4 FIRST as an Astrometric Standard

1 Introduction

As we show in WBHG, all unresolved FIRST sources down to the survey limit have 90% confidence positional uncertainties of $<1\arcsec$; furthermore, the absolute reference frame is tied to the VLB reference frame with an offset of $<0.03\arcsec$ and an rms uncertainty of $<0.15\arcsec$ in each coordinate. This level of precision is unprecedented for such a high-surface-density catalog (e.g., the Guide Star Catalog entries have an rms uncertainty of 0.4" with systematic errors of up to 1"), making FIRST a useful astrometric calibration standard for any astronomical catalog which contains a significant number of FIRST source counterparts. In this section, we use the radio source positions to calibrate and correct the APM intraplate astrometry errors and the plate-to-plate shifts, allowing an improvement by nearly an order of magnitude in POSS-I positions.

The FIRST catalog contains about 90 sources deg$^{-2}$ of which $\sim$15% (see § 5.2) have APM optical counterparts within $2^{\prime \prime }$, yielding $\sim$600 matches per POSS-I plate. This, in effect, provides a dense grid of astrometric standards across each plate. More important, the optical counterparts to FIRST sources are generally faint, and thus complement the astrometric calibration that can be achieved using bright standards such as the PPM and Tycho catalogs that have been used heretofore as the basis of the APM plate solutions.

2 Overall APM Astrometric Offset

Whilst working on radio identifications for the Jodrell Bank-VLA Astrometric Survey (JVAS - Patnaik et al. 1992; Hook et al. 1996; Snellen et al. 2001), it was noted that there was evidence for a shift between the mean Right Ascension and Declination of the APM reference frame and the VLA reference frame. The mean shift was derived by taking the median of the nearest optical counterparts to the survey's flat-spectrum radio emitters and yielded:

• RA$_{APM}$(corrected) = RA$_{APM}$ (original) + 0.35 $\pm 0.05$ arcsec
• Dec$_{APM}$(corrected) = Dec$_{APM}$ (original) + 0.35 $\pm 0.05$ arcsec

We verified that this effect was not intrinsic to the PPM reference frame used for the APM plate solutions at that time by comparing the optical positions of a set of VLBI Radio Reference Frame objects (Johnston et al. 1995) with optical counterparts on APM scans and found that the same systematic shift was present. We then remeasured POSS field 1393 centered on the North Galactic Pole in two orientations: with the normal scanning direction and with the plate rotated by 180 degrees. We found that, after application of the original astrometric alignment using the PPM stars, the derived positions for faint ($>14^{th}$ magnitude) objects differed by $\sim0.7^{\prime\prime}$ in both coordinates. This indicated that the APM measurement system was introducing a systematic shift in the positions of bright objects with respect to fainter ones. The origin of this effect is still under investigation. Our first step in comparing FIRST sources and the APM catalog, then, was to take out this systematic shift of 0.35" in both coordinates.

3 Intraplate Errors

The original POSS-I plates suffer from various distortions resulting from the stress induced by the plate holder, the vignetting of the 48-inch Schmidt telescope, etc., which are reproducible from plate to plate (Irwin 1994). In Figure 4(a), we display the pattern of residuals from the TYCHO reference stars derived by stacking the APM-TYCHO offsets for all such stars on 148 plates. Systematic errors reach 1.3" in some regions of the plate. A correction for this effect has, as noted above, been applied to all APM catalog positions using a correction map derived from the Tycho catalog.

Figure 4: (a) The distortion in raw APM catalog positions as a function of position on POSS-I plates derived by comparison to reference positions from the TYCHO catalog of astrometric reference stars. The lines show the average position shifts in $20^{\prime } \times 20^{\prime }$ regions across the plates (6667 pixels = 1 degree); the scale of the shifts is shown on the right. There are large, systematic distortions that are not modeled by the APM plate solution. (b) The distortion remaining in the APM positions after correction for the distortions shown in (a). These are derived by comparing corrected APM positions to FIRST positions for APM-FIRST matches. The distortions are small near the plate center, but beyond $\sim 2.5^\circ$ from plate center there is a strong, increasing radial distortion. This is a magnitude-dependent position error in the APM catalog; the positions of bright sources (such as TYCHO stars) are properly corrected by application of the distortion map in (a), but positions of faint objects (such as the optical counterparts to FIRST radio sources) are actually made worse by the bright-star correction map.

Figure 5: The radial offset between APM and FIRST positions as a function of distance from plate center; each point represents one of the $20^{\prime } \times 20^{\prime }$ grid cells into which we have divided the plates.

We compared these corrected APM positions with those of FIRST radio sources over the 148 POSS-I plates which subsume the current survey region. Including all sources with optical counterparts having offsets $\le2.0\arcsec$ ($\sim$60,000 objects with a false rate of $\sim5$% -- see § 5) yields the result displayed in Figure 4(b). Residuals within a radius of 2.5 deg of the plate centers are substantially reduced compared with Fig. 4(a), but large radial errors with magnitudes up to 1.2" are now seen around the plate edges (Fig. 5). We believe this effect is a consequence of a combination of telescope vignetting and plate saturation of the bright stars in the Tycho grid: vignetting distorts the stellar images in the radial direction, producing elliptical isophotes whose centroids are systematically shifted toward the plate center. The FIRST counterparts, largely much fainter objects for which saturation is not a problem, reveal this distortion.

