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    Re: Star-star distances for arc error
    From: Douglas Denny
    Date: 2009 Jun 27, 15:13 -0700

    I have not been able to respond on this forum to the points made to or 
    regarding myself until now as I have been moving my boat  which has been a 
    bit of a headache as it is 19 tonnes. 
    This post is fairly long and my spellcheck seems not to be working so excuse typos/spelling.
    I have looked at the responses to my submissions on both potential eye damage 
    and the star distances question, and am amazed at the diversity of  
    discussion this has generated: some of which has little to do with the 
    scientific principles involved and has even brought personalities into it.
    It appear to me there is a strong underlying desire by some who wish to prove 
    others 'wrong' - when someone  tramples upon little feet challenging pet 
    theories.  It was pointed out in my case I am a politician, though what this 
    has to do with it I am not sure except that I am, I confess, immune to 
    receiving a few brickbats thrown around - I am used to that.  What I do not 
    wish to do, or intend to do however,  is send one posting about a subject, 
    and then debate it endlessly,  going around in circles.
    There is too much armchair anecdotal 'evidence' and too little of hard facts. 
    Too much subjectivity for a strictly scientific subject, and not enough 
    objectivity.   And as for sacred cows - I like to think mine are ready for 
    slaughter anytime if my propositions are comprehensively demolished.  I have 
    not, however, seen anything in the responses yet which has caused me to worry 
    about beefsteak appearing on my table.  You my call that arrogance;  I would 
    suggest the another possibility - perhaps it is just that  I am dim and 
    cannot see your devastating points. On the other hand, maybe, you cannot see 
    mine?  Here are some contentious points which I seem to have unwittingly 
    stirred up........
    Bygrave slide rule:
    It was my prompting of Peter Martinez to post about the Bygrave slide rule, as 
    I asked him to help produce a better computer derived scale.  We are of the 
    opinion that although there is room for discussion and both systems can be 
    considered alongside of each other as they reach the same result,  that isd:  
    of logCotan/ logCos scales,  or,   logTan/ logCosecant;
    I am completely convinced Bygrave made an error in his patent description 
    referring to log Cos scale, and that the scales are definitely logTan  and 
    logCosecant.  I do not want to rehearse the arguments here. It is immaterial 
    which you choose;  but what struck me was the unwillingness of some here to 
    even consider this alternative argument.  That is what put the idea of sacred 
    cows into my mind.
    The explanation of Aquino's short-method of logTan / logCosecant  sight 
    reduction (I sent in a copy of the explanation recently from 'Spherical 
    Trigonometry' by J.H. Clough-Smith) makes the argument conclusive for me.  
    Aquino published his method a couple of decades later than Bygrave. In fact, 
    Aquino's method should perhaps really be called Bygrave's method,  as the 
    latter invented the principle of perpendicular from the observer's meridian 
    to the star meridian  firstly,  in 1921 (using a mechanical implementation).
    The solar eye damage argument:
    That solar damage looking through telescopes occurs, (including the real 
    possibility of central vision blindness within seconds) is indisputible;  yet 
    some here seem to me to be trying to suggest looking at the setting sun with 
    a telescope is somehow Ok,  justifying this as reasonable "because they have 
    got away with it", and the sun's strength is weak.   It is true the 
    attenuation at sunset is great and 'getting away with it' is more of a 
    possibility than when it is high, but the principle remains that it is not a 
    good idea to do it,  regardless of the height,  as you cannot be sure of the 
    I/R and U/V content even at sunset.  You can also get away with looking at 
    the sun directly, or through a sextant telescope if you have forgotten to put 
    in the filters,  if the exposure is fleeting; but the principle remains it is 
    dangerous, and damage might occur for the longer term.
    Ask any eye-health care person and they are all of one accord  - don't do it.
    Then there is the telescope light magnification argument brought into the 
    argument about the Sun's intensity at the eye, a subject which has now  
    become littered with red-herrings.  The magnification point is only part of 
    the an issue; there is another point ignored by the afficianados on this 
    forum -  light gathering by  aperture area and the focussing of that energy 
    upon a smaller area at the image.
    You are making the mistake I think of fixating on the comparison of 
    'brightness'  (more correctly 'luminance')  of object and luminance of  
    image, as one would be interested in astronomical photometry with low 
    intensity objects,   without considering focussing effect of a  very intense 
    source and it's increased light flux density onto a small area.
    Let's get back to the maths. (simplified - without considerations of inefficiencies of the optics).
    In optics the apparent luminance of an object using a telescope, divided by 
    the apparent luminance of the object viewed directly,  is proportional to A / 
    M^2  where a is the aperture area and M is magnification. ( N.B.  I use as a 
    source for my references:  'Geometrical and Physical Optics' by R.S. 
    Longhurst ).   
    So the maths is clear, whatever the luminance of the object (L), ( and with 
    the Sun it is very high indeed) the luminance seen through a telescope is 
    equal to L times a constant, times the Aperture area, divided by the 
    Magnification squared.   So "brightness" of the image is a function of two 
    quantities: aperture and magnification.
    Increasing aperture does indeed increase luminance of the image in direct 
    proportion.. Increasing magnification decreases luminance squared.
    Increasing aperture in telescopes like Hubble  to large values enables 
    photographic film or CCD sensors to view deep space objects which would not 
    register with small telescopes. The 'brightness' has been vastly increased.
    Consider aperture size change only with unity magnification:
    As I have already indicated, a magnification of one gives a situation of the 
    sun covering the fovea with full 'brightness' of the Sun and is equal to  (L) 
    This is the situation with the eye and normal eye aperture (of between three 
    and six mm diameter).
