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    Re: March 22 Lunar Observations
    From: Arthur Pearson
    Date: 2002 Mar 24, 19:02 -0500

    As always, peer review in invaluable, thanks for looking over my work.
    By way of follow up to your comments:
    The gross error in moon altitude that Bruce pointed out (30* not 70*)
    was an error in my spreadsheet. My computed altitude formulas
    interpolate between the almanac values for GHA and Declination for the
    hours before and after the time of observation. In this case, I was
    interpolating between GHA1 = 349* and GHA2 = 3*.  My formula was
    inadequate for an interpolation that spanned 360*.  I changed GHA2 to
    363* and my values for everything fell back in line with Bruce's stated
    values (my corrected GMT was 8 seconds slower than Bruce's). This
    illustrates the dangers of automation: the calculations go fast, but
    unseen errors are repeated without re-examination. This is the second
    material error in my spreadsheet to be uncovered by the peer review
    George's comments on the error inherent in using the slope of change in
    computed distance is an important insight. I am still a firm believer in
    graphic averaging as it retains the element of judgment by the observer.
    I think having an approximate slope is better than no slope,
    particularly if we now know from George's discussion that the slope of
    Ds will always be less than the slope of Dc. Perhaps George could
    calculate a rough guide for the delta between the slope of Dc and the
    slope of Ds for different apparent altitudes. Even if we had a delta
    value for every 10* of altitude, it would refine our ability to judge
    the best fit line.
    With respect to the scattering of the distances, I am reasonably
    confident I made some gross error on the two outliers. I am using an
    Astra IIB and its index error has been very stable over the last year
    and I have had two or three series of distances that were less
    problematic than these. However, even without the outliers, this is a
    mediocre series. This was my longest distance by far, and it was very
    cold and windy.  As George points out, that is nothing compared to the
    conditions our predecessors faced, but apparently it was enough to throw
    off a rank amateur.  We'll see if things improve on the next round.
    Thanks for the comments, I look forward to seeing sights by others
    -----Original Message-----
    From: Navigation Mailing List
    [mailto:NAVIGATION-L{at}LISTSERV.WEBKAHUNA.COM] On Behalf Of George
    Sent: Sunday, March 24, 2002 11:15 AM
    Subject: Re: March 22 Lunar Observations
    Here are some comments on Arthur Pearson's lunar of 22 March.
    1. It's good that he has presented us with the complete details of his
    observations, warts and all, instead of just averaging the measurements.
    plot of those lunar distances against time shows just what Arthur is
    worried about. I would regard those measurements as unacceptable even
    my cheapo plastic Ebbco sextant. Presumably Arthur is using something
    better than that.
    He is right to be worried. The scatter is very great. The two highest
    values, which he has rejected (numbers 8 and 9 out of 11 measurements)
    a jump out of the trend-line of the others, by about 7 minutes. It's
    to believe that Arthur could have made such a gross error in his view of
    the "kissing" of the Moon and Sun limbs as they brush past, if they
    be clearly seen, as it would have to amount to a gap between them of
    half a semidiameter. It's not easy to see how a misreading of the scale
    could cause a 7-minute error. And it's not just an error in one rogue
    reading, but in two successive readings. There's something wrong, and an
    explanation should be sought.
    Presumably he is obeying the important rule that the final movement of
    knob before a reading is made is always in the same direction, always
    clockwise or always anticlockwise, whichever, for every measurement (the
    same direction as used for measuring the index error). Even so, I would
    expect the backlash (lost motion) of any sextant to approach anywhere
    seven arc-minutes.
    It's very important that the mirrors are located firmly, by being
    sandwiched between the pressures of an adjusting screw on one side and a
    spring on the other (depending somewhat on the design). It can happen
    in several successive rounds of adjustment, the screws can become
    progressively slackened until one or more allows some looseness of a
    mirror. This should be checked for by gently nudging the mirrors at the
    corners with a fingertip. No looseness or shake should be discernable.
    Even if points 8 and 9 are rejected as erroneous (for some unknown
    the remainder do not lie that well on a straight line. For them, my own
    estimate of best-fit (which does not lie through the point 18:50:00,
    98*34.8, as Arthur's does, more like 98*33.8) has the measured points
    scattering 1' or 1.5' on either side. I would hope for better, for a
    measurement made on land and with a clear sky. The inherent resolution
    the eye is about 1', and the usual telescope has a magnification of
    x2.5 or x3. I would hope for a band of measurements mostly within .5' of
    best-fit line. I wonder what scatter other listmembers manage to achieve
    their measurements of lunar distance, in terms of plotted scatter?
