2.5-m telescope mirror actuator resolution

Sloan Digital Sky Survey Telescope Technical Note 19970912

Contents

• Introduction
• Primary mirror actuators
• Secondary mirror actuators
• Acknowledgement

Introduction

The Sloan Digital Sky Survey (SDSS) 2.5-m telescope incorporates a two-mirror optical design that achieves zero distortion in the imaging mode using two transmitting correcting elements. The two mirrors are positioned in five axes using electromechanical actuators. The resolution of these actuators must be chosen so that the performance goals for the telescope are met.

Telescope operation can be divided into three categories.

• Engineering, e.g., collimation
• Spectroscopy
• Imaging

The resolution of the primary mirror actuators must be fine enough to collimate the telescope optics. The primary mirror will not be moved during imaging. The actuator resolution does not affect astrometry. In spectrographic mode, the primary, along with the secondary, are moved axially to adjust the scale of the focal surface to match that of predrilled plug-plates. This occurs between spectrographic observations. Consequently, primary mirror actuator resolution does not affect spectroscopy.

During imaging, the secondary must be moved both axially and laterally to compensate for gravity-induced sag. Consequently, the resolution of the secondary mirror actuators must be fine enough to not contribute significant astrometric error. Astrometric considerations lead to stringent requirements on actuator linearity as well as resolution. However, we do not discuss this further in this note.

Primary mirror actuators

Primary axial actuator analysis

It is not anticipated that the primary mirror will be moved during imaging. Consequently, the specifications of the primary mirror actuators are not influenced by the goals for astrometric precision. It is anticipated that the primary and secondary mirrors will be moved axially to adjust the f/5 image scale during spectrographic observations to compensate for the thermal expansion of the aluminum plug-plates. Also, the primary mirror will be translated in 5 axes during collimation. This is expected to occur regularly, but not every night. The resolution of the axial actuators for the primary mirror is set by the criterion that the tilt quantization error not produce significant decollimation.

The quantization error of the primary actuators can be compensated by the finer tilt and lateral motions of the secondary actuators. However, this does not appear to be necessary or desirable and we do not consider this further.

Table 1: Parameters for the primary mirror axial actuators.

Parameter	Value	Units	Notes
f/5 scale	60.258	microns/arc sec	1
Axial actuator radius	0.865	m	2
Lead screw pitch	635	microns	3
Number of motor steps	200	steps/rev	4
Linear step	3.175	microns	5
Tilt step	2.447	microradians	6
Maximum tilt quantization error	1.06	microradians	7
Degradation rate	4.906	mas/microradian	8
Maximum image degradation	5.199	mas	9
Degradation allowance for primary tilt	30	mas	10

Notes

1. Nominal scale for the imaging mode.
2. Distance of the axial actuators from the center of the primary mirror.
3. Assumes a 40 thread per inch screw with no reduction.
4. Standard stepper motor.
5.  
6. This is the mirror tilt due to one actuator step. The mirror is assumed to pivot about the other two actuator attachment points. The distance from one actuator to the line connecting the other two actuators is 1.5*(Axial actuator radius). This number gives the shift in telescope pointing due to one step. If this is less than about 5 microradians, the mirror can be adjusted during spectrographic observations without interrupting guiding. However, this is not a requirement.
7. The worst tilt quantization error occurs when the tilt is not in the direction of any of the actuators. It is root(3)/2 larger than the maximum 1-D error, which is half the tilt step.
8. Tolerance analysis by Steve Kent gives 86 microns for the incremental FWHM image degradation in the worst direction for 1 arc min of tilt.
9.  
10. Total for Primary/Secondary alignment/tilt from the February 1996 image degradation budget by Ed Mannery is 100 mas RMS diameter. Contributors to this collimation error include primary and secondary tilt and translation and corrector translation relative to the instrument rotator axis. We allocate 30 mas to this item that affects the resolution of the primary tilt adjustment.

Primary transverse actuator analysis

The primary mirror will be translated laterally during collimation. This is expected to occur regularly, but not every night. The resolution of the lateral actuators for the primary mirror is set by the criterion that the lateral motion quantization error not produce significant decollimation. When the telescope is pointed at the horizon, one actuator acts vertically and the other two act sideways on the vertical diameter.

Table 2: Parameters for the primary mirror transverse actuators.

Parameter	Value	Units	Notes
f/5 scale	60.258	microns/arc sec	1
Lead screw pitch	635	microns	2
Number of motor steps	200	steps/rev	3
Linear step	3.175	microns	4
Maximum lateral quantization error	2.245	microns	5
Degradation rate	1.211	mas/micron	6
RMS image degradation	1.468	mas	7
Degradation allowance for primary tilt	30	mas	8

Notes

1. Nominal scale for the imaging mode.
2. Assumes a 40 thread per inch screw with no reduction.
3. Standard stepper motor.
4. One step causes an 0.58 microradian offset in telescope pointing, i.e., (Linear Step)/(Final focal length). If this is less than about 5 microradians, the mirror can be adjusted during spectrographic observations without interrupting guiding. However, this is not a requirement.
5. Divide by 2 to get the maximum 1D quantization error. Multiply by root(2) to get quantization error along the diagonal.
6. Tolerance analysis by Steve Kent gives 61 microns for the incremental FWHM image degradation in the worst direction for 1 mm of lateral displacement.
7.  
8. Total for Primary/Secondary alignment/tilt from the February 1996 image degradation budget by Ed Mannery is 100 mas RMS diameter. Contributors to this collimation error include primary and secondary tilt and translation and corrector translation relative to the instrument rotator axis. We allocate 30 mas to this item that affects the resolution of the primary translation adjustment.

