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    Re: Lunar distance measurement in ideal conditions: attainable accuracy.
    From: Alexandre Eremenko
    Date: 2013 Jan 7, 00:27 -0500

    Bill,
    Could you tell us the dates of your observations?
    the spreadsheets only say January 2013.
    
    Alex.
    
    > Recent posts have questioned vehemently (2 Jan 13 per Frank Reed) or
    > mildly (31 Dec 12 Alex Eremenko) whether it is or was possible to achieve
    > particular degrees of accuracy and consistency with lunar distance
    > observations. My own interest is in the instruments of navigation rather
    > than the many uses to which they can be put. In 2011 I took up the
    > challenge of the late George Huxtable to see how instruments perform in
    > the real world at sea. I regarded recent statements as a challenge to see
    > what can be achieved for lunar distance measurements  in the best of
    > conditions using the best of sextants. I say nothing about the quality of
    > the observer except that I have spent a lot of amateur time peering
    > through optical measuring instruments of various types, my vision can be
    > corrected to 6/5 (it isn’t at present, as optometrists in New Zealand
    > charge like wounded bulls for new lenses) and, as a copper-bottomed
    > brass-bound , dyed-in-the-wool sceptic  I do my very best not to
    > “tweak” observations despite feeling tempted at times.
    >
    > The sextant used was a 1978 Russian SNO-T sextant whose non-correctable
    > errors in the range of angles measured were under 3 seconds, so I ignored
    > them. As Frank Reed had sung the praises of a x7 telescope, I assumed he
    > had used a prismatic monocular, so I did too for my first series, a
    > vintage Beck Kassel  6 x 30 borrowed from a 1953 C Plath sextant. For my
    > second and third series I used the Keplerian (“inverting”) telescope
    > supplied with the instrument. The mirrors and shades are in perfect
    > condition and, as I overhauled the instrument myself, I am confident that
    > it is in as good a mechanical condition as possible.  While backlash is
    > well within the 6 seconds allowable by the specification, I consistently
    > turn the micrometer drum in the direction of increasing the reading, both
    > when estimating index error and when making observations. To minimise
    > digit preference bias, I glued a paper vernier reading to 0.1 arcmin to
    > the index.
    >
    > Even quite large telescope collimation errors are of little importance in
    > modern navigation, but when taking lunar distances one is trying to
    > squeeze the last bit of accuracy out of the results and the angles
    > measured are often large, when collimation error is most important. For
    > example, if the telescope is mis-aligned by half a degree, the error at
    > 105 degrees is about 20 seconds. The easiest way to check collimation is
    > to use an engineer’s square to see that the face of the objective lens
    > mount is square to the face of the sextant frame. You should not assume
    > that it is safe to transfer a telescope from one sextant to another
    > without checking this. There are several other, more-or-less sensitive
    > ways of doing it. I had to place shims between the Beck Kassel mounting
    > bracket and the body of the monocular to correct a large error, and by the
    > time my patience had run out there was still a small collimation error
    > remaining.
    >
    > If the estimate of index error is wrong, then it is possible to have
    > series of observations tightly clustered about a mean that is wrong. The
    > results will have a small standard deviation (a measure of central
    > tendency, or distribution of the results about the mean), implying good
    > consistency of measurements, but the figure of interest, the mean, may be
    > wildly wrong. For this reason, it is worth while taking great care with
    > the index error. Though several people, Greg Rudzinsky most recently (6
    > Jan 13), have stated a preference for a particular method of estimation,
    > when a few years ago I compared 30 observations each using the horizon,
    > the sun and two stars, I could detect no statistically significant
    > difference of the methods in my hands. On this occasion, I used the ridge
    > of a barn on the horizon 6 km away. It is easier on the neck. For these
    > and subsequent observations I clamped the instrument on top of a
    > theodolite tripod. It makes a big difference to the ease of observations,
    > especially in a 20 km/h wind gusting to 40 km/h. Incidentally, collimation
