ATM More Spheroidalage

[Anthony Stillman <atmer@flash.net>]

In his reply to "ATM Spherical mirrors", Guy Toutant quoted Texereau's "How
to Make a Telescope" who intern referenced A. Couder "Lunettes et
Telescopes"

Couder's rule of thumb f^3=34.9*D^4, as echoed by Texereau contains hidden
within it a tolerable wavefront degradation criteria.  In the preceding
paragraph Texereau implies that a telescope mirror, not blustered over by
an industrial worker needs an un-haphazardly figure of 1/10th wave or
better.  The title of the table printed on page 19 states that the focal
ratios listed will satisfy Rayleigh's Criterion for the associated
diameters.

Couder's rule is derived from the root of a three term expansion of the
difference between the saggitta of a sphere and a parabola.  Specifically
this yields:

min required focal ratio = (1/8) (mirror radius / degradation criteria)^(1/3)

where the mirror radius and allowed profile difference are in like units.
Remember, this is a mirror, 1/8 wave profile error yields 1/4 wave
wavefront error, the so called Rayleigh Criterion.

Reverse engineering Couder's rule of thumb we see that it amounts to one
wave wavefront error.

wavefront error = (1 / 558.4)^3 cm = 574 nm

Not exactly a quarter wave, but then given Texereau's propensities, not
exactly unexpected either.


Anthony

PS  Below is a table of minimum focal ratios required for spheroidal
mirrors.  Differing wavefront criteria (in waves) and diameters (in
millimeters) are given.  Lambda is 574 nm

       wavefront    1/10       1/8        1/4        1/2         1
   diameter

     80 mm          14.0       13.0       10.3        8.2       6.5
    100 mm          15.0       14.0       11.1        8.8       7.0
    150 mm          17.2       16.0       12.7       10.1       8.0
    200 mm          19.0       17.6       14.0       11.1       8.8
    250 mm          20.4       19.0       15.0       11.9       9.5
    300 mm          21.7       20.1       16.0       12.7      10.1
    400 mm          23.9       22.2       17.6       14.0      11.1
    750 mm          29.4       27.3       21.7       17.2      13.7



PPS  I know, a 30" sphere, silly silly, but I was curious.