Sloan Digital Sky Survey Telescope Technical Note 19960214
To avoid significant primary mirror induced image degradation (seeing), the front surface of the mirror must be maintained within a few tenths of a degree of the ambient air temperature. In addition, mirrors of borosilicate glass, a material with low but not negligible thermal expansion, must have certain particularly troublesome temperature distributions controlled to 0.1 °C to avoid significant image degradation due to thermal deformation.
Various temperature control systems, using air as the working fluid, have been designed to meet these criteria. The system used for Apache Point Observatory 3.5-m primary mirror is particularly simple. Ambient air from above the center of the primary mirror is used for ventilation. This ventilating air is drawn down through the center hole of the primary mirror and radially outward behind the mirror before being drawn up into the hexagonal interior voids of the mirror through holes in the back-plate. There, it is drawn upward to the back of the face-plate and is drawn radially inward through a small gap between the perimeter of a hexagonal air baffle and the back of the face-plate, across the back of the face-plate to the center of the air baffle and down an exhaust tube to the interior of the mirror platform. Flexible hose carry it from the mirror platform to the forks. From there, it flows out the base of the telescope and eastward through a duct to exhaust fans near the operations building where it is vented to the outside.
To directly measure the performance of mirror temperature control systems, one requires sensors with relative accuracies of 100 m°C peak-valley spaced at 0.03 m intervals, the characteristic thermal length in the face-plate. In practice, one can argue that less frequent spatial sampling is necessary because of the spatial uniformity of the thermal environment. An analysis of data obtained in 1994 support this view in that thermal features appear to be well resolved by sensors separated by 0.2 m or more. Still, the application requires hundreds of sensors.
Two approaches have been used. In one approach, silicon integrated circuit sensors, e.g., AD590, that produce a current proportional to the absolute temperature, are used ("Temperature measurement system for a 3.5-meter borosilicate mirror", C. L. Hull, W. A. Siegmund and D. Long, Proc. of S.P.I.E., 2199, 1994, p.858). These are low cost devices and 256 sensors can be multiplexed reliably using integrated circuit multiplexors at low cost. Unfortunately, to achieve the relative accuracy discussed above, they must be carefully calibrated. The stability of the sensors is such that periodic recalibration appears to be necessary. Since no in-mirror calibration method has existed, every few months the sensors would have to be removed from the mirror for recalibration and then reinstalled in the mirror. This is a very labor intensive activity.
The second approach uses thermocouples ("System for precise temperature sensing and thermal control of borosilicate honey comb mirrors during polishing and testing", M. Loyd-Hart, Proc. of S.P.I.E., 1236, 1990, p.844). The voltages produced by the thermocouples are inherently stable and proportional to the temperature difference between the thermocouple and the reference junction. The difficulty is in multiplexing the signals. Since the signal must be measured to the microvolt level to meet the resolution criterion, the switches must have low resistance and induced voltage. One solution is to build a custom multiplexor using mechanical reed switches. The output of the multiplexer is digitized by a sensitive commercial voltmeter. It is unlikely to be practical to implement hundreds of temperature sensors in this manner.
No matter how the data are acquired, analysis and visualization of the data is necessary for the system to be useful. The interpretation of hourly samples of the temperature distribution in various portions of the mirror and telescope is a burden on the operations staff of the observatory. This problem is alleviated if the thermal time constant distribution rather than the temperature distribution is considered. The former should be constant if the temperature control system is stable whereas the latter depends on the history of the ambient temperature. Additionally, as shown below, the thermal time constant distribution can be obtained in a straightforward manner that largely avoids the shortcomings of integrated circuit temperature sensors.
The temperature of an element of the mirror, in the absence of heat sources or sinks, conduction within the mirror, or radiative heat transfer, depends on heat transfer from the ventilating air. Because of thermal inertia, its temperature depends on the prior history of the ventilating air temperature. At discrete times , the temperature of the ventilating air is , the actual temperature of the mirror element is , and the estimated temperature of the mirror element is . If the heat transfer coefficient between the air and the element is constant, its response to temperature changes can be characterized by the thermal time constant, . This is the time required for the temperature difference between the element and the ventilating air to decrease by a factor of 1/e with the air temperature constant.
In this form, the equation is not very useful because it requires knowledge of the air temperature (and operation of the ventilating system) extending into the infinite past. Fortunately, it is straightforward to avoid this problem. Using summation symbols, this equation can be rewritten as follows.
Separating the two terms in the numerator and then rewriting the second term,
(1)
where the right-most factor is the estimate of the temperature of the element at time . Now, if k is selected so that the second term is small, i.e., , then the error introduced by using the actual temperature of the element, instead of the temperature estimate, is minimal.
(2)
Substituting eq. 2 into eq. 1,
.
This equation gives an estimate of the temperature of a mirror element at an arbitrary discrete time based on its temperature at an earlier discrete time and the temperature of the ventilating air in the meantime. The infinite series in this equation can be approximated by a definite integral if the temperatures are sampled at equal time intervals, i.e., for all m.
