Copyright © 2005 Joseph Mack
v1.2, Oct 2005, released under GPL
Abstract
How to construct an analemma in the privacy of your own home. Fun for you and your family. You need a clock, a stick, a can of spray paint and a year or so of spare time. What happens when you try to do it at your kid's school.
Table of Contents
Everyone knows that the sun crosses the sky at a lower elevation in winter than in summer. If you were to take the position of the sun at a fixed time of the day (say noon) throughout the year, you might guess that the sun's path would be a straight line between its low position in winter and high position in summer and would retrace the same path on its return in winter. In fact the path is not a straight line, but a figure "8" called an analemma. (http://www.analemma.de/, see defn 2). Here's Dennis DiCicco's famous photo of an analemma (http://www.analemma.de/images/articles/dicicco.jpg) the first analemma photo I know about.
The analemma shape (rather than straight line) is caused by the earth's orbit being an ellipse, rather than circle. The ellipticity of the earth's orbit
- causes the sun to spend a longer time in the northern hemisphere (188 days) compared to the southern hemisphere (178 days)
- causes the sunrise and sunset to not occur at 6am and 6pm (local time) on the equinoxes
- causes the latest/earliest sunrise/sunsets not to occur on the solstices.
- is incorporated into globes of the earth, where the analemma is drawn in the south Pacific Ocean.
All this information was available to me as a high school student, reading the newspaper each day for the time of sunset/rise. Most of this was also known by the ancient Greeks, who also knew the correct size and shape of the earth and moon. The ancient Greeks used nested epicycles to approximate the planet's and the moon's orbits. That the shape of the orbits were in fact ellipses, was determined by Kepler, using Tycho Brahe's meticulously collected data.
Using the geocentric view of the sun's daily position, the sun is closest to the earth (perihelion) about 5th Jan and so whizes past us more quickly. By 6 July, the sun is further away and moves more slowly against the fixed stars (and looks a little smaller). Because of the different speed of the sun against the fixed stars during the year, if you look up at the same time (say noon), some days the sun will be fast (by 17 mins in November) and be slow (by 14 mins in February) (see Sundials on the Internet http://www.sundials.co.uk/equation.htm). The difference between true time (as measured by a clock) and the apparent time (as measured by the position of the sun) is called the "equation of time" and is found on a sundial to allow for correction.
An article in Sky and Telescope ("The Analemmas of the Planets", David A. Harvey, Flandrau Planetarium, Sky and Telescope, March 1982, p 237-9) showed the analemmas and the calculations for the other planets. Since the mid '90's, you've been able to draw analemmas on your azimuthal equidistant projection maps (http://www.wm7d.net/az_proj/az_html/az_help.shtml#analemma). Now everyone is doing it e.g. Astrophotography by Anthony Ayiomamitis (http://www.perseus.gr/Astro-Solar-Analemma.htm) and there is hardly a place on earth that hasn't been analemmaed.
The analemma is the path that the sun would take through the sky if the earth rotated about its axis only once a year (i.e. kept the same side pointed towards the sun). Because the earth instead rotates every 24hrs, rather than once a year, to see the analemma we have to look at the sun at the same time every day. Because the moon keeps the same face to the earth, astronauts on the moon looking up at the earth would see it move through an analemma shaped path every month.
In the times not so long past, an understanding of the analemma was so basic that they were drawn on globes of the earth. The analemma rated up with the equator and the poles. By the time I entered grade school, the dark ages of education were upon us, and teachers mumbled incoherently when asked to explain the analemma. The embarrassement of the teachers on not having an understanding of the motion of the sun possessed by the ancient Greeks, was handled deftly by globemakers who, anxious to retain their customers, removed the analemma from globes [1]. We are now in an era where graduates of the education system leave with a view of the solar system that predates the ancient Greeks. This is a fitting educational outcome in a country where both major political parties agree that our best exports are jobs, subsidised agricultural goods and promisory notes payable by our children.