    What happens with aperture changes with constant magnification?
    If a telescope is used of unity magnification, the maths says increase the 
    aperture area and the luminance of the image increases. If aperture is 
    doubled then L doubles.  If aperture is quadrupled, then L is quadrupled.  
    Use a telescope of unity magnification but aperture area double that of the 
    pupil diameter of say 3mm and the luminance must double to 2L.
    Now consider magnification effect: with constant aperture.
    A magnification of two doubles the Sun's size (and only half covers the fovea 
    incidentally) but luminance ('brightness') diminishes by 1/ M^2  i.e. L 
    becomes a quarter).
    A magnification of four  ( which incidentally has only a quarter of the Sun's 
    disc covering the fovea)  has  luminance diminished by a sixteenth of L 
    luminance of the sun.  
    In all cases, even that sixteenth with Magnification of four is still much too high to be 'safe'.
    So it is clear in my mind, for brightness of image in a telescope there are 
    two parameters working in opposition: aperture increasing directly and 
    magnification decreasing and by a square function..
    If light concentration was not possible then all of the observatories in the 
    world would be out of business.  Telescopes take light from a large aperture 
    wavefront and focus it down to a smaller area; in so doing they concentrate 
    the light flux into that smaller area.   That's it, there's not much more to 
    say about it, or needs to be said.
    I mentioned  the simplest case of a bi-convex magnifying lens burning paper to 
    try to illustrate the effect of concentration of light flux - but it was 
    ignored with other red-herring arguments of magnification, and  'brightness' 
    not being increased with a telescope' increasing magnification; or the eye 
    pupil is too small for the exit pupil of the telescope, or other spurious 
    The case of the magnifying lens burning paper (or your hand)  is all you need 
    to know about it, and illustrates the whole argument of solar radiation eye 
    damage in a nutshell, as the eye acts in exactly the same way.  The 
    experimental method is always best and I have invited you to try it.  You 
    will find a small aperture magnifying lens (say 1" diameter) cannot burn 
    paper very well because the light flux of I/R at the focus is not high 
    enough; all you have is a slowly smouldering brown hole appearing.  Try it 
    with a magnifying lens of the same magnification but an aperture size of say 
    5"  and the increased burning effect is dramatic, with an instant hole,  
    smoke and soon flames appearing.
    I need to correct one small error I made, regarding exit pupil size of an astronomical telescope.
    Mr Huxtable quite rightly picks me up on it, though the effect in the argument 
    is nil as the field of view for the sun is well within the exit pupil with my 
    binocs which have 7 and a half degrees FOV and the Sun is half a degree.
    Douglas Denny wrote-
    "Take my simple cheap binoculars. Magnification ten; objective aperture 50 
    mm;  focal length of objective 195 mm, focal length of eyepiece 19.5 mm. The 
    aperture of the exit pupil(x) is therefore:-
    x/19.5 = 25/214.5
    which  = 2.2 mm.  real, and at 19.5 mm. from the eyepiece."
    I don't follow that. I wonder if Douglas is familiar with B K Johnson's 
    little book "Optics and optical instruments". I attach a relevant page with 
    a ray-diagram. According to that, the exit pupil would be the size of the 
    objective divided by the magnification, which would come out as 5mm, not 
    2.2mm. It seems to make sense to me; but then I'm no expert on optics. If 
    Douglas thinks that Johnson has it wrong, perhaps he will explain why, 
    preferably with another ray-diagram.
    Apologies. Another of my 'senior moments'. I do it all the time these days 
    with calculations; mind racing ahead not noticing silly errors.  I was 
    working with half angles on a ray diagram.
    The exit pupil is 4.4 mm diameter. real at 19.5 mm from the eyepiece.
    The 2.2 referred to earlier is the radius; derived from simple ray diagram and 
    similar triangles. I forgot I was dealing with one side only of the diagram.
    The size of the image of the objective by the eyepiece is given by the similar 
    triangles of sum of the two foci which are coincident in an astronomical 
    telescope; and the focus of the eyepiece, in ratio with the aperture radii. 
    i.e. :-  =  2 *  (f e  *  obj diam) / (f o + f e) 
    I am not familiar with Johnson or his formula. perhaps you can elicit the difference. 
    Johnson give a ratio of magnificatin as the governing factor. This formula as 
    you sate it, is exit pupil diam =  objective diam / Magnification
    which is equal to  Obj diam / (f ' (obj)  / f (eyepiece) )
    which =  Obj diam * f e / f 'o
    Whereas I think it is as above with f e /f o + f e.  I am not sure where this difference arises.
    Finally, I wanted to comment on the Star distance argument by presenting real 
    objective analysis instead of the anecdotal evidence I have been presented 
    with here such as  "I regularly can obtain accuracy to 0.1 of a minute of 
    arc".   Such statements are scientifically meaningless without rigorous 
    experimental evidence and  an assessment of the errors.
    I have a report available which I have scanned from the 'Journal of 
    Navigation'  (Institute of Navigation) of the mid 1960s which  has done this 
    very thing,  and in detail.
    It is called  " The Attainment of Precision in Celestial Navigation"  by 
    Robert Gordon  of Yale University.  The content is of such importance to this 
    whole argument (and will be of great interest here I have no doubt) I shall 
    send a copy in a separate posting to this in due course when I can spend a 
    littel more time to compose more. This is already far too long.
    It mentions specifically measuring star distances for checking sextants.  I do 
    not think you will like the conclusions.
    Douglas Denny.
    Chichester.  England.
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