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    Re: Star to Star Distances taken on a Second Hand Sextant
    From: Brad Morris
    Date: 2019 Dec 10, 20:21 -0500
    If not in one night, Fred, then the temperature change, one night to the next, means you will calibrate each section of the arc under a different temperature regime.  Each material comprising the sextant has a different thermal coefficient of expansion.  For example, a glass mirror is typically is 7.6 parts per million per degree C, while the steel screws that hold the mirror expand at 13 PPM/°C.  This changes the shape of the surface of your mirror.  

    Each arc section, therefore, will be calibrated when the sextant has a different shape/size.  These mechanical variations are a physical property of the materials.   Only a very few, very expensive materials like Zerodur or Invar have invariant sizes when subjected to temperature differences.  That is, their thermal coefficient of expansion is extraordinarily low.

    The only bypass is to either keep your instrument in a temperature controlled environment or to measure in the shortest time regime possible, thus minimizing the thermal effect.  

    When doing professional calibration, we would set up all the equipment, aligning the instruments and objects.  Once we had it ready to go, we would walk away for a minimum of 1/2 hour so the metals could reach ambient temperature again after we touched them.   One screw up and you start all over again!

    Your milage may vary of course.  I am just relating my professional experience in rotary calibration.  Of course, you can always accept the additional uncertainty.  That would be a personal choice.  You may not think it matters for your purposes.  Its rather surprising, however, just how bendy objects really are!


    On Tue, Dec 10, 2019, 7:47 PM Fred Hebard <NoReply_Hebard@fer3.com> wrote:

    Your requirement of doing all the measurements in one sitting is not necessary.  If one could do two sets per sitting, it would be possible to link sittings in a lattice design, also known as an incomplete block design.  

    I narrowed down Ho - Hc after ceasing to do star-to-star sights.  I attribute that to technique improvement. Based on that experience, I don’t think one could get errors in each measurement much below 0.1 to 0.2 arc minutes.  How would that do?

    Here are my best data when I was at the top of my game:

    Block         Date          N(Ho-Hc')     Mean(Ho-Hc')  Std Dev(Ho-Hc')
    254           09/21/2004    3             0.052         0.229
    255           09/22/2004    3             -0.141        0.298
    256           09/23/2004    3             -0.029        0.108
    257           09/23/2004    3             -0.461        0.194
    258           09/25/2004    3             -0.253        0.069
    259           09/26/2004    2             -0.753        0.080
    259.1         09/26/2004    5             -0.414        0.023
    260           09/26/2004    4             0.009         0.138
    261           09/29/2004    3             -0.541        0.160
    262           09/30/2004    5             -0.391        0.151
    263           09/30/2004    6             0.004         0.107
    264           10/01/2004    6             0.025         0.138
    265           10/01/2004    6             0.026         0.099
    266           10/02/2004    6             0.013         0.152

    Date          Block         Ho-Hc'
    09/21/2004    254           -0.08
    09/21/2004    254           0.32
    09/21/2004    254           -0.09
    09/22/2004    255           0.16
    09/22/2004    255           -0.44
    09/22/2004    255           -0.14
    09/23/2004    256           0.05
    09/23/2004    256           -0.15
    09/23/2004    256           0.01
    09/23/2004    257           -0.24
    09/23/2004    257           -0.56
    09/23/2004    257           -0.58
    09/25/2004    258           -0.18
    09/25/2004    258           -0.31
    09/25/2004    258           -0.27
    09/26/2004    259           -0.70
    09/26/2004    259           -0.81
    09/26/2004    259.1         -0.43
    09/26/2004    259.1         -0.39
    09/26/2004    259.1         -0.40
    09/26/2004    259.1         -0.41
    09/26/2004    259.1         -0.44
    09/26/2004    260           -0.10
    09/26/2004    260           -0.06
    09/26/2004    260           0.21
    09/26/2004    260           -0.02
    09/29/2004    261           -0.37
    09/29/2004    261           -0.58
    09/29/2004    261           -0.68
    09/30/2004    262           -0.18
    09/30/2004    262           -0.34
    09/30/2004    262           -0.35
    09/30/2004    262           -0.53
    09/30/2004    262           -0.55
    09/30/2004    263           0.20
    09/30/2004    263           -0.01
    09/30/2004    263           -0.12
    09/30/2004    263           0.02
    09/30/2004    263           -0.03
    09/30/2004    263           -0.03
    10/01/2004    264           -0.06
    10/01/2004    264           -0.05
    10/01/2004    264           0.04
    10/01/2004    264           -0.15
    10/01/2004    264           0.15
    10/01/2004    264           0.22
    10/01/2004    265           -0.08
    10/01/2004    265           0.10
    10/01/2004    265           0.01
    10/01/2004    265           -0.06
    10/01/2004    265           0.18
    10/01/2004    265           0.02
    10/02/2004    266           0.18
    10/02/2004    266           -0.03
    10/02/2004    266           0.01
    10/02/2004    266           -0.08
    10/02/2004    266           0.20
    10/02/2004    266           -0.19