Growing up in the southern hemisphere and being being subject to the Euro/American northern hemispherism of our education system, gave plenty of opportunity for testing our knowledge. To navigate by the stars, you look for Polaris, which as everyone (well the Scout manuals) knows is directly north [2] . The books talked about 4 seasons: summer, winter, spring and fall - if you can believe that. Unbeknown to the small fraction of the world's population in USA and Europe, most of the world has wet and dry, or dry and dry, or wet and wet. We had dry and occassionally wet and when it was wet, there was lots of it, but only once every 10-20 yrs. Sure, sometimes wet comes with cold and sometimes it comes with hot, but plants and animals breed and grow on water, not temperature. Agriculture based on crops and animals that couldn't survive 20yrs of drought was doomed from the start. It was only the willingness of the Australian taxpayer to compensate farmers for decades of unexpected crop losses that kept the system functioning for 200 yrs. Sundials are latitude dependant: in particular in the southern hemisphere the sun crosses the sky from the right to the left. When I grew up in Australia, no-one would be caught dead buying anything made in Australia. The sales clerk would only have to say to my mother "but Modom, this is British" and that would settle it, the article would be snapped up, the purchase assuring the world that my mother recognised only the finer things of life. The problem with a sundial made for a latitude of 60N, transplanted to a garden at 38S, was that it would give a time in the morning when it was the afternoon. Parks and gardens were filled with these sundials installed by credulous burghers wishing to simultaneously declare their love of art and pay homage to the scientific accomplishments of the ancients. Running the gauntlet of the Australian education system was a punishing proving ground for the critical thinking of a young person.
I occassionally visit my native Sydney, Australia to see friends and to walk in the Blue Mountains. On returning from one hike, I bought fish and chips in Penrith, on the banks of the Hawkesbury River. Looking for a place to eat, the proprieter gave me directions to a local park. He said it was a little difficult to find, but to not be discouraged. It wasn't far, but there were no signs to it and I had the place to myself on the banks of the river at sunset at a time when the place should have been jumping with people. Why do they have this nice park if no-one goes there? I guess Australia is such a nice place, that you don't need parks to enjoy the outdoors :-) There to my surprise, on the edge of the river, was an analemmic sundial. An analemma, on the ground, was marked by the date, and was surrounded by an arc of stones marking the hour. A person (of any height) stood on the analemma at the spot for the day and looked at their shadow on the arc of stones. It was apparent that the analemma automatically corrected for the equation of time. I hadn't realised this was possible. I wondered if it was an exact solution or only approximate. Did it work all year? What was the shape of the arc of stones? I did some calculations on my return to USA, but wasn't convinced that I'd done them correctly. On my next trip back to Sydney, I visited the father of a friend of mine, who lived near the park and mentioned the analemmic sundial, how pleased I was to find it, and that I was amazed such a thing existed. He grinned and said "I designed that sundial" and pulled out the local phone book from a few years before. The front cover showed him and the team that build the analemmic sundial, with each person standing behind one of the hour marker stones. He then took me to his garden to show me his own private analemmic sundial. I gathered that he'd done more than just design the park's sundial; he may have proposed and helped raise the funds too.
Many months later, after my return to USA, I remembered that on a previous trip, I'd told him I was going to visit the Mt Annan Botanic Garden (http://www.rbgsys.gov.au/mount_annan_botanic_garden/). He'd smiled and said "make sure you look at the analemmic sundial". Back then, I didn't think it anything special - it was just another sundial. I'd missed the point.
If you can watch the analemma from all of the ancient temples in Greece, then it could be done from the driveway of my house in Durham, North Carolina, USA. What school nowadays can be without an analemmic sundial? The science teacher at my son's school was interested, so we looked for ways to put an analemma in the school too.
a gnomon and a surface to project the shadow: "gnomon" is the greek word for a stick that casts a shadow. The surface is usually flat and horizontal but it can be curved and/or vertical. It just must be illuminated all year at the time of day that you're going to mark the analemma.
At my son's school I picked a some open ground in the middle of a circular driveway infront of the school, using a streetlight as a gnomon and marked the spots by pushing in coloured golf tees. I hoped that no-one would notice the golf tees and pull them up (or worse - move them). The ground wasn't flat or level, but it was available.
