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    Re: Longitude by unequal altitudes - Brent
    From: Frank Reed
    Date: 2020 Jul 6, 21:30 -0700

    David C, you wrote:
    "I wonder if there is a method for finding lat when the sun is near the prime vertical?"

    No matter what algorithms and mathematical manipulations we devise, I try to remember that there is an underlying geometry to every altitude sight. The analytical math --the equations-- don't matter. The algorithm doesn't matter. There's a line of position there. Each sight yields a line of position with some error bars around it, and that's all it is. Picture the line of position for any sight, and you have all that it can provide. And when the Sun is on the prime vertical, that line of position runs north-south. So a single sight of the Sun on the prime vertical, no matter how we massage the math, cannot yield latitude. 

    Of course, there are tricks. We could measure the rate of change of altitude on the prime vertical. A little calculus tells us that the rate of change in minutes of arc per minute of time for any celestial altitude is 15.04 · cos(Lat) · sin(Azm) so on the prime vertical that rate is 15.04 · cos(Lat). That looks very promising! The rate of change yields a rough latitude with no dependence on any other variables. But then again, measuring a rate of change necessarily requires measuring two altitudes, and that is equivalent to two lines of position, one north-south, the other just a little inclined to north-south. And of course those two lines of position cross and produce a rough latitude. The explanation is in the geometry of the lines of position. 

    Example: we measure the Sun's altitude exactly on the prime vertical. Maybe it's 15°. Then one minute later, we measure it again, and it is 10.5' higher with, let's suppose, a relative accuracy of 0.3' (individual sights might be worse, but two in a row should have similar sources of error). That implies the rate of change is in the range 10.2-10.8' with some confidence. Doing the math (above), we find that the latitude is between 44.1° and 47.3°. Ok so far? Now let's treat this as a case of two separate altitude sights. If we have two independent sights instead but with a relative error of 0.3', then given the change in azimuth (which would also be just about 10'), the crossing LOPs yield an error ellipse that is just about 3° long oriented north-south. See? It's all the same, and the geometry of the lines of position covers it. :) 

    Frank Reed

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