(1)
An overview of the guider performance
III. Engineering data on the guider performance
Atmospheric turbulence limitations
Simultaneous guider and SPIcam tests
Past reports of guider performance and comments on guider requirements
Searches for correlations between guide errors and other parameters
The temporal characteristics of the tracking errors
IV. Summary
Past attempts to improve guider performance
V. References
The first section of this report is geared to inform a normal user of the telescope about the general capabilities of the 3.5m guider. We also present information on routines which will allow a user to make better use of the guider data and to monitor its operation.
The guider hardware consists of a Grade 1 thinned, backside illuminated MPP SITe CCD with 24 µm pixels in a 1024x1024 format. This camera was choosen for its high sensitivity, complete lack of image defects, and its large pixels. Without binning, the telescope optics produce an image scale of 0.14 arcseconds/pixel on the guider. The imager is cooled by a two stage TEC. The hot side of the TEC is liquid cooled by an ethylene glycol/water solution. The radiator, pump, and air reservoir for the liquid cooling system are mounted in the instrument rotator along with the guide camera and its filter mechanism. Fans and ducting are included to vent the heat from the radiator out of the rotator box into the ambient air of the dome. The liquid cooling system is necessary owing to the large physical size of the CCD. The choice to mount the cooling system inside the rotator box was made because this was felt to be a better option than attempting to run liquid lines through the rotator cable wrap. The cooling system is sufficient to keep the CCD at -25 C at all times, which is its current temperature setpoint. The CCD utilizes a Photometrics PXL controller. The gain on the camera is 10.5 ± 0.1 electrons/ADU and the read noise is 22.5 ± 0.2 electrons. The readout rate of the camera is fairly rapid at 200 kHz, resulting in a total readout time of about 3 seconds for a full, unbinned frame.
The guider hardware includes a focus mechanism which is independent of the normal telescope focus and a 7 position filter wheel. The operation of the focus and filter mechanisms is described in the TCC documentation found in the normal APO web pages (see the GCamera documentation). We will not duplicate that here. The APO web pages also contain information on the filters which are available for the guide camera.
However, it is worthwhile mentioning here some information on the units of the guider focus mechanism and how this is related to the normal telescope focus. Movement of the telescope focal plane is related to movement of the secondary by the equation
(1)
where 
is the induced axial motion
of the telescope focal plane, 
is the motion of
the secondary mirror, and 
is the ratio of the
telescope focal length to the focal length of the primary mirror. For
the APO 3.5-m we have 
which means that equation
(1) reduces to 
. Equation (1) implies that
as the distance between the secondary and the primary is increased,
the telescope focal plane moves in toward the tertiary. Owing to some
mechanical changes to the guider focus mechanism when the new guide
camera was installed, the units of the guider focus have been
inverted. The controller software has not been changed to reflect
this fact. As a result, increasing values of
the guider focus correspond to motion of the focal plane toward the
tertiary. The software expects units of
microns for the guider focus mechanism. Experience with science
instruments has shown that focus changes smaller than 25 µm
produce neglible changes in the measured stellar profiles. Equation
(1) shows that an equivalent motion for the guider focus is
approximately 850 µm. Focus motions smaller than this are not
normally useful because they produce changes in the stellar profile
smaller than the typical noise of the stellar width
measurements.
We have measured the par-focal behavior of the guider with respect to SPIcam's focus. At 90 degree intervals of the rotator angle (starting with a rotator mount position of 257 degrees), we found the best focus position of SPIcam by movements of the secondary. At the same time, after focusing SPIcam we focused the guider by means of the guider mechanism focus. As a final check against possible temporal drifts of the focus throughout the duration of the focus measurements, at the end of this procedure we repeated the focal measurements at the original starting rotator mount position. As expected, the on-axis focus for SPIcam varied by less than 50 µm. The reproducibility of the first and last SPIcam focus measurements were within 20 µm, which is typical of the noise level of the focus measurements. At the same time, we measured the following best focus positions for the guider:
|
Rotator Angle |
Guider Focus |
Focal Difference |
|---|---|---|
|
257 |
47096 |
0 |
|
347 |
47945 |
850 |
|
74 |
48395 |
1299 |
|
160 |
46578 |
518 |
|
247 |
47578 |
482 |
There is at most a slight drift of guider focus as a function of rotator mount position. 180 degrees from the beginning focus, at a rotator mount position of 74 degrees, a focus difference of 1299 µm was seen in the guider focal position compared with the original measurement at a rotator mount position of 257 degrees. This difference is small enough compared to the noise that its reality is in doubt. The average focal position for the guider when using SPIcam is therefore at 47518 ± 710. This should be the current instrument block setting for the SPIcam guider focus position.
We have measured the guider throughput using several Landolt standards. In a one second exposure through the R filter, a 15th magnitude star produces 956 ± 15 ADUs on the camera. With the gain quoted above, this is equivalent to 10038 ± 157 electrons/sec. This means that with a one second exposure through the R filter, a 21.6 magnitude star will produce one read-noise worth of electrons on the detector. A more realistic detection limit for the guider can be estimated as follows. With 1 arcsecond seeing and 3x3 binning, a stellar profile covers approximately 9 binned pixels. If we require at least 3 times the read-noise in each pixel to define a minimum stellar detection, we are requiring a minimum flux of 608 electrons for the detection of a star. This definition of a minimum stellar detection and the throughput given above imply that in 1 second through the R filter we can detect a star of 18.0 magnitude. The brightness limit for practical guiding is much more difficult to define because it depends on many more variables, such as background variations and the sensitivity of the fitting routines to noise. We discuss this issue below.
We have also used stellar measurements to characterize the 1% ND filter which is currently in use in the guider's filter wheel. By comparing the measured flux of several stars through the R filter and the R filter + 1% ND filter stack, we conclude that the 1% ND filter has an average transmission of 1.55 ± 0.04 % over the R bandpass.
The telescope is currently capable of keeping objects
centered to within an error circle with a 1-
radius of 0.24 ± 0.1 arcseconds anywhere in
the sky with one notable exception. Near the zenith, this performance
will degrade. We have only one measurement of how fast this behavior
degrades as you approach zenith. Users are advised that tracking
within 6 degrees of zenith will produce significant degradation.