Our flat but sloping northern hemisphere driveway faced the south and was illuminated most of the day. I marked the shadow spots with spray paint using a piece of cardboard as a mask. A portable flagpole was made from a 12' long section of EMT (tubing for 120V house wiring) readily available from hardware stores (you can check they're straight, by rolling them on the floor before you buy, but all the tubes were straight). The length of the pole was about the largest that would fit the shadow of the sun on the driveway in winter. Since the driveway was bumpy (gravel in the cement) and sloping, the tube needed a large radius rounded end to sit reproducably on the ground. I also needed a large blob at the top to cast a shadow. I cut a hole in two tennis balls and glued them on each end of the pole. If I accidentally bent this tube, I could easily make another one of identical dimensions. To position the pole vertically for the measurement, I initially used the surveyor's method of balancing it between my closed thumb/forefinger. If the gnomon wasn't exactly vertical, it would start to fall one way or another. This method didn't work on a breezy day, so I glued a 3-axis bubble level (also from the hardware store) to the tube (I only looked at the two horizontal axes). To figure out where to put the gnomon in the driveway (so that the shadow would always be on the cement and not in the lawn) I calculated the maximum length of the shadow (i.e. at the winter solstice) using the length of the pole and the sun's angle below the zenith (my latitude of 38degN plus 23.5deg) and added a few feet for good measure. I found the north-south line by looking at the sun at noon and located a piece of driveway that would hold the vector and wouldn't be under the parked cars. I then spray painted a spot on the driveway for the gnomon.
a time to observe the sun:
At home I chose local mean noon. Since our longitude (-79deg) is 4 deg west of the standard meridian, local mean noon is at 12:16pm (1:16pm in daylight savings time). At my son's school, the science teacher and I decided to mark the ground at 16mins past any hour we happened to be around. The local mean noon was marked with red golf tees, +/- 1hr with orange golf tees and +/- 2hrs with yellow golf tees (rainbow order) producing 5 sets of analemmas at 1 hr intervals. We hoped, that just by doing it when we could, that by the end of a year, we'd get enough coverage to see the analemmas at various times of the day. At home I was going to do it each week on saturday (or sunday if bad weather). Most times I got one reading in the weekend. I co-opted the family to stand in when I wasn't home (most of soccer season). Occassionally the sun's shadow was marked through the day (at home at 30min intervals at solstices and equinoxes, at school, near the 22nd of the month, at 10min intervals) to show the path of the sun through a particular day.
a clock:
For the first year, I used a quartz clock, set to within 15secs of the correct time with ntp (http://www.ntp.org/). The most accurate clock in the house (less than 15secs drift/week) was a really cheap digital clock won as a prize at a fair. The smoothness of the analemma turned out to be much worse than my estimate of 15secs error. About a year into the project, I realised there were problems with this clock. It didn't have a second hand and I had to watch it the whole minute to get the time right. I also had to bring up my dial-up internet connection each week to reset the clock. This was a time consuming nuisance. About the same time I discovered WWVB controlled clocks (http://en.wikipedia.org/wiki/WWVB) courtesy of my newly arrived Celestaire 2006 catalogue (http://celestaire.com) [3]. Most areas of the world (except Australia ;-\ ), have similar transmitters to control household clocks to within 1sec.
Celestaire was selling a LaCrosse Technology WWVB controlled clock. I wanted an alarm clock with a back-lit screen and time in 24hr format and this wasn't it, so I went to the LaCrosse web site and found a contender, but no instructions on turning off the back-light. The battery wouldn't last real long if the back-light was on all the time. (How hard is it to put complete documentation on the website?) I called LaCrosse (no 800 number) to be shunted to a computer answering machine, which assured me that my phone call was very important and they were busy helping other customers and they'd answer as soon as they could and to hang on. After about 5 minutes they cheerfully told me to leave a message and they'd call back just as soon as they could (no option to stay on the line). I left my question expecting a reply in time to order that day. Nope, so about 6 hrs later I called again and got the same treatement. They did reply two days later: I should call back again or send my question by e-mail. It would have been simpler to just answer my question. After the 2nd phone call I realised that LaCrosse was only marketing to people with a high tolerance for bad service. I didn't qualify, so two days before LaCrosse returned my call, I'd already been to Walmart (not my favorite store, for its treatment of its workers, but their service is better than LaCrosse's) where I bought a Sharp SPC325 WWVB controlled alarm clock with 24hr format and a back-light that turns off for half the price (about $10) of the LaCrosse device. It locked onto WWVB the first night and I tried it on the analemma the day after that.