The optimum value of t can now be determined. The error at time between the measured and the estimated temperature of an element is . These errors are valid for the time interval , at least after the beginning of mirror ventilation, to , the end of mirror ventilation. Both the mean and variance of the errors are functions of . The for which the variance is a minimum is optimal in the least squares sense. This is not the linear least squares problem. However, computers are fast enough that the Brent 1-D minimization algorithm (Numerical Recipes, William H Press, Saul A. Teukolsky, William T. Vetterling and Brian P. Flannery, Cambridge, New York, 1994) is quite adequate.
This procedure not only gives the time constant for each sensor, but also the mean offset corresponding to the best fit time constant. This latter value can be interpreted as the offset of the sensor relative to the air temperature sensors.
Good results are only obtained for data that include significant changes in the temperature gradient in time. Otherwise, a temperature offset cannot be distinguished from a change in the time constant.
Data taken during the summer of 1994 were analyzed. Both the face-plate and back-plate temperatures were sampled at 39 locations. Other sensors sampled adjoining portions of the telescope structure.
A graph shows the quality of a typical fit. The points represent the data from four sensors from the night of July 24-25. The lines show the fits to these data. CH#52 is on the back-plate about halfway between the inner and outer edges of the mirror. CH#51 is at about the same location on the face-plate. CH#53 is between the back-plate and the mirror platform and measures the temperature of the ventilating air just before it enters the mirror. CH#102 measures the temperature of the ambient air about 1 m in front of the mirror. It and two other similarly situated sensors are averaged and taken to be the temperature of the air, . Typically, the standard deviation of the fit error is 30 to 50 m°C.
Some of fit error is likely due to undersampling of the air temperature (900 seconds per sample). It is apparent that the sharpest features in the air temperature are unresolved. Undersampling helps to explain why the fits are worse for the temperature of the ventilating air behind the mirror. These have standard deviations about 80 m°C. The short time constants of the air behind the mirror, less than 0.5 hours in most cases, imply that only a couple of air temperature measurements contribute to the temperature estimates. Consequently, undersampling contributes more error than for the mirror where more averaging occurs. This effect is most extreme for the sensors with the shortest time constants. As one would expect, the corresponding fit errors have the largest standard deviations.
Overall, the goodness of fit is very good. This fact, and the lack of any apparent systematic patterns in the quality of the fits with location in the mirror, suggests that the model described at the beginning of this section is an excellent description of the thermal performance of the mirror.
Time constant distributions were calculated from measurements taken on several nights. Each night was analysed separately. Contour maps of the time constants for the face-plate (upper maps) and back-plate (lower) were produced from the data for July 18, 20, and 24 (left to right). Minimal smoothing has been done. Sensors were installed on the right half of the mirror viewed from the front with the telescope pointed at the horizon (the perspective for the contour maps). The time constants shown are for k=4 (1 hour). For k=8 (2 hours), the differences in the time constants were calculated. The standard deviation of these differences is 0.024 hours, i.e., the results are very insensitive to the value of k. This result lends additional credence to the model.
Additional sensors were mounted elsewhere on the telescope to help characterize the thermal environment of the mirror. The following observations can be made.
The approach described herein provides a high quality and easy to use picture of the time constant distribution in the primary mirror and other portions of the telescope. It is straightforward to implement and and is insensitive to the weaknesses of the integrated circuit temperature sensors while taking advantage of their strengths. It assumes that the heat transfer coefficient between the part measured and the air is constant in time, i.e., the thermal time constant is meaningful. This is not true in the case of variable wind-forced ventilation, for example. However, if this is assumption is not true, the fit to the temperature profile with time will be poor. Radiative effects, especially to the cold sky. are assumed negligible. Constant radiative heat loss cannot be distinguished from the sensor temperature offset error (assumed constant) so the temperature offset will include this effect. The time constant will be calculated properly, however. For most telescopes, radiative effects are small since they are coated with low emissivity material. They can be distinguished by closing the telescope enclosure opening.
In the case of the Apache Point Observatory telescope, some improvement in the uniformity of the time constant pattern of the primary mirror would be desirable. While the time constants are much shorter than currently achieved in any meniscus mirror, time constants larger than 1 hour may contribute several tenths of arc seconds of image degradation under unfavorable conditions.The excellent performance along the +y axis is encouraging. However, the non-uniform angular distribution of time constants is not understood. Studies of the SDSS 2.5-m mirror may be helpful in this regard. The design of the 2.5-m mirror PSS incorporates additional exhaust vents beyond the outer radius of the mirror. These can be used to increase the flow of ventilating air to the outermost portion of the mirror.
The time constants of the PSS in the immediate vicinity of the primary are quite reasonable. It is not likely that heat transfer from the PSS to the primary mirror is a problem. The time constants of the tops of the forks are very long and their influence extends to neighboring portions of the telescope. They should be insulated or (preferably) actively controlled.
The relatively long time constants of the secondary truss elements likely accounts for the reported poor performance of the automatic secondary refocusing system during periods with large temperature gradients. Directly measuring the temperature of the truss or applying a filter with a 3 hour time constant to the ambient temperature data before sending it to the focus algorithm would likely improve performance.
It is a pleasure to thank Craig Lewis, Dan Long, Mark Klaene, Jim Fowler and Bruce Gillespie of Apache Point Observatory, and Siri Limmongkol and Patrick Waddell of the University of Washington for their assistance.
Date created: 5/6/96 Last modified: 10/23/97 Copyright © 1996, 1997, Walter Siegmund
