Analysis of the Shack-Hartmann Data
Taken on August 14 and 15, 2000: Confirmation of the Tilt
Coefficients and an Investigation of the Thermal Response of the
Telescope Support Structure
Two nights of Shack-Hartmann data were acquired on
the nights of August 14 and 15, 2000. On both of these nights
complete altitude sweeps were done, taking Shack-Hartmann data at
intervals of 10 degrees in altitude. There were no tilt coefficients
installed in the telescope software at the time that these data were
taken. Both Russet and Cameron should be congratulated on the
acquisition of a very useful and important set of data. If only one
of these nights of data had been acquired most of our current
understanding of these data would have been impossible to
obtain.
We use these data to derive independent estimates of
the tilt coefficients that are used to eliminate coma due to sag of
the telescope truss as it moves in altitude. These results compare
favorably with those found on April 14 of this year.
The measurements of the telescope astigmatism
reported here indicate that the changes which we implemented during
the July 2000 shutdown have been quite successful in reducing the
amount of astigmatism on the 3.5m telescope. Throughout these tests
the maximum astigmatism seen was 0.25" and 0.18" is a more typical
value. There does appear to be a systematic shift of the magnitude
and orientation of this astigmatism as a function of telescope
altitude. At high altitudes the astigmatism is usually below 0.15". I
attribute this behavior to slight pinching of the primary mirror as
it rests on the transverse support system. We have effectively
eliminated astigmatism as the major limitation to image quality on
the 3.5m telescope.
I have also used these data to investigate the
behavior of defocus as a function of telescope altitude. The defocus
data from August 14 were seen to be consistent with that found on
April 14, but the data from August 15 were not. It is highly likely
that thermal changes in the telescope on August 15 were responsible
for the differences seen. The nature of these data have led to an
investigation of the motions of both the truss and the secondary cage
which one might expect based on purely theoretical grounds assuming
the changing gravity loading with telescope altitude and thermally
induced motions. This investigation has lead to the conclusion that
the thermal changes in the primary to secondary distance are driven
by thermal changes in the secondary support rods as well as by
thermal changes in the truss. This is one reason why the thermal
model that we are currently using to describe changes in the
telescope is inadequate. I show at the end of this report the
beginnings of an improved thermal model. However, putting together a
good thermal model of the telescope is likely to be a difficult task.
A partial solution to this problem, which we might want to consider,
is to replace the steel rods on the secondary with a low thermal
expansion material such as graphite.
A second conclusion from this investigation is that
thermal changes in the telescope support structure are unlikely to be
the cause of the changing secondary tilts of the telescope, which
continue to be observed over time. This conclusion depends on the
assumption that the secondary cage itself is solid and it not warping
in its shape. I estimate here that warpage on the order of 2 mm is
required to explain the types of shifts which we have seen in the
alignment of the telescope mirrors. Motions this large in the
secondary cage seem unlikely, but this should be investigated. At the
end of this report I suggest a series of measurements that might help
resolve this issue.
Finally, there are two other things of minor interest
that these data shed some light on. First are indications that the
secondary focus motions have slop on the order of 20 µm. This is
of minor interest only in that this is simply confirmation of
something we were already fairly certain of. Second, the data from 15
August show that there is some transient source of "noise" in the
measurement of the telescope aberrations. This may be either the
signature of small primary mirror oscillations which were seen on the
night that these data were acquired or it might be the signature of
rapid cooling of the telescope at the beginning of the night. Weather
conditions indicate that it is unlikely that this "noise" was the
result of telescope windshake. The noise appears almost exclusively
in the x-tilt measurements of the telescope coma. This is quite
unusual and is currently unexplained.
