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    Re: October Lunar
    From: Frank Reed
    Date: 2008 Oct 07, 00:00 -0400

    I wrote previously,
    "The short distance is not an issue (two reasons: the altitudes are
    calculated so their accuracy is not a problem, and also we don't have to
    interpolate between geocentric distances three hours apart, which was an
    issue historically but not today)."
    And George, you replied:
    "I don't understand the relevance of "the altitudes are calculated so their
    accuracy is not a problem" to the acceptability of a short lunar distance.
    Anyway, even if the altitudes had been measured rather than calculated, that
    measurement can be made to quite sufficient accuracy."
    The accuracy of the altitudes is critical for short distance lunars. The
    error in clearing a lunar distance arising from errors in the altitudes that
    enter the clearing process is given approximately by:
     errLD = (1/60)*[errH2*cos(H2)/sin(LD) - errH1*cos(H1)/tan(LD)],
    where errLD represents the error in clearing the distance, LD is the
    observed lunar distance, errH2 is the error in the altitude of the other
    body (Sun, star, or planet), H2 is the altitude of the other body, errH1 is
    the error in the altitude of the Moon, and H1 is the altitude of the Moon.
    For "typical cases" of lunar observations, the error in the lunar clearing
    process is about a tenth of a minute of arc for a five arcminute error in
    either altitude, which gives plenty of leeway. As I've noted a number of
    times before, the most interesting thing here, a bit of a "miracle", is that
    the error in the Moon's altitude becomes insignificant when the LD is close
    to 90 degrees --you can be wrong by a degree or two and it will do no harm.
    At the other extreme, as in this specific case, when the lunar distances are
    small enough, sin(LD) and tan(LD) are approximately given by LD when LD is
    "in radians". If also, the altitudes are nearly the same (as they are to
    some degree whenever the LD is small enough, and as they were in practice in
    this specific case), then cos(H2)=cos(H1), and then we have:
      errLD = (1/60)*(errH2 - errH1)*cos(H1)/LD.
    Supposing that we measure LD in degrees, and assuming cos(H1) is nearly one,
    this reduces to approximately
      errLD = (errH2 - errH1)/LD.
    This means that if we have an error of 2 minutes of arc in either altitude,
    which is well within the realm of possibility for measured altitudes, and
    the LD is 12 degrees, then the resulting error in the clearing process is
    about 0.15 minutes of arc (an important difference in clearing lunars). Also
    notice that a COMMON error in both altitudes is harmless. If, for example,
    there is some odd refraction affecting both bodies by the same amount or we
    have miscalculated the dip, or the IC is wrong for the instrument used to
    measure the altitudes, it leads to no error in the clearing process.
    Nothing about the above formulas for the error in the clearing process
    resulting from an error in the altitudes depends on the altitudes being
    measured. They also apply when the altitudes are calculated. We calculate
    altitudes from latitude and local apparent time (two inputs yield two
    outputs). If you think through the various cases, an error in either of the
    inputs will generally affect both altitudes by about the same amount when
    the lunar distance is short. And as noted above, a common error does not
    affect the outcome of the clearing process for short distances. So if we
    calculate the altitudes, the whole problem of getting accurate altitudes for
    short distance lunars evaporates.
    To summarize, if you intend to shoot short distance lunars, you need to be
    aware that the altitudes have a critical effect on the clearing process. The
    altitudes either have to be measured simultaneously to an accuracy of a
    minute of arc or better (more so if the distance is shorter than in this
    case) or they should be calculated. Naturally for a modern observer, it is
    convenient to calculate them.
    And you wrote:
    "that left unaddressed the other problem about short-distance lunars; that a
    planet, even though never far from  the Moon's orbital plane, can get way
    out-of-line with the Moon's direction of travel on near approach."
    I left that 'unaddressed' for two reasons. First and foremost, that is
    completely irrelevant to this observation and the accuracy question arising
    from it. Second, as it happens, in this particular case, the Moon and Venus
    were only moderately out of line. This degree of misalignment was acceptable
    historically (by the standards of the historical Nautical Almanacs, as
    "As we have discussed before, the estimated longitude error, stated in
    Frank's reduction program, doesn't allow for that angular offset, so paints
    an over-optimistic picture in such a situation. He is considering a
    revision, which would be useful."
    No, I'm not considering a revision 'per se'. I have considered adding a
    details page for people with interest in possibilities, but at present I'm
    happy with what it does, as is. Ya see, the users of this site consist of
    two very different groups. The first group is much larger and consists of
    people who are just learning about lunars and who are interested in shooting
    a lunar once or twice and getting a general idea of what it means. The
    "approx longitude error" number is for them. They've shot a lunar, and they
    say to themselves 'ok, that was fun, but what would this have meant
    historically?' They don't care if the particular lunar geometry of the sight
    they've taken is optimal for finding longitude. They just want a general
    indication of how well they would have done with that same level of accuracy
    in a traditional lunars case. Rather than over-stating the accuracy, I've
    frequently felt that the number under-states it since we normally quote
    navigational accuracy in miles rather than minutes of longitude.
    The second group of users consists of people with some expertise in lunars
    like the lunarian members of NavList, including myself. This group is
    perhaps ten or even fifty times smaller in number, but we're much more
    likely to be repeat customers. For us, there are many possibilities. Just a
    few ways we might use lunars:
    1) We have a historical lunar distance observation from 1809. It includes
    the observed distance and the observed altitudes. We enter the data, and
    adjust the position until all of the observations have zero error. This is a
    simultaneous solution of the lunar problem and the time sight. There is no
    single error in longitude.
    2) We have a lunar observation from 1850 serving as a check on a chronometer
    which is probably nearly correct. In this case, we really should be talking
    about the error in GMT though the equivalent error in longitude is
    fundamentally the same thing. Here we don't worry about the time sight, and
    this case is closest to the "pure" case where an error in the observed
    distance gives a direct error in the resulting longitude.
    3) We're measuring lunars simply to test our sextants and our skills and the
    longitude is quite irrelevant.
    4) We have measured a lunar distance at known GMT and we are trying to
    determine a "lunar distance line of position". In this case, an error in the
    measured distance leads to an error in the LOP (typically 60 times larger
    than the error in the LD) but dependent on the geometry. There IS a
    sensitivity to longitude, a corresponding "longitude error" in this case,
    but in general these LOPs are not aligned with lines of longitude so it
    would be misleading to give a pure longitude error in this case.
    5) And all the other things I haven't thought of right now. :-)
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