Within 6 degrees deviations of up to 0.75 arcseconds are caused by
the inability of the rotator to keep up with the sidereal motions. We
also know that there are times when or places on the sky where the
behavior of the guider appears to improve to give a 1-
error circle of approximately 0.10 arcseconds.
We currently do not know the causes nor the systematics of these
variations. We will return to this topic in section III of this
report.
The guider camera sensitivity is sufficient to make guiding feasible nearly anywhere in the sky without worrying about preparing finding charts for guide stars ahead of time. We have been able to guide on stars as dim as 18.9th magnitude using 20 second guider exposures. A star this dim with moderately poor seeing (approximately. 1.5 arcseconds) produces a profile with a peak of about 40 ADUs above the background on the guide camera. This profile was successfully identified by the star finder software and the error circle of guider centroids did not significantly increase compared to the distributions obtained with stars 10 and 100 times brighter. From star counts near the galactic pole, we estimate that there should almost always be a star brighter than 19th magnitude in the field of view of the guider at all times.
For moderately bright stars (< 15th magnitude), the guider software is not very sensitive to the background conditions. Tests of the guiding at twilight show that a sky background as large as 3 times the peak guide star intensity has little or no effect on the guiding. Details of this test are shown in section III of this report.
The guider keeps a ring-buffer of its 20 most-current images stored on tycho.apo.nmsu.edu in the directory /export/images/guider under the names gImg**.fits where "**" runs from 01 to 20. As the names imply, these are stored as FITS images. The most recent image can be determined by reading the file "last.image" which is found in the same directory. In this directory you will also find a null file called "gImg.ready". When this file is present in the guider images directory you may safely open and read "last.image" or any FITS file present. When it is absent you should not attempt to open these files. GImg.ready will disappear from the guider images directory only momentarily while the guider Mac writes out a new file. In this way a user can tell when its OK to transfer files between systems.
There are three facilities that users should be aware of which make the guider easier to use. Two of these scripts allow the user to see the data that comes from the guider and the third script allows a user an easy way to focus the guider.
We have Eric Deutsch to thank for the authorship of this routine. The guider web page routine converts the FITS images to GIFs and then displays them on a web page. This allows almost real time display of the guider images. There is an inherent lag of approximately 30 seconds between the display and the acquisition of the most recent image. To use this facility connect to:
http://www.astro.washington.edu/deutsch-bin/latestguiderimage
This page should be self-explanatory. The page is set to reload every 20 seconds by default, although the reload time can be adjusted from the page itself. Note that the processing system is not on all the time; if the field "Guider Auto-Fetcher" is OFF, please ask the Observing Specialist to turn it on for you (via the cloudcam interface). Questions/problems/suggestions about this display system (but not about the guider itself) may be sent to deutsch@astro.washington.edu.
Eric has also put together a second application which plots up parts of the guider data stream as it comes out to the TCC. This application is written in IDL and therefore will require this commercial software if you want to run it. In addition to the most current guider image, this application includes a plot of the X and Y differences between guider centroids as a function of time, a plot of the FWHM of the guider stellar profiles as a function of time, and a plot of the integrated stellar intensities as a function of time. As an added convenience, this application also displays the latest cloud monitor image. If you are interested in running this application on your home machine, you are encouraged to contact Eric at the address given above. The primary weakness of this application is that it does not allow you to do stretches of the guider image, nor can you do any quantitative analysis of the guider data. These needs are addressed by the next routine.
The next facility comes to us from Al Diercks, but don't blame him if things don't work. I have hacked his code to remove a security bug that the original code contained. It is an IRAF script which grabs a guider image and displays it in an IRAF display window. This is useful when one needs to make quantitative use of a guider image. The routine is called glg which,I believe, stands for "get last guider (image)". Al is a concise fellow! The guts of this routine requires that you have a "Perl" interpreter on your local machine. Perl stands for "Practical Extraction and Report Language". It is a script processing language which runs on any Unix machine. You can find a copy of Perl and a description of it at the following site:
http://www.cis.ufl.edu/perl/
All users, please note that an original version of this routine exists which attempts to retrieve the guider images directly from the guider Macintosh rather than from tycho. This technique has a strong tendency to crash the guider, especially if rapid guider exposures are being made. Please make sure that you have the current version of this routine! Any copy of glg.cl and gsnag.pl which do not have the version labeled in the first three lines of the routine, or any copy earlier than version 2.0 should be discarded.
If you wish to install a copy of this, then proceed as follows:
The third routine is an mcnode script called gfocus5.tcl. As its name implies, this is a TCL script which will run in an mcnode window. Tcl stands for "Tool Command Language". It is a script processing language similar to Perl. A copy of this routine is kept in the home directory of the visitor1 account on tycho. You must have an mcnode window open to use gfocus5.tcl. After opening an mcnode window on your home machine, follow these directions to use gfocus5:
As an example of a call to this routine, if you wanted to take 10 second exposures, moving the guider mechanism focus 2000 µm between each exposure, centered around a focus position of 48000 µm, then the call to this routine would be: gfocus5 10 g 48000 2000.
As with any TCL procedure, all of the command parameters are position specific. You may default any of the command parameters. However, once a parameter is defaulted, all of the following parameters must also be defaulted. This routine takes five guider images and measures the widths of the stellar profiles in each image. It then does a least-squares fit to a second-order polynomial to this data. If the polynomial has a minimum within 2 * inc of the middle focus position, then the routine sets the telescope to this position. Otherwise, it returns the focus to the starting position. The routine will fail gracefully if it is unable to find your star in each of the five images taken. Note that the default behavior of this routine is to use the guider focus mechanism rather than telescope secondary motions. For more detail on the behavior of this routine, see the comments documented at the beginning of the script. In the same folder, there is also a routine called gfocus3.tcl. This routine is called with the same parameters as gfocus5.tcl. It is faster in that it fits the polynomial to the data from 3 images. In general, I recommend the use of gfocus5.
