GPS:Global Positioning on a Shoestring

How do you figure out where you are on the planet?

These days you can find your address on a mapping website, or maybe switch on a GPS. But before these things what could you do? When I got into astronomy I also started looking at celestial navigation techniques. I got a cheap student sextant ($35) and a nautical almanac and taught myself how to take a sight and convert it to a line-of-position on a map. With a few lines plotted, I could pinpoint myself within a handful of miles. But it was complicated and I was left asking: If I know what star is right above me, isn't that enough? That same star isn't above any other place on the planet, so that's the only measurement I should need.

So I made an experiment. This is where the shoestring comes in. I lay down on something soft on my driveway and hung a shoestring from a basketball hoop. OK so it was a regular piece of string. I got it wet so that it would hang straighter. I mentally extended the line of the string into the sky and noted the point that it hit and also noted the time. Drops of water got me directly in the eye, so I knew I was well aligned. Then I dragged out my star hopping skills and found the spot on my star chart. The declination of that point is more-or-less the latitude of my driveway. (If I did my math correctly, the non-roundness of the earth causes the measurement to be 5 minutes (= 5 nautical miles) lower than my actual latitude).

The right ascension is not my longitude of course because the earth spins. The longitude is given by a formula that I worked out by trial and error using data from www.nao.rl.ac.uk and is not super-accurate (within a half mile or so for 2006) and will probably be pushed into the online version of the newsletter by my editors. It goes as follows, where Tx is the number of hours since midnight, Jan 1, 2006, Greenwich mean time (subtract 8 hours for the number of hours since midnight PST or 7 hours for PDT), and 6.700555556 was the Right Ascension of the sky over Greenwich at that moment, and Rax is the Right Ascension measurement from the driveway, and Lyear=365.256363, the length of a year, and frac returns the fractional part of a number.

Longitude = 15*(Rax - 6.700555556 - 24*frac(Tx* (Lyear + 1)/Lyear)/24))

Negative numbers are W latitude. If you get a number larger than 180, subtract or add 360. I had to do the experiment a few times to get it right. On the last try, I put in the time to the nearest minute and was only 5 miles off in longitude. Then I corrected it with the time to the nearest second and I was only 1 mile off! This is unbelievable accuracy...literally. That this was pure luck is evidenced by my 3 tries for latitude which were between 15 and 55 miles off. Hey, it's a big planet! And what did you expect for a shoestring!?!

How useful is this? Well... Good luck using it on a rocking boat. And on land, if you don't know where you are within 55 miles, you're pretty lost. The accuracy could be improved by using some magnification instrument. A design using a spinning vat of mercury (highly toxic) is quite appealing (and far from being executed). I'll leave it as an exercise to the reader to determine the point of all this. If you try this technique yourself, the usual warnings about laying down on driveways in the dark apply.