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    Re: How flat do sextant mirrors need to be?
    From: Gary LaPook
    Date: 2009 Jan 30, 17:24 -0800

    
    You can check the flatness of the mirror with a Foucault test setup used 
    for telescope mirrors.
    
    gl
    
    Geoffrey Kolbe wrote:
    > in [7190] George wrote:
    >   
    >> Clive, in [7074], and Geoffrey Kolbe, in [7187], have tackled Bill's
    >> question, about how flat a sextant mirror needs to be. I suggest there are
    >> errors in both those analyses, and ask them to reassess.
    >>
    >> First, neither of those contributors appears to have allowed for the fact
    >> that tilt of a mirror surface produces a doubled tilt in the image.
    >>     
    >
    > Thankyou for that George, I did indeed forget about that.
    >
    >
    >   
    >> Second, as there are two mirrors involved, the defects of which will
    >> probably add together in some way, that reduces further the defect that's
    >> may be tolerable  in each one.
    >>     
    >
    > That I did take into account, by halving the allowable single mirror 
    > defect. That was probably too crude - a reduction of 1 over root 2, rather 
    > than a straight 1/2  is probably more appropriate, given the random way in 
    > which the mirror defects may add or cancel each other out. This hauls back 
    > a good part of the error you pointed out above.
    >
    >
    >   
    >> Geoffrey tells us that he first considered an unmagnified naked-eye image
    >> with just a peep tube, and attempts to assess the fuzziness caused by
    >> surface defect in a 25mm mirror. But he has forgotten that the eye pupil
    >> will only have an aperture of 5mm or so, so only a pencil of light of that
    >> width is relevant; the rest won't enter the eye and contribute to fuzz. The
    >> mirror size is irrelevant, in that case.
    >>
    >> However, even for the naked eye, there's a more important factor than
    >> fuzziness, which is this- You may have checked for index error with the
    >> image centralised  in the mirrors. Then, when you make an actual
    >> observation, you might well be looking at the star, reflected in another
    >> part of the mirror, away from the centre. You don't want that image to shift
    >> about, relative to the direct view, affected by lack of flatness of either
    >> mirror. That sets a limit on the curvature that can be tolerated in the
    >> mirror.
    >> Once a telescope has been added, then a larger area of the mirrors, perhaps
    >> 30mm diameter or more, becomes relevant, the size of the objective. Any
    >> curvature in the mirrors over that aperture will contribute to fuzziness of
    >> the image in the 'scope. The sensitivity to image displacement remains as it
    >> was with the naked eye, with one big proviso, which is that when using
    >> magnification, you will be expecting a correspondingly higher precision, so
    >> will be prepared to tolerate correspondingly less displacement error.
    >>     
    >
    > Yes George. You are right to make the distinction between the "low 
    > magnification" case, where only a part - which may randomly vary - of the 
    > mirrors are being used at any one time, and the "high magnification" case 
    > where all the mirror surface is contributing to the image. Talking about 
    > "blurring" of the image in the naked eye case, as I did, was not correct. 
    > Though in defence I should say that I was laying out my stall to consider 
    > the case when a telescope is put in the optic train - and the analysis of 
    > how much mirror defect is tolerable before precision is affected is the 
    > same in both cases.
    >
    > I should also add that due to the way in which mirror flatness was 
    > specified, the mirror size in the "low magnification" case is not entirely 
    > irrelevant. Where a "saucer shaped" mirror is being considered, a 25mm 
    > mirror that is flat to one wavelength is not as flat as a 50mm diameter 
    > mirror that is flat to one wavelength. Cutting two 25mm mirrors from a 50mm 
    > blank of one wavelength flatness will give two 25mm mirrors that will be 
    > flat to something nearer to a half wavelength. This is the main reason why 
    > large flat mirrors are considerably more expensive than small mirrors of 
    > the same specified flatness. Given their small size, the cost of having 
    > your sextant mirrors polished to tenth wave flatness and front coated, for 
    > example, may be considered a reasonable. Here in the UK, it is possible to 
    > have this done for about �150.
    >
    > In conclusion, I think my numbers remain pretty good. Where the usual 3x 
    > Galilean telescope is being used in a standard sextant where the mirror 
    > size is of the order of one inch (25mm), then the mirrors need to be flat 
    > to about one wavelength of light. Where a higher power scope is being used, 
    > the allowable flatness error decreases proportionately.
    >
    >
    > Geoffrey Kolbe
    >
    >
    >
    >
    >
    > >
    >
    >   
    
    
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