The goals of this section of the report are four-fold. First, we will discuss the theoretical limitations to guiding and a few results that have been obtained from others. Then, we would like to convince the reader that the signal and position sensitivity of the current guider are not limiting factors. Third, we wish to show that we need to be doing a better job if ever we want to take advantage of sub-arcsecond conditions at the site. And finally, we will conclude this section by discussing our attempts to understand what are the root sources of the tracking errors that we currently experience when guiding.
There are several things that can become the fundamental limitations to guiding accuracy. Poisson errors of the photon flux can create inaccurate centroid positions for very dim guide stars. But, it can be easily be shown that we are not typically limited by this effect with the current 3.5-m guiding. The easiest way to observe this is to measure the guiding errors as a function of guide star intensities. These errors will decrease with the square root of the integration time. We see no such effect in the current guider errors.
Atmospheric turbulence represents another fundamental
limitation to guiding. Turbulence is the root cause of telescopic
"seeing". It is common to relate FWHM measurements of seeing to the
Fried parameter, 
, effectively the phase coherence length through
the atmosphere. For long exposure images (anything greater than a few
tens of milliseconds) on telescopes greater than a very modest size
(approximately 5" in diameter) its relationship to telescopic seeing
is well known and is roughly given by the relationship:
(2)
where 
is the 1-sigma seeing in radians and 
is the wavelength of light observed (Roddier,
1981). Often, the factor of 1.27 in equation (2) is dropped owing to
the approximate nature of the relationship. We will follow that
convention here. The relationship between the seeing in FWHM units
and in sigma units is given by the relationship 2.35 * = FWHM.
Image motion caused by turbulence is a more complex phenomena which
depends on many atmospheric conditions. Despite its complexity, we
can estimate the effects of turbulence on the image motions based on
known average atmospheric conditions and theory.
Martin (1987) describes how to calculate the image
motions caused by atmospheric turbulence given knowledge of several
key atmospheric parameters. The phenomenon depends on the seeing
(
), telescope diameter (
), wind speed (
), wavelength of light observed (
), thickness of the turbulent layer one looks
through (
), exposure integration time (
), and direction with respect to the wind in
which you measure the image motion (
). In his article Martin shows calculations
appropriate to a small aperture telescope which did not include the
effects of a finite thickness of the upper atmosphere layer of
turbulence. I include here the extension of those calculations
appropriate to the 3.5-m telescope. I also include here the effects
of the finite thickness of the upper atmospheric turbulent layer.
This is necessary when large (greater than about 4" diameter)
aperture telescopes are considered because the effects become
dominant under these conditions.
Following Martin, for given atmospheric conditions
and telescope size the variance of image motion caused by turbulence
is a function of the exposure time in which it is measured and the
direction on the image plane in which it is measured. It decreases
with exposure time and it decreases if the image motion is measured
in the direction of the wind. With the convention that 
= 0 is the direction of the wind, the variance
of image motion in
arcseconds2 is
given by
(3)
where 
is the first order Bessel function and 
and 
are dimensionless
spatial frequencies corresponding to the outer and inner scale
lengths of turbulence in the atmosphere. Their relation to the outer
and inner scale lengths of turbulence are given by the
equations
(4)
Here 
and 
are the inner and outer scale lengths.
Hufnagel (1978) gives an excellent discussion of important atmospheric parameters which affect turbulence and the range of variation that can be expected to be seen in them. The inner scale length is the dimension at which turbulent motions dissipate into frictional heating of the atmosphere. Typically this length is on the order of 1 cm and is small enough that its effect on the calculations can usually be ignored. The outer scale length is the upper size limit to turbulent eddies. Its size depends considerably on the environment the turbulent layer is found in.
In the context of image motions the equations given by Roddier (1981) and Hufnagel (1978) indicate a very important thing about atmospheric turbulence (Cf. 6-28 and 6-29 in Hufnagel). Image motion is affected more strongly by turbulence near the ground than it is by upper atmospheric turbulence. The opposite is true for scintillation! It is often noted in the literature that Hufnagel shows that the largest contribution to turbulence comes from a shear layer near the top of the tropopause where average wind velocities are around 27 m/sec and the outer scale length varies between 10-100 m, independent of observing location. For scintillation, this is an important simplification, but it is not appropriate to apply these conditions to image motion owing to its sensitivity to ground layer turbulence. However, both Hufnagel and Roddier point out that even in ground layers turbulence has been shown to be confined to shear regions which have scale lengths which vary from 100-200m. In summary, when it comes to image motion, ground wind speeds and outer scale lengths of 100-200m are approprate to use in equation (4). It is also likely, but not proven, that outer scale length is a function of wind speed. Here we will simply assume that these are independent quantities.
In
Figure
1 I show the image motion calculated by
equation (3) for three different situations. The line marked by
circles shows the 1 standard deviation image motion expected in the
direction perpendicular to the wind vector (
=90) when the outer scale length is 200 m, which
is about the largest expected. The line marked by squares shows the
image motion in the orthogonal direction under the same conditions.
The line marked with diamonds shows the image motion expected if the
outer scale length is near the smallest expected. The abscissa is a
dimensionless parameter that depends on the local wind speed (
), the exposure integration time (
), and the telescope aperture diameter (
).
As can be seen from
Figure
1, theory predicts that image motion will
increase as wind speed decreases. This is perhaps counter-intuitive,
but it is the correct conclusion. It helps to understand that image
motion is driven by the larger scale turbulent features in the
atmosphere. Equation (3) is based on Taylor's "moving screen"
approximation in which it is assumed that the turbulent cells are
essentially stable in size during an exposure and move with the wind
velocity across the telescope field of view. The effect of the wind
motion is to average over the large scale structure in the direction
of the wind, nullifying their effects on the integrated exposure.
This is also why the maximum image motions are predicted to be in the
direction perpendicular to the wind (
=90) rather than in the direction of the wind.
The image motions do not go to zero when the wind velocity is zero
because it is assumed here that the turbulent motions are independent
from the bulk wind velocity.