    It's often useful to repeat the measurement of index error after the
    measurement as well as before it. In a lunar, where precision is
    all-important, this gives confidence that the sextant is stable and
    repeatable results.
    On land, there are lots of opportunities for checking the scatter of a
    series of angle measurements using a sextant. You can measure the
    horizontal angle between the gable-ends of two distant houses, or
    poles. Or the angle in the sky between two bright planets. As these
    alter perceptibly during the measurement, any variation is in the
    or in the instrument.
    Arthur should find, as he gains confidence and becomes more aware of the
    testing task that faces him, that his skill improves and any scatter
    less. His respect for his maritime predecessors, who had to do the same
    thing in rough weather and flying spray, may have increased already.
    2. On the graph showing the sextant lunar distance observations, Arthur
    plotted the rate of change of predicted lunar distance obtained from the
    almanac, in order to get some guidance as to the slope of the trend-line
    fit to the points (as I see it). But these lines are expected to show
    different slopes, so one is not really a useful guide to the other. The
    reason for the difference is the evil effect of the rapidly-changing
    The Almanac positions, 1 hour apart, are given as if the Moon was being
    viewed from the centre of the Earth, so they are unaffected by parallax,
    is the resulting rate-of-change of lunar distance over that hour.
    the observations, made from a point on the Earth's surface, are affected
    the changing parallax of the Moon as it passes across the sky, as viewed
    a real observer. As the effect of this parallax change is always to
    the motion of the apparent Moon, one has to expect that the slope of the
    trend-line between the observed points will ALWAYS BE LESS than the
    calculated from the Almanac. Sometimes, when the Moon is very high in
    sky, the slope can be very much less: under some circumstances it can be
    halved, roughly speaking. This is the "parallactic retardation" effect
    has been discussed previously.
    The effect I describe has worried some people. How can it be, they ask,
    that the changing parallax always works the same way, to always slow the
    apparent Moon in the sky? Sometimes, surely, it would work the other
    to speed it up? It seems unphysical.
    That's true, it does. It's a very reasonable question. The answer is,
    part of the time, changing parallax does speed up the motion of the
    apparent Moon. Unfortunately, those times are when the Moon is
    being on the other side of the Earth. If the Earth happened to be
    completely transparent, so we could still see it when it was below the
    horizon, its apparent motion against the star background would then be
    speeded up. I hope that's an acceptable answer.
    3.  Arthur has taken the time, and his position, from a GPS receiver.
    an exercise, to see how well the lunar distance can be measured and
    plotted, that's an acceptable thing to do. But Arthur, and other
    need to bear in mind the limits of this exercise. The problem is that
    value of GMT, taken from the GPS receiver, hasn't just been used to
    the accuracy of the final answer. It has also been used, in the course
    the correction of the lunar distance, to choose the time at which to
    calculate the appropriate altitude of the Moon. The lunar-distance
    correction is rather sensitive to that altitude, because of the way this
    damned lunar parallax affects the lunar distance (yes, "parallactic
    retardation", once again).
    So Arthur has used his privileged position of knowing what the final
    is going to be in order to obtain the lunar altitude. In real life, of
    course, it couldn't be like that. He would have no prior knowledge of
    time (if he had, why would he be bothering with lunar distance?). The
    navigator would make his best-guess at the GMT, and calculate
    This would give a more accurate time, so he could plug that in as a
    value and repeat the operation, as many times as necessary., getting
    and more accurate until the results converge toward the correct value
    GMT. And as I have warned, in "about lunars, part 4", that convergence
    be a slow one. The reiteration is something a computer could handle with
    trouble, but would be unacceptable to anyone calculating by hand or by
    So anyone who carries out such an exercise should be aware of the
    artificial situation they are creating. In presuming the answer when
    deciding on lunar altitude, they are bypassing one of the trickier
    of the lunar distance method. In real life, the best way is to measure
    George Huxtable
    >Here are the numbers from a set of lunar distances off the sun on March
    >22, 2002.  I would welcome any comments on my procedures and the
    >problems I encountered, and if anyone is inclined to work the
    >calculations on their own, I would be curious how they come out.
    >All times reflect GMT from my GPS unit.
    >Latitude and longitude are also from GPS.