To my very pleasant surprise, the chalk line produced with this clock had barely any radial jitter (less than 1/4 of the line produced with the previous clock). I attribute this to being able to hold the WWVB clock next to the pair of bubbles in the same field of view and being able to count down the seconds to the exact minute. Only then do you look up, to the location of the shadow on the ground and walk over and mark it. Clearly the previous method of looking up every few seconds (in case the next minute rolled over) was adding jitter to the shadow.
a way to show the analemma for a photograph:
Spray paint shows up well on walls and the side of trains, but didn't photograph well on the driveway. I needed something with a bit of verticality, coloured, light and cheap to mark the spots for a photograph of the whole analemma. I chose plastic drinking cups.
Here's the spray painted dots at home.
Figure 1. Analemma spray painted dots looking south.
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The spots closest to you were done 9 months earlier and are a bit faded.
Figure 2. Analemma spray painted dots looking north.
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The spots closest to you were done 3 months earlier and are relatively bright.
The gap on the left side, below the cross over point, occurred during our summer vacation - we were away for 2 weekends. Note the miss-aligned point on the right side of the lower loop - I did that one myself - so I can't blame the family, I have no idea what went wrong there. There is an inflexion on the lower left side with a concave section. This shouldn't be there. In general the analemma is not as smooth as I'd hoped :-(
Figure 3. Taking a reading.
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The cups indicate previous readings. The red cups are the noon analemma.
My son in the foreground is holding the gnomon. The 3-axis bubble level is the yellow device 1/3 the way up the gnomon. I'm in the back holding the marking chalk in my right hand. The can of spray paint is on the ground to my right, mostly occluded by the gnomon. The camera is looking north (east is to the right of the picture). The yellow cups are marks taken in the morning, the blue cups are taken in the afternoon. This photo was staged, and was taken about an hour before local mean noon (note: the shadow of the pole passes the nearest yellow cup and not the red cups as it did an hour later; and both our shadows are not parellel to the axis of the analemma).
Figure 4. Looking east.
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The E-W arcs of cups are the shadows at 30min intervals during the day. The yellow cups are the marks taken before noon, the blue cups are the marks taken after noon. The red cups are the noon analemma.
The analemma was started in Sep 2004 after returning from our summer vacation. It didn't occur to me till May 2005 to take marks at any other time than noon. This was about the time when the sun is at the analemma's node. The three E-W arcs are taken (L-to-R) about 3 days before the fall solstice (Sep 2005), the (May 2005) analemma node and the summer solstice (Jun 2005). Note that the solstice line is (almost) straight, while the other two lines are increasingly hyperbolic. I'll do one at the winter solstice in Dec 2005.
Figure 5. A reading close up:
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spray paint; the cardboard mask for spraying (only the large hole was ever used); the gnomon (pole on right, laid on ground) with 3 axis bubble level (vertical bubble not used, the two horizontal ones at the top of the yellow frame are used). The bubble level is attached with silicon glue - the rope is just incase the glue comes undone. A line of pink chalk marks are across the image below the cardboard, and opposite the yellow bubble level frame. The driveway concrete is a mixture of mafic (dark) and felsic (light) material.
The photo here shows about 15mins of chalk marks (seen better in the enlarged photo). on a partially cloudy day It's Sep 24 (northern hemisphere fall), just after the sun has crossed the analemma's node. The image is looking away from the sun (north, towards the winter loop of the analemma). Local mean noon (due to daylight savings) was at 1:16pm. The data (a line of pink chalk marks across the image from left to right, towards the yellow bubble level on the gnomon) start about 1:00pm at 1 minute intervals. There is a gap a 1:07 and 1:08pm and another for 1:16 and 1:17pm, due to cloud. The position of the sun at 1:16 is interpolated from the position at 1:15 and 1:18pm. The calculated position at 1:16 is marked by pink chalk lines above and below the calculated position. The radial jitter in the line corresponds to about 15secs of time error (my initial reproducability estimate). This position of the mark differs from last year's by by about 1/2 a shadow diameter (if they were both good readings, they'd be on the same line). Presumably the mark I collected last year (at the beginning of the project) was not a good one. The new point, with 15mins of pre-data, fits the overall curve better than last year's point.