    Fred Hebard

    On Dec 10, 2019, at 17:16, Brad Morris <NoReply_Morris@fer3.com> wrote:

    Hi Fred

    Do not worry about the confusion over which Morris you thought you were talking to.  It has happened enough times that I am well over it.  It doesn't bother me at all.  I just try to gently correct the mistake.

    I remain unconvinced that your star to star distance experiment can be put down to "bad technique".  The requirements comprise such a high bar that it may prove nigh impossible to accomplish.  The need to take repetitive, precise measurements in rapid fire short order would exhaust anyone.  Add the requirement that even one graduation (0.1 arcminutes) of error in measurement has a significant impact on the standard deviation.  That the two star points really cannot be superimposed but just flutter by each other.  That the sextant arc error should be measured to the single arc second (to achieve a good result) when the instrument doesn't possess that graduation. 

    The repeatability requirement alone is the real terror.  This is imposed because a correction must have a tiny error band associated with it.  The tiny error band is the repeatability.  No matter the magnitude of the correction, it must be repeatable, to three standard deviations, within your tolerance for error.   You don't need a correction that reads 0.7°, +/- 0.7' 3 sigma.  That error band on the correction is 1.4 arc minutes (+/-0.7) wide.  Ouch, not usable.  But to get it to read 0.7°, +/-0.1', 3 sigma means that one standard deviation in your data set was 0.033' (2 arc seconds).  Put another way, that essentially means you hit 5 of 6 star to star distances on the nose, and the last one you were out a smidge.  

    Its not practical Fred.  Its simply beyond human ability.   

    Yes, we can use Cmdr Bauer's table, or other similar tables, to confirm that the sextant is okay after dropping it.  That is perfectly viable.  A simple check here or there on the arc will convince even the hardest boiled egg that the sextant can still be used (or not).   Bauer is not wrong.  If his intention is to provide a sanity check, then his table accomplishes that goal.

    I am asserting that the requirements imposed are too severe to accomplish by even the most determined of individuals.  I mark this idea down to fable and wishful thinking.  It seems feasible, right up until you try it.  


    As to Michael Bradley's recent data publication, there is no evaluation of the error band associated with the correction.  Whilst I do see Michael's data as a very serious attempt to complete the task, I find the result incomplete.  I have no idea if the correction is 0.8' +/- 1 degree, 3 sigma or 0.8' +/-1 arc second, 3 sigma.  That is a seriously different result, three orders of magnitude different!  Unfortunately, all we understand is the 0.8', without an error band.   I am not disparaging Michael's efforts or result.  It achieved his stated result of determining if an arc was good enough for his use.  He states that he accomplished that, to which I emphatically state: Bravo!  However, I would never consider using it for arc correction.

    On Tue, Dec 10, 2019, 2:47 PM Fred Hebard <NoReply_Hebard@fer3.com> wrote:
    Hi Brad,

    The only publicly available collimation equipment of which I am aware is at the Frieburger factory in Germany, and I don’t know they are still active.  So most of us are stuck with star-to-star distances.  My efforts to use star-to-star distances for calibration did not give good results, due to poor technique.

