The ideal, direct way to determine astrometric performance is simply to compare star positions determined with SDSS with those determined externally. Since positions of objects in the r' photometric bandpass are required, it is necessary to compare positions of stars which are fainter than 14th magnitude (and indeed down to 19th to fully satisfy the requirement). There are no wide-area astrometric catalogs available for stars fainter than 14th which are accurate enough to verify SDSS astrometry.

However, the Naval Observatory is presently constructing an all-sky CCD catalog ( The USNO CCD Astrograph Catalog or UCAC) which reaches fainter than 16th magnitude. UCAC appears to have systematic errors of no worse than about +/- 15 mas, and the precision ranges from 20 to 70 mas for stars between 14 and 16th magnitude.

At present, the survey is underway in the Southern hemisphere and ome of the SDSS equatorial scans covered regions where UCAC observations have been made. For example Run 745 was an equatorial parked scan conducted UT March 20, 1999. SDSS positions were matched against the UCAC preliminary catalog and the results are shown in the following figure:

These results are extremely encouraging, but we should realize that such a run represents a best-case scenario for the following reasons:

* The telescope was parked and clamped on the equator at the meridian during the observations so that no telescope motion was involved: telescope pointing or tracking errors were not a consideration.

* A number of astrometric calibration fields lie along the equator (so-called FASTT fields, dense astrometric grids of stars produced by the USNO for the purpose of calibrating the SDSS imaging camera geometry), so that a well-calibrated focal-plane model, generated from the very data being reduced, was available to be used for the astrom pipeline reductions.

For a worst-case scenario, we turn to Run 1449, which was observed recently (UT May 3, 2000) on SDSS' southern-most stripe (Stripe 1). A comparison of astrometry from that run with 5555 UCAC star positions show a mean offset in RA of -11 mas, rms of 119 mas, and in DEC 7 +/- 139 mas. Frankly, this run was not particularly good from an astrometric viewpoint. The atmosphere was particularly troublesome. This is probably due to how far south the stripe is (DEC of -22.5), which means that the observations must be conducted at an airmass of 2.0. The following figure shows data from one of six Camera Columns for Run 1449. Note the significant excursions traced by the astrometric residuals. We believe that these are due to atmospheric fluctuations. The grossest such excursions are fitted and removed by the astrom pipeline, and the points here show the residuals after such fitting.

A possible further contributing factor with run 1449 is that the telescope was required to track a great circle which changed very little in azimuth or elevation during the run. The altitude axis rate ranged from only -0.1 arcsec/sec to 1.5 arcsec/sec, while the azimuth axis rate ranged from 1.3 to 1.4 arcsec/sec. Such slow rates may be difficult to sustain uniformly.

 We expect that the absolute astrometry may also suffer when the telescope is required to track at very high axis rates (i.e. near the zenith). In such circumstances, it is difficult to keep the instrument rotator positioned so that stars track straight down the camera CCD columns. Unfortunately there are no calibrated astrometric catalogs above the equator which reach faint enough magnitudes yet, and so we cannot do a direct comparison.

We can, however, compare SDSS-determined positions of stars which appear within the overlap regions of adjacent or interleaved strips. Run 1345 is notorious (at least among the astrometrists) for its high tracking rates (the azimuth rate exceeded 50 arcsec/sec at the beginning of the run). Comparing the positions of 953 stars from this run with matching positions from Run 1331 (an interleaved strip of the same Great Circle) yields a mean error in RA of -22 +/- 103 mas, and in DEC of 60 +/- 120 mas.

In summary, the absolute astrometry consistently meets the Drop-Dead requirement, surpasses the Science Goals Requirement when extreme conditions are not being encountered, and approaches the Enhanced Goals requirement under ideal conditions.

Relative Astrometry Performance

The relative astrometry between bands currently being delivered consistently betters both the Drop-Dead and Science Goals requirements and is flirting with the Enhanced Goals requirement in all but the u' band, even for the aforementioned run 1449, as shown in the following table. (The numbers here are taken directly from the calibrated object files for stars brighter than 19th magnitude.) Relative (CCD-to-CCD) Astrometry

   Run 1449: Relative (CCD-to-CCD) Astrometry (mas)

