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    Re: A noon fix inland with an peri-sextant
    From: Antoine Couëtte
    Date: 2017 Dec 21, 05:18 -0800

    Re : Dolkas-dec-2017-g40944

    Dear Paul,

    Altogether with the excellent “Folded Paper” technique (see : FrankReed-dec-2017-g40950), your ”Equal Altitudes” technique is a very good one if you do not have computational power at your finger tips. You certainly know its limitations since you are naturally taking the appropriate steps to avoid them:

    -    Not observing too far from culmination since the height curve vs. elapsed time becomes no longer “symmetric”, with the higher the culmination height, the smaller the “valid” observation time interval around culmination. For practical purposes, observing heights within 15’ (minimum) and 30’ (maximum) from Culmination Height should adequately resolve that matter. And :

    -    Not observing too close from culmination since – for such “manual” methods - accumulated observation errors too close from culmination may become be detrimental to culmination time accuracy determination.

    Paul, you also indicated that “The average time between the first shot you took and this last one is the time of local noon”From your description, I would rather feel that the “time” derived through your procedure here-above is rather the “culmination” time, which can be quite different from the “Meridian passage” time (e.g. from the “local noon” time).

    *******

    In the same subsequent post Frank indicates that there are cases when both times can be significantly different.

    QUOTE - FrankReed-dec-2017-g40950

    Note: at sea almost always, and on land when not near the solstices (not a worry at this time of year), you have to correct this symmetry axis time for relative motion between the observer and the Sun. This is easy: either a simple table or a short calculation. But if we neglect this it can throw off the longitude by many miles.

    UNQUOTE

    *******

    The difference between culmination and local noon times may be given – at first order – by the following formula (given from memory here, so please anybody check for its correctness, or be so kind as to check my results, thanks in advance):

    tnoon - tculm = 48/π * (λ – d) * tan (Obs Lat) – tan (Body Dec)

    With:

    tnoon : local noon time (i.e. when Body on the Observer’s meridian)

    tculm : culmination time, with :

    tnoon - tculm : given in seconds of time, and :

    λ : Observer’s speed in Latitude in Arc minutes per hour

    d : Body change in declination in arc minutes per hour

    Obs Lat : Observer’s Latitude

    Body Dec : Body Declination

    As an example, for the Sun at equinoxes, for a steady observer at 53°N (which is David’s “steady” latitude in DavidPike-dec-2017-g40941):

    λ = 0 (Steady Observer)

    d = -0.97’/ hour

    Obs Lat : 53°N

    Body Dec : 0°

    Hence tnoon - tculm = 19.7 seconds of time, which translates into 4.9 arc minutes in longitude. Here noon time occurred after culmination.

    In addition to the above example if the Observer were moving just 5 knots to the South (i.e. λ = -5.0’/hour), then tnoon - tculm would be equal to = - 81.7 seconds, translating into 20.4 Arc minutes in Longitude, with noon time occurring before culmination in this example.

    Last notes

    1 - The formula given here-above is [only] a 1st order one accurate to only 90% let’s say. Actual “tnoon - tculm ” value may differ by up to another 3 or 4 seconds of time under extreme cases.

    2 – In order to determine the Observer’s latitude, in both the “Folded Paper” and the “Equal Altitudes” methods, one makes the implicit assumption that Local Apparent noon height is equal to the recorded Culmination height, which introduces another small – and most often totally inappreciable – systematic error.

    3 – In spite of their potential “dangers” (keying in so many numbers without any error) Numerical “Least Square” Methods do have an advantage over the “manual” methods here-above. Numerical Methods can yield [much] more reliably culmination and local noon times, as well as local noon heights. They do not also require recording the exact culmination height itself. They certainly remain the best performing ones if one needs to get the most of any series of observations, even if quite often their results are marginally better than “manual” methods for most practical purposes. The most widely used Least Square Methods are the now classical 2nd order “Parabolic” ones. There are other more sophisticated and more versatile higher order methods which can yield more accurate results.

    4 – All methods can be also be easily used for lower culminations and also for other bodies than Sun : Venus, Moon, or even stars whenever visible. Long ago I once shot a “Southern bright star” (maybe Achernar ?) lower culmination Local Apparent Noon in the Southern Indian Ocean, with observations time span exceeding 2 hours (in my archives at home).

    Antoine M. "Kermit" Couëtte

     

       
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