ASTROMETRY and PHOTOMETRY for the HDF(N)

The data presented for the HDF(N) include:

1. Radio: MERLIN+VLA 1400 MHz
2. IR: ISO 7 and 15 micron
3. Optical: HST I-band
4. X-Ray: CHANDRA

See AVO Demo Data Sets (and Catalogues) for HDF(N) for details

Astrometry

All the data were previously calibrated as well as possible using observed standard sources and the known characteristics of the various instruments. As we are only interested in the extragalactic sources, there is no proper motion on scales detectable in these data (in fact to the best of my knowledge there are no FR2's nor other sources likely to have detectable proper motion even by a second epoch of VLBI or HST data).

The ICRF is defined with respect to 212 extragalactic radio sources. The positions of compact extragalactic radio sources can be found with greater than 1 mas accuracy with respect to the ICRF, as in the case of the phase reference sources used for the radio HDF observations.

In the unaberrated field of view¹ radio interferometry images are linear and do not suffer from rotation or distortion. The position accuracy of well-calibrated data depends on the position accuracy of the phase-reference source and its distance from the target, the beam size (i.e. freq and max. baseline), the signal-to-noise ratio (snr) and the accuracy with which the antenna positions are known. All these factors differed for the MERLIN and the VLA observations of the HDF(N), so comparison of the positions of the brightest compact sources detected by both arrays is a measure of position accuracy. The MERLIN-only data and EVN data were also used to confirm the positions of the most compact sources. This showed the data are aligned with the ICRF to better than 15 mas, consistent with an analysis of the factors contributing to interferometry position accuracy². The error is less than one pixel (40 mas) in the combined MERLIN+VLA images and is a systematic error.

Most of the radio sources in the HDF are extended over 1" or more. The positions of such sources are found by fitting Gaussian components, with a typical accuracy of beamsize/snr (Condon et al. 1998, NVSS home page), giving ~15 mas for a 60 microJy source in the MERLIN+VLA images.

Aligning the HST frames with the ICRF via CFHT and radio data

This is described in Muxlow et al. (2002, 2003). The combined MERLIN+VLA data were used for the radio data. The HST data were aligned with the radio data in a two-stage process. The single CFHT field (9' arcmin on a side, Barger et al. 1999) and the HDF/HFF fields (Williams et al. 1996) were obtained with as good astrometry as could be obtained in isolation (and were fully calibrated and presented as a single image per field). The individual HST WFPC2 frames are assumed to have axes parallel to the ICRF and to be undistorted, but require linear translation and/or rotation for accurate alignment. However these corrections could not be performed directly as the individual frames do not contain enough sources also detected in the radio.

The CFHT image also suffered distortion but sufficient radio sources were detected to allow corrections to be derived. Suitable matched sources chosen as follows:

1. Choose radio-optical pairs with well-defined peaks.
2. At each frequency, measure the sides of triangles composed trios of the selected sources and compare the radio and optical distances. If there is a significant difference (indicating the brightest part of the source does not coincide at the two frequencies), reject the source responsible.

36 of the optical sources with bright radio counterparts were selected in this way.

The six potential parameters are

1. x translation,
2. y translation,
3. rotation,
4. stretch,
5. correction for non-parallel axes and
6. associated further stretch.

It was found that only the first four corrections are needed since the CFHT image axes are perpendicular to better than 0.01 degree (less than one pixel at the edges of the radio images of the 10' field). These were derived as follows:

1. Plot vectors of offsets of optical from radio positions.
2. Take 6 regions of similar offset and derive the average vector offset for each region (magnitude and direction).
3. Derive the gradient of the components of the vector in each region.
4. Use this to interpolate across regions.
5. Perform a final correction where there were significant residuals after applying the above corrections.

Stages 2 and 4 were performed by eye. Stage 2 could be automated and further iterations of the process or a more refined version might remove the need for stage 4 although we should at least offer the user a check/chance to refine at the end.

This process was used to deduce the appropriate rotation and translation at any given point of interest (assuming a constant stretch over such ~100" or smaller regions, so that it is incorporated into rotation and translation) and applied to cut-outs of the CFHT image around sources with radio counterparts.

We should be able to store such corrections e.g. as a matrix of values of the required parameters at intervals of e.g. 100" plus the gradients of each value) as some kind of look-up table or mask which could be interpolated to any region, and applied to cut-outs; the inverse correction would be applied to requests for flux density measurements given on the ICRF and then the correction applied to the results.

