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Geoffrey Kolbe replied to an initial question from Kieran Kelly. I will add
my own comments, as annotations to a copy of Geoff's mailing, below.

Geoffrey Kolbe said-

>Kieran.
>
>Flatness of an optical surface is determined by measuring the maximum
>variation of the optical surface in question from a theoretical flat plane
>placed on the optical surface.
>
>The flat plane will sit on the high spots of the optical surface and the
>maximum distance from the deepest trough on the optical surface to the
>theoretical flat plane determines the flatness of the optical surface.
>
>Classically, the optical surface in question will be compared to a known
>optical flat in an interferometer using monochromatic light of a known
>wavelength. Variations in the flatness of the optical surface in question
>show up as interference fringes, which act like the contours on a map. By
>counting fringes, you can say how many wavelengths of light by which the
>optical surface is in error.
>
>You say your mirror is flat to seven wavelengths, but that will be over the
>whole mirror. It may be flat to half a wavelength over most of its surface
>and have one bad section. But in any case, how 'bad' this is depends on the
>diameter of the mirror, which you do not tell us.

I agree with everything Geoff has said so far, particularly our need to
know the dimensions of the mirror involved.

>Let us suppose your mirror is 14cm in diameter,...

I doubt if it will be so large as that, but we will see. It might need to
be that size if it was used, placed with its adjustments, down on the
ground, with the observer standing up with his sextant. But I think it's
more likely to be used closer to the sextant, either on a firm tripod to
raise it to a more convenient level, or else with the observer squatting on
the ground to get nearer to a low-level mirror. In either of these cases, a
3-inch (76mm) mirror, as Peter Ifland illustrates in figure 104 of "Taking
the Stars", should be quite adequate.

>... but due to the diameter of
>the object lens in the telescope on your sextant, you will only be using a
>section of the mirror around 2cm in diameter at any given time. The
>flatness of that section of mirror will be around one wavelength of light
>(around 5x10-7 metres) on average. Your limit on resolution (which equates
>directly to the angular error about which you were asking) of this section
>of mirror will be roughly equivalent to a perfect mirror about 1cm in
>diameter, which is about 1/10th of one minute of arc.

Here I disagree with Geoff's argument, to some extent (though I may not
disagree much with his conclusion). He treats it as a question of
resolution, but here it's really a question of angular discrepancy. What's
important is that the part of the mirror that is being looked through is
exactly parallel to (or better, exactly the same as) that part of the
mirror that was tested for horizontality by a sensitive spirit-level.

The simplest way in which the mirror surface may deviate from flatness is
if it's slightly dome-shaped or alternatively slightly saucer-shaped. Let's
assume one or the other for now: it doesn't matter much which, just that
it's slightly spherical rather than flat. For our mirror, assumed circular,
take its radius, r, to be 38mm, corresponding to the 3-inch diameter
instrument referred to above.

Assume that the spirit level used will span across most of the mirror, so
that effectively it's checking the mean slope at or near its centre, and
that slope has been adjusted to zero.

Then if the dome shape causes the surface to fall away by 7 wavelengths of
light toward its edge, that will correspond to a slope at the edge of 2 x 7
wavelengths in 38mm. Agreeing with Geoff a wavelength to be about .0005mm
(it depends a bit on the colour of the light) we get a slope at the edge of
.007/38 radians or 0.63 arc-minutes. That becomes the maximum error in
slope, compared with the zero-slope at the centre of the mirror. At the
opposite edge of the mirror would be 0.63 minutes in the opposite
direction. Of course, such a change of slope implies a double-that change
in the angle of the reflected light, but you may recall that sextant
readings using a reflected horizon are then halved again, which cancels out
that doubling.

An observer is unlikely to put the image of the observed body right at one
adge or the other of the mirror, but instead will keep it much nearer to
the centre. If the image is no more than half-way out to the edge, then the
maximum angular error due to mirror curvature will be reduced to 0.16
minutes (it goes as the square). So as long as the observer takes some care
about the positioning of the image in the mirror, I would then agree with
Geoff's conclusion, as follows-

>I would say that the flatness of your mirror (or lack of it) will not
>contribute significantly to any sextant altitude error.
>
>Standard optical flats can be purchased and are generally flat to 1/10th of
>a wavelength of light across their surface.
>
>Geoffrey Kolbe.

George adds-

However, there are other matters which could contribute to angular error.
It's important that the spirit levels are very sensitive and are reversed
in direction, for both North-South and East-West adjustments, to cancel out
any errors in those levels.

It's also important that the glass plate sits securely on three adjusting
points without any constraining force which could cause a bending tendency.

And it's important that the levels are small enough, and light enough, and
that the support tripod arrangement, right down to the ground, is rigid
enough. Otherwise, having levelled the plate, the act of removing the
weight of the level might cause it to spring back slightly, to a position
where it was no longer exactly level. To check for this, with the level
remaining in position, add an extra weight corresponding roughly to the
weight of the level, close to the position of that level; there should be
no discernable effect on adding or removing that weight.

There are also questions of resolution to consider, as Geoff has done, and
this limits the accuracy that can be obtained by a fuzzifying of the image,
without adding any definite error.

It's also possible for a more complex distortion of the reflecting surface
to occur, such as if it was corrugated rather like a tin roof. This could
cause severe local perturbations in angle, without adding up to much
non-flatness when measured in wavelengths over the whole surface.

George.

===========================


>At 10:51 AM 11/2/03 +1100, Kieran Kelly wrote:

>>Gentlemen,
>>
>>For many years I have used a Zeiss Artificial Horizon in the bush for
>>celestial navigation and position fixing. It has a standard three- legged,
>>adjustable mount and came supplied with a dark, machine-ground glass
>>reflector, for sun observations. Needing to do star observations I had made
>>a front-silvered mirror from a piece of float glass. This was obviously not
>>machined but as the glass was floated when molten, it should have been
>>reasonably accurate due  to the force of gravity on a liquid. To do star
>>sights I simply removed the dark glass plate from the frame and inserted the
>>front silvered mirror. This arrangement has given great service over the
>>years and I am normally not out by more than 2 nautical miles from known
>>positions on land (arithmetic calculation errors being the exception).
>>
>>However, I have often wondered just how flat and accurate is the front
>>surfaced, glass mirror. I recently took it to an optical engineering
>>workshop here in Sydney who assessed it as being accurate to 7 wavelength's
>>of light. They advised that while this was not perfectly level anything
>>under 10 wavelengths of light is so accurate that it would not add any
>>meaningful error to that already produced by someone holding a hand held
>>sextant. It was their belief that even after index error and instrument
>>error was accounted for the, minute deficiencies in the sextant would still
>>produce greater error than the 7 wavelengths of light in the horizon. Also
>>it is clear that the levelling process for the horizon itself i.e. defects
>>in the bubble levels that I use and operator error eg in reading the bubbles
>>would produce error.
>>
>>Does anyone on the list have a comment. What is the error in a ground piece
>>of glass? The engineering workshop told me they could get it down to
>>practically zero wavelengths. What is a 7 wavelength's error? Can this be
>>translated into minutes of arc? Can the error in the horizon mirror be
>>eliminated through adjustments in the sight reduction process?
>>
>>Your advice would be appreciated.
>>
>>Kieran Kelly


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Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
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