    I was advocating using one pair of stars for perfecting one’s technique.  Once that were accomplished, it might be possible to make sets of measurements on four or more pairs of stars in one night.   It also would be possible to assess the variation from night to night in the reduced distances, which remain the same.  Setting up an incomplete block experiment to calibrate a sextant over multiple nights might be feasible using observations on two or more pairs per night.

    Yes, I noticed that you were neither David nor Bill Morris after sending my last missive.   Sorry for the mixup.



    Fred Hebard

    On Dec 10, 2019, at 11:16, Brad Morris <NoReply_Morris@fer3.com> wrote:

    Hi Fred

    You wrote
    It would seem that star-to-star distances are preferred for sextant calibration because of their lack of day-to-day variation

    Yes, under identical pressure and temperature regime.  The temperature and pressure changes affect the effect of refraction. The apparent star to star distances are not identical day to day.  Further, their position in the sky changes day to day, which also affects the distance due to refraction.

    Additionally, temperature changes the different materials in a sextant in different ways, due to the different thermal coefficient of expansion associated with each material.  You are strongly advised to not spread it out over several days due to this uncontrolled aspect.  

    Star to Star distances may be preferred by some, but what is really preferred was the calibration equipment possessed by the National Physical Laboratory (or the NIST traceable equipment in my possession).  This places the sextant in a temperature controlled regime to be compared against "perfect" angles.  This costs real money however.   The results are in seconds of arc error, far finer than any other method I know of. Traceable to national standards and repeatable calibration.  To me, that's far more preferable. 

    So why isnt this available? Why am I not offering this?   I have the critical equipment, the Nikon 6D autocollimator and stand, and the Ultradex angle standard.  However, the tooling to hold a sextant without deforming the sextant costs real money.  I am very unlikely to invest thousands of dollars in tooling to obtain a few hundreds of dollars of return on that investment.  There's a reason the NPL and others faded from service.  No demand for service, non viable.

    So I guess that makes star to star distances the only viable method for calibration, for now.

    Brad Morris

    I am also often confused with Bill Morris, who lives in New Zealand.  I'm Brad Morris, who lives in New York.  How lucky for us that we share an engineering background and the same last name! :)

    On Mon, Dec 9, 2019, 1:48 PM Fred Hebard <NoReply_Hebard@fer3.com> wrote:
    Hi Brad,

    .  Thus one could easily compare sets of measurements between days.  It took me a long time to get mean Ho - Hc for position lines down below 0.1’, most of the time.  The standard deviation was a bit larger, maybe 0.1’ to 0.2.

    The keys for my improvement were threefold.  First was  accurate assessment of tangency in an artificial horizon.   I improved that by measuring semi-diameter of the sun and comparing the observed value to that in the Nautical Almanac.  Improvement of SD measurement also led to an accurate index error, the second key.  The third key was accurate timing of a sight.  Each second of timing error leads to a 0.25’ deviation in Ho, IIRC.

    The suggestions I offered below are an equivalent program of technique improvement for star-to-star distance measurements.  I estimate it would take me several months to achieve proficiency.  Only then could one start calibrating a sextant.  Using stars of appropriate magnitude, as suggested by Peter Monta, would be an an important component.

    Sorry for calling you David; he lives in Alabama, not New Zealand.

    Fred Hebard

    On Dec 8, 2019, at 02:16, Brad Morris <NoReply_Morris@fer3.com> wrote:

    Hi Fred

    The arithmetic for finding the standard deviation is readily available on the internet.  I'm quite sure that absolutely anyone who can reduce an observation, can also determine the standard deviation.  In industrial robotics, we cared about a 3 sigma result, to wit, that 997 times out of a thousand, the angle desired would be the angle obtained.

    To help you through this, go out and pick out any observation.  The best illustration of the technique is likely a meridian crossing.  The sun hangs there, and it won't strain your neck.  Its a comfortable observation, no?  So take 6 observations in a row, taking care to record each value.  Please check your index error.  Post them up.   

    We'll go from there!  


    Owners of vernier sextants need not worry about approaching always from one side.  The only possible lost motion is in the index arm bearing surfaces, which would only occur after substantial use.  The S-N failure curve for a fairly low low load (index arm) on metal bushings will run to the millions before failure.  In consideration that most vernier sextants will never see that level of use, the index arm v arc has very, very little lost motion.  Virtually insignificant for our purposes.