Finally, many sources in each WFPC2 frame could be matched with CFHT sources and, using their corrected positions, simple translation was applied to align the WFPC2 frames with the ICRF. This was done by chamging the header position of the reference pixel, so that the correct scale and rotation angle were preserved. After this process, the residual radio-optical position differences were <50 mas in the HDF itself (the inner ~3'), rising to 150 mas at ~5' radius from the pointing centre. This accuracy has proved crucial in finding (or ruling out) radio-optical associations, for example different properties of merging systems, or identification with faint red objects close to brighter sources.

Homogenising the data sets

In order to make full use of the data for the demo they have to be not only on the WCS but on the same pixel scale with respect to the same overall reference position (12 36 49.40 +62 12 58.0). Every care has been taken to avoid introducing errors during resampling but the following are potential hazards:

• Transformation between sine and tangent projections. This produces a position error of up to 0.02% of the radius from the individual frame reference position, which is < 0.1 pixel.
• Redistribution of flux. This can produce a discrepancy of < 0.1 pixel in the peak position measured by Gaussian fitting and of a few percent or less in the peak flux depending on the source structure. The change in an aperture of 1" - 2" diameter is <1% for a sample of sources tested.
• Unnatural scales. This is a warning not an error; the resamping scales were chosen so that the data are smoothed rather than oversampled where possible. The exception is the ISO data where pixel scales have been reduced by a factor of 2 or 6 (7 and 15 micron data respectively) so it therefore does not make sense to make measurements in apertures less than 2 or 6 pixels in diameter from these data.

In any case the demo methods cannot provide optimum source recognition from most of these data, as compared with the best published catalogues. The latter involved, for example, comparing CHANDRA data in the high- and low-energy bands; the use of the PRETI method on the ISO data and examining radio data for MERLIN and the VLA separately as well as in combination. However the 'one-size-fits-all' methods of the demo will be superior for comparing the flux densities at a given position in a suitable-sized aperture. The masks prepared from the published catalogues provide templates for such extraction.

The MERLIN+VLA data were mapped in ~1' squares and mosaiced into an 8192x8192 field with 0".0625 pixels, i.e. an 8'.533 square. Images from the three HST WFPC2 chips for the HDF were obtained with 0".03985 pixels, and the 8 pointings of the trio of chips for the HFF had 0".09896 - 0".09946 pixels. After completing astrometric alignment and removing the outer rim of bad pixels, the data from each chip were different sizes at different orientations. All HST data were transformed to the IRCF as defined by the MERLIN+VLA data, with Right Ascension and Declination along the x and y axes, using the same reference point and pixel scale, and mosaiced into an aligned 8192x8192 field.

The CHANDRA and ISO data were provided as single fields, and were already aligned with the ICRF via matched radio sources. The MERLIN+VLA data were resampled to the CHANDRA pixel scale (0".492) and then the CHANDRA data were transformed to the radio orientation and reference position and a 1024x1024 image was cut out of each data set to maximise the overlap, giving an overlapping region of 8'.09 x 8'.36. This is less than the total potential field of view in each case but covers >2/3 of the reasonably sensitive regions. The HST data were transformed to this orientation, size and scale as described above. Small parts of 4 out of the 24 flanking fields are lost. The ISO data were transformed to this scale and reference and occupy the central few arcmin.

Photometry

See note and warning above about potential errors of up to a few percent in peak flux densities and pixel positions, caused by resampling; aperture measurements should be more accurate. More serious inaccuracies may be caused by averaging over wide bands, for sources with steep spectral indices - another reason why the published source lists should be used for high precision analysis in a particular regime. The fractional bandwidths are given below to allow an estimate of how serious this might be. However these data provide the only way to compare data from GHz to keV (EHz, 10¹⁸) frequencies in a reasonably consistent way.

Conversion to consistent flux units We assumed that the data supplied were already photometrically calibrated in the appropriate units for each instrument. What follows describes how the units were transformed to Jy/pixel as required by SExtractor.

• MERLIN+VLA: Input units Jy/beam, linear scaling to Jy/pixel for the appropriate output pixel size.
  The fractional bandwidth is ~6% (not completely filled - non-contiguous bandpasses were combined, to avoid interference).
• ISO: Input units mJy/arcsec², linear scaling to Jy/pixel.
  The fractional bandwidths are 53% and 34% at 7 and 15 micron respectively.