Some historical background on this data
set
The data from the nights of 14 and 15 of August 2000
are some of the very first data acquired after the July 2000
shutdown. Two significant changes were made to the telescope at this
time. First, the air skirt, which once sat underneath the primary
mirror, was removed. Second, the plenums, which fit on top of the
telescope exhaust tubes that extend inside the primary mirror hex
cells, were removed. These changes were made in an effort to minimize
the stresses that are placed on the primary mirror. Earlier
Shack-Hartmann measurements indicated that there was a significant
amount of astigmatism in the optics that caused approximately 0.3"
blurs in the telescope stellar profiles. Raising the mirror by
adjusting the location of the primary mirror hard points reduced this
astigmatism to about 0.2" at the price of moving the mirror out of
the center of the axial piston's range of motion. The air skirt was
designed to force the airflow of the telescope exhaust fans to enter
through the center hole in the primary. This was done in an effort to
create a laminar flow over the surface of the primary. This laminar
flow was desired to inhibit the formation of surface convection on
the primary, which is thought by some to be a significant source of
"mirror seeing".
The plenums, which were once installed on top of the
telescope vent tubes, were meant to keep the flow of the exhaust air
over the backside of the mirror front faceplate. It was thought that
by keeping the airflow close to the front surface of the glass, you
could more rapidly bring the mirror into thermal equilibrium with the
ambient air. The force that the bellows on top of the vent tubes
applied to the plenums when keeping them up against the front
faceplate of the mirror was measured to be a fraction of a pound. In
isolation this is an insignificant force on the mirror, but in
aggregate we were worried that these forces could become a
significant factor in distortion of the mirror. These considerations
have motivated us to remove the plenums. Needless to say, we were
also influenced to do so by the fact that their removal greatly
simplifies the installation of the primary mirror!
A proper investigation of the effects of removing the
skirt and plenums should include studies of the effects that these
changes have had on the telescope optics, a study of the effects that
these changes have had on the thermal cooling of the primary mirror,
and a study of the effects that these changes have had on the typical
seeing experienced by the telescope. I report here only on the first
of these three investigations. I am also currently working on a study
of the effects that these changes have had on the thermal cooling of
the primary mirror. I hope to finish that report within a few weeks
time. Unfortunately, we will most likely never know with much
certainty the effects these changes have made on the telescope seeing
because it will be very difficult to separate out the effects of
telescope seeing from the changes in the telescope optical
quality.
Telescope tilt coefficients: Coma
Figures
1a and 1b show the measurements of the
secondary tilts required to correct for flexures of the telescope
truss and the transverse flexures of the secondary mirror mount as
the telescope moves in altitude. A comparison of these data with
those obtained on
14
April 2000 shows very good agreement in the
tilts required to correct for transverse flexure of the truss and
secondary supports. All of these tilt values are therefore confirmed
by the altitude sweep measurements made on
14
June which were done to check the tilt which
were implemented after the April measurements. The work done during
the shutdown in July has not significantly altered the tilt
coefficients.
Sometime during the third week in August, the tilt
coefficients seen in Figure 1a were applied to the telescope
software. On August 1, right after the July shutdown, the TCC tilt
coefficients were set to SecXTiltCoef 70.0 0.0 0.0. These were the
tilts that applied when the data reported here were acquired.
Sometime around August 20, when time was available to redo the
telescope pointing model, the tilt coefficients were set to
SecXTiltCoef 14.0 21.0 75.0. As shown in the report on the April 14th
data set, these coefficients were derived by adding the fit values
shown in Figure 1a to the coefficients that were active at the time
the data were taken.
Two other things are worth noting in regards to the
telescope tilt coefficients. Note that in Figures 1a and 1b there is
an offset to the curves. The optimal x-tilt at 60љ in Figure 1a is
very close to 0". But in Figure 1b, the optimal x-tilt is
approximately -12". This offset is not well explained by the
increased noise seen in the August 15th data. It appears to be a real
variation of the zero point in the optimal tilt curves. This is one
indication that the mirrors are moving with respect to each other on
a nightly basis. We will revisit this notion when we discuss the
defocus errors. A tilt of 12" is equivalent to a differential motion
of 92 µm in one of the secondary actuators with respect to the
other two! The differences seen in the zero points of the optimal
tilt curves in Figures 1a and 1b represent substantial changes in the
mirror tilts.
