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    Length of Day and Latitude
    From: Frank Reed
    Date: 2010 Mar 1, 20:13 -0800

    I've found a nice concise formula for the length of day in low latitudes (latitudes within 30 degrees of the equator):

    LOD = 12:08 + (Dec/6.3)*Lat

    That is, under the usual assumptions of an observer at sea level with standard refraction, we take the mean length of the day to be 12 hours and 8 minutes (the actual mean should be closer to 12:06:40 but this gives slightly better results as an approximation). Then we take the Sun's declination in degrees and divide by 6.3. Multiply that by the latitude in degrees. The result is a certain number of minutes that you then add/subtract from 12:08. You can use the usual sign conventions for Dec and Lat or you can remember that if the Dec and Lat have the same name you add, contrary names, you subtract. The result is accurate to +/- one minute on average with extreme errors of about three minutes. Real-world variability of the length of the day is comparable to this, so not too shabby for such a simple formula.

    How would you use this?

    Well, for starters, it's so simple that you could probably commit it to memory and use it in a practical case. Imagine sitting around the lunch table with friends while on vacation in Oahu in the last week of January next year. Someone says, "I wonder how long the day is here?" You know that the Sun's declination is about -19 degrees, and your latitude is about +21 degrees. So 19/6.3 is 3... 3*21 gives 63 minutes... And we subtract that from 12:08 since Dec and Lat have opposite names... that gives the length of the day as 11 hours 3 minutes give or take a minute. That lunch question may never happen in practice, and even if you could pull off this calculation, most people will assume you googled it anyway, so better just keep your mouth shut and enjoy the salad... ;-)

    Another use for this simple formula is to gauge the sensitivity of the length of the day to a change in latitude. Pick a day, any day. Suppose the Sun's declination is 6.3 degrees. On that day, LOD differs from the mean value by one minute for each degree of latitude. That is, the formula simplies to LOD=12:08+Lat. If you can only estimate the length of day to the nearest minute, which is probably about right, then you can only determine latitude from length-of-day observation to the nearest degree (quick reminder: this formula has the stated accuracy only within 30 degrees of the equator). And clearly, if we pick another day of the year, and the Sun's declination is, let's say, 12 degrees, then formula simplifies to LOD=12:08+2*Lat (nearly) so a one minute change in the length of day correpoonds to a half-degree change in the latitude. That's the best estimate of latitude that you could expect on that day of the year from an observation of the length of the day.

    Although not too accurate for calculations, the general behavior is similar even at higher latitudes. Up to 45 degrees away from the equator, the average error rises to 5 minutes with extreme errors around 20 minutes. There is of course an intermediate equation that would work over a wider range of latitudes (replacing Lat by tan(Lat) and changing the constants) but if we're gonna dig out the trig, there's no reason not to go straight to the complete spherical calculation.

    -FER


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