In another thread, Pierre Lemay is describing the Design and construction of his 20" ball scope. The 20" mirror is conical (f/3.9) and is centrally supported through a hole in the center. When vertical, on the test stand, the mirror is essentially parabolic. However, when pointed at the zenith, the mirror sags slightly toward the edge because the support is small ring under and just outside of the hole.
In post #22 of that thread, Pierre describes and presents the results of a finite element analysis (FEA) done on this mirror. The deformation at the very edge (which is only 1/2" thick) was calculated to be approximately 1/3 wave! Pierre would like to know how the shape changes due to this deformation (although he has been informed that it should still be very close to a parabola).
This thread attempts to answer the question quantitatively.
Please beware that I cannot guarantee that these results are correct. It would be great if someone could verify the computations or find errors!
We will assume that the mirror's central support on the test stand works very well and that there is little in the way of flexure because the mirror is supported at CoG on the stiffest plane and the outer parts of the conical shape are thin thus light while the inner parts are thick thus stiff.
The question is just how parabolic is the amount of deformation in the horizontal position (since a since parabola subtracted from another is still a parabola with slightly longer focus).
Since the mirror is parabolic (when the mirror is vertical) the height of the surface above the vertex (which is virtual because of the hole) at any radius r is
height = -r^2/8f + sagitta
If the FEA deformation were parabolic, then it must follow a quadratic formula:
deformation = a*r^2 + b*r + c
where a, b, c are some very small constants representing a parabola with an extremely long focal length.
In the case of parabolic deformation, the full formula for the surface height when the mirror is horizontal is:
deformed height = height - deformation = -r^2/8f + sagitta - (a*r^2 + b*r + c)
or rearranging terms:
= (-1/8*f - a) r^2 + (-b)*r + (sagitta - c )
which demonstrates that if the deformation is parabolic, the mirror surface will still be parabolic because this formula is a quadratic.
So, we want to know how much the FEA deformation deviates from an ideal parabolic deformation:
deformation = a*r^2 + b*r + c
We must chose values for a, b and c. This means picking an ideal parabolic deformation for a "best focus" parabola. One choice for "best focus" is a parabola that coincides with the FEA deformations at the outer radius of zones 1, 9 and say 6. (There is a unique vertical parabola through 3 non-collinear points.)
A better method could be used to find a best fit. One could use the least squares method to weigh the deviations according to the area of each zone. A brute-force, iterative program could determine such a best fit for a, b and c. However, just to get a rough idea of the deviation from a perfect parabola, this guess at "best focus" is probably acceptable. A fully optimized best-focus parabola might give somewhat better results (less deviation) than will be derived by matching a parabola exactly to just 3 of the zones.
To be continued in the next post...