Telescope Tracking Smoothness
Sloan Digital Sky Survey Telescope Technical Note
19920608-03
Walter
Siegmund
Contents
Introduction
The SDSS project requires relative astrometry (over the 3°
plug plate diameter) of about 70 mas RMS according to our fiber
positioning error budget (SDSS Technical Note 910903-01). Of course a
lot of interesting science is possible if one can do better than
this.
The SDSS NSF proposal points out that Hipparchos stars can be used
to define the trajectory of the SDSS telescope. Such stars will be
imaged about every 44 seconds on average. Each measurement will have
an accuracy of perhaps 30 to 40 mas RMS. If the tracking performance
of SDSS 2.5 m telescope is better than this over the "relevant low
frequencies", several successive measurements of Hipparchos stars can
be averaged to reduce the effect of seeing errors on the individual
measurements.
I assume that the relevant frequency range is 3 to 300 mHz.
Hipparchos stars occur at a 23 mHz mean rate. Thus the telescope
trajectory is well defined by Hipparchos standards at frequencies
below 3 mHz. Frequencies above 300 mHz are filtered out by the
astrometric CCD integration time of 7 seconds. (Work at the Naval
Observatory suggests that an integration time of 7 seconds or less
will strongly limit astrometric accuracy because atmospheric image
motion is not averaged out in that time. I think this implies that
the atmosphere rather than the telescope is likely to be limiting at
frequencies above 300 mHz.)
The implication of this argument is that if telescope tracking is
better than 30 to 40 mas RMS over this frequency range, we can
improve the astrometric accuracy of the survey by averaging over
Hipparchos stars. If tracking is only 30 to 40 mas RMS from 20 to 300
mHz, the astrometric accuracy will be limited to this level. However,
this is more than adequate for the core science of the survey.
Telescope performance
We propose the following tracking error budget for the telescope.
See Table 1. This error budget applies to scales
of approximately 0.01° to 1° of axis motion. (The actual
range of axis motion that corresponds to 3 to 300 mHz will depend on
the particular stripe being imaged.) It assumes that significant
sources of repeatable error have been removed. Each source of error
is assumed to be independent and to add in quadrature. (This
assumption is a bit dubious in the case of the azimuth disk. Error in
the radius of this component causes both azimuth axis wobble and
encoder error.)
The largest components in the error budget are the following:
- Drive disk high frequency radius error. The WIYN azimuth drive
disk is expected to be better than 20 µm peak-valley as
ground on a hydrostatic bearing turntable at WestTech Gear in Los
Angeles (subcontractor to L&F Industries). I assume that the high
frequency (greater than 4 cycles/revolution) RMS radius error over
1° is one percent of the peak-valley error or 200 nm.
- Guide roller radius error. Little information exists regarding
what can be expected for this component. It should be possible to
make a small part more accurately than the large drive disk and
the number used, 50 nm RMS, reflects this. As an example of
commercial practice, individual Grade 10 balls for (very high
quality) ball bearings are specified at 125 nm radius peak-valley.
This also suggests that 50 nm RMS is possible, but difficult. The
manufacture of ball bearings is nearly state of the art.
- Servo error. The servo error for each axis is assumed to be
one encoder count RMS. This is probably conservative in the
absence of wind-induced tracking error. Charlie Hull reports one
encoder count peak to valley for the 3.5 m instrument rotator.
This is about 0.2 encoder counts RMS. A better value for this
effect for the 3.5 m telescope main axes should be available in
the near future.
