Sloan Digital Sky Survey Telescope Technical Note 19980805
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 optician's calculations and measurements 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.
Jim Gunn circulated a CAD drawing (MIRCELL.DXF) on 7/24/98 showing the 2.5-m primary mirror and the locations of actuators (Figure 1) proposed to control the astigmatism of the primary mirror surface. Two rings of 12 actuators at two different radii are proposed. A similar pattern (green crosses in the Figure) was analyzed because it was simpler to model; the patterns are so similar that the results should be almost the same.
A second proposal is to modulate the pressure of a subset of the existing primary axial support pneumatic pistons. These are arranged in two rings. The outer ring contains 30 pistons and the inner ring 18. The ring radii are approximately the same as those of green crosses. The model should apply to this proposal also.
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 crosses of the Figure. Three cases were analyzed. Linear superposition should be valid for these results since the deflections are small and stress stiffening is not present.
Figure 1: CAD drawing showing the 2.5-m primary mirror and the locations of actuators suggested by Gunn (blue circles) These actuators apply forces to correct astigmatism. Two rings of 12 actuators at two different radii were suggested. Forces given by the expression 10*cos2ø N were applied at locations near those of the proposed pattern (green crosses).
Figure 2: Surface deflection in meters calculated for astigmatism forces applied to the outer ring of green crosses 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 2 with astigmatism removed.
Figure 3: Residual surface deflection in meters for Figure 2 with astigmatism removed.
Figure 4: Surface deflection in meters calculated for astigmatism forces applied to the inner ring of green crosses shown in Figure 1.
Figure 5: Residual surface deflection in meters for Figure 4 with astigmatism removed.
The results of the calculations are shown in Figures 2 and 4 for the two astigmatism cases. 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 and 5 for the two cases. The parameters for the astigmatism fits are given in Tables 1 and 2. A coefficient (as defined in the Tables) of 143 nm/m2 corresponds to the Zernike coefficient found by the optician. Consequently, a force amplitude of 11.0 N and 19.6 N is required for the outer and inner ring of astigmatism actuators respectively.
Table 1: Surface error fit parameters for astigmatism forces applied to the outer ring of Figure 1. prim94 uz = a1 + a2*r2*cos(2ø) + a3*r2*sin(2ø) i ai sigma 1 129.9 nm 0.161 nm 2 110.9 nm/m2 0.126 nm/m2 3 0 nm/m2 0.213 nm/m2 Initial error (RMS) 156 nm Residual error (RMS) 5.2 nm Table 2: Surface error fit parameters for astigmatism forces applied to the inner ring of Figure 1. prim95 uz = a1 + a2*r2*cos(2ø) + a3*r2*sin(2ø) i ai sigma 1 73 nm 0.161 nm 2 62.8 nm/m2 0.126 nm/m2 3 -0.2 nm/m2 0.213 nm/m2 Initial error (RMS) 88 nm Residual error (RMS) 10.2 nm
prim94
uz = a1 + a2*r2*cos(2ø) + a3*r2*sin(2ø)
i
ai
sigma
1
129.9 nm
0.161 nm
2
110.9 nm/m2
0.126 nm/m2
3
0 nm/m2
0.213 nm/m2
Initial error (RMS)
156 nm
Residual error (RMS)
5.2 nm
Table 2: Surface error fit parameters for astigmatism forces applied to the inner ring of Figure 1.
prim95
73 nm
62.8 nm/m2
-0.2 nm/m2
88 nm
10.2 nm
The two support rings of Figure 1 are at approximately the same radii as the existing primary axial support pneumatic pistons. One implementation approach consists of replacing some of the pistons in the outer ring with modified pistons that would allow a force to be subtracted from the nominal support force. This will reduce the mean force in the outer ring. Consequently, an analysis of prim98, a case with -1 N forces applied to the inner ring and +1 N forces applied to the outer ring, was performed. Prim 98 can be linearly combined with prim94 or prim95 to understand the effect of subtractive forces on the mirror figure.
The deformation for prim98 is conical, i.e., linear with radius (Figure 6). However, it is fit reasonably well by a quadratic (Figure 7). This is equivalent to a change in the radius of curvature of the mirror but does not introduce significant image aberrations for small changes. The departure from the quadratic fit is troublesome, however. Also shown is a r4 fit. The fit is even worse. However, Figure 8 shows that the deformation can be viewed as a combination of change of the radius of curvature and spherical aberration.
Figure 6: Surface deflection in meters calculated for -1 N applied to the inner ring and +1 N applied to the outer ring (see green crosses shown in Figure 1). Figure 7: Quadratic and r4 fit to the data of Figure 6. The deformation is conical, but the quadratic fit removes much of the deformation. The r4 fit is not good, but shows the extent to which spherical aberration can be imposed by means of these forces. Figure 8: Quadratic and r4 fit to the data of Figure 6. The combination of r2 and r4 fits the deformation well.
Figure 7: Quadratic and r4 fit to the data of Figure 6. The deformation is conical, but the quadratic fit removes much of the deformation. The r4 fit is not good, but shows the extent to which spherical aberration can be imposed by means of these forces.
Figure 8: Quadratic and r4 fit to the data of Figure 6. The combination of r2 and r4 fits the deformation well.
Two patterns of actuators that apply astigmatism correcting forces to the 2.5-m primary mirror were examined. The residual error for the outer ring is less than that of the inner ring, so it is to be preferred. The residual error for the outer ring is so small that the addition of the inner ring of actuators appears to be an undesirable complication. The force amplitude that is required is approximately 11.0 N.
The addition of the inner ring would allow comatic surface error to be reduced. However, comatic surface error is not readily distinguished from aberrations due to decollimation of the optics. Also, the mirror support system is not expected to be a source of coma. Consequently, it is not likely that the inner ring of actuators will be useful for this purpose either.
Modulating the force applied by every other or every third existing axial support piston in the outer ring should be feasible. This would require modifying 15 or 10 pistons respectively. Further analysis will indicate if the cos2ø force modulation should be changed because all actuators are not at the same radius.
It has been proposed that modulating the force applied to the mirror in the outer ring be accomplished by subtracting astigmatism correcting forces. This has the effect of reducing the mean force on the outer ring by 11 N. Scaling the results from prim98, the maximum departure from a quadratic fit is about 11 nm. The surface error is 6.8 nm RMS. These errors are small enough to be acceptable.
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: 8/5/98 Last modified: 2/5/99 Copyright © 1998, 1999 Walter A. Siegmund Walter A. Siegmund