By the the good planning of Pope Gregory XIII (http://www.geocities.com/calendopaedia/gregory.htm), the day of the week (e.g. a saturday) advances by 1 (or 2 for a leap year) days each year, so if I keep doing this on saturdays, I will get new points each year, till the 5th (or maybe 6th) year when I'll be get to redo my original bad points again.
Initially taking a reading was a two person job; one to keep the gnomon vertical (possibly in the breeze); the other to mark the shadow. Often there were manpower problems on the day, so a one-person method was developed. The chalk was put on the ground near the last shadow spot (as a coarse reference), you'd hold the gnomon vertical, then in turn look at the bubble levels, then hoping you didn't move the gnomon, look at the clock and the shadow on the ground, scanning for a convenient feature in the random noise of gravel in the concrete driveway to act as a reference. Since the shadow didn't move all that fast, you had a pretty good idea where it would be at the exact minute. When the moment came, you'd walk over keeping your eye on the reference spot and mark it with the chalk. The shadow of the tennis ball took about 3 mins to pass over a spot on the ground. When I later changed to the WWVB clock with a seconds display, I held the clock next to the bubble levels and only looked up to the shadow on the ground at the exact minute. This method gave a lot less scatter.
Initially I would just take one reading at the exact time. If the moment was cloudy, I'd just forget about taking a reading for that day. It didn't take much thinking to realise that you can't throw away a good reading, just because it's cloudy - you won't get that data point again for another year. I started taking one minute interval marks leading up to the exact moment, and depending on the cloud cover, might have to interpolate or extrapolate to the required moment. I expected a nice straight line with 15secs of scatter, but found that the reproducability was a lot worse than my initial tests. The errors didn't account for some of the really bad points, but were about the right size (1 min of time error) for many of the bad points. I then decided to take about 5 mins of reading for any one mark.
The clock setting error was 15secs. Initial reproducability tests gave a positioning accuracy corresponding to about 15-30secs of time error (about 1/8th the diameter of the shadow). The actual analemma as it developed over the months was much lumpier than expected, with an error corresponding to about 1min of time error - about 1/2 the diameter of the shadow. As well there were the occassional obvious outlyers.
While I can attribute bad data at the beginning of the project, to lack of practice, even after 9 months of collecting data, I still was getting bad data e.g. data taken at the summer solstice (there are a couple of spots not in the line, one on the right about 4 dots from the summer solstice and one at the bottom left just after the summer solstice - the loop on the left side should be convex and not have a concave section).
With 30 yrs of experimental science under my belt, including a paper on estimating errors, I was greatly chagrined to find that the experimental errors were much worse than I'd estimated and I didn't really know what caused them. I have these clues
- I came to trust the clock, which rarely needed week to week resetting, and wouldn't check if I was in a hurry. It turns out that occassionally (like after a change in temperature) the clock would be out by upto a minute in a week.
- While I could acheive 15sec reproducability in positioning the shadow, I could only do this after multiple readings. When I'd do 15mins of readings in a row, it would take about 5mins for the radial distance to the shadow to converge on a value. One reading didn't do it.
I'm expecting most of the errors to disappear with the new WWVB controlled clock that I can hold right next to (and in the same field of view as) the bubbles and that displays seconds.
Although teachers are glad of interested parents, I found the administration less welcoming. I had previously tried to introduce the National Archery in the Schools program into the school, for those kids who didn't want the standard sports. This is competitive archery using Olympic competition rules. I'd arranged for the equipment to be sponsored (about $2,500) and for certification as a coach. Although there's never been an accident in this program, I arranged for $1,000,000 insurance for each kid. The cost to the school? - $0. All I needed was a field or gym to do it. The school wasn't interested; no explanation was given or thought neccessary. It took multiple calls to even get a reply to my proposal.
