Astrometry

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Introduction

A detailed description of the astrometric calibration is given in Pier et al. (2003). Portions of that discussion are summarized here.

Note that the default astrometry in DR8 has substantial errors, particular northward of around 41 deg in declination. Please see the description of the astrometry caveats below. We have fixed the astrometry for DR9 and the DR9 version of the astrometry is available through DR8 now.

The r photometric CCDs serve as the astrometric reference CCDs for the SDSS. That is, the positions for SDSS objects are based on the r centroids and calibrations. The r CCDs are calibrated by matching up bright stars detected by SDSS with existing astrometric reference catalogs. One of two reduction strategies is employed, depending on the coverage of the astrometric catalogs:

Whenever possible, stars detected on the r CCDs are matched directly with stars in the United States Naval Observatory CCD Astrograph Catalog (UCAC2, Zacharias et al. 2000), an (eventually) all-sky astrometric catalog with a precision of 70 mas at its catalog limit of r= 16, and systematic errors of less than 30 mas. There are approximately 2 - 3 magnitudes of overlap between UCAC and unsaturated stars on the r CCDs. The astrometric CCDs are not used.
If a scan was not covered by UCAC2, then in DR8 it was calibrated against the USNO-B catalog. This occurs northward of about 41 deg in declination. This choice introduced a large systematic shift in the declination direction (about 200 mas). See the caveats below for details.

The r CCDs are therefore calibrated directly against the primary astrometric reference catalog. FRAMES uses the astrometric calibrations to match up detections of the same object observed in the other four filters. The accuracy of the relative astrometry between filters can thus significantly impact FRAMES, in particular the deblending of overlapping objects, photometry based on the same aperture in different filters, and detection of moving objects. To minimize the errors in the relative astrometry between filters, the u, g, i, and z CCDs are calibrated against the r CCDs. Each drift scan is processed separately. All six camera columns are processed in a single reduction. In brief, stars detected on the r CCDs , are matched to catalog stars. Transformations from r pixel coordinates to catalog mean place (CMP) celestial coordinates are derived using a running-means least-squares fit to a focal plane model, using all six r CCDs together to solve for both the telescope tracking and the r CCDs' focal plane offsets, rotations, and scales, combined with smoothing spline fits to the intermediate residuals. These transformations, comprising the calibrations for the r CCDs, are then applied to the stars detected on the r CCDs, converting them to CMP coordinates and creating a catalog of secondary astrometric standards. Stars detected on the u, g, i, and z CCDs are then matched to this secondary catalog, and a similar fitting procedure (each CCD is fitted separately) is used to derive transformations from the pixel coordinates for the other photometric CCDs to CMP celestial coordinates, comprising the calibrations for the u, g, i, and z CCDs.

Note: At the edges of pixels, the quantities objc_rowc and objc_colc take integer values, contrary to standard practice.

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Important caveats on DR8 calibration

The DR8 astrometric calibrations are substantially degraded relative to the DR7 astrometric calibrations, particularly at declinations northward of about 41 deg. These errors have a much smaller, but non-zero, effect on the DR8 proper motions. We have completed and made available ahead of schedule in DR8 the DR9 version of the astrometry.

DR7 (but not DR8) was recalibrated astrometrically against the second data release of UCAC (UCAC2; Zacharias et al. 2004) plus a supplemental set of UCAC results covering the Northern Celestial Pole at declinations northward of about 41 deg (an internal USNO product known as "r14"). While the systematic errors for UCAC2 are not yet well characterized, they are thought to be less than 20 mas (N. Zacharias, private communication). UCAC2 also includes proper motions for stars with δ < +41º. For stars at higher declination, proper motions from the SDSS+USNO-B catalog (Munn et al. 2004) were merged with the UCAC2 positions. With these improvements, DR7 astrometry has statistical errors per coordinate for bright stars of approximately 45 mas, with systematic errors of less than 20 mas.

