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    Re: Lunars with SNO-T
    From: George Huxtable
    Date: 2004 Oct 27, 14:52 +0100

    I wish to return to Alex's lunar distances, if he doesn't object. They
    have, by now, been rather thoroughly analysed, but there may be a bit more
    juice to be squeezed out, perhaps instructive to others, including me.
    My argument (and I think Fred's also) is with Alex's insistence that all
    his lunar observations MUST follow a monotonic sequence, always increasing
    or always decreasing, and if any fail to do so, in his words-
    "On my opinion, one should REJECT such
    an observation in such a series, without any hesitation.
    That looks just evident to me, without any scientific arguments."
    I ask him if he can quote any authority in support of such a dogmatic attitude.
    Pursuing this matter, I have asked whether the rejected point in his first
    lunar set would still have been rejected if the discrepancy had been an
    arc-minute less-
    > Alex: if his point-4 had been 51d 23.4',
    > would he then have rejected it on
    > the grounds that it breaks his monotonic rule?.
    His reply-
    "I am not sure. Depends on the purpose of the observation
    I would certainly take all results into account (as a precaution)
    but in computing the average I'd reject 51d22.4'
    but probably not 51d23.4 to compute my chronometer correction."
    That shows a welcome flexibility creeping in, but the picture I get is that
    Alex is very reluctant to abandon his self-imposed rule. However I think he
    would be wise to do so. It has no basis in logic.
    Let me ask another hypothetical question. If his point-4 had been read as
    51d 24.4', how would he have treated it, and the following points, then?
    It is, of course quite true that the lunar distance itself MUST change
    monotonically, and any measurements that indicate otherwise do so as a
    result of deficiencies in the measurements. But that's no reason for
    rejecting some, and not others, of those measurements, which may all share
    those same inherent deficiencies.
    Usually, the Moon moves through the sky at an apparent speed of about 30
    arc-minutes in an hour, though it varies a lot (for Alex's second set, it
    appears to have been about 20' per hour). With a note-taker's help, an
    experienced observer can measure lunars at one-minute intervals, in which
    case the step in distance between his observations will be on average only
    0.5 arc-minutes (and in some Moon geometries, down to 0.3 arc-minutes).
    Would Alex still insist, under those circumstances, that the observed lunar
    distances must change monotonically? If he did, then even with his own
    precise balcony measurements, he would be discarding the baby (a large
    fraction of his valid observations) with the bathwater (a few blunders, if
    And that's just the observational scatter that Alex finds on his balcony.
    Go to sea, and it's another world; especially in a small craft. The
    precision decreases, and the scatter increases, as a result of the vessel's
    motion. Alex's perfect straight-line plot of points would instead start to
    look more like the distribution of currants in a bun. Under those real-life
    conditions, Alex's dogmatic rule would be quite untenable.
    I ask Alex to consider this further consequence of applying his rule. For
    increasing lunar distances, it will always tend to reject observations that
    are below-the-line, so skewing the average toward a higher value, and of
    course the converse when lunar distances are decreasing.
    I recommended a scientific attitude, in stopping to question doubtful data,
    rather than just unthinkingly reject them.
    Alex appears to reject this approach, and recently said-
    ">In science you are exploring the Unknown. And you have a chance
    >to discover something new. In measuring the Lunar distances with
    >a sextant you follow a routine established for centuries
    >and you are not likely to discover anything new:-)
    >This is the difference."
    Let me quote an example, from just this correspondence about his lunars, in
    which the habit of "stop; question; ponder" has paid off in practical
    terms. Alex had said, about his rejected point-4.-
    "I suppose the reason of that blunder was incorrect time recording.
    My (easily solvable) problem is that I have no watch that I
    can read in the darkness and without eyeglasses.
    So I am just thinking of buying a good large diameter
    split-second-hand stop watch,..."
    Well, since then, just by stopping and thinking about that rejected
    observation and its sequencing, we have deduced with certainty that it can
    not possibly be blamed on the timekeeping. It has removed one reason for
    him to buy a new stopwatch. Indeed, lunars are VERY undemanding as far as
    timekeeping is concerned. 10-second, even 20-second, accuracy should
    Finally, I asked Alex this question about his second set of lunars-.
    >> One minor puzzle arises, however.
    >> If I sum the 5 values of ERDIST, and
    >> divide by 5, the average comes out at -0.02, not -0.1.
    >> And, more strangely,
    >> if I sum the 5 values of ERLONG, and divide by 5,
    >> I get an average of +1.6
    >> minutes, not -2.1 as stated.
    >> Not that it matters; all these values are well
    >> within the error range expected from a lunar.
    >> But what causes a simple
    >> averaging to go wrong?
    Alex replied-
    >I also noticed this. I think this is a question to Frank Reed:
    >how his program really works. I introduced the row data and
    >printed what his program returned. Then I introduced the average
    >of my row data and printed the program output.
    >So I am only responsible for computing the
    >averages of GMT and DIST,
    >which I computed by hand (and then verified with my calculator:-)
    I credit Alex, as a mathematician, with the ability to correctly average
    his raw data of GMT and DIST, even for sexagesimal quantities. But it would
    be interesting to know exactly what were those averaged values that were
    fed back into Frank Reed's program, and whether any rounding had taken
    That discrepancy in calculated longitude, between -2.1' for the longitude
    of averaged distance/time, and +1.6' for the average of longitudes, is
    well, well within the expected precision of any lunar. So there's no
    discredit to Frank's program. But it's interesting nevertheless, and I
    can't break the habit of a lifetime, in wishing to ask further questions
    about it. After all, discrepancies are far more interesting than
    Such a result might well indicate that we are approaching, but not yet
    reaching, the limits of the simple averaging procedure, as we have
    discussed in such detail recently on this list. That procedure relies on
    quantitities changing linearly, showing a straight line when plotted
    against time. Or nearly so, anyway; with significant curvature causing the
    assumptions to break down.
    Alex's second set of lunars lasted over a period of 18 minutes or so, and
    in that time the apparent lunar distance changed by about 6', in a lunar
    distance of over 70d. Without doing calculations, it seems hard to credit
    that there can be any significant non-linearity in the true lunar distance
    over that small range, though we know it can become important when the
    distance gets really small.
    What about curvature in the corrections, however; particularly in the Moon
    parallax correction, which in some circumstances can be as much as 1
    degree?  We know that it varies by cos (altitude), which is itself
    non-linear. And under some circumstances the Moon's altitude can change by
    15d in an hour, though it must have been less than that in that Florida
    observation. I would put my money on changes in the slope of the Moon
    parallax correction, over the 18 minutes of the observation, giving rise to
    the small discrepancies that we see. But in practical terms, such small
    discrepancies are negligible when weighed against the inherent inaccuracy
    of a lunar, so there's nothing that needs "fixing".
    Frank may have a simpler explanation.
    contact George Huxtable by email at george@huxtable.u-net.com, by phone at
    01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy
    Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.

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