[ATM] Analysis results for a 30" mirror on edge sling

["Robert Houdart" <robert.houdart@cruxis.be>]

Below a summary and discussion of some analysis results for a 30"
parabolic mirror resting on an edge sling.

This discussion has started on the bigdob list, but since the results
are quite relevant for large telescope makers, I've decided to post them
also on the ATM List.

Thanks to Mauro Da Lio for initiating the topic and sharing his results
in previous posts on the bigdob list. Mauro's and my results have been
obtained independently and are in very good agreement.

The post is relatively long, but I feel it's important to place the
numbers in the correct context.

Description of 30" Mirror Model
===============================
The analysis simulates a 30"x2" (762 x 50 mm) f/5 parabolic Pyrex
mirror. Total weight of the mirror is about 46 kg (100 lb).
The edge load is taken by a 180° cable sling over the lower half of the
mirror; this is slightly different from Mauro's model which uses a full
sling over the height of the mirror. The difference doesn't affect the
global deflections of the mirror surface.

How Surface Errors are processed
================================
Tilting and focusing errors are removed using a best-fitting parabolic.
The central zone of 100 mm diameter (4") is not taken into account as it
lies in the shadow of the secondary mirror.

The most relevant result is the RMS surface error. The conventional
"diffraction limited" or "1/4 wave" criterion corresponds to 1/27 wave
RMS or about 20 nm RMS surface error.

Case 1: Sling at center of gravity
==================================
The edge sling is put at its perfect position, the center of gravity
(COG) of the mirror (22.7 mm from the bottom).

Results:
- Surface error RMS:  3.0 nm or 1/183 wave
- Surface error P-V: 19.5 nm or 1/28 wave

This is in perfect agreement with Mauro's result ("about 20 nm PTV").
Most of the very small surface error is spherical aberration and some
trefoil. There is no trace of astigmatism in the result.

Clearly there is no real negative effect from a well-placed sling; the
"myth" that a sling generates astigmatism because it pinches the sides
of the mirror is evidently not valid.

Case 2: Sling 2.5 mm above center of gravity on 18 point cell
=============================================================
In this case the edge sling is put 2.5 mm above the COG of the mirror,
at height 25.2 mm.
The 18-point cell uses the optimal support positions computed by Plop: 6
inner points at radius 149 mm, 12 outer points at radius 306 mm. The
offset of the edge sling is compensated by opposite forces on the 12 top
points and 6 bottom points.

Results:
- Surface error RMS: 12.3 nm or 1/45 wave
- Surface error P-V: 59.9 nm or 1/9 wave

The error is mostly astigmatism and coma. Depending on the orientation
of the 18 point cell some deformations change sign but the overall RMS
error remains the same.

The RMS surface error is quite acceptable for a 30" mirror looking at
the horizon where the air turbulence will usually create much larger
wave errors. The 1/45 RMS surface error will reduce the Strehl ratio of
the optical system by 0.08. 

Note that this error does not only occur at the horizon. For a telescope
pointing at 45° altitude the RMS error would be 71% of the previous
value or about 1/60 wave RMS, and the Strehl ratio would drop by 0.04.

Case 3: Sling 5 mm below center of gravity on 18 point cell
=============================================================
Do the surface errors double when the offset doubles? Yes!

Results:
- Surface error RMS: 23.9 nm or 1/23 wave
- Surface error P-V:  134 nm or 1/4 wave 

As expected, the surface errors are twice as big.
At this point the optical performance will really begin to suffer from
the careless edge support.

Conclusions
===========
The results show the importance of correctly placing the edge support at
the center of gravity of a large mirror, a fact that has been known in
the ATM community for quite a long time.

The result also makes you wonder what happens when the mirror is glued
on the cell instead of using a sling. The edge load would then be
applied at the bottom, more than 22 mm below the COG. An interesting
case...

Clear skies,
Robert Houdart
www.cruxis.com/scope/scope1100.htm


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