Mirror edge supports
A case study of the sensitivity to forces out of the mirror plane
by Nils Olof Carlin, written june 10, -03
(one day later, my idea of a Newton is revised ;-)
(next day again, with the help of Jim Burrows, how to find the center of gravity)
(Feb 26, -04: I should have been more explicit about deformations - they refer to the surface, and the errors on the reflected wavefront are twice as large!)
Thin mirrors for Dobsonian type telescopes are traditionally supported by a floating cell arrangement supporting the back of the mirror, and a sling covering the lower 180 degrees of the mirror's edge, or else 2 support points on the lower edge, most often +-45 deg from vertical. (A third possibility is a whiffle-tree arrangement where the force is applied equally at 4 points on the lower half of the edge.)
PLOP is a program by David Lewis, building on work by Toshimi Taki and others, employing a finite element model to calculate the deformation by forces perpendicular to the plane of the mirror for a certain cell configuration, as well as optimize the parameters of such a cell. The deformations caused by the edge support and its forces in the plane of the mirror are not calculated in PLOP.
However, even if edge supports take up the largest part of the force in the plane of the mirror, a small part of the force can exist perpendicular to this plane. For instance, if a sling leaves the mirror edge on its way to the support bolt at a small angle, the relative force is proportional to the tangent of this angle. Or in the case of a point support, a certain static friction coefficient between support and mirror edge (often only roughly ground!) can transfer a part of the force in the perpendicular direction, given inevitable sag in the mirror support.
If this small part can be approximated with a point force, it could be added in PLOP as 2 extra points along the edge of the mirror. The forces to these 2 points can then be varied, and PLOP will show the total RMS deviation of the surface (and as is often desired, any induced focus shift and mirror tilt can be cancelled).
For this study, I chose a 22" (559 mm) f/4.5 mirror, with an edge thickness of 1.5"(38.1 mm), mounted in a 18 point cell (20% dia. central obstruction, and default Pyrex). This is a fairly thin mirror, and about as large as can be well supported (with a generous safety margin!) by a 18 point cell.
First I did a test run, optimizing the 18 pt cell (with equal loads on all points):
Next, I locked the cell parameters thus found, and added 2 extra rings of support at radius 1 (and 0.99, to avoid some difficulty in PLOP), with relative force 0.1 (of the force on each of the other 18 points), varying the angle from vertical. (at 0, there is one point, with relative force = 0.2)
|Angle deg||RMS error nm||Main aberration (Zernike coefficient)|
|no extra points||1.96|
|+-90||19.2||"common" astigmatism (Z6)|
|+-45||6.8||coma (Z7), trefoil astigmatism (Z10)|
|0 (rel force=0.2)||22.1||"common" astigmatism (Z6)|
This revealed not unsurprisingly that "lift" at the edges by a sling can cause astigmatism - even if I believe this problem has not been given its deserved attention. Somewhat more surprising is the fact that the +-45 deg point support was essentially insensitive to astigmatism, the main error here is coma (causing some small shift of collimation error - if the coma accounts for 5 nm error, the shift is approx. 0.5 mm).
From left to right, PLOP color plots of: +-90, +-45, 0
What forces are involved? The mirror weighs about 15 kg. At an altitude of 45 deg, his means a total of 104 N in each plane - or 5.8 N by each of the 18 points, and + 0.58 N by each of the extra points at rel force=0.1. This is what it takes to lift 60 g! Also, at 45 degrees, the deformations are approx the ones listed *0.707 (=sqrt(2)).
A total of 20 nm deformation is enough to lower the Strehl ratio by 0.20, and transform a perfect mirror to a barely acceptable one - leaving no margin at all. At 45 deg, we have about 14 nm, and the Strehl is lowered by 0.1 instead - less, but still potentially serious.
In the case of a sling, how large an error angle is this? Arctan(0.58/52)=0.6 degrees!
In the case of the 2 point support, what is the maximum friction? Here, we can accept 3 times the force of the previous example for 14 nm error: The force normal to the edge of the mirror is sqrt(2)*52 N or 74N, and the maximum friction coefficient is
(3*0.58/74)=0.024 - this corresponds to a force error angle of 1.3 degrees!
For this fairly large and thin mirror, a well designed and built 18 point cell will give quite negligible deformation. However, even very small forces that are not in the plane of the mirror will have disastrous consequences for the performance - a fact that I feel have not been given the deserved consideration in amateur literature, despite frequent reports of visible astigmatism. So, while in theory, the common 180 deg sling is very close to ideal, these results show that it is extremely sensitive to adjustment, and an error angle that is far from obvious can have disastrous effects on performance.
While the +-45 deg 2-point support seems a bit less sensitive, it is not much less so, and the friction coefficient between common materials will be far too large to ensure adequate performance.
What could be used safely? Given that the mirror edge is accurately perpendicular to the plane of the mirror, ball-bearings at +-45 deg from vertical, touching the mirror at the level of the center of gravity, would seem possible.
The piano-wire supports as described by Frédéric Géa might be an attractive solution, at least if it is possible to choose a size of wire that is stable enough not to fold, yet pliant enough not to transfer significant undesirable forces. I have no idea how to do the calculations needed. Note however that the alignment of such a support is almost as critical as that of a sling!
Cementing the mirror to the support points using flexible silicone RTV has been used - it might work, but differential thermal expansion between glass and the material used in the supports can introduce forces that are difficult to estimate.
Clearly, whatever method you use, you should do your best to make sure that you avoid out-of-plane forces.
The mirror's center of gravity
To cause least deformation, the forces of the lateral support should be applied at the plane (parallel to the mirror) of the center of gravity (COG). To calculate its position, you need to know E, the thickness of the glass by the edge, and C, the thickness at the center. C is not so easy to measure, but it can be easily calculated: S, the sagitta (depth of curve) can be calculated from D, the mirror diameter and F, the focal ratio, as S=D/(16*F). This known, C=E-S. Then the distance from the back side to the COG is given by a very cute formula that Jim Burrows supplied at my request:
If you do a division (by a common factor E-C that is zero for a flat mirror) you get:
Take the example above (all distances in inches): E=1.5; D=22; F=5; this gives S=0.275 and C=1.225. The calculator then does the rest: the distance from backside to COG is 0.684 ( with more decimals than practical!).