Where's the Sun?

"Class, let's get started. 

"Let's assume you live in the northern hemisphere -- somewhere near 40 degrees north latitude.  Take a camera.  Find someplace convenient nearby where your view of the sky to the south is unobstructed.  Mount the camera pointing directly south, with an angle about 50 degrees above horizontal.  Fix the camera in place, so it can't move.  Set the camera for a very short exposure -- maybe 1/2000 of a second or so, but it depends on the film.  Get yourself a very accurate timer for the camera -- one that can run for a year or so and still keep its accuracy to within a second or so. 

"Ok, does everyone have their materials for the assignment?  [Rustling noises from the classroom, but no hands raised.]  Good.  We'll proceed. 

"Set your camera to take a picture at exactly noon, ok?  Yes, in the back...."

"Is it daylight savings time yet?"

"Ah, I love teaching a smart class.  No Jennifer, not yet.  And we'll ignore the effects of daylight savings time for the rest of this exercise.  DST is a plot by the government to get you up earlier in the mornings, just like water fluoridation is a plot to impurify our precious bodily fluids.  Thanks for bringing that up. Now, take your picture at noon, standard local time.  Go develop it or push the right buttons or whatever you have to do.  Ok, what does your picture show?"

[Several people answer out loud.]  "The Sun!"

"Excellent.  I assume it's more-or-less in the center of your exposure?  Good.  Those of you who don't have the Sun in the frame of your photo, go check your setup.  If the weather is bad, just wait until one day when it's clear.  OK?  Let's take a break while everyone gets things set up. 

[Several days pass.]

"Are we all set up now?  God, it's so hard to get class started when we take a break.  I think we're going to have to conduct the rest of this exercise as a thought experiment.  How's that?  [Groans from the class.]  Gang, we have to finish this before the end of the semester -- sorry.  Now let's get back to picture-taking, or at least the idea of picture taking.  Set up your camera so that it takes a picture, say, once a week for say 8 weeks, still at exactly noon.  Take the pictures on the same piece of emulsion; you digital folks, superimpose your 9 exposures on top of each other.  Now, develop your film, or print your exposures.  What do you see?"

"The Sun!"

"Um, yeah!  Good!  But we established that you'd seen the Sun earlier.  Is there anything unusual about your exposures, other than the occasional cloud, airplane, streaker, or the like?  Yes, John, up front...."

[John, in a farm-boy drawl:]  "Well, the Sun is lower in the sky in winter and higher in summer, so it's going to move up and down."

"Excellent: the Sun is not in the same place in the sky from day to day.  You folks are good.  I like teaching smart people: it's so easy and fun.  Anything else?  [Sussurations of unsurety from the class.]  Hm?  Anyone?  What's going on?  Alice?"

[Alice, in a prim, yet unsure voice:] "Well, I thought my setup might be wrong, but I checked it and I can't find anything."

"And why do you think your setup is wrong?"

"Well, the Sun is in different places, but it's not in a straight up-and-down line.  It curves a little bit." 

"Ah, but it moves!  And you can find no error in your setup.  Anyone else with this problem?  [Mumbles of assent.]  You all have this problem?  Hm.  Well, you're all smart kids: maybe it isn't a problem.  Did you all take your pictures at exactly noon, each time?  Yes? 

"Well, what is noon, anyway?  [Dead silence.]  No, I'm not off my rocker, at least not yet.  What is noon, other than lunchtime?  [Uncomfortable silence.]  Surely it's not just what some clock says, right?  We didn't always used to have clocks, at least not the mechanical or electrical kind.  So what is noon?  [Several people look at their watches.]  Ok, lets get at this a little differently.  I'm sure some of you are either media-savvy or well-read:  How many of you saw Master and Commander, or read any of the source novels by Patrick O'Brien?  [Sporadic murmured assent.]  Didn't they also observe the Sun?  Remember, they had no clocks.  What were they doing?"

[Volunteered from an indefinite location:] "Finding noon."

"Excellent! Finding noon is very important on a sailing ship of that era: it helps you figure out where you are when you have no landmarks.  But how did they do it? [Unsure rumblings.]  I'm sorry -- did someone say 'south'?  It's true they had compasses, but magnetic compasses are unsteady and don't necessarily point exactly north/south. [Mild, scattered noises of disbelief.]  I guess that's for another lecture.  Were it night, they could establish north and south pretty closely by looking at the stars, but the nighttime stars aren't visible during the day.  By the way, they'd have to look north for the Sun if they were south of the equator.  So anyway, what were they doing?  John, is your hand up?"

