every globe."... Ooops... that seems not to be so.
It came from a lesson in trig on simple harmonic motion. We were talking about things that demonstrated sinusoidal behavior, and one bright young man suggested that the height of the sun at noon would be an example. I sort of agreed with a comment about "not exactly at noon.. but" and then the little guy was confused.. "You know, I said, like today ."(it was Feb 12) "I think the sun was about 12 minutes late or so."
Slow looks at each other, then back to me... the three letter word look,,,,,"Huh?"
"You know, that's what the analemma is for, telling if the sun is early or late.".....
Same look, compounded by the wild eye..."HUH?"
"On the Globe." Some frustration now... ".. the little figure eight... in the Pacific Ocean usually.. " Now the look has turned to fear... Will this be on the test? What is going on in the pacific? We were up doing homework, we don't watch the news... Help...and then, God answered their prayers...the bell rings... and they RUSH to exit...
So this morning, I came in and put out a call to the staff, "...someone loan me a globe, my kids don't know what an analemma is ". The first offer was just down the hall, cool, I'll walk down and get one and bring it in and when they come in I'll show them what it is and they'll say,... "Ohhhh, That thing." But it wasn't there... and it wasn't on the second globe offered, or the third... or ...gasp.. any of them.. the analemma has, it seems, gone the way of the two-dollar bill... So today I went in and set aside the basics of simple harmonic motion and we talked about a little astronomy, and why we have a 24 hour day.
First I pulled the graph of an Analemma off a Plus-Math on-line article from Cambridge, and checked some notes... (I was wrong about the globe..don't want to trust my aging memory)...
"The difference between the mean solar time and the actual solar time is called the equation of time. The Sun is furthest behind GMT around 12th February, when it is about 14 minutes 20 seconds slow. It is furthest ahead around 3rd November when it is about 16 minutes 23 seconds fast. The Sun's position coincides with GMT on four days of the year: 16th April, 14th June, 2nd September and 25th December," Later I found a cool picture taken by Dennis DiCicco, and editor with sky and telescope who spent two years taking pictures of the sun at noon once a week for a year (the second year was after he found out that the first had been slightly off center and cut off part of the bottom and had to start over...patience).
The Plus sight even a nice reminder that the "mean" in Greenwich Mean Time is really an average. This was a little more difficult for them to explain.
When I showed it to them not one reported ever seeing one. I asked them how long it took for the earth to spin on its axis and they mostly knew the memorized value...23 hrs 56 minutes (they always leave off the 4 seconds)... but they never thought about it.
Hmm, I ask.... "so if the sun reaches its highest point today at noon, and we turn around in 23 hours 56 min.. then tomorrow the high sun will be at 11:56..YES?" They conceded the four minutes possible error.
"And in two more weeks, the high sun would be about 11 am?????.. the hour is troublesome, but the math seems incontrovertible..so they mumble assent.
"And after a month,.... at ten am.... and in three months... The sun will reach its highest point at Six AM?????... Have you noticed that happening???" They were a little confused, but finally were ready to admit that the 23 hours 56 minute day must be a mistake..... Until one kid finds a reference.. "MR. Ballew, it says right here.. (finger poking into the book...we got that old guy now....."It says RIGHT here that the ..." and here we get a really bad attempt at "Sideral day or rotation period."
So we walk slowly through the idea that as the earth spins the very distant starts rotate in the sky and come back to their rightful position in 23 hours, 56 minutes and four seconds... but in that day, the earth also wandered part way around its elliptical path around the sun (god I will so love using this again when they are working with conics.. SURELY they will remember). And so the sun (a not so distant star) will have moved, and the earth will have to turn a little longer...about four minutes longer and till it is back to its rightful place high in the sky.... cool... but the figure eight thingy????
Well, remember Kepler??? "Equal areas in equal periods of time." But when we are farther from the sun, we move more slowly.. and so we turn too far.. the high sun comes early.... and when the sun is closer we move faster... and the sun shows up a little late... But the MEAN in Greenwich Mean Time...(we are big on GMT around here, not because we are in England, but this is a military base... Military Zulu time IS GMT) is really, the average of the time the sun will be at its highest. It is when noon comes around..."on average." Some times it is early, some times it is late, but on average a day from highest sun position to the next will be 24 hours.
We did manage to review a little for the trig stuff coming up, but they actually seemed to be thinking more than normal, offering conjectures and (too often) wild guesses. But mostly they were thinking, and doing a little math... and we all had a pretty good day...
Now.. can we Please get the analemma back on the globe?