Unfortunately, in DR8 these improvements were not properly incorporated. In particular:

Northward of a declination of about 41 deg, we did not calibrate to the UCAC r14 catalog, but instead to the USNO-B catalog. There is a substantial shift between the UCAC and USNO-B systems, of about 250 mas in the declination direction. The UCAC system is in much better agreement with the Tycho-2 system.
Color terms were not included in the transformation from position on the detector to right ascension and declination. This causes 10 to 20 mas systematic errors with color in catalog positions. Systematic errors of similar size are introduced in the measure of position offsets between filters; the errors are somewhat smaller in i and z, and somewhat larger in u and g.
The UCAC2 proper motions in right ascension were incorrectly applied, introducing systemic errors in right ascension of 5 to 10 mas. For declinations above 41 deg, the SDSS+USNO-B proper motions were not applied at all, introducing systematic errors in both right ascension and declination of typically 20 to 40 mas, and as high as 60 mas.
Previous SDSS data releases based the catalog right ascension and declination values on the catalog objc_rowc and objc_colc frame coordinates. These coordinates use the r-band centroid for unsaturated stars brighter than r=22.5, but for stars that are saturated in the r filter but unsaturated in another filter, or fainter than 22.5 in r but better exposed in another filter, uses the centroid from an optimal filter. For DR8, the right ascension and declination values use the r-band centroid for all stars. This increases the statistical error for some stars fainter than r=22.5 over that in earlier data releases. For stars saturated in r but unsaturated in other filters, it can introduce systematic errors of up to 100 mas compared to previous data releases.

We have performed a detailed comparison of the large-scale astrometry errors relative to the UCAC2 and r14 catalogs. In the regions not covered by UCAC2 (starting above around 41 deg declination) the DR8 astrometry is offset 240 mas to the North and 50 mas to the West relative to the r14 catalog. On scales of about 0.25 deg, the scatter around this offset is about 80 mas in the declination direction and 94 mas in the right ascension direction. Some of that scatter is coherent on larger scales; if we unsharp mask by subtracting off the residual field smoothed with a Gaussian (FWHM = 3 deg), the remaining scatter is about 60 mas in either direction. A similar analysis in the regions covered by UCAC2 finds very small offsets (less than 10 mas), with the expected level of scatter (40 mas), and with no large-scale coherence to the scatter.

The figure below shows the DR8 astrometry and proper motions versus the DR7 versions as a function of position on the sky (DR8 minus DR7). The difference between the region covered by UCAC2 and the regions northward of 41 deg or so is clear: the DR8 astrometry is systematically 250 mas northward of the DR7 astrometry. There are also a few areas of extreme outliers between DR7 and DR8 in that region, which turn out to be catastrophic errors in the DR7 astrometry (in certain areas of runs 3358, 4829, 5960, 6074, and 6162).

	RA offsets	Dec offsets
Position
Proper Motions

Southward of 41 deg in declination, the systematic errors introduced in DR8 are typically smaller than or comparable to the 45 mas systematic errors that characterize the SDSS astrometry for brighter stars. Given these problems (which we plan to fix in a future data release), we recommend that users interested in precise astrometry, especially statistical studies of star positions at the less than 0.1 arcsec level, use the DR7 results. For many applications, however, the quoted positions should be acceptable.

Note in particular that the proper motions tabulated in the CAS are only mildly affected by these problems. The primary effects on the proper motions are to introduce an additional systematic error with color of order 0.5 mas/yr, and to introduce an additional source of error for stars with declinations above 41 deg, of order 1 mas/yr.

DR9 improvements now available

For DR9, we have rerun the astrometry for the full imaging data set, fixing the errors described above. These new data are available within DR8 in the special CAS tables: astromDR9 and properMotionsDR9. These tables can be joined with photoObj using objID to recover more correct astrometry and proper motions for DR8 objects.

To use the DR9 proper motions, simply replace properMotions in your SQL query with properMotionsDR9. For example, you can try that using the proper motions sample query.