"My grandma told me that, in the old days, they used to find noon and south by looking for the Sun at its highest point over the top of town hall.  So everyone had south down pretty well, but noon varied a little bit from town to town if you paid attention and had a good watch."

"Very good.  Thank you, John.  I think we can see a benefit of a rural education here: people have to observe things they wouldn't otherwise need to attend to.  In fact, the whole problem of clocks and calendaring may originally have been driven by agricultural concerns.  People wanted to know whether they were just having a warm spell, or if spring had actually arrived: if you plant your crops too early and have them freeze in the ground, you might starve.  But back to noon: people established noon as the time when the Sun was highest in the sky that day.  It's mid-day, as you all already knew.  [Groans.]  Yes, it makes sense now, doesn't it?  But it looks like our series of exposures don't have the Sun at its highest point.  Take some more exposures around noon one day -- say every minute or so.  The Sun is always highest directly south of us, but our clocks don't always say that's noon.  What's going on?  Yes, go run check with your cameras.  We'll take another break.  [Stampede out of class.]

"So is everyone happy that their clock disagrees with what the Sun says about noon?  What does that mean?  How much are we off by?  Are we ever right?  Do the Sun and the clock ever agree? 

"Well, it turns out that they do.  Has anyone taken a full year of exposures yet?  Betty, you did? [Betty nods.]  What did you get?"

"Kind of a figure-8."

"Betty, great.  That's exactly what I anticipated.  Can you pass your exposure around class?  Any of the rest of you get that?  Pass your pictures around too.  Thanks.  Now, what does this mean about the Sun and the clock?  Sometimes the Sun gets behind or ahead, right?  But then it always catches up or slows down.  This cycles around over the course of a year.  Sometimes the Sun gets maybe 15 minutes ahead or behind.  Of course the Sun isn't wrong; neither are our clocks.  The clock is just running ahead or behind the Sun.  The discrepancy accumulates for a while, then starts running in the other direction.  But a clock day -- 24 hours, or 1440 minutes, or 86,400 seconds -- is sometimes shorter, sometimes longer than a solar day -- a day measured by the Sun.  The Sun and a clock agree four times a year: when the Sun is due south at clock noon, which are the points down the center axis of your figure-8.  Yes Jack?"

"I've seen this figure-8 before somewhere.  I think I've seen it on a globe.  It's not exactly symmetric: the top and bottom halves are different sizes." 

"Good point, Jack.  The figure-8 isn't exactly an 8-shape: it's off a little bit.  And it used to be commonly printed on globes.  It's called an analemma; it's a visual representation of how much the Sun and a clock disagree by.  The amount of the disagreement has what I think is a really cool name: the equation of time.  When you made your time-lapse photos, you created your own analemmas: graphical representations of the equation of time. 

"You don't have to make an analemma this way.  You can also use a sundial.  A sundial is typically aligned on a north-south axis.  The shadow falls to the north of the sundial's pointer or gnomon in the northern hemisphere, and to the south in the southern hemisphere.  The shadow is at its longest at sunrise and sunset, and is at its shortest at local noon.  If you mark a dot on the ground at the end of the shadow at clock noon every day for a year, you will be drawing a kind of analemma on the ground.  You'll also discover that your sundial is not indicating noon when your clock does.  But we already established with your pictures, because you people are smart.  Smart enough to be let go now.  Class dismissed!" 

Comments

[Eric J]

Just this weekend, Mitchell was looking at a globe and asked about the analemma. I kind of understood it, but didn't have a really good explanation for him. This was a good explanation for me, but I'm still not sure if I can render it in terms he'll understand.

[Joe]

Well, that's a bizarre coincidence. That's what happens when you live in the same house for 7 years, I guess. :) I think I got the idea of writing this in mind sometime while my PC was dead. The equation of time was at a minimum about then: looking it up, the minimum was Feb. 11th, when the Sun was about 14.26 minutes behind local mean time.

Maybe this Friday your son will ask for a hot dog or something. ;)

[Joe]

Tell him that the Sun and a clock don't always match up because clocks are not complicated enough and too regular: the analemma is the adjustment.

[Jack]

Damn well written, Joe! My time on-line is currently very precious since I'm having to use an internet cafe these days (it's not the money, but the annoyance of having to listen to twenty kids screaming and yelling while they play Counter Strike ... if I have hear the expression "The Terrorists Win" one more time, I swear that I'm gonna shoot somebody). Still, I was absolutely enthralled by both the concept and the way you wrote this. Kudos!

[Joe]

Thanks Jack -- it pretty much just squirted out of my head. I had the idea, but didn't really know how I wanted to write the thing. I was also testing my ability to type with a new keyboard and my maybe-better brain.