• HST: Input units in the HDF were counts/sec. In the HFF the input units were counts and were scaled to counts/sec using the exposure times for each field taken from the HDF observing logs (see total time per HST frame³). The scaling factor to erg/cm²/Angstrom is given by PHOTFLAM, and converted to Jy (PHOTFNU⁴ at the central frequency. Finally the units were scaled to Jy/pixel for the appropriate output pixel size. The fractional bandwidth is 19%.

• CHANDRA: Input units were counts, converted to counts/sec for an average observing time of 977520 sec in the data for the central fields downloaded via the CHANDRA Deepest Fields. We asumed the effective energy of the image was 1 keV, the peak sensitivity of the band, and took an average source spectral index⁵of -1.7. This was used to scale the data to Jy.
  The total bandwidth was ~0.4 - ~8 keV, giving a fractional bandwidth of 180% at the nominal central frequency.

Notes

1. The field of view of a radio interferometer is determined by the integration time per visibility, the channel width and the primary beam width. For MERLIN this is typically ~1000 synthesised beam widths in wide-field mode. For the HDF observations more than 90% of the radio flux should appear at its 'true' position within a central region of ~2' radius. At larger distances from the pointing centre sources are progressively more distorted and the noise level increases; in the case of these data the FWHM of the Lovell primary beam, means there is an additional decrease in sensitivity at >5' radius. However sources are detected out to ~20' from the pointing centre albeit with progressively reduced accuracy.

2. The position accuracy of phase referenced interferometer observations is determined by the error in the position used for the phase reference source, the synthesised beam width (depending on the frequency and baseline lengths) and signal to noise ratio, the target - phase-reference source separation on the sky and the uncertainty in telescope positions. The MERLIN HDF observations used the compact 0.4 Jy phase reference source J124129+622041, ~34' from the centre of the HDF(N). Its position is known to 0.5 mas accuracy. The observations were made at 1.4 GHz with a synthesised beam size of ~180 mas at maximum sensitivity, giving a noise rms of 5.9 microJy/beam in the HDF and ~12 microJy/beam on J124129+622041, so the position errors due to noise are negligible for the latter. As the target and phase-reference source are so close, and the observations were made over a total of 18 days, the error in phase correction transfer due to atmospheric differences is <5 mas. The remaining source of error is the uncertainty in baseline length, thought to be 1-2 cm, introducing an error of 10-20 mas, consistent with the uncertainty deduced observationally.

   To achieve this accuracy the curvature of the sky must be allowed for in making the images. This is straighforward and produces images in sine projection, so care may be needed when comparing these with images in other projections.

   In general a radio interferometry observation from a single array, or data supplied as combined from 2 or more arrays, should have positions determined in this way, tied to the ICRF, with an error estimate based on these factors. If a more accurate position for the phase refernece source is later determined, a linear correction can be made. (In some cases more accurate antenna positions are later derived - a more complex problem for future consideration!).

3. The total time per HST frame, in sec, was:
   HDF 123600
   SE 2500
   OE 3000
   NE 2500
   IE 5300
   IW 5300
   SW 2500
   NW 2500
   OW 2500
   

4. The scaling factors from counts/sec to Jy for each of the WFPC2 wide field chips was PHOTFNU
   PHOTFLAM PHOTFNU
   2.484e-18 5.2235e-6
   2.480e-18 5.2151e-6
   2.545e-18 5.3518e-6
   

5. The CHANDRA source average spectral index was taken as alpha = -1.7 in the convention A(E)=K E^alpha, where A(E) is the energy across the band and K = 9e-12 is the scaling factor from counts/sec to erg/cm²/sec.
   1 keV = 2.42e17 Hz
   The band reached from 0.4 to 8 keV above the 10% sensitivity limit.
   The flux across this band is given by the integral of [E A(E)], i.e. of [K E^-0.7] = 3.688 K in keV.
   The flux density is then given in Jy by
   [(9e-12 * 1e-3 * 1e26) / (3.688 * 2.42e17)] * counts/sec.

Thanks to the providers of the websites listed here and at AVO Demo Data and to Mike Watson (Leicester) for supplying data and assistance in performing these conversions.

-- AnitaRichards - 04 Dec 2002 original 18 September 2002