To use the DR9 astrometry, you must use an SQL join to obtain the better astrometry. For example, the clean photometry sample query could be altered to:

SELECT top 10 PhotoObj.objID, PhotoObj.ra as ra_dr8, PhotoObj.dec as dec_dr8, g, clean, astromDR9.ra, astromDR9.dec
FROM PhotoObj
JOIN astromDR9 ON astromDR9.objID = PhotoObj.objID
WHERE CLEAN=1

In order to obtain the DR9 astrometry, we have applied the following corrections:

We used the UCAC2 and UCAC r14 catalogs across the entire survey, removing the systematic shift at northern declinations.
We included color terms in the same manner as included in DR7.
We corrected the application of proper motions to the UCAC stars used for astrometric calibration. Note that although we have new proper motions available for most of those using SDSS itself, we retained the UCAC proper motions for consistency. The UCAC astrometry was taken around the year 2000; for images taken late in the SDSS survey (up through 2009), this difference yields about 20 to 30 mas difference in the overall calibration.
We used objc_rowc and objc_colc for the astrometry.

The resulting residuals relative to DR7 are shown below. There are several catastrophic outliers, in the runs 3358, 4829, 5960, 6074, and 6162, which are errors in DR7.

	RA offsets	Dec offsets
Position (DR9)
Proper Motions (DR9)

In addition, there are remaining proper motion discrepancies at low Galactic latitude (the "SEGUE stripes"), particularly in the declination direction. These errors are due to DR7 photometric catalog errors that cause a number of stars to be misclassified as galaxies in the reductions for these runs (in rerun 648); since galaxies are used as the reference for the proper motion calculation, this error has the effect of making the stars appear to have slower proper motions than they actually do.

The DR9 values can be shown to be correct by comparing to photometric UV-excess quasar samples, which should return zero. Using the low-redshift (z < 2) quasars from Bovy et al. (2011), we find proper motions as a function of position to be nearly zero across the sky (see below). There is some declination proper motion detected at the 1-2 mas/yr level at low declinations, similar to that reported by Bond et al. (2010) and likely due to color-dependent refraction offset errors either in the USNO-B or SDSS data.

	RA PM	Dec PM
QSO Proper Motions (DR9)

Finally, note that when changing the astrometry, a large number of other photometric parameters change in principle (for example, sizes, surface brightnesses and position angles). In practice, these changes are very small, and for this reason the special DR9 tables included in DR8 do not have new versions of these parameters. The full DR9 data set will have these parameters updated to be consistent with the new astrometry.

Proper Motions

The SAS and CAS include proper motions for objects derived by combining SDSS astrometry with USNO-B positions, recalibrated against SDSS (Munn et al. 2004). These are given in the ProperMotions table in the CAS, and in the external catalogs directory in SAS.

As noted above, the proper motions in DR8 are slightly degraded in quality relative to DR7. However, improved DR9 proper motions are now available through CAS in the ProperMotionsDR9 table. As shown above, these proper motions are superior to both those in DR7 and in DR8.

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Calculating Errors for Individual Objects

The calibrations are performed in great circle coordinates. The estimated errors in the calibrations are given on a per-frame basis. The calibration errors in great circle longitude and latitude are given by the attributes muErr and nuErr, respectively (in arcseconds). These are in the photoField files in the SAS. These should be added in quadrature with the centroiding errors for individual objects to give the estimated total error in the position of a given object. The centroiding errors in great circle longitude and latitude are given by the attributes objc_rowcErr and objc_colcErr, respectively (in pixels; these should be multiplied by the focal plane scale of 0.396 arcseconds/pixel to convert to arcseconds). These attributes are in the photoObj files in the SAS.

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Calibration Equations

Astrometric calibrations are generated as a separate set of equations for each frame, converting frame row (x), frame column (y), and star color to catalog mean place great circle longitude (μ) and latitude (ν), in degrees:

for color < (color)₀:

x' = x + g₀ + g₁ y + g₂ y² + g₃ y³ + p_x color
y' = y + h₀ + h₁ y + h₂ y² + h₃ y³ + p_y color

for color > (color)₀:

x' = x + g₀ + g₁ y + g₂ y² + g₃ y³ + q_x
y' = y + h₀ + h₁ y + h₂ y² + h₃ y³ + q_y

μ = a + b x' + c y' ν = d + e x' + f y'

However, note that in these equations, for DR8 we did not account for the color term at all, which results in 10 to 20 mas systematic errors.