For a typical ground wind speed of 10 mph (4.47
m/sec) and a 10 second exposure 
. Under these conditions, Figure 1 indicates that the
maximum image motion due to turbulence is about 0.066 arcseconds and
that it should vary by as much as a factor of about 40%, depending on
the direction in the image plane in which it is measured. Under the
best of conditions (smallest value for 
), the maximum image motion measured on the image
could be as small as 0.054 arcseconds. Shorter exposures or differing
wind speeds can cause image motions to rise up to about 0.082
arcseconds. Equation (3) shows that these results depend on 
. Therefore, if the seeing degrades to 2
arcseconds (
m) then the expected image motion
will rise to a maximum of 0.15 arcseconds. From this analysis we can
see that the guider is not usually being limited by atmospheric
turbulence. But, we will see that at times we are reaching this
theoretical limit.
Figure
2 shows a sequence of measurements that were
taken where guiding was initially off, turned on, and then again
turned off. Only the x-centroid data are shown, but the y coordinate
had statistically similar behavior. Each 3x3 binned pixel is equal to
0.42 arcseconds. At this time the 1- scatter during guiding was 0.28 pixels, or 0.12
arcseconds. The drift in the telescope during the periods when
guiding was turned off is evident. During time intervals of 20
minutes or greater the telescope drift is often more complicated than
a simple linear motion. This is illustrated by the data shown in the
first 20 minutes of
Figure
1. If the slow telescope drift is removed by
subtracting a polynomial fit to the data, then one finds that the
scatter of the guider centroids is approximately equal to the scatter
seen during guiding. In this case a scatter of 0.15 arcseconds was
measured before and after guiding was turned on. This is not the
lowest scatter which we observed on this night. Data from the same
night shows measurements with a scatter as low as 0.09 arcseconds
during guiding. In all of these guiding-on/guiding-off tests we have
observed that the scatter of the points both before and after guiding
is approximately equal to that observed during guiding. This
indicates that the scatter is not caused by the addition of the
guiding control loop to the telescope software, but is of a more
fundamental nature. The seeing at the time that all of these
measurements were taken was 1.2 arcseconds, which from my earlier
discussion equates to a maximum expected scatter from turbulence of
0.10 arcseconds. From this it is seen that at times we are in fact
approaching the best performance possible out of the guider.
Figure 3 shows the results of two guide-on/guide-off tests that were taken at nearly the same telescope coordinates. Test 3 was taken at the coordinates: Azimuth=223, Altitude=62, and Rotator Angle=215. Test 7 was taken 5 hours later using a different star at the coordinates: Azimuth=225, Altitude=60, and Rotator Angle=219. The wind conditions differed between the two tests. Test 3 was taken when the wind velocity was 8 m.p.h. and in a direction which was 29 degrees from the telescope enclosure opening. During test 7 the wind velocity was higher (14.3 m.p.h.) and directed more closely to the slit opening (only 13 degrees away).
We will return to the issue of the varying wind
conditions later. Here, I wish to point out the similarities between
these two tests in the telescope drifts during the times when the
guiding was off. Its easy to see in the figure where guiding was
turned on during each test. For test 3, guiding was turned on about
21 minutes after data acquisition started. For test 7 guiding was
turned on about 20 minutes after data acquisition started. In the
x-direction both tests show a steady downward drift with a magnitude
of about 1.5 arcseconds over the guide-off period. The drift in the
y-direction over the same period of time was smaller (about 0.75
arcseconds) and in the upward direction in both tests. Marked on
Figure
3, are the 1-
errors measured during the guide-on portion of
each curve. It can be seen that the tracking errors measured during
both tests were also quite similar.
The guide-off drift at random places in the sky varies considerably and on time scales as fast as 2 minutes. The drifts seen in Figure 1 are good examples of such behavior. By way of contrast, the shape of the drift curves in Figure 1 are much different from those seen in Figure 3 during tests 3 and 7. In general, this is what has been found, very few of the drift curves look alike. From the similar shapes of the drift curves seen in Figure 3, I conclude that the drifts are not random and are primarily a function of telescope position. This is consistent with the hypothesis that the drifts come primarily from small scale errors in the telescope pointing model.
Figures 4 and 5 show a sample of some of the data which we have analyzed in order to characterize the guider's performance in terms of its effects on on-axis imagery with a science instrument. For these observations SPIcam was mounted on-axis. A star of moderate intensity (peak of 360 ADUs) was found and used for guiding with 20 second exposures. Simultaneously, 10 second images were taken with SPIcam. Guider centroid positions were taken from the TCC log and IRAF's "imexam" was used to compute the centroid positions on the SPIcam exposures. The mean x and y centroid positions were subtracted from each centroid and the difference from these means are shown in the figures. The solid squares show the SPIcam centroid positions and the open circles show the simultaneously acquired guider centroid positions. Even though we used shorter exposures on SPIcam, these data are more sparse than the guider data owing to SPIcam's long read time.
When the rotator is at an object angle of 0°, the -x axis on the guider is always aligned towards the West, which is the direction of stellar motion. North lies in the +y guider axis direction. The parallactic angle is the direction from North towards the local zenith. The orientation of the parallactic axis with respect to the guider axes is illustrated on Figure 4. Motions of the telescope's altitude axis always move the image along the parallactic axis. Motions of the telescope's azimuth axis always move the image perpendicular to the parallactic axis. While the direction of tracking is always in the -x direction, the proportion of altitude and azimuth motions that compose this tracking motion change with the parallactic angle. In the case of Figures 4 and 5, the parallactic angle was +22 degrees and most of the tracking motions were being done by the azimuth axis of the telescope.
The x-axis is also the axis which points in the direction of the rotator motions. It is important to remember that the TCC does not actively control the instrument rotator when guiding. Motions of the field owing to imperfections in the rotator are degenerate with motions owing to imperfections in the positioning of the altitude and azimuth axes. Simultaneous observations of multiple stars are required to break this degeneracy and correct for rotator errors along with errors in the altitude and azimuth axes. Requiring that multiple stars be available in the guider field of view at all times is too restrictive to guarantee guider operations at all positions in the sky. Because of this, the guiding software assumes that the rotator is a perfect mechanism and attempts to keep the guide star stationary on the guide camera by moving only the altitude and azimuth axes of the telescope.