Second, it is curious that the tilt curves show zero
point changes only in the x-axis. The zero points of the y-tilt
curves were quite reproducible. It would seem likely that these
changes are somehow related to the "noise" that was seen only in the
x-tilt data on August 15. This might imply that we have a significant
problem with the secondary actuator A. However, the Heidenhein
encoder data on the secondary mirror is very clear on one point: we
see more slop and vibration in actuators B and C than we do in
actuator A. The noise and zero point drift in the x-axis would then
seem to indicate that we have either coordinated noise in secondary
actuators B and C (unlikely!) or motions of the primary mirror in
sector A. Primary mirror oscillations were reported during the
observation of August 15 and this may offer the best explanation of
the "noise" seen in the August 15th data. A correlation between the
PMSS oscillations and the noise seen in these data might be worth
looking into.
It was originally thought that the increased noise in
the August 15 x-tilt data could be attributed to the presence of wind
on the telescope. However, the TCC weather logs do not support this.
During the acquisition of the August 14 data, the wind speed was
between 6 and 8 m/s (13-17 mph) and it had a direction of 20љ (where
0љ is due South and 90љ is East). At this time the telescope was
pointed between 170 and 190љ, at least 150љ away from the wind
direction. During the acquisition of the August 15 data, similar
conditions prevailed. The wind speed was between 7 and 8 m/s (15-18
mph) at a direction of 40љ. The telescope was pointed between 150 and
250љ.
The main environmental difference between the two
data sets was the time of night at which they were acquired. The
August 14 data set was acquired near the end of the night after the
telescope had reached equilibrium with the ambient temperatures. But
the August 15 data were acquired right after the telescope was opened
for observations. By way of illustration of this point I offer the
following observations. At the time of the August 14 data acquisition
the truss was cooling at a rate of only -0.19 C/hr. But during the
August 15 data acquisition the truss was cooling at a rate of -1.26
C/hr. Its possible that the noise in the August 15 data could have
been generated by cooling of the mirrors, but it seems unlikely that
such a mechanism would be so clearly confined to the x-tilt
motions.
Astigmatism and Spherical Aberration as a Function
of Telescope Altitude
Figures
2a and 2b show the telescope spherical
aberration as a function of telescope altitude for the nights of
August 14 and 15. To compare these results with the coma shown
indirectly in Figure 1, its useful to know that a mirror tilt of 25"
is equivalent to coma of about 0.30" in the image plane. Figure 2
shows that at the time of the August 14 measurements spherical
aberration was negligible compared to coma. But at the time of the
August 15 measurements the telescope image quality was dominated by
spherical aberration. In a separate (as yet unwritten!) report, using
data from several other nights I will show that this variation of
spherical aberration is the result of the time of night at which
these two measurements were made. The August 14 data were taken at
the very end of the night after the telescope had cooled to
equilibrium with the ambient atmosphere. The August 15 data were
taken near the beginning of the night, right after the dome had been
opened. As reported in Figure 2, the telescope and mirror cooling
rates during the acquisition of the August 14 and 15 data sets
differed by about a factor of 6.
Figures
3a through 3d show the telescope astigmatism
as a function of telescope altitude for the nights of August 14 and
15. Both nights show identical behavior of the magnitude and
orientation of the astigmatism. As the telescope moves toward the
horizon, the astigmatism increases from about 0.12" to 0.22" and the
orientation rotates about 25љ in a counter-clockwise fashion. These
data represent a significant step forward in the reduction of the
telescope astigmatism. Even at altitudes as low as 30љ astigmatism
remains a minor contributor to the image blur. Its possible that the
current increases in astigmatism as a function of telescope altitude
is the result of uneven support of the primary mirror by the
transverse posts. But, at this level it's not something we need to
worry about.
Defocus as a Function of Telescope
Altitude
Figures
4a and 4b show the telescope defocus as a
function of time and telescope altitude on the night of August 14.
The first graph illustrates the effects of manually changing the
telescope focus on the data. Figure 4b shows that on this night, the
telescope focus had only a very minor dependence (10 µm) with
telescope altitude. Focus changes this small are quite acceptable and
will not seriously degrade the image quality of the telescope.