The transformation from (x, y) to (x', y') corrects for optical distortions (which, in TDI mode, are a function of column only) and differential chromatic refraction (DCR). For u and g frames, DCR is modeled as a linear function of color (u-g for u frames, g-r for g frames) for blue stars [(color)₀ = (u-g)₀ = 3.0 for u frames, (color)₀ = (g-r)₀ = 1.5 for g frames], and a constant for redder stars. For r, i, and z frames, DCR is modeled as a linear function of color (r-i) for all stars [(color)₀ = (r-i)₀ >> 1]. (The DCR corrections are mis-stated in Pier et al. [2003], where [r-i]₀ appears in the equations rather than the correct [color]₀, and where the wrong value for [color]₀ is given for u frames.) The corrected frame coordinates (x', y') are then transformed to catalog mean place great circle coordinates (μ, ν) using an affine transformation.

The calibration coefficients may be found in the photoField files in the DAS, where the attribute names are different than given in the transformation equations above; (color)₀ is called riCut; g₀, g₁, g₂, and g₃ are called dRow0, dRow1, dRow2, and dRow3, respectively; h₀, h₁, h₂, and h₃ are called dCol0, dCol1, dCol2, and dCol3, respectively; p_x and p_y are called csRow and csCol, respectively; and q_x and q_y are called ccRow and ccCol, respectively.

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Transformation from Great Circle Coordinates to J2000 Celestial Coordinates

The calibration equations above yield catalog mean place in great circle coordinates. To convert these to J2000 celestial coordinates you need to know the right ascension and inclination of the ascending node of the scan great circle with respect to the J2000 celestial equator. These are given as the header keywords "NODE" and "INCL", respectively, in the "photoField" file. The celestial coordinates are then

tan(α₂₀₀₀ - α₀) = [sin(μ - α₀)cos ν cos i - sin ν sin i]/[cos(μ - α₀)cos ν]
sin δ₂₀₀₀ = sin(μ - α₀)cos ν sin i + sin ν cos i

where μ and ν are great circle longitude and latitude, α₀ and i are the right ascension and inclination of the ascending node of the great circle with respect to the J2000 celestial equator, and α₂₀₀₀ and δ₂₀₀₀ are J2000 right ascension and declination.

Astrometry QA files

We have implemented a new astrometry quality assurance system in order to understand the errors in the SDSS imaging astrometry better. The astromqa data model fully describes all of the files that are produced in this process. Here we outline the major results and plots that are produced, both the global plots and those appropriate for each run.

The astrometric QA is defined with respect to a set of reference catalogs:

DR7 where available (including the proper motions)
2MASS
USNO-B
UCAC2 plus r14 (the latter an internal USNO product filling in UCAC at declinations greater than 41 deg)
UCAC3

The astrometry QA results are summarized in the following way:

A description of the astrometric quality of each field is produced, the astromQAFields file, which can be downloaded directly (note, it is large at arouund 2.5 Gbytes). This file contains for each field in DR8 an estimate of the DR8 position offset relative to the set of reference catalogs and an indication of whether the field is catastrophically bad.
There is a global picture of the offsets relative to the reference catalogs. The top row of this table shows on the left the position offsets induced by fact that DR8 ignored color terms when applying the photometry; on this stretch these offsets do not appear large. On the right side of the top row are the DR8 proper motions as a function of position.

The subsequent rows show position offsets for each reference catalog, and proper motion offsets for SDSS DR7. The examples below illustrate the differences between the DR8 astrometry and the UCAC2 plus r14 reference catalog. Note that above 41 deg, DR8 is offset by a large amount relative to UCAC; this is because DR8 calibrated to USNO-B in this region.

Similar errors are seen in the comparison to 2MASS and to SDSS DR7. There are some additional outlying regions in the comparison to SDSS DR7, which are due to catastrophic errors in the DR7 astrometry.
For each run, we produce plots of the comparisons to each reference catalog. For example, consider run 752. We provide overall offsets for each camcol, as well as the distribution of matched star offsets for each camcol as a function of field number. For DR7, we include a comparison of the proper motions. For some runs, DR7 did not include them so those are blank.

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