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    Re: How to pre-calculate for negative altitudes?
    From: Frank Reed
    Date: 2017 Oct 23, 07:22 -0700

    It's just the usual spherical triangle "cosine formula" solved for hour angle:

    HA = cos-1[sin(h) / cos(Dec) / cos(Lat) - tan(Dec) · tan(Lat)].

    This comes from:

    sin(h) = sin(Dec) · sin(Lat) + cos(Dec) · cos(Lat) · cos(HA),
      or re-arranged:
    cos(HA) = [sin(h) - sin(Dec) · sin(Lat)] / [cos(Dec) · cos(Lat)].

    Plug in -0.833° for h, and you will recover the hour angle for standard sunrise/sunset tables. Convert that angle to hours and minutes, apply equation of time and zone correction, and you'll find exactly the times as normally published for sunrise and sunset. As I noted in an earlier message, these times are a convenient fiction. 

    You say that you want to generate the "correct" tables using -51' for the altitude, setting h = -0.850° above. Even if there is some sense in which this is a "correct" number, and I don't think there is, your tables won't have much merit unless you travel back in time. There are no sailors adrift in lifeboats in the year 2017. There aren't even lifeboats anymore! :) Of course, calculating for entertainment --calcutainment!-- is good, clean fun, no matter what. :) And maybe you do have a time machine?? I've been working on one myself. It also includes a pocket-sized gravitational wave detector... ;)

    Frank Reed

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