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    Re: Lunars.
    From: Steven Wepster
    Date: 2001 Jul 04, 8:27 AM

    Earlier I wrote
    > >> the altitudes of the bodies need not be taken very accurate. <<
    Herbert replied:
    > As a matter of fact, the altitudes don't have to be taken at all, as they
    > can be computed. However, the problem with this is that actual refraction
    > might differ from the tabulated values and hence measuring the altitude
    > might result in higher accuracy. Has anybody ever investigated this?
    The altitudes of the bodies are needed for the Clearing of the Distance,
    i.e. to get rid of the effect of parallax, refraction, and dip in the
    measured LD. The spherical trigonometry formula (e.g. Krafft, Borda) that
    you use to actually compute the true LD depends not so much on the
    altitudes (it is sufficient to have them to a few minutes accurate) but
    rather on the _difference_ between true and apparent altitudes. These
    differences must be known as accurate as possible. One of the contributing
    factors is the refraction; normally there is no other way to get it than to
    take it from a refraction table. So, wether you use measured or computed
    altitudes, an error in the refraction will have the same effect. The same
    applies to dip.
    Since it is important to have very accurate differences (0.1' is necessary
    and sufficient) it is wise to include a correction for non-standard
    atmosphere, table A4 in Nautical Almanac.
    As I said, it is not necessary to have the altitudes of the bodies
    themselves to a very high precision. That's why it is at all possible to
    make use of computed altitudes. Remember that to compute an altitude you
    need to know your lat and long fairly precicely.
    On the practical side however, I presume that an Old Salt would prefer to
    take the altitudes standing on deck with an octant in his hand, than to
    compute them using log-trig tables. But there are circumstances that it is
    not practicable to measure them.
    > As to the question of an algorithm itself, this depends on the purpose of
    > the algorithm. For historical research, often the original methods have
    > be emulated and the then available ephemeris be used. For modern use, the
    > fastest and safest algorithm would be right in the spirit of St. Hilaire:
    > heuristic and iterative. The distance is a function of time. Starting
    > a reasonable "assumed time", compute the corresponding "computed
    > distance", compare it against the "observed distance", make a correction
    > and iterate until the difference between computed and observed value is
    > small enough.
    That is more or less the procedure that Tobias Mayer proposed to use, when
    he submitted his Lunar Tables to the Board of Longitude in 1754. Quite a
    cumbersome method, because his tables were for _calculating_ the position
    of the moon from epoch + mean motion + 14-plus 'equations' (perturbation
    terms). Imagine that you have to _iterate_ ... It took Maskelyne up to 4
    hours to compute a Lunar when on the way to St. Helena in 1761.
    I don't see what Marc St Hilaire has to do with it. I associate him with
    the intercept method; quite something different. But maybe he has made
    other contributions to astronavigation?

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