To correct for this effect, we have chosen to calibrate the errors using all FIRST counterparts with $15.0 < E < 19.5$; the bright limit is chosen such that the magnitude-dependent errors described above will be unimportant (and only excludes 6% of the matches), while the faint limit is imposed in order to include only those objects for which the random errors are small. The correction proceeds iteratively. We calculate a mean offset (in both the radial and tangential directions) for all sources in the stacked image in cells $20^{\prime}$ on a side (20$\times$20 cells cover the plate). We then apply the derived offsets to all APM objects in the catalog based on their $x$-$y$ plate positions, rematch to FIRST, and then recalculate the mean offset for the cell. The process is repeated until it converges ($\sim$4 steps). An iterative approach is required because the APM position shifts are comparable to the $2\arcsec$ APM-FIRST matching radius. The resulting APM-FIRST position errors are much improved: the maximum residual mean offsets are less than 0.18" with an rms of only 0.06" over the whole plate.

This intraplate correction is strictly valid only for the range of magnitudes used in the calibration ( $15.0 < E < 19.5$). An examination of the offsets for fainter counterparts suggests that the corrections work well to the plate limit, although the positional uncertainties increase slightly owing to the statistical fluctuations which are inevitable in deriving positions for objects near the detection threshold. For the brighter objects, however, the derived positions of candidate counterparts to FIRST sources suffer from the same type of distortion evident in the Tycho-based solution, presumably in a magnitude-dependent way. Rather than attempting to derive the functional form of the magnitude dependence from the small number of bright FIRST counterparts, we recommend that for very bright objects either the separation for an acceptable match be increased or the offset be computed using the APM astrometry without the FIRST astrometric correction. For example, in the FIRST Bright Quasar Survey (White et al. 2000) we include all sources with optical matches closer than $1.2^{\prime\prime}$ in either astrometric frame, a procedure which adds 24 objects to the 1214 selected using the FIRST astrometric solution alone. Since the typical discrepancy between the two solutions is $<1\arcsec$ and our minimum matching criterion is $\ge 1\arcsec$, no genuine matches will be missed if this procedure is followed. However, since the overwhelming majority of optical counterparts are faint, we have used only the matches with the FIRST corrections applied when compiling the match statistics in this paper.

This use of FIRST sources as astrometric calibrators assumes, of course, that there are no systematic errors in the radio positions. In Figure 6, we display the analogous plot to Figure 4(b) for the FIRST catalog: the $x$-$y$ positions of each FIRST source in the radio ``plate'' (i.e., coadded image -- see BWH) are extracted and the mean offset from the APM matches to these sources (binned in $2^{\prime}$ cells) is calculated. The uniformity is excellent and the magnitude of the errors is very small: they are all $<0.13\arcsec$ with an rms of 0.05" over the whole field. This result, coupled with our global astrometric calibration described in WBHG suggests our confidence in FIRST astrometry is justified, and that our radio catalog can be utilized routinely to aid the astrometric calibration of other catalogs.

Figure 6: The distortion in FIRST catalog positions as a function of position within the FIRST coadded images. The scale of the vectors is shown on the right; 1 pixel = $1.8^{\prime \prime }$. No systematic errors in the FIRST positions are evident, indicating that we have successfully corrected the geometric distortions in the FIRST images.

4 Plate-to-plate Shifts

Having established the global APM offset and calibrated out the reproducible intraplate errors, we are left only with translations of individual plates with respect to our radio reference frame. The mean FIRST-APM offsets for each of the 148 plates in the survey region are shown in Figure 7. These offsets were computed using an iterative approach similar to that used to derive the plate distortion. There is no obvious trend with Right Ascension or Declination, but shifts remain at levels ranging from 0.01" to 0.70"; the distribution of errors is roughly Gaussian in both coordinates with an rms of $\sim 0.16^{\prime\prime}$ and an apparent tendency for large offsets in one coordinate to be matched by those in the other. We remove these shifts by simply subtracting the shift computed for a plate from the APM positions of all objects on the plate. Note that for plates which fall completely within the boundaries of the FIRST survey, this calibration is final; however, for those plates only partially covered by the existing survey, the global offset will be redetermined as new data accumulate. The size of any subsequent correction is expected to be no more than $\sim$0.1". The current values, along with the intraplate correction matrix are available on the FIRST Website.

Figure 7: The mean shifts for each APM plate in the FIRST survey area derived by comparing APM positions (corrected for the intraplate distortions in Fig. 4) to FIRST positions. Each point shows the RA and Dec shifts for a plate with $1\sigma$ error bars.

After correcting for the plate shifts, we have recomputed the plate-based distortion maps (Fig. 4) to determine whether the systematic plate shifts affect them. The changes in the distortions are small ($<0.1\arcsec$).

Having completed the astrometric calibration, we can now proceed to generate a catalog of optical identifications for the FIRST survey.