    On Sat, Dec 7, 2019, 9:34 PM Fred Hebard <NoReply_Hebard@fer3.com> wrote:

    I shared your problems and never felt my observations of star-to-star distances were adequate.  A good place to start might be in perfecting one’s technique for using star sights to measure index error.  Another important piece is that star pairs be vertically or horizontally oriented.  Otherwise, the difficulty of holding the sextant is overwhelming.  Finally, I believe getting one good set of observations on one pair of stars in one night would be a good first objective.

    Hopefully David Morris will continue to guide us in data reduction, which is also an important component of the endeavor.  


    Fred Hebard

    On Dec 6, 2019, at 15:46, David Pike <NoReply_DavidPike@fer3.com> wrote:

    Well there was only one thing to do; go out and try it.  Wednesday, Thursday, and Friday we were down in the low light pollution area of Norfolk, so with great fortitude and determination and the temperature down to 2 degrees Centigrade, Mrs P and I drove the car onto Kelling Heath (N52.92789 E1.13489 on Wednesday evening 4th Dec, taking along ‘Navigator’ on a Netbook, and my Hughes Three Circle Mates Sextant No 25410.  First star pair was Deneb at roughly Hs= 48.55 and Az= 286 and Vega at Hc= 27.17 and Az= 299 and a tilt of about 30 off the horizontal. 

    First thing I found is, with bifocal eyeglasses and only a 2.5x telescope, checking index error against a nice bright star like Vega is nothing like as easy as checking against the sea horizon or a roofline many blocks away in daylight.  All I could say was that index error didn’t seem too far off the sextant’s historic index error of zero.  Next came how to hold the sextant.  I ended up holding it handle side up at a tilt of about 30 degrees to the horizontal.  I found this rather awkward.  No one ever tells you that your carefully attached lanyard will get in the way of just about everything, and to be prepared for an index shade to fall under gravity and leave you wondering where the second star has disappeared to. 

    After that comes bringing one star onto the other from zero.  Well that might be OK for the ‘Experten’, but for yours truly it explains why I was never any good at ball games.  The ‘wimp’ way to do it is to set the sextant about half a degree lower than what you’re expecting; point at the brightest star though the horizon glass and look for the other star in the mirror.  The half degree was to prove to myself that if, I could only see one star they weren’t, though some sort of fluke, on top of each other.  There turned out to be little chance of that I can assure you.  In fact, I found finding the second star so hard that in the end I unscrewed the telescope and just looked through the hole, even then, I had to cheat a bit at times by keeping both eyes open.

    After about 15 minutes practicing like this, I felt confident enough to put the telescope back in and do the job properly.  I found the best method was to rock the sextant ever so slightly like rolling the Sun about the sea horizon and gradually increasing Hs until the reflected star crashed though the index glass star.  I then increased Hs further until the stars separated again.  Going back onto crashing into each other, the best I could manage with the 2.5x scope and my not very special eyesight were values within a couple of minutes of Cdr Bauer’s values.

    After all that messing around with Vega and Deneb, Orion and the Twins were becoming nicely visible and with Betelgeuse at roughly Hc= 17.54 Az= 102 and Pollux at Hc=17.42, Az=067, I had two stars horizontal and at a low Hc.  (I’m not sure if this is the worst case or not.  True there’s a fair amount of refraction, but the XP lines are almost parallel at low altitudes, so that might cancel out the refraction effect a bit.)  This was a lot easier.  With handle down I could get greater support from the car roof and steady the telescope with my left hand.  Also, Betelgeuse is a bit more yellow, so I could see which star was which (In my Universe all major stars would be colour coded).  Once again, with my eyes, spectacles, and sextant, all I could say was that my measurement was within a couple of minutes of Baur’s values.  Therefore, my conclusions so far are that for a rough check it does work, but there must be easier ways of checking a sextant, even inland.  However, you need good eyesight, decent magnification, a strong, steady wrist, and most of all you need to be in good practice.

    My next job is to work out some 2019 values for the stars used and record my results properly, but I wanted to get something written down while this topic was still hot.  DaveP

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