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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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