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    Re: LAN curve parabola?
    From: Frank Reed
    Date: 2019 Nov 22, 15:03 -0800

    Almost any quantity that reaches a maximum (or minimum) and then falls away again (or starts to rise again, if from a minimum) can be approximated by a quadratic curve --a parabola-- over some range of its independent variable. The altitude of the Sun approaching and following noon is neither a parabola nor an ellipse, but if we look for a short enough period of time, it's an excellent match for a parabola. So how short is "short enough"? That can be a tricky question.

    In mid-latitudes, if you observe the Sun for 45 minutes or so centered on local noon and plot the Sun's altitude as a function of time, you will get a symmetric curve that is almost indistinguishable from a parabola. You can use a quadratic-fitting routine and expect excellent results. How is this curve different from a parabola? Of couse you can just calculate some altitudes and compare using the standard spherical trig relationship, or you can do a series analysis and find out how large the variation is from a purely quadratic expansion. Another way to gain some insight is by looking at extreme cases.

    Imagine being on the equator the day before an equinox. The Sun's maximum altitude will be anywhere from 24' to perhaps 48' less than 90°. Suppose you observe and record altitudes every minute for 45 minutes centered on the time of LAN. The Sun's altitude will climb steadily, linearly for about 22 minutes before noon. And after a couple of minute of turnover right at noon, peaking about 89° altitude, it will fall steadily, linearly for about 22 minutes through the end of the observing period. If you graph these sights versus time, you will have an almost straight line rising at a rate of 15' per minute of time before noon, a small curve (almost parabolic?) right at the top, and then a straight line falling away after noon. The curve will be basically an inverted V with a small curved section near the peak altitude. But it all comes down to asking what interval of time is "short enough" again. If instead of 45 minutes we shoot sights every ten seconds for three minutes right around noon, we would once again find a nice little parabola-like curve.

    In general curve-fitting in a spreadsheet is probably a bit of over-kill for this sort of thing. You can do a good job finding the peak altitude by inspection and some thoughtful analysis of your sights, and you can get the axis of symmetry by folding the graph. I invented and developed this folding trick over a decade ago, and it works. I teach it often. Simply plot your sights of altitudes versus time on thin graph paper, loosely fold the graph in half and look through it with a bright light behind (smartphone flashlights are great for this if you don't have sunlight). Slide the paper back and forth until the before noon and the after noon sights overlap to produce a smooth curve. When aligned, hard fold the paper. Then read off the UT of LAN from the crease. Calculate GHA for that UT, and there's your longitude. Well, you have to correct for vessel motion relative to the Sun's position, but that's just a quick table lookup or a calculation of k·v·[tan(Lat) - tan(Dec)], which is easy enough (the value of k is a secret, guarded by a dragon in a deep dark cave...). Just to emphasize: there's nothing wrong with doing this in a spreadsheet, especially if the process provides entertainment by itself. But you might be surprised how well you can without a spreadsheet.

    Frank Reed

       
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