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Sunday, May 1, 2011

Longitude or Entropy

The following is a consolidation into a single post of a recent series of posts, which detailed the testing of an inexpensive mechanical watch for use with celestial navigation.
The posts were consolidated at the request of my wife, whose critique was something like "ten posts about how to wind a watch, really?".


Longitude is time, time is longitude. Within the context of navigating on (or above or below) the surface of the earth, time and longitude are one and the same. At the equator four seconds of time equals one mile of distance east or west. At the latitude of Seattle it's close to six seconds equals one mile, but it is still readily apparent why precise timekeeping is so critical. If my chronometer is off by 12 seconds my position on the earth is in error by two or three miles. The larger the chronometer error the larger the position error, the two are inseparable.

In order for my chronometer to actually be a chronometer, it doesn't have to keep accurate time. Every timepiece, including the most sophisticated atomic clocks, gain or lose some amount of time. What makes a chronometer a "chronometer" is the fact that it gains or loses time at a predictable rate.

Tomorrow I will describe in more detail what precisely I am hoping to prove with this inexpensive mechanical watch experiment, and the methodology I'm using to do so. This isn't a trivial consideration; the advent of quartz watches and GPS plus the loss of LORAN has completely changed the balance and importance of marine celestial navigation for routine and emergency applications, and the role (if any) of a non-electronic timepiece which is cost-consistent with a low-end handheld GPS receiver.

For now, the test watch has been rewound and reset at 2300 tonight, ready to roll in the morning.

More to come..


So, take-two of the testing of the mechanical non-chronometer.

The manufacturer recommends winding the watch twice daily, but this proved insufficient. Ideally, as with any mechanical chronometer, the watch should be wound at the same time(s) every day, but this is probably not realistic for me this week given my variable work schedule. So instead I'm just going to wind it every several hours or so as I think to do so, understanding that this is less than optimal. Also standard protocol with a mechanical chronometer is to wear the watch constantly, as your wrist provides a better heat-sink than most other things do to prevent the mechanisms from being exposed to temperature extremes. This is reasonable but the very inexpensive watch-band isn't especially comfortable, so I'm just wearing it as much as I can. On the other hand, it's spring (finally!) in Seattle, so the temperatures shouldn't be very extreme in any event. He said.

In my initial test, tracking the watch from one initial winding, it started out losing about a second an hour. Toward the end of the day this had increased to about a second and a half per hour. However, today I've been winding the watch every several hours, and it's keeping remarkably accurate time doing so. 18 hours after resetting the watch, it is only two seconds slow per Android GPS. We'll know in a few days how well this continues to track, but so far it is significantly more accurate than I would have anticipated.

A note on timekeeping; I'm using the GPS on my Android phone as a time reference for this simply because I always have it with me. WWV would probably be a better reference, and I happen to have that on my speed dial as well (cough), but the possibility of routing delays with the phone systems makes direct GPS at least as accurate as a cell phone WWV "radio" time-tick.

What am I trying to learn from this? Specifically for emergency navigation, in the absence of any electronics (such as may result from the electromagnetic pulse from a lightning strike) how would an inexpensive mechanical watch compare to other methods of deriving Universal Time (and hence longitude), such as lunar distances or lunar altitudes. And also to see if, just like the plastic Davis Mark 3 sextant, it is possible to achieve comparable results to far more expensive instruments, simply by taking all of the steps recommended (but rarely done) for the more expensive instruments. So far with this test, I am encouraged.


Last night around 10pm the watch stopped even though it was still wound. Tapping the face started it again, which is fine for routine use but worse than useless for the sole source of Universal Time in an emergency. Rewound and retesting.

Meanwhile, the $20 Casio quartz watch is tracking like a champ. But it wouldn't survive the EM pulse from a lightning strike. Which brings me to the topic of Faraday Cages.