Collecting data for the analemma was problematic. You're on someone else's turf and obeying the rules of people, who have higher priorities than helping kids tackle large problems with simple tools. I had hoped that the analemma would be constructed dot-by-dot on the playground blacktop, where the kids would see the analemma developing and maybe ask questions about it. Having the kids mark the spots would increase their involvement, otherwise the project would just be another of the things that adults produce out of the blue to take up the kid's time. However kids aren't allowed to duck out of the classroom unsupervised. I had hoped that I'd get some photos of the kids with the analemma, but I'd need a legal release from the parents and the school.
The first year, a request by the science teacher to mark the playground blacktop was greeted with "you're not to deface the school property" and "if the school board wanted an analemma, you would already have one". The school doesn't have an analemma, but it does have a sundial mounted prominently above the south facing entrance. Everything in a school is for educational purposes, right? Well you'd be run down by a bus if you stood in a place to look at the sundial; the science teacher wasn't consulted on its purchase, so it wasn't expected to be part of the curriculum; there is no correction for the equation of time; except for October, when it gives correct time, the sundial is off by upto 30minutes. Presumably only the kids have noticed.
kid: Mommy, what's that?
Mommy: It's a sundial, one of the first scientific instruments. It's been used for thousands of years to tell the time. The daily passage of the sun through the sky, marked by the shadow of a stick on the ground, is so regular, that it was the original definition of time.
kid: so how come it's off by 30mins?
Mommy: It isn't a real sundial. It's art. It's to show you that there are more important things to learn at school than your teacher's lessons.
kid: like what?
Mommy: Like when you spend other people's money, you don't have to ask anyone if it will work. Watch out for the bus.
Since the school is 16 minutes offset from the standard meridian, it's likely that the sundial is displaying local time rather than standard time. This is a reasonable thing for a sundial to do, but it's one step more complicated to explain than a sundial displaying standard time. The sundial should overlook the playground not a spot where a kid risks instant death. It's unfortunate that the administration doesn't recognise the educational value of a sundial and installed it as a non-functional piece of art. It seems that sundials continue to be a universal tool for testing a kid's critical thinking.
That was the first year.
The next year the science teacher secretly marked the driveway infront of the school, where the kids are picked up, using the school flagpole as a gnomon. Black marker pen on blacktop looked like oil drops. No-one at the school knew what was happening (even the kids), but if you knew what to look for, you could see a nice analemma. I was reminded why secret societies and martial arts like karate (empty hand) existed: to propagate knowledge when rulers controlled the masses by keeping them ignorant and powerless. You encode your knowledge in pictures on cards and tell everyone that it's a game.
When classes started the next year and the pattern revealed, some of the kids would start asking questions. But after a year of data collection and with a nice analemma displayed for all to see, the school repainted the driveway during the summer break, getting rid of all those nasty oil marks.
That was the 2nd year.
We've just started (at the summer solstice) on the 3rd year, I pushed golf tees into the ground on an otherwise unused piece of ground infront of the school, using streetlights as gnomons. The ground is like concrete and I use a pin-driver and hammer to knock a hole in the ground before whacking the golf tee in the last 1/4inch. The ground's people were informed of the golf tees, flush to the ground, but were concerned that they'd damage the mowers. One day I came along to find many of the tees ripped out. Then later after inserting about 100 golf tees, I came along to see that the ground had been dug up with an aerator and only a few tees were left in the ground.
I gave up. 3 years is enough.
It seems that the teachers' idea of an intersted parent and the administration's idea are quite different. The administration's idea of interest must be donations of money without any strings attached. If you want to know why american education is lagging, you only have to look at the treatment interested parents and enthusiastic teachers get. How much harder must it be for the students?
[1] A child's idea that the continents were once joined, as seen from the obvious fit of South America and Africa, were dismissed too. It was just coincidence - it had been proposed before by a crackpot meteorologist who was quite rightly laughed out of academia by the world's best geologists.
[2] Polaris is always below the horizon in the southern hemisphere.
[3] Celestaire sells lovely brass sextants. I can't afford any of them, so I bought their plastic Davis sextant. Over the range Maine to Florida, it's only accurate to about 1 degree, which is not all that great, but if you need a sextant (and who can be without one nowadays), I was happy with Celestaire. I like reading their catalogue too. If I had an ocean going yacht and time and money to sail it, I'd have a lot more neat stuff from the Celestaire catalogue.