    >Angles are expressed XX* YY.Z for degrees and minutes or YY.Z' for
    >minutes alone.
    >Index Error = 2.63' ON (determined by Bruce's averaging method)
    >Latitude:       N 42* 21.7
    >Longitude:      W 71* 15.5
    >I took eleven distances as follows:
    >GMT             Ds
    >18:40:46        98* 29.9
    >18:43:41        98* 32.2
    >18:44:47        98* 32.4
    >18:46:10        98* 31.0
    >18:47:16        98* 31.2
    >18:48:16        98* 34.6
    >18:49:28        98* 34.8
    >18:50:45        98* 39.8
    >18:52:25        98* 41.4
    >18:53:37        98* 34.8
    >18:54:51        98* 36.2
    >As you can see, I had considerable trouble getting consistent
    >I may have misread one or more of these, or I may have bumped the knob
    >inadvertently. Bruce's book emphasizes the sensitivity of long angles
    >parallelism of the telescope, so I tried to make these contacts in the
    >center of the scope. Needless to say, I have lots of room for
    >My next step was to graph these distances against time and then plot a
    >reference line that shows the slope of the change in comparing
    >from the hour before to the hour after the observations. Here are the
    >Nautical Almanac data I used:
    >Sun GHA 18:00   88* 17.4
    >Sun GHA 19:00   102* 17.6
    >Sun Dec 18:00   0* 46.1 North
    >Sun Dec 19:00   0* 47.1 North
    >Sun SD          16.1'
    >Moon GHA 18:00  349* 11.9
    >Moon GHA 19:00  3* 37.4
    >Moon Dec 18:00  24* 40.0 North
    >Moon Dec 19:00  24* 40.2 North
    >Moon HP         57.8'
    >>From this I calculated:
    >D1 =    97* 55.9 (Comparing distance at 18:00)
    >D2 =    98* 26.9 (Comparing distance at 19:00)
    >Slope = +5.16' every 10 minutes during this hour (for ease of plotting
    >the reference line)
    >I plotted this reference line alongside the graphed distances. I
    >the highest two distances based on my belief that they were spurious.
    >visually established a "best fit" line parallel to the reference line
    >through the 9 remaining valid distances. A lot of judgment here, this
    >a pseudo-scientific process, but it allowed me to select a point on the
    >best fit line as follows:
    >Selected GMT of observation:            18:50:00
    >Ds corresponding to selected time:      98* 34.8
    >As I am landlocked, my altitudes were first calculated based on my
    >selected time and my latitude and longitude. From the calculated
    >altitudes, I work backwards to get apparent altitudes (altitudes
    >for all but parallax and refraction). Bruce has tables for this, I have
    >to fiddle around with a sight reduction spreadsheet to come up with the
    >corresponding values.
    >Sc =    40* 41.7
    >Sa =    40* 42.8
    >Mc =    70* 27.7
    >Ma =    70* 8.2
    >With all the above in hand, I can clear the distance and determine GMT
    >of the observation. Here are the intermediate values and final results:
    >Moon's refraction and parallax =        19.53'
    >Sun's refraction and parallax   =       1.06' (I use Sun HP = 0.0024*)
    >Moon's augmented semi diameter =        16.0'
    >Sun's semi diameter =                   16.1'
    >Ds =    98* 34.8
    >Da =    99* 4.3
    >D  =    98* 19.2 (the cleared distance)
    >D1 = 97* 55.9
    >D2 = 98* 26.9
    >GMT per observation =   18:45:07, March 22, 2002
    >This would suggest that my "watch" is fast by 4 minutes and 43 seconds.
    >However, as I assume time by GPS is quite accurate, the error is in my
    >observations and calculations. Presuming the calculations are accurate,
    >I measured the angle as smaller than it actually was. Bruce has tables
    >to calculate the error in the distance, I have to fiddle with my
    >spreadsheet to determine that I should have measured Ds = 98* 37.3, an
    >error of about 2.5'. When I go back to how I placed my "best fit" line
    >on the graph of my sights, I might have gotten closer to the mark if I
    >had included the two distances I discarded as spurious, but they were
    >clear outliers in the series, so I would not have used them in practice
    >at sea.
    >So it's back to the sextant to shoot some more. If anyone finds errors
    >in my calculations or problems with my procedure, please let me know.
    George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    Tel. 01865 820222 or (int.) +44 1865 820222.

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