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    Re: Quicker Ex-Meridian Formula for Slide Rule or Calculator
    From: Frank Reed
    Date: 2016 Dec 1, 10:31 -0800

    Greg Rudzinski, you wrote:
    "Cos of Altitude is used in place of Cos (Latitude +/- Declination).  This is a good alternative method but doesn't allow the navigator to precalculate [...]"

    I think you meant cos(altitude) in place of sin(lat +/- dec)? Note that you can still precalculate, so long as the instructions are clear. The altitude that should be used in the calculation is the true noon altitude. That formulation also makes it easier to understand the +/- choice and usually means that you don't have to think about that choice. The result is not very sensitive to an error in this altitude for "typical" noon observations so you can use a predicted noon altitude based on the DR, and it's also possible to use the altitude as observed close to noon, which would, as you say, imply that you would have to do the calculation after taking the sight (not really a significant cost). For noon observations where the altitude is high, roughly above 60°, the error from an error in the estimated altitude increases linearly with altitude. Of course, when in doubt it's possible to re-run the calculation with the better estimate of the noon altitude after the first pass through the ex-meridian computation. The real uncertainty in these calculations is the time from local apparent noon. That error obviously increases quadratically with time from noon. Given that the calculation is short, it seems to me that a user should always be advised to run this twice, varying the input parameters by reasonable estimates of the errors in those quantities, especially the time from noon.

    It's also important to remember with ex-meridian calculations that this is just a calculation method. Ultimately, every altitude sight always yields a line of position whether we get around to plotting it or not. A sight near noon corresponds to a line of position that runs nearly east-west, skewed a bit. Inputting a time from noon in the ex-meridian calculation, is equivalent to drawing a longitude through that slightly skewed line of position and picking off the corresponding latitude where they cross. Just as with any case where we cross two LOPs, the error in the resulting latitude will depend on the crossing angle between the lines, and in the case of a sight near noon, that depends on the actual azimuth of the Sun. 

    Frank Reed

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