A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2019 May 10, 19:02 -0700
You wrote regarding the series expansion: "It was fairly clear how to derive up to the quadratic term in an ad hoc fashion".
Those ad hoc analyses can be useful ways of motivating or justifying an equation (without qualifying for anything like mathematical proof), and I think there's an audience that can benefit from that sort of thing. Naturally, it's a very small audience -- a subset of a subset of that small group of people interested in lunars, but nonetheless, I would be interested in any ad hoc derivations you might have. Just for a specific case, how would you motivate the largest of the quadratic corrections, that proportional to the square of the Moon's altitude correction?
You wrote: "The problem requires eliminating the azimuth difference, Z ".
I don't see the necessity of this, but maybe it's required from a particular point of view? I remember when you sent me this file many years ago. I remember thinking at the time that I should look into this "Cagnoli's equation" that you had found, and it was on my list of things to get back to until that entire topic dropped off the bottom of an ever-climbing "to do" stack. Ah well... Thank you bringing it up again.
I'm curious why you wouldn't derive the series simply by applying the standard two-dimensional Taylor expansion?
LDc = LD + dhm · (∂LD/∂hm) + dhs · (∂LD/∂hs) + quadratic and higher terms...
Finding slick derivations of all the partials takes some work, as I recall, but if we're careful to define the corner angles at the Moon and Sun, which I lately call μ and σ (mu and sigma), then things go quickly. For example, we find that the first partial is that corner cosine of μ and similarly for σ:
∂LD/∂hm = cos μ,
∂LD/∂hs = cos σ,
...and so on.
And how was it done in the 18th century? It seems that the quadratic terms were already considered relatively common knowledge by the time Mendoza Rios was writing his long review article (published in 1797 in the Transactions of the Royal Society). How were they derived originally? Geometrically?? And who published them first?