In Figures 4 and 5 it can be seen that the image motions on the guider are mirrored by image motions on SPIcam. For example, when an Easterly excursion is seen in the guider, an Easterly excursion of similar magnitude is observed on SPIcam. There are a few notable exception to this. For example, on Figure 4, the SPIcam centroid at 16.5 minutes into the measurements differs by 0.4 arcseconds from the corresponding guider centroid, but for 80% of the points on all similar measurements that we have made, the agreement is within 0.1 arcsecond or better. The image excursions on SPIcam are in phase with the guider image motions. There are no apparent delays between the motions seen on the two cameras. This mirrored, in-phase behavior I call "common-mode" motion. It is a strong indication of four things.
First, the common-mode motions show that the errors observed in the guider are directly related to the errors that will be present in the science instrument. This means that from guider data alone we should be able to deduce the effects of tracking errors on the PSF of the science instrument.
Second, the common-mode motions imply that there are no problems with the centroiding algorithms nor significant noise sources which are specific to the guider. The software used to analyze the SPIcam and guider images are different, the sky backgrounds and noise levels of the two cameras are very different and yet the agreement between the SPIcam and guider centroid positions is good. We have verified that the guider software gives centroid positions that are in good agreement with other software by taking raw guider frames and computing the centroids for these frames with IRAF imexam. We have found the imexam centroids to agree with the guider code centroids to within 0.02 pixels. This represents agreement to within 2.8 milliarcseconds, which is two orders of magnitude smaller than the reported guide errors. Thus, the centroid software does not appear to be a problem. There currently is 3.3 kHz noise present in the guider images, but we do not believe that this noise is a significant contributor to the guide errors. This 3.3 kHz noise generates fluctuations in the guider images of 7.8±2.5; ADUs peak-to-peak. This noise level is not significant when compared with the flux levels obtained with the bright star tests that we mention above (thousands of ADU in the peak). Since the residuals that we see are present at the same level even when guiding on bright stars, we conclude that the 3.3 kHz noise is not a significant contributor to the current guiding errors. In addition to this, the common-mode motions imply that at most, errors caused by sky background or camera noise contribute less than 0.1 arcseconds to the tracking errors.
Third, the common-mode behavior of the tracking errors argues that the rotator is not the cause of the tracking errors that we measure. Errors in the rotator motions will show up preferentially as differences between the SPIcam x-axis centroids and the guider x-axis centroids. This is not seen in the data. The agreement between the x-axis data is as good as that for the y axis. Rotator errors will also tend to exhibit a phase lag between the motions seen on the guider and those seen on the science instrument. No such lag appears in the data.
Fourth, the common-mode behavior is one indication that variations in the seeing are not the cause of our large scatter in the centroid positions. The stars observed by SPIcam and the guider are separated by 0.31 degrees. Their fields of view experience different seeing variations. Seeing variations would cause differential motion between the science instrument and the guider. Common-mode motions are indicative of errors in the telescope tracking.
The 1-
tracking errors for each
axis are labeled
on Figures
4 and
5.
Since there is no apparent correlation between the x- and y-axis
positions, the total tracking errors are just the quadrature sum of
both the x- and y-axis tracking errors. In the case of Figures 4 and
5, we have a total tracking error of 0.23 arcseconds on the guider.
This represents a contribution of approximately 0.53 arcseconds to
the FWHM seeing.
On the surface, these results appear to be worse than those quoted from a measurement of guiding performance which was done back in September 1995 by Al Diercks et al. In 1995, using less sensitive, poorer quality guide camera made by SpectraSource, Diercks et al. did similar tests and found an rms deviation of 0.15 arcseconds while guiding. As we pointed out above, we have made single measurements of the guider performance which have rms deviations equal to or smaller than those made by Diercks et al. But the average performance from approximately 16 measurements is worse. The earlier tests were rather limited in extent (a single run) and they could easily have been taken at a position or time when the telescope tracking was above average. Similarly, early tracking tests on the 3.5-m indicated a best performance of 0.1 arcseconds peak-to-peak (Figure 10.2 of the SDSS proposal, volume 1). However, there they also noted that the data shown was taken from a period when "...the tracking was particularly smooth." All of these results are consistent with the viewpoint that the average closed-loop guiding of the telescope has not changed significantly since the telescope was built. But past reports of its behavior have been somewhat optimistic.
It should be noted that the comments above are focused exclusively on our closed-loop tracking and this behavior is somewhat different from open-loop tracking and pointing of the telescope which have seen recent improvements. Although these issues are not completely seperate, the open-loop tracking errors are dominated by errors in the pointing model for the telescope whereas this is not the case for closed-loop tracking.
We need better performance from the guider if we are
to ever make good use of sub-arcsecond seeing conditions at the site.
The current errors are not sufficiently small to eliminate the
telescope tracking from contributing a significant increase in the
widths of stars on the science instrument. Normally the seeing is
reported in units of FWHM of a stellar profile on the science
instrument. The conversion between the FWHM of a Gaussian profile and
the 1-
radius of this same profile is given
by 
. If the seeing is producing stellar
profiles with 1 arcsecond FWHM, this is equivalent to a 
arcseconds. If we assume that the tracking
errors are random, then the seeing profile will add in quadrature
with the tracking errors. For example, if we assume 1 arcsecond
seeing and a 
of 0.24 arcseconds for the tracking
errors while using the guider, then we will have a stellar profiles
with 
arcseconds. This is equivalent to
1.14 arcseconds FWHM. Under these conditions it can arguably be said
that the guider only slightly degrades the stellar profiles. But, in
sub-arcsecond seeing conditions we would have significant
degradation. If the seeing drops to 0.6 arcseconds FWHM, a 0.24
arcsecond tracking error produces a 0.82 arcsecond FWHM stellar
profile. If we assume a goal of only a 20% degradation in the stellar
PSF when the seeing is 0.6 arcseconds, then we require the tracking
errors to be less than 0.17 arcseconds.
