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    Re: How Many Chronometers?
    From: Gary LaPook
    Date: 2009 Sep 24, 00:12 -0700
    There has been some discussion on the effect of changes in temperature on the accuracy of inexpensive quartz watches. Let's look at a hypothetical voyage to assess the effect that changing temperature will have on the daily rate of the watches.

    I did an experiment and established that the change of rates (for the watches I tested) due to change in temperature was .02 ppm times delta tempº C squared which is half of the commonly quoted rate. We will use this rate and this standard formula and will assume that the quartz crystals in the watches are cut  to the standard 25º C temperature where the rates of the watches will be zero seconds per day.

    We will be making a voyage from New Zealand to Tahiti.  Jimmy Cornell says this is a distance of 2220 nm and he  recommends making this voyage in March. Lets assume we will make about one hundred miles a day so the voyage will take 24 days. Referring to the U. S. Navy Climatic Atlas of the World, Volume 5, South Pacific for March we find that the average temperature in New Zealand is 16º C and in Tahiti it is 28º C so we will encounter a change of 12º C. Looking at the isotherms on the chart for surface air temperature we see that they are approximately equally spaced so we can using the simplifying assumption that we will spend an equal number of days crossing each one.

    Is it valid to use these average temps for this analysis? Maritime climates feature very constant temperatures controlled by the sea surface temps which are very stable. A table shows that offshore of New Zealand that only 10% of observations are between 12º and 14º C, 80% are between 14º and 19º C, and only 10% are between 19º and 20º C so the average of 16º C will predict the temps to be encountered very accurately.

    It is similar near Tahiti, 10% 24º-27º, 80% 27º-29º and 10% 29º-31º. We will keep the watches in an insulated box with a bottle of water to supply thermal inertia so that they will not respond to daily swings in temperature, only the day to day change.

    Since there are 86400 seconds in a day and the constant we are using is .02 ppm, the standard formula simplifies to .001728 times delta Tº C squared. ( If you want to use the .04 ppm constant then every thing doubles.) We will spend one day at 16º C, two days at each of the intermediate temperatures and one final day at 28º C, a total of 24 days. Since the watch rate changes the same amount for temperature deviations both above and below 25º C we can combine high and low temps for this computation. We will spend one day at 16º C which is 9º from 25º C; 4 days 1 degree from 25º C; 4 at 2; 3 at 3; 2 at 4; 2 at 5; 2 at 6; 2 at 7; 2 at 8 and 2 at right at 25º C. Running the formula for the four days when we are one degree away from 25º C, produces a daily rate of .001728 times four days at this rate equals a .006912  second contribution to the total drift of the watches during the voyage. The first day, when the temp is 9º less than the standard, the watch will run .139968 seconds slow. Totaling the contributions from each day we find that the watches should be running .877824 seconds slow at the end of the 24 day voyage.

    While we were in New Zealand we checked the rates of our watches and they each lost .139968 seconds per day, the rate at 16º C. Using this rate we calculated that the watches would lose a total of  3.359232 seconds during the 24 day voyage. But they will only lose the .877824 seconds that we just calculated so we will have overestimated the loss by 2.481408 seconds. This overestimation would cause an error in longitude determined by celnav of .620 minutes which is .596 nm at the latitude of Tahiti, or just 1104 meters.

    Enough to really worry about?

    Interestingly the error is not the same on a voyage in the reverse direction. This is because the rate you had determined for your watches while in Tahiti at 28º C is only .015552 per day since you are only 3º away from the 25º C standard. Twenty four days at this rate would show the watches running .373248 seconds slow which subtracted from the actual drift of 2.481408 makes an error of  only 2.10816 seconds causing a longitude error of .527 minutes or .397 nm at the latitude of New Zealand, just 736 meters.

    Anabasis75@aol.com wrote:
    My $30 Timex Ironman watch has been reset twice in the last 8 months and had an error of about 1.5 seconds from the HF radio timetick in that time.  It has worked quite well for my Celnav traveling from Japan to the East Coast of North America (the long way).  You certainly don't need an expensive watch to do quite well with navigation.
    In a message dated 9/15/2009 12:10:31 P.M. Eastern Daylight Time, lunav@abelhome.net writes:

    A fascinating experiment and experimental confirmation of my long-held belief that even the cheapest digital watch makes a superb chronometer.

    Now a challenge for anyone wanting to repeat the experiment or extend the results:

    1.   Take the "watch board" and simulate motion.   Okay, maybe we can't take it to sea but maybe drive it around town with us?

    2.   Vary the temperature.   Maybe put the board outside in direct sunlight (and rain and maybe even snow).

    Remember, Harrison's biggest challenge was not making an accurate chronometer (that had been done already) but rather making one that remained accurate despite the motion and temperature changes experienced at sea.

    Whoops, as I write this I'm looking at my digital watch on my wrist and watching it bounce around.   Maybe experiment #1 isn't required.   In fact, the very nature of digital watches should make them motion-insensitive (excluding relativistic effects :-P ).

    Lu Abel

    Gary LaPook wrote:
    Based on our discussion, I became curious about the accuracy of digital
    watches and their suitability for use as chronometers so I went to my
    local TARGET store and purchased three identical watches for $17.00
    each, the cheapest that they had. I set them and let them run for a few
    days and, as I expected, they each had different rates. Based on this I
    labeled them "A", "B", and "C" in the order of their rates starting with
    the slowest. I then reset them to UTC at 0121 Z on May 28, 2009. I
    checked them against UTC from WWV eleven days later on June 8th and
    found that they were all running fast by 2, 4 and 7 seconds respectively
    and I worked out their daily rates as .1818, .3636, and .6363 seconds
    per day, respectively.
    On July 11th, 44 days after starting the test, the watches were fast by
    9, 17 and 28 seconds. Using the rates determined in the first 11 days
    the predicted errors would have been 8, 16 and 28 amounting to errors in
    prediction of 1, 1, and 0 seconds. If using these three watches for a
    chronometer we could average the three errors and end up with only a .66
    second error in the UTC determined by applying the daily rates to the
    three displayed times after 33 days from the last check against WWV
    which took place on June 8th.
    I determined new rates now based on the longer 44 day period of .2045,
    .3864 and .6363 seconds per day, respectively.
    On September 15th at 0800 Z (per WWV), 110 days after starting the
    test,  I took a photo of the watches which I have attached. The photo
    shows the watches fast by 21, 41 and 69 seconds but by carefully
    comparing them individually with the ticks from WWV the estimated actual
    errors are 21.5, 41.8 and 69.0 seconds. Using the 44 day rates, the
    predicted errors are 22.5, 42.5, and 70 seconds giving the errors in the
    predictions of 1.0, 0.7 and 1.0 seconds which, if averaged, would have
    caused a 0.9 second error in the computed UTC after 66 days from the
    last check against WWV on July 11th.
    If, instead, I used the 11 day rates then the predicted errors would
    have been 20.0, 40.0, and 70.0 seconds which would result in errors of
    prediction of -1.5, -1.8, and 1.0 which, if averaged, would cause and
    error in the computed UTC of -0.6 seconds after 99 days from the last
    check against WWV which would have been on June 8th in this example.
     From this experiment it appears that fifty one dollars worth of cheap
    watches would give you a perfectly adequate chronometer.

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