Similar measurements of the defocus as a function of
altitude were made on
April
14. At that time, the telescope focus was
measured to change by approximately 50 µm over a 60љ range in
telescope altitude. Shortly after those measurements, corrections
were made to the TCC piston coefficients. The smaller variation
measured here should be an indication of the improvements that those
changes in the piston coefficients have made. However, the
measurements which were then made on August 15 show that there are
significant complications to this picture and we are not doing as
well as these data seem to indicate. I will show below that the
differences between these data and that measured on August 15 are
probably the result of thermal variations in the telescope.
Figure
5 shows the telescope defocus as a function
of telescope altitude as measured on the night of August 15.
Unfortunately, several manual focus changes were applied to the
telescope during the acquisition of these data and it is not possible
to simply throw out a few points as I did in the analysis of the
August 14 data set. This complication certainly adds significant
uncertainties to this data set. However, it should be immediately
obvious from comparing Figures 4 and 5 that the variation of
telescope defocus with altitude was measured to be nearly 7 times
greater than that seen on August 14! If the magnitude of the defocus
variation with telescope altitude had been similar to what was
measured earlier, the increased noise from the manual focus shifts
would have been sufficient to completely mask the variation. However,
the effect is clearly real and points to significant complications in
the interpretation of the defocus data.
The cause of the large difference between the defocus
changes seen on August 14 and 15 is thought to be temperature
variations in the secondary support rods. The first clue to this
comes from looking at the data in Figure 5 as a function of both time
and telescope altitude. This comparison is shown in
Figure 6. The first part of
this graph shows what looks like a clear sign of temporal variations
in the defocus data. I will assume here that the most likely causes
of a slow temporal variation like that seen in Figure 6 are thermal
variations in the telescope structure. This idea is supported by the
fact that the telescope structure was experiencing major cooling
rates at the time the August 15 data were acquired, but not while the
April 14 nor August 14 data were acquired. This is indicated by the
truss temperatures and cooling rates that are shown in Figures 4 and
5.
Figure
7 clearly shows that the truss temperatures
continue to monotonically cool throughout the entire time period
covered by the data in
Figure 6. From Figure 7 we
can estimate that the cooling time for the truss was approximately
1.41 hour. You cannot explain the second half of Figure 6 by
appealing to temperature changes in the telescope truss because the
truss temperatures change too slowly and they were monotonic at the
time of the defocus reversal seen near 3:30 UT. This is the first
reason to suspect that something other than the truss is driving the
defocus changes seen in Figure 6.
In the next section I will show that theoretically
one expects to find that the secondary support rods are as important
as the truss in determining the spacing between the primary and
secondary mirrors as a function of ambient temperature. This leads us
to focus attention on the thermal behavior of the secondary support
rods.
Back in May of 1996, Karen (of Loomis fame! Yes, we
still miss you, Karen!) was kind enough to measure the temperature of
the secondary support rods as the telescope cooled. She measured the
temperature and tension on 4 of 8 of the support rods. Her
measurements started right after the enclosure was opened and lasted
for about 3 hours. Figure 8a shows the
temperature data which she recorded for one of the 4 rods measured.
The data are fairly well described by an exponential fit. From this
data we can estimate an upper limit of 1 hour for the thermal time
constants of the rods. The plot of the air temperatures on this same
graph shows that the rod time constant could easily be (and probably
are) significantly shorter than 1 hour.
Figure 8b shows temperature
differentials between secondary support rods at the time of the data
shown in Figure 7a. Accept for very early in the cooling curve, the
rod temperatures are all the same. The differentials seen early in
the evening are significant, but below I will focus mostly on the
common-mode temperature changes in the secondary support rods.
Thermal Effects on the 3.5m Telescope
In the final section of this report I investigate the
primary-secondary mirror spacing as a function of the telescope
temperatures. This investigation is driven by an appreciation that
focus errors of only 10 µm are sufficient to degrade our image
quality by 0.19". The defocus measurements reported above show that
we are not currently capable of maintaining telescope focus to this
accuracy for large portions of each night. I will show here that
thermal changes in the secondary support rods are as important as
thermal changes in the truss in determining the mirror spacing. The
rods are almost certainly responsible for the telescope's current
sensitivity to rapid temperature changes during the night.
