# Astrometry

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

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 the UCAC astrometric reference catalogs.

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), which has a precision of 70 mas at its catalog limit of r= 16, and systematic errors of less than 30 mas. UCAC2 extends up to around a declination of 41 degrees north. Outside the UCAC2 area we use an "internal" UCAC data release known as "r14". Together UCAC2 and r14 cover the whole sky. There are approximately 2 - 3 magnitudes of overlap between UCAC and unsaturated stars on the r CCDs. The astrometric CCDs are not used.

The r CCDs are 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.

## 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 "PM"external catalogs directory in SAS.

The proper motions in DR9 and later ("DR9+") correct an error in DR7 proper motions for stars at low Galactic latitude; see the QA discussion below for more details.

## 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.

## 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 + px color
y' = y + h0 + h1 y + h2 y2 + h3 y3 + py color

For color > (color)0:

x' = x + g0 + g1 y + g2 y2 + g3 y3 + qx
y' = y + h0 + h1 y + h2 y2 + h3 y3 + qy

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

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. However, for DR9 and subsequent releases, we accounted for this color term correctly.

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; px and py are called csRow and csCol, respectively; and qx and qy are called ccRow and ccCol, respectively.

## 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 identify 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 for DR9 and subsequent releases ("DR9+")
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 around 2.5 Gbytes). This file contains for each field in DR9+ an estimate of the DR9+ 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 left side of the top row is blank by design, and just exists to verify that the color terms in the astrometry described above are applied properly. On the right side of the top row are the DR10 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 DR9+ astrometry and the UCAC2 plus r14 reference catalog, showing good agreement.
α offset from UCAC
δ offset from UCAC
• There are some outlying regions in the comparison to SDSS DR7, which are due to catastrophic errors in the DR7 astrometry (in certain areas of runs 3358, 4829, 5960, 6074, and 6162).
• Additionally, there are some differences with respect to the DR7 proper motions at low Galactic latitude; this results from an error in DR7 runs processed through rerun 648 causing a number of stars to be misclassified as galaxies and contaminating the reference frame used for the proper motions. We believe that the DR9+ proper motions are superior in these areas.
• Finally, the proper motions values in DR9+ have been checked for photometrically identified "XDQSO" quasars from Bovy et al. (2011), in the bottom row of the astrometry QA page. These results are consistent with those found by Bond et al. (2010), who found a small (mas/yr) offset at low declination that may be related to refractive effects in the USNO-B catalog for these very blue sources.
• 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.

## Caveats on DR8 astrometric calibration

The DR8 astrometric calibrations were 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 recommend using the DR9+ astrometry and proper motions, which are available in the DR9+ releases, or in the astromDR9 and properMotionsDR9 tables of the DR9+ CAS.

A detailed description of these errors was released as part of the DR8 Paper Erratum, and is also available on the DR8 version of this algorithms page.

Note in particular that the proper motions tabulated in the CAS were 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.