Analysis of temperature measurements of telescope mirrors
Sloan Digital Sky Survey Telescope Technical Note
19960214
Walter
Siegmund and Charles
Hull
University of Washington
and Bruce Barnes
Central Washington State University, Ellensberg, WA
98926
Contents
Introduction
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.
Analysis
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.
![](eq01.gif)
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.
![](eq02.gif)
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.
![](eq06.gif)
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.
![](eq08.gif)
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.
Results
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.
![](front_sensors.GIF)
![](back_sensors.GIF)
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.
![](fig03.GIF)
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.
- The time constant patterns for the face-plate and back-plate
are quite repeatable from day to day. This provides further
support for the model.
- The time constant distribution can be obtained in a single day
as long as the temperature gradient with time is not constant.
This is short enough that it should be useful for evaluating
potential improvements to the mirror ventilation system.
- The time constant patterns are similar on the face-plate and
back-plate. Since it is differences in the temperature pattern of
the face-plate as compared to the back-plate that cause the
largest figure errors, the effect on mirror figure should be small
in the current system.
- Time constants increase with radius and as the distance from
the source of the ventilating air increases. A notable exception
is the +y axis (up in the Figure). Four sensors along this radius
on both the face-plate and back-plate indicate little dependence
of time constant on radius. It should be possible to reduce the
radial gradient where it is large.
- The pattern has a six-fold angular symmetry. This is not
understood but is presumably associated with the six-fold angular
symmetry of the primary mirror or its axial supports.
![](fig04.GIF)
Additional sensors were mounted elsewhere on the telescope to help
characterize the thermal environment of the mirror. The following
observations can be made.
- Mirror platform. The mirror platform is that portion of the
primary support structure (PSS) that provides axial support for
the mirror. The time constants of the upper plate of the platform
are 0.7 to 1.7 hours. The time constants increase with radius. The
mirror platform is a hollow steel weldment that acts as a
ventilating air exhaust plenum. This air has a time constant of
about 0.9 hours.
- Mirror cover brackets. These are located just beyond and above
the edge of the upper surface of the primary mirror. The time
constant for the bracket closest to the fork is 1 to 2 hours. Away
from the fork, the time constant is less than one hour.
- Earthquake bumpers. Four equally spaced bumpers are just
beyond the edge of the back-plate of the primary mirror. Located
at 45°, 135°, 225°, and 315° (0° is to
the right facing the primary mirror front surface with the
telescope pointed at the horizon), they provide lateral seismic
constraint for the mirror. During a seismic event they act to keep
stresses in the mirror to a safe level. The time constant for the
bumpers is 1.3 hours.
- Fork top. The time constants are 10 to 20 hours on the massive
fork top, but decreases to about 5 hours near the bottom of the
altitude bearing.
- Fork instrument mounting surface. The time constants are 1.5
to 2 hours.
- Fork front and rear surfaces. These are the left and right
surfaces as you face the fork. The time constants are 2 to 4
hours.
- PSS outer surface. The PSS is a large weldment that surrounds
and supports the primary mirror. Viewed from in front of the
primary mirror, it is octagonal. Sensors were located near the
upper and lower edges of three of the outer octagonal faces (the
directions correspond to the telescope pointed at the zenith). For
the upper edges, the time constants are 0.5 to 1 hour except near
the altitude bearing where they are greater than 5 hours. For the
lower edges, the time constants are 1.5 to 2 hours. ·
- Secondary truss elements. The lower 2 m of two elements were
instrumented. The time constants are 3 to 3.5 hours.
Conclusions
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.
Acknowledgements
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
siegmund@astro.washington.edu