Chasing the Sun

As an incomplete response to Ben’s long-ago challenge, “Long Walk After the Sun”, I found 2 websites that gave me enough information to start calculating the answer.

The first is the Naval Oceanography Portal, a veritable plethora of astronomical almanacs, calculators and errata. In particular, their Sun Altitude/Azimuth Table Calculator is particularly useful. Plug in a date and town in the United States and it will calculate for you at down to minute resolution, exactly where in the sky you will find the sun from sunup to sundown.

Using their software to generate a table of headings, you would then know exactly which direction your hypothetical man would be traveling in at any given time of day.

The next useful online calculator I found was on the Federal Communications Commission’s website, of all places. It has a calculator which will determine “Terminal Coordinates Given a Bearing and a Distance”. You plug in a starting latitude and longitude, as well as a distance to travel and your bearing, and it will spit out the latitude and longitude you’ll end up at.

From Wikipedia, humans walk at a pace of about 5km/h, or about 83 meters per minute. Given a set of starting coordinates, you could look up a heading from the Azimuth/Altitude chart, and set the distance traveled to (83 meters x the number of minutes walking) and get a new location. If you had the patience to calculate a very large number of items, you could trace the path from sunrise on January 1st, 2009 to sundown on December 31st, 2009.

Or someone could write a scraper. Or better yet, look up the algorithms these sites use to calculate these sort of things and not hammer their servers for curiosity’s sake.

I did a few calculations by hand (and by “by hand” I mean I copied and pasted numbers into web forms) and here’s the track of a person following the sun today, November 17th, 2010 from sunrise at 5:50 AM until he gets tired and passes out at 8:00 AM

Time		Latitude		Longitude		Bearing (E of N)
05:50		38 38  0 N		90 15 0  W		105.8
06:00		38 37 53 N		90 14 27 W		107.3
06:10		38 37 45 N		90 13 54 W		108.8
06:20		38 37 36 N		90 13 21 W		110.2
06:30		38 37 27 N		90 12 49 W		111.8
06:40		38 37 17 N		90 12 17 W		113.3
06:50		38 37 06 N		90 11 45 W		114.8
07:00		38 36 55 N		90 11 14 W		116.4
07:10		38 36 43 N		90 10 43 W		118
07:20		38 36 30 N		90 10 13 W		119.6
07:30		38 36 17 N		90  9 43 W		121.3
07:40		38 36 03 N		90  9 14 W		123
07:50		38 35 48 N		90  8 45 W		124.7
08:00		38 35 33 N		90  8 17 W		126.5

Something that stands out is that in the northern hemisphere, the sun sweeps an arc from something like 60 to 300 degrees E of N in a single day. That’s during the summer. Winter months the sun is lower in the sky (spends less time above the horizon, so less time walking), and sweeps a smaller arc, but one still mostly centered south. Also, the bearing of the sun changes slower around noon. I’d venture to guess that the time you’d spend walking east and west wouldn’t be as long as the time you spent walking mostly south. In a single day, a person wouldn’t cover very much ground over the surface of the earth. There are over 100 kilometers between lines of latitude on the earth, so even traveling in straight lines, it’d take close to 2 ten-hour days to drop a line of latitude. Over the course of an entire year, though, that might add up to some serious displacement.

I’m now very interested in this problem, and as a shot-in-the-blue guess, I’m going to say that our wandering man would end up somewhere on the line where the sun traces a perfect East to West path overhead, as north of it he’d be drawn south and south of it he’d be drawn north. I’m not sure that’s the equator, though since the earth is tilted on it’s axis. Ah, more problems already brewing and I haven’t even properly finished this one yet. Time to stop typing.

A few other websites I stumbled across researching this post that provide some illuminating links:
The Griffith Observatory in Los Angeles, in particular this page which turned me on to the possibility of not having to derive all the math myself because humans have been looking at the sky since the dawn of their existence and who would’ve thunk that at least one of those dirty apes thought to write down what they saw.
This website’s FAQ has a tip for a dead-tree book with Astronomical Algorithms in it.