Camera	r'-u'		r'-g'		r'-i'		r'-z'
Column	RA	Dec	RA	Dec	RA	Dec	RA	Dec
1	+4.2 +/- 48.8	19.0 +/- 75.3	+5.3 +/- 62.3	-26.6 +/- 54.4	+5.4 +/- 34.6	-5.6 +/- 46.9	-11.0 +/- 48.5	14.8 +/- 49.5
2	-4.9 +/- 53.1	18.7 +/- 75.3	-9.1 +/- 65.3	-22.1 +/- 58.5	-0.6 +/- 41.6	-2.7 +/- 52.1	-13.3 +/- 52.2	13.6 +/- 56.4
3	-1.1 +/- 46.4	11.1 +/- 66.0	-9.8 +/- 42.3	-15.4 +/- 51.3	-4.4 +/- 34.1	-0.6 +/- 48.2	-5.6 +/- 34.8	11.5 +/- 52.3
4	0.1 +/- 51.4	-2.1 +/- 66.1	-9.5 +/- 58.0	-15.6 +/- 52.0	-9.3 +/- 38.5	3.1 +/- 46.4	-10.9 +/- 46.7	8.5 +/- 49.1
5	6.9 +/- 48.1	-8.4 +/- 66.3	-3.9 +/- 61.0	-22.6 +/- 65.6	-3.0 +/- 35.8	1.2 +/- 47.2	-3.4 +/- 49.2	8.5 +/- 57.4
6	8.4 +/- 43.8	5.0 +/- 73.6	-4.1 +/- 35.4	-27.1 +/- 51.5	-6.8 +/- 34.7	-1.9 +/- 50.1	0.1 +/- 34.1	-1.9 +/- 52.3

It is worth noting, however, that while these formal results are encouraging, they hide much that is interesting if not troubling. The next figure shows the distribution of offsets in Right Ascension and Declination between the r' and g' CCDs in column 3 for run 1449. The distribution of errors appears to be nicely behaved.

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 However, the next figure shows the same offsets as a function of RA for about one degree (in RA) of the scan. While Run 1449 was observed on a great circle with a modest amount of inclination to the equator (22.5 degrees), changes in RA correspond quite closely to changes in clock time at nearly a sidereal rate (1 degree of RA is nearly 4 minutes of scanning). Note the swings on the order of +/- 100 mas over short time scales. We believe, though are not 100% convinced, that this is the atmosphere at work. We continue to investigate these problems.

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The astrometric pipeline has changed rather significantly as a result of the commissioning phase of the Survey. While we hope that the pipeline is now quite stable, we are not naive enough to think that our job is done. As we encounter different observing conditions we may well need to make additional changes.

In addition to unanticipated modifications, there is one area where we need and intend to make changes in the next few months. As mentioned above, a fundamental problem in attempting to evaluate astrometric performance is that there is no "truth table." There is no wide-area astrometric catalog which goes faint enough with which we can directly compare SDSS positions. What we can do, however, is to compare the positions of objects determined on adjacent scans to get some measure of the internal consistency, if not the accuracy, of SDSS astrometry (there is a fair amount of overlap throughout the Survey regions, something like 40% of the Survey area will be imaged twice, not counting any duplicate scans). We can also compare SDSS astrometry with that of other wide-area surveys (e.g. digitized scans of Palomar Observatory Sky Surveys, 2MASS, etc.), though none of these surveys can claim sufficient astrometric precision to do more than assure us that our errors are not gross. At present, we are performing such tests in an off-line fashion. We need to integrate such QA tools into the pipeline for routine execution.

A possible means of improving the astrometry might be to change the transformations. The astrometric solution is modeled as a linear transformation from pixel coordinates to celestial coordinates, one transformation per CCD per frame. High-order terms model optical distortions (as a function of column) and DCR as a function of color. Various systematic effects cannot be completely removed by the adopted model. Typical seeing introduces systematics of order 20-50 mas. The rms error measure is, in fact, dominated by systematic errors due to seeing. DCR in the g' filter can introduce systematic errors of up to 20 mas with color. Optical distortions can also introduce systematics of the same order. CCD charge transfer inefficiencies are also problematic and are currently under active investigation. We are currently considering whether it would be worthwhile to adopt a non-linear transformation to cope with these systematics.

When faint, all-sky astrometric catalogs, such as UCAC, become available, it will be possible to determine the transformation from photometric CCD row/column to celestial coordinates directly, by-passing the mapping step from astrometric CCDs to photometric. Atmosphere aside, it is the astrometric CCD mapping which is the largest contributor to our error budget. This change should decrease our astrometric residuals for at least two reasons: * the density of stars increases dramatically with magnitude so that the new catalog contains many times the number of reference stars thus allowing us to better track atmospheric excursions and,

* the differences in integration time for the astrometric and photometric CCDs will cease to be an issue.

When UCAC is completed in the Northern hemisphere (this should happen in 2003), it will be a straight-forward exercise to re-derive the astrometric transformations from the new catalog (software is already in the pipeline to do this). Since the positions of objects in the databases are stored in device (CCD row/column) coordinates, it is a simple matter to generate new celestial coordinates by merely changing the transformation coefficients which are applied to CCD row/column to generate celestial coordinates.