There are numerous possible causes for the tracking errors which we report above. At a fundamental level, we will always be limited in the accuracy of our guiding by the strength and stability of the stellar point spread function (PSF). As I implied above, we do not believe that the errors that we are reporting are intrinsic to the signal strength of the guide stars themselves because we get good agreement between simultaneous measurements of the telescope position made with the guider and with SPIcam. Also, the errors reported have been shown to be independent of both guide star brightness. Using 2 and 5 second guider exposures (with peak guide star fluxes of 300 ADUs) produce results similar to those obtained with 20 second guider integrations on the same star. In addition, tests with fainter stars (peak fluxes of only 30 ADUs) give similar error residuals.
We have extensively looked into the possibility of systematics in the tracking errors that we have observed. There are no obvious correlations of the tracking errors with position in the sky, parallactic angle, nor velocity of a particular telescope axis (including the rotator). Early in the process of collecting guiding data it appeared that the tracking errors might be correlated with wind velocity. However, subsequent tests seem to disprove this hypothesis.
Figure
6 shows a plot of the wind velocity versus
the measured tracking errors. The numbers above each point in Figure
6 show 
, the difference in degrees between
the wind direction and the slit direction. The initial dataset did
not include the points marked with solid circles. If one ignores
these points and the point with the largest tracking error, then a
trend might be indicated in the data. We noticed that the large error
point was initially unique in the data set in the sense that it was
the only point measured while the wind direction was directed
straight into the dome slit. This spurred us to test the wind
correlation hypothesis by taking the data points marked with solid
circles. The new points appear to disprove the hypothesis. The
tracking error curves for the data points with 
and 
have already
been displayed in
Figure
3. Other plots of the tracking errors as a
function of position on the sky (not including the region near the
zenith!), rotator position, azimuth position, altitude position, and
the associated axis velocities show even worse correlations than that
which is seen in Figure 6.
Guiding tests during twilight were done to measure the sensitivity of the guiding to changes in the background sky level. A bright star was chosen at a moderate zenith angle and within 1 hour of the meridian in the southern sky. Guiding was turned on and measurements were taken until significant variations were noticed in the guider data. Figure 7 shows the sky background measured on the guider as twilight approached. The arrow on the figure shows the point at which the sky level became equal to the peak intensity in the guide star profile. The brightness increase shows a logarithmic behavior, perfectly consistent with theoretical expectations (Tyson and Gal 1993). The sky level had risen to approximately 3 times the peak stellar intensity before the test was terminated.
Figure
8 shows the x and y guider centroid positions
measured during the period covered in
Figure
7. The 1-
deviations for each
axis are shown on the figure. No evidence for effects of the rising
background level can be seen in the centroid data. The total
dispersion of the data points (
arcseconds) is
lower than our average quoted tracking error, but quite consistent
with tracking errors measured earlier on that same night.
Figure 9 shows the stellar widths as measured by the guider software during the twilight sky test. The average seeing during this test was 1.29 arcseconds. The variations in measured FWHM widths are, for the most part, quite typical of the variations seen during normal nighttime guiding. But, at the end of the plot, starting near 56 minutes after starting data acquisition, the effects of the rising background level can be seen. The onset of errors in the width determination are quite sudden and catastrophic. The third to the last point was obtained when the background level was 2.7 times the stellar peak intensity. By the time the second to the last and last points were obtained, this ratio had risen to 2.9 and 3.0, respectively.
A similar story is seen in the stellar peak measurements which are shown in Figure 10. If the last three measurements in Figure 10 are ignored, the mean stellar peak was measured to be 21163±3376; ADUs. This dispersion represents a 16% fluctuation in the measured peak values. The integrated intensity was found to be 219164±10698, representing a 5% variation. These are not good photometric conditions, but they are not unusually bad. It is a well known phenomenon that seeing deteriorates at the onset of twilight due to the associated increase in atmospheric turbulence. However, the results here are more consistent with failure of the fitting software to handle the large increase in sky background. At about 56 minutes into the data acquisition, the stellar widths begin to rise. This is what one would expect from deteriorating seeing. But, the stellar peak intensities and the integrated stellar flux both begin to drop to levels far below the measured statistical fluctuations. Since the stellar flux is a constant, the decrease in integrated flux is a clear sign that the fitting routine began to do a poor job of estimating the stellar profile at this time.
Finally, we have also investigated the temporal and statistical characteristics of the tracking errors in an effort to shed light on their possible causes. Many of the tracking error measurements do not appear to come from segments of randomly varying data. Figure 11 is one example. This is the power spectrum of the data shown in Figures 4 and 5. The x-axis data show a strong peak at 0.00637 cyc/sec (corresponding to a period of 157 seconds) while very little is seen on the y-axis. A random distribution will have evenly distributed power at all frequencies. This power spectrum is reproducible only in a very crude sense. Just 1 hour prior to when the data of Figure 11 were taken, in a close part of the sky, a data set taken under identical conditions shows a spectral peak in the x-axis at 0.0075 cyc/sec (133 seconds), while again showing almost nothing in the y-axis. In other words, if two measurements of the power spectrum are made close in time, then they are similar, but not identical.
The power spectrum of Figure 12 gives us insight as to the true source of the long period (approximately 140 seconds) variations and insight into why this low frequency component seems to be variable in time. We will show below why we believe that this long period variation is simply an artifact of the data acquisition while the true noise source lies at higher frequencies, somewhere above 0.02 cyc/sec.
The total power contained in the low frequency spectrum of Figure 11 can be explained quantitatively as the result of aliasing from higher frequency components. Immediately after the data of Figure 11 were taken, using the same star with a higher sampling rate (and shorter guider integration time), the power spectra of Figure 12 was measured. Most of the power in the high frequency spectrum of Figure 12 is contained in harmonic multiples of the peak seen in the low frequency plot. Marked on Figure 12 are the locations of the first 13 harmonic multiples of the peak seen in Figure 11. Most, but not all, of the harmonic multiples have peaks associated with them. In particular, peaks associated with the 7:1 and 10:1 multiples are apparently missing. This shows qualitatively that the power in the low frequency spectrum could be coming from aliasing of the higher frequency components. But we will show that the absolute values of the two power spectra are also consistent with this picture.
