Sloan Digital Sky Survey Telescope Technical Note 19920608-03
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.
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:
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.
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.