I believe that an improved thermal model of the
telescope is required if we are to hope to make efficient use of the
improved telescope optics. I show the beginnings of such a model at
the end of this report. However, we do not currently have enough data
to support the implementation of a new thermal model. Future
Shack-Hartmann measurements will be key to improving the telescope
thermal model.
The calculations done here offer another reason to
change the current design of the secondary cage. Owing to subtle
asymmetries in the current design, the secondary cage drifts axially
with temperature changes. This drift can be removed either by
redesigning the cage to make the support rods truly symmetric or by
changing the materials used in the current support rods. The current
rods are made of 303 (Se) stainless steel. Replacing these with
graphite rods would offer about a factor of 2 improvement in the
thermal stability of the telescope.
The Thermal Expansion of the Truss
I use here a very simple model for the thermal
expansion of the telescope truss. I start by giving values or a range
of values for the pertinent telescope variables. For the 3.5m
telescope we have:
- L = M1-M2 separation = 6.03 m. Note that this
number includes the length of the main truss (4.76 m) plus the
length of the primary mirror cell fork (1.27 m). I assume here
that the steel of the fork is similar to the steel of the truss
tubing.
- =
Coefficient of Thermal Expansion of 1018 steel = 11.7 µm/m-C.
Note that the truss tubes consist of a carbon steel. The thermal
expansion above was taken from the TCC software and this value is
typical for normal carbon steels. For carbon steels, values as low
as 10.8 are possible.
- =
2.8 µm/m-C = CTE of E-6 (low expansion) glass. This is a
fairly well known number and applies to both the primary and the
secondary mirrors.
- =
mean thickness of M1 = 0.39 m
- =
mean thickness of M2 = 0.15 m
If we only need to worry about thermal expansions of
the truss and the glass, then the change in the M1-M2 separation from
the thermal expansion of the truss and the mirrors is given
by:
The current (as of 9/19/00) temperature coefficient
in the TCC software is +43 µm/C. The difference between the
expansion predicted above for the truss and that in the TCC software
appears to be significant.
Discussions with Russet on the behavior of the
telescope focus as the telescope cools down at the beginning of the
night do not help explain this discrepancy. She reports that at the
beginning of the night one must often set an instrument focus below
what one expects it to be later in the evening. But, this behavior is
not consistent from night to night. In other words, she reports an
inconsistent positive piston is required as the telescope cools. As
shown above, the truss coefficient is a positive quantity, which
shows that the truss expands when it warms up. In the 3.5m software a
positive piston command moves the secondary closer to the primary.
Therefore, as the truss cools and contracts we expect to make
negative piston moves. Positive piston moves as the telescope cools,
like those reported by Russet imply that the real telescope
temperature coefficient is even lower than the current setting of 43
µm. This is inconsistent with the range of CTEs found in carbon
steels.
The Thermal Expansion of the Secondary Support
Rods
I must begin this discussion with a simple
description of the secondary cage geometry. This geometry drives the
thermal behavior of the support rods.
Figure 9 shows one quarter of
the rod and cage geometry. The rods are 1/2" 303(Se) stainless steel.
They are held in tension (approximately 5000 lbs.) by turnbuckles
near the truss blocks, which are not shown in Figure 9. There are a
total of 8 support rods, two on each corner of the square truss
assembly.
As the telescope moves from the horizon to the
zenith, the support rods sag under the weight of the secondary cage
load. This is the normal cause of telescope defocus as a function of
altitude. The TCC software compensates for this motion by pistoning
the secondary out as a function of telescope altitude. The current
piston coefficient for the secondary rod sag in the TCC software is
400 µm. One might question how accurately this is known and
whether this sag has any temperature dependence. I will show below
that the answer to these questions is "not very well" and "no". The
differential motion of the secondary cage as the telescope moves in
altitude is not temperature dependent. However, the point from which
this differential motion starts is. This is a somewhat subtle
distinction that primarily affects the form that one chooses for a
thermal model of the telescope. Below and in Appendix B I refer to
the "absolute" secondary position when talking about the thermal
motions of the secondary cage in an attempt to make this distinction
clear. I hope this term does not prove to be confusing.