To compare the absolute values shown in the two power
spectra you must take into account both the differing sampling rates
as well as the differing averaging. In the first case, 20 second
exposures were taken with a sampling rate of 24.7 seconds. In the
second spectra, 2 second exposures were taken with a sampling rate of
5.5 seconds. The absolute units of FFT power spectra shown are
(arcseconds/sample
time)2. With
differing sample rates, the easiest way to compare the absolute
values of the power spectra is by looking at the standard deviations
of each data set, which are directly related to the integral power in
the power spectrum in the following way. If 
is the standard deviation of a data set and 
its power spectral frequency components,then the
relationship between these is given by:
(5)
where N is the total number of points used to compute the FFT. It is thus possible to compute the "sigma" that would have been generated by only a portion of the frequency spectrum by using only those frequency components in the sum.
The varying exposure times of each data set
correspond to different averages of the input signal. An exposure is
equivalent to multiplying the signal with a boxcar function of the
duration of the exposure, which we will call 
. In Fourier space, this is equivalent to
convolving the input signal by a sinc function of the form:
(6)
where 
is the Fourier frequency
variable in radians/sec. The first zero of this sinc function is
found at 
. Only a few percent of the total
integral of the sinc function is found outside its first zero. We can
therefore approximate the effects of this sinc function as a boxcar
in Fourier space with width 
. In this
approximation we simply assume that the input signal is zero beyond a
frequency of 
cyc/sec. For the case of a 20
second exposure, this corresponds to 0.05 cyc/sec.
We are now prepared to quantitatively compare Figures 11 and 12. We will do so by using Figure 12 to predict the amount of power that we should have measured for the conditions of Figure 11. In the case of Figure 11 the power spectrum is plotted from 0 to 0.02 cyc/sec, owing to the sampling rate, but the 20 second exposure times mean that these measurements are sensitive to variations which extend all the way out to 0.05 cyc/sec. If we sum the power in Figure 12 from 0 to 0.05 cyc/sec, this gives us an equivalent averaging to the measurements of Figure 11. Applying this sum in equation (5) yields an equivalent standard deviation of 0.20 arcseconds. This is in reasonable agreement with the measured deviation of 0.23 arcseconds found in Figure 11. A repeat of these two measurements taken later in the night yielded similar results with the high frequency spectrum predicting a deviation of 0.22 arcseconds while the actual deviation measured in the low frequency spectrum was found to be 0.26 arcseconds. In both of the tests the actual low frequency deviation was approximately 15% higher than the predicted deviation. I believe these discrepancies are the result of the approximations used above, but they could also be due to slow temporal variations in the noise source itself. In either case, I contend that the low frequency power spectra are quantitatively consistent with a noise source that was roughly constant in spectral distribution and amplitude over the whole period it took to measure both low and high frequency spectra (ie. over approximately 45 minutes time). Small changes can be seen between the two sets of data. Therefore, over periods of 2 hours (the time between the starts of these two data sets), noticeable variations are present.
It is worth noting here that the data of Figures 11 and 12 are consistent with aliasing in one other way. The low frequency peak that is so prominent in Figure 11 appears in Figure 12, but with a much diminished absolute power. In fact, if equation (5) is used to compare the power in the same frequency intervals, we find a deviation of 0.23" between 0 and 0.02 cyc/sec in Figure 11, but a deviation of only 0.056" over the same interval in Figure 12.
The picture that emerges from these spectra is that the peaks seen in the low frequency power spectra are almost completely the result of aliased high frequency structure which is slowly (on a time scale of 2 hr.) changing with time. In fact, Figure 12 implies that there is almost no significant power at the lower frequencies. A high frequency measurement of the guider errors was also made right after we took the spectrum in which the 0.0075 cyc/sec peak was found. Similar results were obtained when comparing this low frequency spectrum with its matched high frequency measurement; i.e. 9 out of 11 of the harmonic multiples have power spectral peaks associated with them, the absolute powers of the low and high frequency measurements match when corrected for variable sampling and averaging, and there is very little power in the low frequency portion of the high frequency spectrum. I need to emphasize here that the two high frequency spectra differ in a way that is consistent with the drifts seen in the low frequency peaks. In other, words there are no fixed high frequency peaks in common between the two high frequency measurements. This made acquiring an understanding of these data sets particularly difficult!
We conclude from this spectral analysis that the
tracking errors are not random in nature and that noise with temporal
frequencies in the range of 0.02 to 0.05 seconds are what are
contributing the most significant part of the tracking noise in 20
second guider exposures. Unfortunately, we cannot conclude from these
data that the noise is being generated at these frequencies. In fact,
the high frequency spectra show that the noise increases as you
decrease the guider exposure time. The ratio between the short and
long exposure deviations is 1.38. The exposure times here differ by a
factor of 10. Therefore, the noise would appear to vary only weakly
with the exposure time. The measured ratio implies a variation
proportional to 
.
Figure
1 shows that for the right conditions this
might be consistent with turbulence being the source of the noise.
The slowest variation in Figure 1 is proportional to 
, but as was mentioned before, its very hard to
reconcile the absolute values of the errors with this
hypothesis.
It is possible that the noise source has its fundamental frequency well above what we have measured. From this analysis we can only say that the noise is being generated at frequencies above 0.02 Hz. Acquiring further guider data is unlikely to shed more light on this aspect of the problem because we cannot acquire the guider data much more quickly than we already have. And, even if we could, we would soon run into seeing limitations if we were to sample more rapidly. We also conclude from this analysis that the frequency spectrum of the noise being generated is not stable over a long time scale (i.e. for periods longer than approximately 1 hour). However, if we are dealing with only high harmonics of the true noise source, then its possible that the amount of temporal drift that we are seeing is consistent with a fairly constant frequency source of noise. Under these conditions, small frequency variations of the fundamental cause much larger variations of the higher harmonics.