    > error does not apply to index error estimations, as the reading is made at
    > zero.
    >
    > In the spread sheet file “Lunar 1”, Cell B25 gives the mean of 20
    > index error estimations and, for what it is worth, cell B26 gives their
    > standard deviation.
    > Before making the lunar distance observation, I first achieved coincidence
    > by holding the sextant in my hands. I used no shades and proceeded from
    > the moon to the brighter object, the sun, using the sun’s glare as a
    > guide to when I was approaching coincidence, at which point  I put a dark
    > shade in place. I then mounted the sextant on the tripod again and
    > adjusted the index shades so that when the images overlapped, the moon was
    > visible through the sun.  I found it easier to determine contact of the
    > limbs by bringing them apart. To minimise the effects of collimation and
    > other optical errors, it is important to keep the objects as close to the
    > centre of the field of view as possible.
    > I used Frank Reed’s lunar calculator to determine what the result should
    > have been for the position and conditions, and the differences between the
    > calculated and observed results are given in column H
    >
    > Cells J13 and J14 give the means and standard deviation (SD) for ten
    > observations with the monocular, while cells J23 and J24 give the results
    > using the inverting telescope. The latter seems to give the better results
    > and we do not need to do a Student’s t-test to see that there is a
    > significant difference in the means of the errors. For a first attempt,
    > these are not poor results.
    > As the following day was also fine, I did another set of observations,
    > this time checking the index error using the same telescope, the
    > inverting, that was to be used for the observations. These are shown in
    > the file “Lunar 2”. Cell C25 gives the mean of 20 IE observations. The
    > dispersion of results about the mean was somewhat closer than before. Cell
    > G30 shows the mean error of 25 lunar distance observations, 0.032 arcmin
    > or about 2 seconds, while the standard deviation in cell G31 tells us that
    > about two thirds of the errors fall within the range of +11 seconds to -7
    > seconds.
    >
    > Thus, by taking care with adjustment of a high quality sextant  and
    > averaging many results, it does indeed seem possible to achieve high
    > accuracy on land with the instrument steadied on a tripod. The errors are
    > likely to be much greater on land with a hand-held instrument and still
    > more so at sea. The file “Cape St Diego” gives results of observations
    > by the astronomer Richard Green aboard the Endeavour in January 1769. We
    > can see that three sets of three observations were made and that readings
    > were to the closest 30 seconds.  Jesse Ramsden’s first dividing engine
    > had been completed only two years previously, not to his satisfaction, and
    > his second engine, which revolutionised precision angular measurement, did
    > not become operational until about seven years later. It is likely that
    > Green used a Hadley’s quadrant, probably of 15 inches radius, divided by
    > hand with a vernier reading to half a minute.
    >
    > To put these observations in context, the astronomer William Wales later
    > wrote:”…the longitude…is near a degree too great; which is not at
    > all surprising, if we consider, that although the air was extremely clear
    > when these observations were made, yet the sea ran so high that it filled
    > the quarter deck three times while they were observing  and the motion of
    > the ship was so great that Captain Cook did not attempt to observe.” My
    > own errors using a metal sextant of 8 inches radius with a vernier reading
    > to 30 seconds, made some time between 1790 and 1810, are shown at the
    > bottom of file “Lunar 2”. The sextant was steadied on a tripod, no one
    > was playing a fire hose on my legs and the ground was not rocking and
    > shaking, yet my errors are of the same order as those of Green.
    >
    > Bill Morris
    > Pukenui
    > New Zealand
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    >
    > Attached File: http://fer3.com/arc/img/121731.lunar-1.xlsx
    >
    > Attached File: http://fer3.com/arc/img/121731.lunar-2.xlsx
    >
    > Attached File: http://fer3.com/arc/img/121731.cape-st--diego.jpg
    >
    >
    > View and reply to this message: http://fer3.com/arc/m2.aspx?i=121731
    >
    >
    >
    
    
    
    
    

       
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