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    Re: Question re. A and B factors (corner cosines) lunars
    From: Magnus Sjoquist
    Date: 2012 Jan 25, 13:49 +0100
    Tks alot. I think the last coin went into the slot.

    ----Ursprungligt meddelande----
    Från: FrankReed@HistoricalAtlas.com
    Datum: 2012-01-25 05:35
    Till: <NavList@fer3.com>
    Ärende: [NavList] Re: Question re. A and B factors (corner cosines) lunars

    Magnus, you wrote:
    "Fog lifting, Clear views 360 degrees."

    Glad to hear it!

    And you added:
    "When, or rather if, I regain some self confidence I will ask You what the Q factor is"

    Now, hey. You hush about "self confidence". Ask whatever you feel like asking. I enjoyed answering your question. :)

    As for the Q term, there is no "easy" way to see why it exists, but here's an attempt to explain it... In the terrestrial distance case that I mentioned, we had an exact distance from Chicago to Paris. And we knew that the great circle route to Paris was on a bearing of 50 degrees. We travelled north 60 nautical miles to Waukegan and calculated the change in distance as 60*cos(50). But the bearing to Paris actually changes a bit as we move north. In this case, it might be 49.95 degrees at that more northerly point. This change can be accounted for by going to the next level in the series expansion: the "quadratic term" or "Q". In the case of the geometry of lunar observations, this small difference was usually tabulated separately so it was an easy "look up". And because it is inversely proportional to the tangent of the measured distance, for lunars near 90 degrees, it is insignificant. Often it could be ignored.

    For the lunarian math fans out there, there is another way to account for the change in bearing (position angle of the Sun relative to the Moon) that does not require a separate table for this "Q" correction. The idea is that you calculate the linear terms dh1*A+dh2*B (the altitude corrections multiplied by the "corner cosines") for the observed altitudes and then you do it again for the corrected altitudes. This is less work than it sounds since most of the terms in the calculation are the same, and, since dh2 is always small, we really only need to work the calculation twice for the Moon term. Then you average the results. The quadratic correction cancels out. This averaging approach was first proposed way back around 1805 by... a well-known French mathematician whose name escapes me at the moment (I will go dig it up later). This averaging trick did not gain much traction as an approach to clearing lunars until late in the game, but it was proposed many times during the 19th century in mathematical papers (not practical navigation manuals). By the end of the 19th century, when lunars were very rare, most navigation manuals dropped the special tables that had previously been included for the "Q" correction. At this point, the averaging trick found a certain popularity, and in fact the last method for clearing lunars that was included in the British "Abridged Nautical Almanac" was based on Airy's method which used the averaging trick.


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