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    Re: Star-star distances for arc error
    From: Douglas Denny
    Date: 2009 Jun 23, 21:45 -0000
    ----- Original Message -----
    From: "George Huxtable" <george{at}hux.me.uk>
    Sent: Tuesday, June 23, 2009 1:02 PM
    Subject: [NavList 8768] Re: Star-star distances for arc error

    In [8767], Douglas Denny describes one of his sextants, in a way that
    puzzles me a bit-

    "My Hughes sextant - beautifuly built, brass frame, with platinum divided
    scale,  with drum/worm drive has a vernier to measure to 10 seconds of arc
    but the certificate indicates accuracy of max error of 1 minute 30 seconds
    at points on the scale."

    His reference to a "platinum divided scale" implied to me that it was a
    Vernier sextant; but then the word "drum" goes with micrometer sextants
    (which sometimes have their own little Vernier for interpolating between
    drum divisions). My guess is that he has a Vernier instrument with a Hughes
    patent endless-tangent knob, but not a micrometer drum (I've never come
    across the two combined, but it might be possible). If it's a Vernier
    instrument, divided to 10", then judging by my own such sextant, I would
    expect the scale to be readable, reliably, to well within 20",  but without
    being certain to 10".
    The sextant has a platinum inlaid scale for the degree markings.  The micrometer drum (with worm thread on the frame tangent thread ) is divided as normal into minutes, but it has a small vernier included for the fiducial markings of six marks from zero to 60 seconds, enabling a vernier reading to 10 seconds of arc.
    Reading to 10 seconds is quite useless of course, as the frame screw thread is where the accuracy lies and that is one minute 30 seconds 'out' along the scale at the measured certificate points. 
    Zeiss, however,  realise one minute reading accuracy is what one normally needs in practical navigation, and adequate expectation even with best conditions on land.
    The point which seems to have been missed entirely in my quoting those figures is that one sextant reads to 10 seconds, but has a tangent thread worse by eight times that;  and the other sextant reads to one minute of arc but has a tangent thread cut to an accuracy better than half that.  Just because the readings on the sextant say you can measure to 20 seconds of arc - as the song goes: it ain't necessarily so.
    No doubt someone given the two sextants as if new to consider buying (but without looking at the certificates) most would choose the former thinking it had "better accuracy"  by reading to 10 seconds of arc; whereas in fact it is a quarter less or worse in absolute terms.  Is this why they put the small vernier there when a single fiducial mark would suffice or am I being too cynical?

    Douglas defends his earlier claim-

    "Attempting to use star separations to try to determine scale accuracy for
    example would not be possible due to the variables in the measurements
    themselves - including the refraction component even if calculated.
    Also, combining of two stars overlapping, with the inevitable abberrations
    seen in the telscope image itself preclude this to being within one minute
    at best.
    Also, the measurement can only be done to an accuracy of the divisions of
    the sextant which in most cases is one minute of arc. "

    Frank Reed and I often differ about questions of precision, but in this case
    I would endorse all that he has said (and also on another matter, of the
    effect of a telescope on brightness, but that's for another posting).
    Douglas has got a lot wrong, in my view.

    I was accused of not trying star separations.   "A sure indication that you have never tried it!"   was thrown my way.
    Well having tried it I am quite sure it is not a practical method of obtaining accurate enough information about the state of a sextant tangent screw accuracy.
    That is my opinon.
    The main point I wanted to express is that to try to accurately assess errors in the sextant divisions when the accuracy of the measurements being made is only of the same order is clearly not good practice.
    Right from my schooldays many years ago in a science lab, it became clear
    that if markings on a scale were sharp and clear, and were far enough apart
    that eye-resolution wasn't a problem, and any movements were smooth and
    reproducible (all of which apply to the drum reading of a good sextant) then
    anyone could interpolate between them, just by eye, to fifths, and a careful
    worker, with a bit of skill and practice, to tenths. A vernier helps a bit,
    but isn't necessary.

    Reading the vernier is not where the accuracy is impaired, making the co-incidence of the stars is.  Even with a clamped sextant (which is what I did) one can only get a resolution of co-incident stars to about one minute of arc and any better requires better equipment than a simple sextant. Use a theodolite and you are now talking turkey. Sub-minute acuracy is then possible (but more difficult as separate viewings of the stars are required with change in HA).
    In fact the resolution of the eye itself varies for different acuity tasks.
    Letter acuity is in the order of one minute of arc (so-called 20/20 vision in feet; or 6/6 when metric in Metres).  Grating acuity (distinguishing between balck and white stripes) or spacial acuity .. and this is what is near to two star  separation acuity.... is in the order of 30 seconds of arc.
    Vernier acuity (the distinguishing of a break in a line is the most accurate with something in the order of 4 to ten seconds of arc.  A point source at the retina occupies about 48 seconds of arc due to diffraction effects in the eye.
    Hence distinguishing between two point sources (represented by grating acuity) can only be done to about 30 seconds of arc anyway no matter how good the sextant.  It can only be enhanced with a different 'target' - a line (against another line preferably) which is why theodolites use them.

