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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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