To: atm{at}shore.net
From: MLThiebaux 
Reply-To: MLThiebaux 


On flats that needn't be so flat:

I'll go through a numerical example based on geometric optics showing the
effect of a slightly spherical diagonal in a Newtonian.  Inches throughout
(although Canadian, I don't mind). Suppose our telescope has

        D = primary diameter = 16
        F = focal length = 80
        d = diagonal minor diameter = 4
        L = primary axis - image plane distance = 9

Suppose the diagonal is concave spherical with a sagitta across the minor
diameter = e = 1 wavelength.  So

        e = 1/50000

The curvature on the flat corresponds to a focal length d^2/16e = 50000. 
At 45 degrees the two oblique principal focal lengths are 50000*cos45 and
50000/cos45 or 70711 and 35355.

Think of the image of a star.  After a little fiddling with the lens
formula we find that if one of the oblique focal lengths is called  f, then
the point image formed in its principal plane is closer to the primary than
it would be if the diagonal were truly flat by the amount  L^2/f.  Hence
there is a separation of the two images by the amount  s =
81/35355-81/70711 = 0.00115. This is the longitudinal astigmatic focusing
error.

Now we'll figure out how much astigmatism in the primary mirror would
produce this same longitudinal focusing error.  The two principal focal
lengths of the mirror would differ by  s, and so the edge of the mirror
surface has a peak error  +/- s*D^2/32F^2 = 1.4375E-6 relative to a truly
parabolic mirror.  The peak-to-valley error would be twice this, and the
surface rms error = peak error divided by sqrt(6). Choose your favorite.

Summary:
        Diagonal error =  1 wavelength
        Equivalent astigmatic primary surface error in wavelengths
                Peak:  1/14
                PV:    1/7
                RMS:   1/34

Martial Thiebaux
Rawdon Hills, Nova Scotia

--- BBBS/NT v4.00 MP