    Douglas writes- "If accuracy is required, the measurements are difficult to
    achieve in the first place unless the sextant is clamped.".  It's true that
    measuring star-star separations is difficult, requiring a steady hand and a
    sharp eye. At my age, I have neither, sufficiently. One difficulty is in
    identifying the correct star-pair, as one star looks much like another. And
    having done so, superimposing star images is rather tricky, that's true. But
    you don't have to superimpose them, you can brush one past another, to put
    them just side by side, and cross them over (tricks of the trade).

    You appear to be agreeing with me here that it is difficult.
    Also, suggesting putting stars "side by side" is an admission of the finite size of the images of the stars in the telescope being sufficiently large to make accurate superposition difficult - which it is if accuracy is wanted.  Tried it - with not much difference in accuracy.

    Because of the stars' motion across the sky, clamping a sextant can be
    counterproductive, and gets in the way of such sweeping-past. An equatorial
    mounting on a pillar stand, as some land geographers used in the days of
    lunar longitudes, would work better. But the genius of the sextant is that
    the sextant itself doesn't need to be aligned with anything, as long as both
    objects can be got into view. To my mind, Douglas is overstating the
    difficulties in such on-land observations, although skill is certainly
    called for.
    I do not think so.  Some form of rest at the very least is needed, or a clamp with some degree of freedom. The stars dance about too much for accuracy otherwise.
    Remember, generations of mariners found their longitudes from
    lunar distances, from a vessel at sea, with a rolling deck underfoot, aiming
    at angular precisions of less than a minute. By comparison, measurement from
    on-land is a doddle.

    And less than a minute accuracy at sea is practically impossible.  I have tried that too; in any seaway, jammed in as near as possible to centre of gravity of the boat's motion, it is not at all easy to get down to one minute of arc let alone less than this.
    I do not think you have a correct appraisal of the accuracy of 'Lunars'.   Lunar distances were very poor indeed.
    Sir  Frank Dyson,  Astronomer Royal, in the forward of Gould's 'Marine Chronometer':-
    ".... in spite of the attention devoted to the lunar theory by some of the world's greatest mathematicians, .......even with perfect tables, it is found that the most skilful navigator cannot obtain a very accurate postion in this manner. With great pains and sometimes elaborate calculation he cannot be correct to within 20 miles"
    And this means with use of the reflecting circle too - more accurate than a sextant - eliminating some of the inherent errors of the sextant.
    The Lunar method was too impractical on grounds of inadequate tables; great difficulty in measuring and far too complex a method of calculation for ordinary use by seamen. It took at least an hour for the astronomers on board ship when the method was tested to 'clear the distance'  to find longitude,  and thus it was rejected for the longitude prize when Tobias Meyer put his tables before the Longitude Board around 1759.

    It surprises me that an optics professional brings in "the inevitable
    abberrations seen in the telscope image itself". Yes, of course, all
    telescopes suffer from aberrations, which degrade the image of a
    point-source star to some extent, ending up with a resolved image disc of
    finite size. But those aberrations affect both star images exactly the same;
    they don't cause any shift between one and another. Then the eye has to
    distinguish the shift of one such disc with respect to another. Is Douglas
    really claiming that the eye can't detect such a shift of less than 1', when
    using, say, a 6x scope? In which case, what sort of resolving power does he
    expect that telescope to have?
    You are again bringing in red herrings into the argument.  The resolving power of the telscope is not the issue here,  but the ability to superimpose star images - accurately. In other words: eye acuity is the issue.  Already dealt with as above.

    About refraction, Douglas complains-
    "and you are suggesting it is OK to measure two stars where the refraction
    error alone can be greater than the measurement."

    No problem about that, none at all, if the refraction is predictable. And,
    as long as the lower few-degrees of altitude are avoided, it is VERY
    precisely predictable, which is what makes precision astronomy possible.

    As he states, the effect of refraction at an altitude of 40º is of the order
    of 1'. If no correction were made for meteorology, even in extreme
    depressions that wouldn't vary by more than 0.1'. And if only roughly
    corrected for local temperature and pressure, no more than .01' variation
    would remain. Would that satisfy Douglas Denny? Far from being "unknown
    effects" of barometric pressure, the effects are precisely known and easy to
    measure and correct for. The effects of humidity are quite negligible, as
    Frank has pointed out..

    Low altitudes should certainly be avoided for this purpose, but there's no
    need whatsoever to use low altitudes.
    Once again: My point is:  that it is not good practice nor good science to attempt to use a method for obtaining accurate measurements,  where the errors involved in the measuring process are comparable with (and can in this case exceed) the measurement accuracy itself.
    Sextants are not a precision optical instrument like a thodolite for example, or astronomical instrument in an observatory.  They are a working tool for seamen which have a working accuracy of about one minute of arc. A good one in skilful hands will achieve perhaps 20 seconds absolute accuracy some of the time, mostly just better than a minute; but to ascribe magical properties to them is ludicrous.
    I am beginning to detect there are many sacred cows here which are being threatened with slaughter, causing distress to their owners, so perhaps it is time for me to desist waving my knife.  I am finding it is not useful to put forward my suggestions or ideas forward where there is an environment of reluctance to consider them seriously.  I object for instance to being dismissed out of hand with an offahnd and glib:  "clear indication I have not tried it" - when I have.  So I think it time to leave.
    Douglas Denny.
    Chichester.  England.

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