Astigmatism correction for the 2.5-m primary mirror: II. edge
forces
Sloan Digital Sky Survey Telescope Technical Note
19980823
Walter
Siegmund
Contents
Introduction
The SDSS 2.5-m primary mirror is supported in a manner similar to
the Apache Point Observatory 3.5-m telescope primary.1
Separate systems support the axial and transverse components of the
2.5-m mirror weight vector. The axial support is provided by 48 air
pistons. The pistons have the same diameter thereby applying equal
forces with the same air pressure. The 48 pistons are divided into
three groups of 16 each. Each group supports a 120° sector of
the mirror. The portion of the weight of the mirror that is not
supported by the air pistons is sensed by load cells located near the
center of each 120° sector. Each load cell controls the air
pressure to the pistons in its sector so that the force on the load
cell is about 10 N. This force is too small to cause significant
distortion of the mirror.
The transverse support is provided by 18 air pistons, supported by
cantilevers from the mirror cell, that act on the mirror local center
of gravity surface when the mirror is lowered into place. Each piston
pushes on a steel force spreader that, in turn, pushes on four
nickel-iron alloy blocks bonded to the mirror ribs. All transverse
air pistons are the same diameter. A single load cell senses the
unsupported mirror weight and controls the air pressure to the
transverse support pistons. The remaining rotational and lateral
degrees of freedom are constrained by two links to the mirror cell.
With the mirror pointed at the horizon, these are located near the
top and bottom edge of the mirror. They are attached to the back of
the mirror and act horizontally.
During figuring and testing, slurry skirts, pressure seals and
tangent rods were attached to the mirror to control contamination,
allow pressurization of the inside of the mirror to prevent tool
pressure-induced dimpling, and to constrain the mirror on its
supports. It is believed that that one of the seals or skirts applied
forces at one or more points along either the inside or outside edge
of the mirror. Upon removal, the mirror relaxed and a small amount of
astigmatism appeared. Testing by the optician2, after the
slurry skirts, pressure seals and tangent rods were removed,
indicated that the Zernike coefficient of surface astigmatism,
R22 was approximately 230 nm
(R22(r/r0)2cos2ø), i.e., four
measurements of the mirror at three different angles relative to the
axial supports ranged from 190 to 280 nm.
The calculations and measurements by the optician suggest that
four forces with a magnitude of approximately 15 N equally spaced on
the circumference alternating positive and negative will correct the
measured astigmatism. I present calculations using the finite element
method that extend these results.
Calculations
In Astigmatism
correction for the 2.5-m primary mirror: I. axial supports
(SDSST Technical Note 19980805), rings of 12 actuators at two
different radii were analysed. These rings are at approximately the
same radii as the existing primary axial support pneumatic pistons.
The results of that analysis suggested that attaching force actuators
to the outer edge of the mirror might be effective. (Figure 1).
The finite element model is described in 2.5-m
primary mirror transverse support system (SDSST Technical
Note 19980713). One-quarter of the mirror was modeled. Symmetric
boundary conditions were applied on the x=0 and y=0 planes. Forces
given by the expression f = 10*cos2ø N were applied to the
nodes corresponding to the green circles of the Figure.
Figure 1: CAD drawing showing the
2.5-m primary mirror and the locations of actuators (green
circles) These actuators apply forces (10*cos2ø N) to
correct astigmatism.
Figure 2: Surface deflection in
meters calculated for astigmatism forces applied to the green
circles shown in Figure 1. (The 9 colors in the legend do not
correspond to the 9 colors in the plot. However, the maximum and
minimum are annotated on the plot and the color order is green,
yellow, red, orange, olive, turquoise, cyan, purple,
magenta).
Figure 3: Residual surface
deflection in meters for Figure 4 with astigmatism removed.
The results of the calculation is shown in Figures 2. The
expression uz = a1 +
a2*r2*cos(2ø) +
a3*r2*sin(2ø) was fit to the results and
the residual deflections at each node were calculated. These were
plotted in Figures 3. The parameters for the astigmatism fit is given
in Table 1. A coefficient (as defined in the Table) of 143
nm/m2 corresponds to the Zernike coefficient found by the
optician. Consequently, a force amplitude of 11.0 N is required.
Table 1: Surface error fit
parameters for astigmatism forces applied to the locations of
Figure 1.
|
pria11
|
uz = a1 +
a2*r2*cos(2ø) +
a3*r2*sin(2ø)
|
|
|
i
|
ai
|
sigma
|
|
1
|
149.2 nm
|
0.161 nm
|
|
2
|
129.6 nm/m2
|
0.126 nm/m2
|
|
3
|
0.3 nm/m2
|
0.213 nm/m2
|
|
Initial error (RMS)
|
180 nm
|
|
|
Residual error (RMS)
|
3.9 nm
|
|
Conclusions
The performance of actuators that apply astigmatism correcting
forces to the outer edge of the 2.5-m primary mirror was examined.
The force amplitude that is required is approximately 11.0 N to
correct the astigmatism reported by the optician. The results are
excellent. The resulting deformation departs from pure astigmatism by
only 4.3 nm RMS.
References
1. W.A. Siegmund, E.J. Mannery, J. Radochia and P.E.
Gillett, "Design of the Apache Point Observatory 3.5-m telescope II.
deformation analysis of the primary mirror", Proc. S.P.I.E. 628,
pp.377-389, 1986.
2. "Fabrication of the 2.5 m primary mirror for the Sloan
Digital Sky Survey Telescope, Final report", Optical Sciences Center,
University of Arizona, Nov. 4, 1997.
Date created: 8/25/98
Last modified: 8/25/98
Copyright © 1998, Walter A. Siegmund
Walter A. Siegmund
siegmund@astro.washington.edu