My analysis leads me to the conclusion that for reasonably faint stars (at least down to 13th magnitude) the guider is producing measurements of the stellar centroids with accuracies that exceed our current control capabilities. We appear to have a noise source which is increasing the image motion during guiding by a factor of about 2 over what we would expect to see from atmospheric turbulence alone. The noise is not random in nature but it does vary with time. The noise appears to be introduced into the system at frequencies above 0.02 cycles/sec. At times the telescope guiding does appear to meet our needs, but on average it is falling short. Press et al. (1988, pg. 440) point out that the variance of power spectra density measurements taken from a single spectrum as we have done with Figures 11 and 12 are very large (100%!). This is the main reason for taking any conclusions based on those spectra with a (large!) grain of salt. The solution to this is, unfortunately, taking much longer data sets in order to get multiple measurements of the power spectrum of the noise. If we continue to measure average noise levels that are around 0.24 arcseconds, then it would be worthwhile to obtain two very long (>2 hr.) sets of guider data with quick exposure times (5 secconds). Both of these data sets should be taken through similar ranges of telescope coordinates. This would allow for the spectral averaging that is necessary to beat down the intrinsic noise levels in any power spectral analysis.
Atmospheric turbulence fits some of the qualitative behavior of the noise such as its weak dependence on exposure time. But, turbulence does not appear to explain magnitude of the errors observed. Also, turbulence offers no explanation for the non-random nature of the error signals that we have measured.
We should investigate the possibility that the telescope servo control is not capable of reliable motions of the telescope below 0.1 arcseconds and the possibility that there are round-off errors in the control software.
It should be pointed out here that in the first week of January, 1998 a software bug was found and fixed in the TCC code which deals with the guider control loop. The error was in how the TCC was computing a running average of the tracking errors that the guider was providing. At the time, the weighting that was being applied to in the running average was such that it did not appear that the error would have much impact on the guide corrections that were being made. The fix was to remove the code that computed the running average. At the same time code which enables the user to control the magnitude of the corrections that the TCC applies was included. At this time the guide_gain parameter was added. This was done primarily to enable the guide-on/guide-off tests reported here. Since this fix was applied, only one night, January 9, 1998, of appropriate guider data has been acquired and analyzed.
On January 9, 1998 six tracking error measurements were made. They show an average tracking error of 0.17±0.02 arcseconds, which is consistently below the average tracking error found from the entire data set. As can be seen in Figure 6 the dispersion of the tracking errors is fairly large. So, it is not yet known whether the most recent data truly indicate improved guider performance. Several nights worth of data are needed to tell. This data can probably be acquired "parasitically" by simply waiting for users to use the guider for long (>1/2 hour), uninterrupted periods of time and then culling the data from the TCC logs. The night assistants should be forewarned of the importance of these opportunities and asked to let me know when such data sets are available.
One last comment should be made concerning the 3.3 kHz noise in the guider images which was noted in section III. Since the time of the early guider measurements, efforts have been made to eliminate this noise in the guider. It appears that these efforts have been successful. Originally, it was thought that putting the liquid coolant pump to the guider on the telescope UPS power was a bad idea. It was feared that the pump would contaminate the UPS power. An isolation transformer was placed in the power line to this pump and it was put on normal, unconditioned line power. When fans were added to cool the coolant heat exchanger, the 3.3 kHz noise increased dramatically. It was noted that these fans were powered from the same line as the coolant pump. We have since placed the pump and fans on UPS power. This appears to have decreased the 3.3 kHz noise back to the original levels by cutting a ground-loop circuit. The isolation transformer now serves to keep the fan and pump noise out of the UPS power line. This noise has not been entirely eliminated, as evidenced by the background noise seen in Figure 13, which we introduce below.
In an effort to help monitor the throughput of the APO 3.5-m telescope, we have measured the R magnitudes of several stars in the open cluster RNG0188, which is at a declination of nearly 85 degrees. It is hoped that by periodic monitoring of this cluster, we can sense the general throughput of the telescope. Owing to its high declination, this cluster is available for observation at APO all year. It will always remain at an airmass of about 2.0, which should simplify corrections for airmass. We present in Figure 13 a finding chart for the stars measured. In this figure north is up and east is to the left. The stars in the field are labeled in order of decreasing brightness. In Table 1 we give the R magnitudes and coordinates for each star as seen in Figure 13. The R magnitudes shown in Table 1 are corrected to an airmass of 0. The flux measurements are uncorrected (ie. the flux measured at 2.05 airmasses) and are given in ADUs. The R band extinction correction at the time of these measurements was found to be 0.09 ± 0.01 magnitudes/airmass. The measurements were taken at an airmass of 2.05. The guider was binned 3x3 and a 10 second exposure was used to acquire the data in Figure 13. Aperture photometry was done using the IRAF routine qphot. The fluxes quoted come from integrating the sky subtracted flux within an aperture diameter of 3.8 arcseconds (9 binned pixels). SPIcam was also used to measure this same field on the same night in the R band. For stars in Table 1 brighter than 16.0 magnitudes the SPIcam measurements agreed with the guider measurements to an error of 4%. For the dimmer stars, the guider measurements quoted here are less accurate owing to the limited fluxes measured on the guider. This standard field is centered at the coordinates RA 00:43:57.9 DEC 84:58:05 (1950).
Hufnagel, R.E. 1978 "Propagation through atmospheric turbulence", The Infrared Handbook, ed. W.L. Wolfe and G.L. Zissis, (Washington: U.S. Government Printing Office), p. 6-1, 6-56.
Martin, H.M. 1987 "Image motion as a measure of seeing quality", P.A.S.P.,99, 1360-1370.
Press, W.H. et al. 1988 Numerical Recipes in C, The Art of Scientific Computing, Cambridge University Press.
Roddier, F. 1981 "The effects of atmospheric turbulence in optical astronomy", Progress in Optics, XIX, ed. by E. Wolf, p.281-376.
Tyson, N.D. and Gal, R.R. 1993 "An exposure guide for taking twilight flatfields with large format CCDs", Astron. J., 105, 1206-1212.