A Faraday Cage is simply a wire mesh (or just metal, it doesn't matter) enclosure which prevents electromagnetic energy from entering or exiting. Lining a small cabinet with chicken-wire and sticking a spare radio, flashlight, GPS, wristwatch and batteries? Good idea. Sticking the aforementioned into your microwave or conventional oven? Very, very bad idea. They may survive the lightning strike, but won't do so well when your shipmate later preheats the oven, or sticks their coffee into the microwave to warm it.

So, how is the mechanical watch doing now?

Currently 2 seconds fast after 9 1/2 hours. Hopefully I can keep it running long enough to start getting some real data. If I can't, then that's real data too.


Now well into the second day with the watch running fine. So far losing about 1 second every 2 hours. I'm chalking up the first two failed runs to operator error. I read a number of reviews of this particular watch saying that it didn't work at all; now I'm wondering how much of that was the result of a generation unused to mechanical watches. Will continue to test, but still cautiously optimistic.


Three and a half days into testing, some interesting preliminary results.

For the first two days the watch was losing about 9 seconds per day, ending up around 20 seconds slow at midnight the night before last. Then yesterday it started gaining time rapidly, and as of right now is about 25 seconds fast.

Will see if this oscillation continues or if it has decided to continue to run fast now. I'm winding it every couple of hours or so when I'm awake, and just before and just after sleeping.

So, for those who are following along with this, the standard of accuracy I'm hoping to obtain is the ability to determine my Universal Time to within 2 minutes after 20 days. These numbers are not arbitrary; 2 minutes of time is about the best accuracy one could expect to obtain from traditional lunar distances from the deck of a small boat at sea (and that may be overly generous), and 2000 nautical miles is realistically the farthest from inhabited land any vessel would likely ever be under even the most extreme circumstances, and a vessel covering only a hundred nautical miles per day (averaging 4.2 knots) would cover that distance in 20 days. Worst case scenario (at the equator) 1 minute of time error is 15 nautical miles, so two minutes is 30 nautical miles, so the land would need to be at least 200 meters high in order to be seen from a small boat at that distance. This is admittedly extreme (the center of all of the overlapping bell-curves here would probably be less than 10 nautical miles inaccuracy), but if your life depended on it you would want to allow for that margin of error.

This is also far more accurate than any means available of determining longitude with the sun alone, with or without an accurate chronometer.

It should be noted that for this to be in any way scientific I'd really need to test dozens of mechanical watches from the same and different manufacturers. But it would only take ten of these watches to equal the cost of a good mechanical chronometer, and if I were going to spend that much money I'd rather just buy a good chronometer. Again, the point of the exercise is to determine whether or not it is possible to have an entirely non-electronic method of routine celestial navigation which costs no more than a cheap hand-held GPS receiver. The other two parts of the equation are the sextant and the almanac and sight reduction tables; will discuss these presently while we're waiting for more watch data.


I've just reached a very critical juncture in this test. Right now, at around 0900pdt five days into the testing, the second hand and minute hand are officially 30 seconds out of synch with each other. At the time GPS read 08:46:00 the watch read either 08:46:52 (which fits well with the existing data trendline) or 08:47:52 (which is an obvious outlier). It is possible that in this situation at sea without an electronic time source that I would be so confident in my record keeping that I would recognize the outlier even without an electronic comparison and then remember to add or subtract a minute for however long is necessary until the second hand became 90 seconds out of synch with the minute hand, and so on. But the room for error here is enormous.

The better practice would be to discard the second hand altogether for this application, and simply read minutes and estimate tenths of minutes on the minute hand. This may even help eliminate some of the oscillating error I've been seeing, will from this point start tracking both.

If this pans out (and based on the very small amount of data I've collected so far, it may) it will be certainly more difficult than using either an electronic or mechanical chronometer. But compared to computing lunar distances, it's making mud pies.

Also if this pans out, it emphatically does NOT eliminate the necessity for a second hand on the watch. You'll need it for many other applications, such as timing lighthouses and estimating speed for dead reckoning. It just won't be used for celestial navigation.