Secondary actuators

Secondary axial actuator analysis

It is expected that the secondary mirror will be moved during imaging to maintain collimation. Generally, this will be needed to compensate for telescope flexure. Occasionally, it may be necessary to compensate for temperature changes, but the small negative thermal expansion coefficient of the carbon fiber reinforced plastic truss elements should make this infrequent. Because these motions occur during imaging, the specifications of the secondary mirror actuators affect the astrometric precision of the telescope and this sets their resolution.

The axial sag of the secondary due to gravity is expected to be less than 200 microns for a 15° zenith angle change at a zenith angle of 45°. The maximum zenith angle rate during tracking is 15° per hour. The corresponding secondary mirror axial rate is less than 83 steps/minute.

Table 3: Parameters for the secondary mirror axial actuators.

Parameter	Value	Units	Notes
Final focal length	12.5	m	1
Secondary - focal surface distance	4.396	m	2
Axial actuator radius	0.32	m	3
Lead screw pitch	635	microns	4
Speed reducer ratio	80		5
Number of motor steps	200	steps/rev	6
Linear step	0.04	microns	7
1-D tilt step	0.083	microrad	8
RMS 2-D step error	0.031	microrad	9
Image motion sensitivity	144.902	mas/microrad	10
RMS image motion	4.426	mas	11
Tracking error allowance for secondary tilt	20	mas	12

Notes

1.  
2. Distance of secondary mirror vertex to the f/5 focal surface.
3. Mounting radius of the axial actuators on the secondary mirror.
4. 40 threads per inch.
5. Reduction ratio of the harmonic speed reducer.
6. Standard stepper motor.
7. Linear step size for each actuator (Lead screw pitch)/((Speed reducer ratio)*(Number of motor steps)).
8. This is the mirror tilt due to one actuator step. The mirror is assumed to pivot about the other two actuator attachment points. The distance from one actuator to the line connecting the other two actuators is 1.5*(Axial actuator radius).
9. Multiply by 0.37 to convert p-v to RMS. This is from a Monte Carlo calculation where points were placed randomly on a rectangle containing 3 equally spaced reference points (representing the discrete tilts of the secondary mirror. The minimum distance from the reference points was calculated.
10. This is 206*2*(Secondary mirror to focal surface distance)/(Final focal length). The factor 2 is necessary because the deviation of the reflected ray is twice the tilt of the optic. Multiply by (Secondary mirror to focal surface distance) to get the physical displacement on the focal surface. Divide by the final focal length to convert to radians on the sky. Multiply by 206 to convert to milliarcseconds.
11.  
12. While it is desirable that this item be about 6 mas, this number comes from measurements of the 3.5-m secondary mirror motions and represents the performance that has been achieved. See item 24 of Table 1 of 2.5-m Telescope Tracking Error Budget (SDSS Technical Note 19970523). Changes have been made to that design to reduce the effect of friction between the nut and lead screw.

Secondary transverse actuator analysis

The resolution of the lateral actuators for the secondary mirror is set by the criterion that the lateral motion quantization error not produce significant degradation of the astrometric accuracy. The lateral actuators act at 45° angles to the vertical plane containing the telescope optical axis. Because of symmetry, the secondary will sag in this plane. Consequently, both lateral actuators must act together to compensate for this motion.

About 520 microns of lateral motion will be necessary from 0° to 60° zenith angle. About 1.8 steps/minute are needed if the telescope tracks at the sidereal rate in zenith angle. For a zenith angle change of 10°, the transverse motion of the secondary is 105 microns. Uncompensated, the image degradation is 100 mas if the telescope is initially collimated.

Table 4: Parameters for the secondary mirror transverse actuators.

Parameter	Value	Units	Notes
Final focal length	12.5	m	1
Secondary focal length	3.597	m	2
Secondary - focal surface distance	4.396	m	3
Number of actuators	2		4
Lead screw pitch	635	microns	5
Lever ratio	3.1		6
Number of motor steps	200	steps/rev	7
Linear step	1.024	microns	8
RMS step quantization error	0.296	microns	9
Total transverse quantization error	0.418	microns	10
Image motion sensitivity	20.142	mas/micron	11
RMS image motion	8.423	mas	12
Tracking error allowance for secondary displacement	5	mas	13

Notes

1.  
2. Distance of secondary mirror vertex to the f/5 focal surface.
3. Mounting radius of the axial actuators on the secondary mirror.
4.  
5. 40 threads per inch.
6. Reduction ratio of the harmonic speed reducer.
7. Standard stepper motor.
8. The step size is the peak-valley quantization error.
9. Multiply by root(3)/6 or 0.289 to convert peak-valley to RMS.
10. Multiply tilt error of one actuator by root(2) to account for the two actuators. The angle between the actuators is 90°. They act along diagonals of the secondary frame so both must be adjusted to compensate for the flexure of the optics support structure.
11. This is 206*(Secondary mirror - focal surface distance)/(Secondary mirror - focal length)*(Final focal length). Multiply by the Secondary mirror - focal surface distance over the Secondary mirror - focal length to get the physical displacement on the focal surface. Divide by the final focal length to convert to radians on the sky. Multiply by 206 to convert to milliarcseconds. See Figure 2.7 and accompanying text (Daniel J. Schroeder, Astronomical Optics, Academic Press, San Diego, 1987, p. 16) for the geometry and relationships.
12. Item 10 times item 11.
13. See item 24 of Table 1 of 2.5-m Telescope Tracking Error Budget (SDSS Technical Note 19970523).

Acknowledgements

We wish to thank Ed Mannery for the image degradation error budget and Steve Kent for the optics alignment tolerance analysis.

Date created: 9/12/97
Last modified: 11/13/97
Copyright © 1997 Walter A. Siegmund
Walter A. Siegmund