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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. A
plot of those lunar distances against time shows just what Arthur is
worried about. I would regard those measurements as unacceptable even with
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) show
a jump out of the trend-line of the others, by about 7 minutes. It's hard
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 could
be clearly seen, as it would have to amount to a gap between them of nearly
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 the
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 not
expect the backlash (lost motion) of any sextant to approach anywhere near
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 that
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 reason),
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 of
the eye is about 1', and the usual telescope has a magnification of about
x2.5 or x3. I would hope for a band of measurements mostly within .5' of a
best-fit line. I wonder what scatter other listmembers manage to achieve in
their measurements of lunar distance, in terms of plotted scatter?

It's often useful to repeat the measurement of index error after the lunar
measurement as well as before it. In a lunar, where precision is
all-important, this gives confidence that the sextant is stable and giving
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 telegraph
poles. Or the angle in the sky between two bright planets. As these won't
alter perceptibly during the measurement, any variation is in the observer
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 gets
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 has
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 to
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 lunar
parallax.

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, as
is the resulting rate-of-change of lunar distance over that hour. However,
the observations, made from a point on the Earth's surface, are affected by
the changing parallax of the Moon as it passes across the sky, as viewed by
a real observer. As the effect of this parallax change is always to oppose
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 slope
calculated from the Almanac. Sometimes, when the Moon is very high in the
sky, the slope can be very much less: under some circumstances it can be
halved, roughly speaking. This is the "parallactic retardation" effect that
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 way,
to speed it up? It seems unphysical.

That's true, it does. It's a very reasonable question. The answer is, that
part of the time, changing parallax does speed up the motion of the
apparent Moon. Unfortunately, those times are when the Moon is invisible,
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. For
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 observers,
need to bear in mind the limits of this exercise. The problem is that the
value of GMT, taken from the GPS receiver, hasn't just been used to confirm
the accuracy of the final answer. It has also been used, in the course of
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 answer
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 the
time (if he had, why would he be bothering with lunar distance?). The
navigator would make his best-guess at the GMT, and calculate accordingly.
This would give a more accurate time, so he could plug that in as a better
value and repeat the operation, as many times as necessary., getting more
and more accurate until the results converge toward the correct value for
GMT. And as I have warned, in "about lunars, part 4", that convergence may
be a slow one. The reiteration is something a computer could handle with no
trouble, but would be unacceptable to anyone calculating by hand or by
tables.

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 aspects
of the lunar distance method. In real life, the best way is to measure the
altitudes.

George Huxtable

=============================

>Gentlemen,
>
>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 distances.
>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 to
>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
>improvement.
>
>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 distances
>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 ignored
>the highest two distances based on my belief that they were spurious.  I
>visually established a "best fit" line parallel to the reference line
>through the 9 remaining valid distances. A lot of judgment here, this is
>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 correct
>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.
>
>Regards,
>Arthur

------------------------------

george@huxtable.u-net.com
George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
Tel. 01865 820222 or (int.) +44 1865 820222.
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