For those interested in the details, I show my
derivation of the secondary sag in
Appendix
A. The total sag that the secondary will
experience as the telescope moves from the horizon to the zenith is
given approximately by
where
- L = the average length of the secondary support
rods
- W = the weight of the secondary cage (including
the mirror!)
- Ar=
the cross sectional area of a support rod
- E = elastic modulus (i.e. Young's modulus) of the
rod steel
- a = the average angle that the rods make with
respect to the telescope radial direction.
I show in Appendix A that the approximation above is
good to about 10%. I also show there that this
result is independent of the rod temperatures even if the rods are at
different temperatures. All that is required
for this equation to hold is that the rod temperatures stay constant
during the altitude change. For the 3.5-m telescope we have
- L = 83.65" = 2.125 x
106
µm
- F = 400 lbs.
- Ar
=
3.14159(0.25)2
= 0.196 in2
(for 1/2" diameter rod)
- E = 2.8 x 107
psi (for 303 stainless steel)
- a = 15.5º
Putting these values into the equation above gives
= 271 µm.
The current sag coefficient in the TCC software is 400 µm. This
represents a difference of about 30% from what is calculated here. It
is unlikely that this difference can be explained by variations in
the expansion of the steel or in the assumed elastic modulus. These
parameters are too well known. The most likely source of error in
this calculation is in the assumed weight of the secondary cage.
Errors in the TCC software coefficient can come from the fact that
these parameters are determined from a multi-parameter fit to data.
Even if you assume that the data that were used for these fits had no
errors, you can still have errors in the fit coefficients based on
the fact that many of the parameters that are being fit are not truly
orthogonal. At the end of this report I will show that a thermal
model that uses the lower sag coefficient calculated above can
actually give a better fit to at least a limited data set. This does
not prove that the calculations above are a better estimate than what
is currently being used in the TCC code, but it does illustrate how
multi-parameter fits have the potential for hiding many errors in the
assumed coefficients.
While it is true that the differential motion of the
secondary as the telescope changes altitude does not depend on
temperature, the absolute position of the secondary does. This
temperature dependence is entirely due to the asymmetries in the
current design of the secondary cage. In
Appendix
B I show that to an accuracy of about 10%,
the absolute position of the secondary depends on the temperature of
the support rods as follows
where
- =
the axial motion of the secondary due to thermal effects
- L = the average length of the support rods =
2.125 m
- Cte=
the coefficient of thermal expansion of the rod steel = 17.3
µm/mºC
- a1
= the angle of the bottom support rod with respect to the
telescope radial direction in radians = 0.237 (=
13.6º)
- a2
= the angle of the top support rod with respect to the telescope
radial direction in radians = 0.318 (=18.2º)
- =
the temperature change of the secondary rods. (Note that this
change is from some currently unknown base temperature!)
For the 3.5-m telescope this translates to
A Provisional Thermal Model of the
3.5-m
I have shown here the unexpected result that the
secondary cage axial position is just as sensitive to the temperature
of the secondary rods as it is to the temperature of the main
truss!
This is a very important for it suggests that the
thermal model of the telescope that is currently in the TCC software
is never going to do a good job of correcting for telescope thermal
effects. The current thermal model simply pistons the secondary based
on measurements of the main truss temperature. The analysis above
shows that you must include measurements of the rod temperatures if
you hope to keep the telescope in focus during thermal
changes.
The thermal analysis presented here suggests that a
better model for the TCC thermal behavior might be achieved by
assuming the following functional form for the secondary piston as a
function of temperature
In Figure 10 I show how such a
model is consistent with the defocus data taken on August 15. In
Figure 10, the functional form above was assumed for the secondary
sag and the defocus data were used to fit for the thermal time
constant of the rods. The rod thermal time constant so derived was
about 22 minutes. Figure 8 showed that the rod thermal time constant
must be significantly less than 1 hour. The defocus data and the
theoretical thermal expansion coefficients are shown to be consistent
with this expectation.