Continuing to test mechanical watch using only the minute hand. So far the results are looking pretty good, and I've determined the most accurate means of obtaining star sights using it. Instead of estimating the decimal of a minute visually, have one person observe the watch and mark the moment the minute hand passes a minute mark, and have the other person with the sextant standing by with the star properly aligned.

This violates the Starpath protocol of not having two people working together on celestial sights (better for accuracy, worse for relationships!), but in this case you need all of the accuracy you can get.

The other drawback with this method is that it severely limits the number of sights which can be taken within the time that both the star and the horizon are visible, but it is unlikely that you would actually get more than one sight per minute anyway.


My original plan was to run the watch test for a full 20 days. However, the actual uncorrected inaccuracy is now large enough that it's becoming a minor nuisance to use the watch for my day-to-day timekeeping, and I'm actually pretty satisfied with the data I've collected after 15 days of testing.

My original parameter for "success" for the watch to be a suitable backup to a GPS was to be able to determine Universal Time within two minutes after 20 days. However, using the methods detailed below it appears that you should be able to obtain Universal Time within 18 seconds more or less indefinitely. Which means that if all of the rest of your celestial navigation is done correctly, you should never be more in error than 4.5 nautical miles at the equator, or 3 nautical miles at the latitude of Seattle. So in even the worst possible case, from the cockpit of a small sailboat you could successfully navigate any distance across any ocean and make landfall on any island in the world, however small and however low above sea level. That counts an an unqualified success, and far better than I imagined possible.

The watch consistently gained six tenths of a minute, or 36 seconds, each day; at the end of the experiment the watch was running more than nine minutes fast. For the record and for comparison, the Casio quartz watch gained 4 seconds total over the entire time of the experiment. Which is to say, the Casio completely uncorrected was more than four times more accurate than the mechanical watch with all corrections. But, the Casio is as vulnerable to electromagnetic pulse as any other electronics on board. And the mechanical watch gained time at a consistent and predictable rate, which makes it a chronometer by any reasonable definition.

In practical terms, what this means is that for use as a secondary means of routine ocean navigation, you would of course use a quartz watch as your chronometer, and meanwhile maintain the mechanical watch and log its error daily. Then, if due to a lightning strike or any other reason you should lose electronic means of timekeeping, simply switch to using the mechanical watch and continue with routine celestial navigation and dead reckoning.

The proper method for using the watch to attain this accuracy is as follows:

Wear the watch on your wrist at all times, utilizing your body temperature as a heat-sink and also your flesh to dampen the effects of vibration.

Wind the watch obsessively, with one finger only to avoid over-winding. Really obsessively, every time you think of it, when you wake up, before you go to bed, before dinner, whenever. This undoubtedly introduces some error, but it would be much worse for the watch to stop. And it will.

Use the minute hand only, unless you're really sure you're only going to need to navigate for a couple of days. Use a set-and-wait method, marking the time for the sight at the moment the minute hand aligns with the minute mark on the watch face.

That's really all there is to it.

If anyone is actually interested I'll be happy to email them scans of the raw data. This is obviously not a scientifically rigorous test, but simply a demonstration that, in principle, a modern inexpensive mechanical watch can be used as a functional chronometer for celestial navigation.

Having established that the least expensive plastic sextant and the least expensive mechanical and quartz chronometers are more than adequate for routine and emergency celestial navigation, the third part of the equation is an ephemeris and a method of reducing the celestial data into a fix which can be plotted as a position on a chart. The next post will compare three inexpensive alternatives for this.


  1. Your post titles are occasionally brilliant. The other times, I just assume I'm not getting the joke. You should write questions for Jeopardy.

    I just got the "Junk Navigation" one yesterday. Shame, shame.

  2. I figured some of the physics geeks would find that funny. Rudolph Clausius rocked.

  3. He did, and then Newton got the credit, posthumously. I never understood that.