Table 1: Tracking error budget
Component Component error (nm RMS) Effect (mas RMS)
Az axis wobble
Drive disk high freq. error 200 17.17
Guide roller error 50 8.58
Lower bearing high freq./nonrep. error 50 4.29
Alt axis wobble
Bearing high freq./nonrep. error 50 4.86
Az encoding error
Drive disk high freq error 200 4.59
Encoder capstan error 50 5.73
Encoder error 1.41
Az servo error 10.00
Alt encoding error
Drive disk high freq error 200 5.83
Encoder capstan error 50 7.28
Encoder error 1.41
Alt servo error 10.00
Rotator bearing error 400 6.59
Rotator encoding error
Drive disk error 400 1.54
Encoder capstan error 100 0.39
Encoder error 0.10
Rotator servo error 0.52
2ry actuator high freq. error 3.5 1.45
1ry actuator high freq. error 3.5 1.85
Total 28.41
In Table 1, the error values are for
individual components, e.g., each azimuth guide roller, each mirror
actuator, etc. In estimating axis drive disk and axis bearing errors,
I assumed that the high frequency/non-repeatable error was one
percent of the peak-valley error for the component. For the azimuth
and altitude axis bearings, I used the 5 µm peak-valley error of
a RBEC class 5 bearing. The accuracy requirements for the large
diameter instrument rotator bearing, the rotator drive disk and the
encoder capstan were relaxed from from the level of other similar
components. The accuracy required for the rotator is much lower than
for the main axes. I used the specifications for the 3.5 m telescope
encoders, Heidenhein ROD 700, for the 2.5 m encoders. To reduce the
effect of encoder and encoder capstan error, I assumed two encoders
per axis. The error for the mirror actuators is one percent of the
bearing and screw once per turn errors. I used the specifications for
the precision ground screws selected for the 3.5 m telescope. An
article on the Keck telescope (Proc. SPIE 428) gives measurements
showing that roller screws have resolution of 4 nm suggesting that
the 3.5 nm used here is plausible.
Data relevant to this error budget are mostly unavailable so the
estimates given herein are quite uncertain. However, the budget is a
powerful tool that indicates which components that are most likely to
limit tracking performance and should be emphasized during
fabrication and inspection. The data (for a limited time interval)
shown in Figure 1 corresponds to a performance of
better than 30 mas RMS for 330 seconds of time. This level of
performance in a telescope for which extreme care was not taken in
the manufacture of critical components suggests that we may do
somewhat better than the error budget proposed herein.
Fig. 1: An interval of excellent tracking
performance of the APO 3.5 m telescope as measured from the centroids
of star images. Images were obtained at 15 second intervals. The
integration time was 1 second.
Appendix: Effect of errors in the encoder roller
and drive disk on tracking.
Definitions:
Angular position of the telescope axis
ø Angular position of the encoder
'
Angular position of the telescope axis measured by the encoder
e
Angular position error of the telescope axis
R()
Radius of the drive disk
r(ø) Radius of the encoder roller
,
Mean radii of the encoder roller and drive disk
The reduction ratio between the telescope axis and the encoder is
defined as follows.
(1)
A small change in the angular position of the telescope axis
causes a change in encoder angle according to the following
equation.
We can integrate this equation over an angle
to find the angular position of the encoder. Dividing this by the
encoder reduction ratio gives the angular position of the telescope
axis as measured by the encoder.
(2)
The radius of a roller or disk can be expressed in terms of the
mean radius of the roller plus a function of roller angle giving the
departure of the radius from the mean. The ratio in the integrand can
rewritten as follows.
The roller errors, r
and R,
will be small compared to the radii ,.
Therefore, the last equation can be simplified by neglecting terms
that are second order in r
and R.
The error in the angular position of the telescope axis as
measured by the encoder can be found by substituting this last
expression into eq. 2 as follows.
(3)
In principle, R and r can be expressed as harmonic series.
These expressions can be substituted into eq. 3 and the result
integrated. Eq. 1 allows
to be eliminated from the final equation.
This result indicates the amplitude of the tracking error due to a
radius error of a given magniture is the same whether it occurs on
the drive disk or the encoder roller, although the frequency of the
effect will be different by the factor n. It is plausible that this
should be true since errors on the encoder roller average to zero in
one revolution of the roller.