What is shown here is clearly not proof of the model
above, but it is encouraging. In Figure 10 it can be seen that the
residuals from the model fit are no larger than about 25 µm. The
defocus data in Figure 6 represents the residuals of the current TCC
thermal model from the optimal focus position. These data are
therefore equivalent to the residuals between the model fit and the
defocus data shown in Figure 10. A look at Figure 6 shows 60 µm
deviations from the current TCC thermal model. The fit in Figure 10
therefore appears to represents a factor of two improvement over the
current TCC thermal model.
Can Miscollimation Be Caused by Thermal Effects on
the Secondary Cage?
I have focused above on understanding the sag of the
secondary as a function of temperature and altitude. However, I have
also considered the effects that thermal changes might have on the
tilt of the secondary mirror. My conclusions here are that it is
unlikely that thermal stresses are responsible for any significant
tilts of the secondary. This conclusion is based on two facts. First,
tilts require thermal differentials between either the secondary
support rods or the main truss supports. When these have been
measured, they have been small as expected. Figure 8b is a good
example. Maximum differentials in the rods appear to be on the order
of 0.5º C. Differential rod temperatures of 0.5º can cause
motions on the order of 30 µm, but these differentials are very
transient. I suspect that thermal differentials of the truss supports
will also be small, but someone might want to investigate
this.
Second, tilts of the secondary cage must be very
large before they will show up as increased coma. The reason for this
is not completely obvious. This follows from the fact that coma may
be caused by both translation of the secondary mirror as well as by
tilts of the mirror. There is a position, called the "neutral point"
by opticians, about which the secondary mirror can rotate without
causing significant coma owing to the fact that when the mirror is on
this surface the coma induced by its translation cancels that induced
by its tilt. The location of the neutral point for the 3.5-m is shown
by the center mark in
Figure 9. If the cage is
rotated about its geometrical center by an angle
a1, then the
mirror is rotated in space by this same angle. However, with respect
to its neutral point it also rotates about an angle
a0, which is
related to a1 by
the equation
where r is the distance from the center of the cage
to the vertex of the secondary mirror and is the distance from the center of
the cage to the neutral point. When you measure the mirror tilts by
measuring the coma with the Shack-Hartmann sensor you are actually
measuring (a1 -
a0). Almost all
cage tilts caused by motions of the rods will have their center of
rotation near the optical neutral point. For example, a tilt of 10
arcseconds about the center of the cage results in a differential
angle (a1 -
a0) ~ 1 arcsecond.
10 arcseconds of tilt corresponds to a differential temperature on
the rods of approximately 1.5 ºC, and a linear change of the
rods of about 50 µm. To effect a coma change corresponding to a
10 arcsecond tilt therefore corresponds to a cage tilt of about 100
arcseconds or a linear stretching of the rods of about 0.5 mm.
Over the course of several nights we are seeing
collimation changes on the order of 50 arcseconds. This corresponds
to approximately 500 arcseconds of secondary cage tilt. This requires
motions of the rods on the order of 2.5 mm. One might speculate that
such tilts might come from slippage in the joints in the secondary
cage. This possibility probably deserves some investigation. Because
of this, I have the following suggestion for measurements that we
might want to make on the 3.5-m.
A Suggestion for Engineering Measurements of the
Secondary Cage Position
I would like for us to measure the stability of the
secondary cage as the telescope cools down. I suggest that one way to
do this would be to fix a small flat mirror to the bottom of the
secondary cage. I would then bounce a laser off of this mirror while
the telescope is at a fixed position. By measuring the position of
the reflected laser beam and knowing the geometry between the laser,
the flat mirror, and the reflected beam we might be able to monitor
changes in the tilt of the secondary cage. Some very rough
calculations suggest that we might be able to measure tilts of a few
arcseconds in this manner. Making measurements before and after
opening the dome would be good and making measurements in the
presence of a wind would be worth while. Having a digital camera set
up to make measurements of the laser beam would be a good idea. There
are cameras available here at UW that we could use for these
measurements.