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I am currently working on improvement of an Android app that calculates solar eclipses. For many days now I have not been able to solve a mathematical problem concerning solar eclipses, so I decided to try to ask for help.

I am working with the book "Elements of solar eclipses 1951-2200" by Jean Meeus. Calculating the central line of an eclipse from given Besselian elements already works good, so I am able to calculate the path of the central line of total and annular eclipses and the curves of equal magnitude.

My problem now is that Meeus only describes the calculation of the central line, so for a given time, that means for given Besselian elements I can calculate the corresponding point of the central line on the globe. Using these algorithms I can not calculate the line of maximum magnitude for partial phases of total and annular eclipses or for partial eclipses. So my question is:

How can I calculate latitude and longitude of the point on the globe with maximum magnitude with given Besselian elements for a time when eclipse is partial, that means, when no central line exists. As far as I understand the problem, this should be the point on the globe that is nearest to the shadow axis, which in the case of a partial eclipse or partial phase of an eclipse does not intersect the globe.

I would very much appreciate any help.

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  • $\begingroup$ I'm afraid I don't have time to delve into this in detail, but a quick look through Jean Meeus' "Astronomical Algorithims" appears to touch on this on page 351 where he describes an eclipse which partially grazes the pole but the central axis doesn't intersect the globe. Not sure if this is helpful or not. $\endgroup$
    – AstroShannon
    Aug 27, 2020 at 17:11
  • $\begingroup$ Sorry, but page 351 in this book is about 'phases of the moon'. Do you have the right page and chapter number for me? $\endgroup$
    – Werner Harnisch
    Aug 27, 2020 at 17:28
  • $\begingroup$ In my copy of "Astronomical Algorithms", chapter 52 is dedicated to eclipses, both solar and lunar. The publishing date is 1991 with a red cover (I think it's the latest edition but I'm not sure). I wish I could post an image here since it talks about the Besselian elements you are working with. In my copy, Chapter 52, page 351 the section title is "Solar Eclipses" though the text goes back and forth between lunar and solar since some of the principles are the same. I wish I could send you the images of these two pages, they might prove helpful. Muees' books are the best for sure. $\endgroup$
    – AstroShannon
    Aug 28, 2020 at 1:26
  • $\begingroup$ I tried to type it out, but there isn't enough room. Meeus describes a central axis of the Moon's shadow (and cone size) and how close it must be to a point on the Earth before the shaow cone intersects Earth and becomes a "cone graze", annular, or total eclipse. The graze sounds like what you are interested in - where the central axis of the shadow doesn't touch the Earth but cone brushes the poles. Wish I could be more help but it's tough in such a small area to type. I had some luck searching GitHub for "Meeus" but it's hit and miss. $\endgroup$
    – AstroShannon
    Aug 28, 2020 at 2:06

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Jean Meeus, Morsels 2, chapter 69 talks about the Nelder-Mead simplex algorithm for minimising a function of two variables - in this instance magnitude as a function of latitude and longitude. Maximising the magnitude equates to minimising the negative of the magnitude. Apparently the method first appeared in magazine Byte in the 1960s, there are various references on-line.

Maximum magnitude always occurs near the polar circle, so try starting with three points equally spaced round the 60th parallel. They would have to be converted to Cartesian X Y with centre at the pole (giving an equilateral triangle), and the result converted back to latitude/longitude at the end. The sign of the element Y will make it obvious which polar circle to use.

Because the shadow is conic, not cylindrical, the resulting point will not (in general) be exactly on the horizon.

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