2.3.1. General calibration and imaging

Bandpass, flux density scale, on-axis leakage, and initial gain calibration are determined using the flux standard of PKS B1934 $-$ 638, which is normally observed once each day or for each observing configuration. For on-axis leakage correction, it is assumed PKS B1934 $-$ 638 is completely unpolarised. Rayner et al. (Reference Rayner, Norris and Sault2000) report $\sim +0.03$ % fractional circular polarisation for PKS B1934 $-$ 638, though as noted by O’Sullivan et al. (Reference O’Sullivan, McClure-Griffiths, Feain, Gaensler and Sault2013) the variable nature of circularly polarised emission means this may have changed in the time since that measurement was made. Once bandpass and initial gain calibration solutions are applied to each science observation, data from each beam are self-calibrated over three loops, independently. Each loop decreases the CLEAN threshold during the imaging step to generate a deeper field model. Self-calibration normalises gain amplitudes, creating a phase-only–equivalent calibration self-solution.

Figure 4. Example (u,v) coverage for a single central beam of the RACS-mid observation (SB21663, field RACS_0812-28, beam 15, red) close to zenith, compared to a similar individual beam from the first epoch of RACS-low (SB8576, field RACS_1618-25A, beam 0, light-grey). The left panel shows the full (u,v) coverage, and the right panel shows the inner $\sim 1.5$ k $\lambda$ . The blue circles on the right panel enclose the (u,v) range corresponding to angular scales of 3 (dotted), 5 (dot-dash), 10 (dashed), and 30 (solid) arcmin.

Imaging with deep deconvolution then follows for each beam independently. The ASKAPSoft imager makes use of a w-projection gridding algorithm, with multi-scale CLEAN and multi-frequency synthesis (MFS) deconvolution. An equivalent of ‘Briggs’ (Briggs, Reference Briggs1995) robust 0.0 weighting is achieved through preconditioning (Rau, Reference Rau2010) of the data. As part of the MFS deconvolver, the sky brightness distribution is expanded into two Taylor terms to account for the normal power law spectral dependence of sources in total intensity and the spectral-dependence of some instrumental features, e.g. the PSF and primary beam. As the effective fractional bandwidth is small ( $\sim 10$ %), the most significant spectral effect is the primary beam FWHM variation across the band (see Figure 2) which is accounted for with the second Taylor term. At this stage, both Stokes I and V continuum products are produced for each beam. The same number of Taylor terms are used for Stokes V imaging, though circularly polarised emission mechanisms have more variation.

For Stokes I, the final CLEAN run uses up to ten major iterations with a minor cycle threshold of 45% and minor cycle gain of 30%. The major cycle stopping threshold is 0.75 mJy PSF $^{-1}$ , with a final minor cycle threshold of 0.5 mJy PSF $^{-1}$ . With fewer sources, for Stokes V deconvolution we use a maximum of only three major iterations with a 30% and 20% minor cycle threshold and gain, respectively. The ASKAPsoft imager does not restrict where CLEAN components can be found except in the final major cycle, where CLEAN components below the major cycle threshold can be found if they lie within pixels of the model generated during the previous major cycles. For both Stokes I and V images, peak positive residuals after CLEANing are $\sim 0.6$ –0.8 mJy PSF $^{-1}$ , though vary depending on beam and field. The individual restored Stokes I and V beam images are convolved to a common resolution prior to mosaicking.

The ASKAPsoft imaging of Stokes V results in the sign of the circularly polarised emission to be consistent with the IAU standard (right-hand circularly polarised light is positive, and left-hand circularly polarised light is negative). This is opposite to the convention generally adopted in pulsar astronomy (see e.g. van Straten et al., Reference van Straten, Manchester, Johnston and Reynolds2010).

2.3.2. Extended sources, the Galactic Plane, and large-scale ripples

Figure 4 shows the (u,v) coverage for a nominal RACS-mid observation compared to a similar observation from RACS-low, highlighting the minimal (u,v) coverage as angular scales increase beyond 3 arcmin. Both example observations represent the highest-elevation pointings in RACS-mid and RACS-low. The Galactic Plane poses a significant challenge in imaging even with good (u,v) sampling for most modern instruments (e.g. the MWA, Hurley-Walker et al. Reference Hurley-Walker, Hancock and Franzen2019, Reference Hurley-Walker, Galvin and Duchesne2022; Tremblay et al. Reference Tremblay, Bourke and Green2022; MeerKAT, Heywood et al. Reference Heywood, Rammala and Camilo2022; and deep ASKAP observations, Umana et al. Reference Umana, Trigilio and Ingallinera2021), more so for the snapshot observations described here.

Issues arise from incomplete sampling of the inner (u,v) plane: flux density on large angular scales is not well measured, and artefacts around bright, extended sources can dominate images. Multi-scale deconvolution is used to help in modelling extended sources, though this can result in ‘ghost’ sources appearing due to the CLEAN algorithm misinterpreting artefacts and source sidelobes as real sources. In Figure 5, we show that despite this, multi-scale deconvolution is still an appropriate choice to ensure extended sources are modelled well during deconvolution; however, the residual ghost sources are not local to the extended source, and can appear throughout the full image. The center panel of Figure 5 highlights an example of the ghost sources, in this case originating from nearby radio galaxy Fornax A. In this instance, a single ghost source has an absolute integrated flux density of up to $\sim 0.5$ Jy, compared to the $\sim 6$ Jy integrated flux density of a single lobe of Fornax A from the same image.

Figure 5. Comparison of the field containing Fornax A (RACS_0329-37, enclosed with the black, dashed ellipse) without multi-scale deconvolution (left), with multi-scale deconvolution (centre), and with multi-scale deconvolution after application of the $(u,v)<75$ m cut to the data (right). Another miscellaneous extended source is highlighted in the black, solid circle.

Despite their prominence during visual inspection of the image, ghost sources are typically not detected and characterised by the selavy (Whiting & Humphreys, Reference Whiting and Humphreys2012) or aegean (Hancock et al., Reference Hancock, Murphy, Gaensler, Hopkins and Curran2012, Reference Hancock, Trott and Hurley-Walker2018)Footnote ¹⁰ source-finders used in this work, as they use a position-dependent noise and $5\sigma_{\text{rms}}$ -thresholding and are not optimised for detection and modelling of faint extended sources. Other source-finders such as PyBDSF (Mohan & Rafferty, Reference Mohan and Rafferty2015, as used in Paper II) may detect them, depending on user-settings. In the Fornax A example shown in Figure 5, no ghost sources are detected by selavy or aegean.

To help reduce the number of ghost sources, we set a minimum (u,v) cut corresponding to 75 m baselines during imaging for a selection of affected observations. This removes large-scale ripples and other sidelobe features, reducing the number of ghost sources. An example of the effect of the (u,v) cut is shown in the right panel of Figure 5. While this (u,v) cut does not improve imaging quality directly at the location of the extended source, it helps the reduce artefacts in the sky within a few degrees of the source, reducing the mean root-mean-square (rms) noise over the image (from 191 to 183 $\mu$ Jy PSF $^{-1}$ across the Fornax A image) and local rms noise at the affected locations (e.g. from 267 to 170 $\mu$ Jy PSF $^{-1}$ around a ‘ghost’ source in the Fornax A image). Most selected observations are within or near the Galactic Plane, though we also select a small number of extra-Galactic fields like the field containing Fornax A. Some additional extra-Galactic fields are also affected by solar interference, which results in a similar problem along with large-scale ripples and a (u,v) cut is used for those observations as well.

The (u,v) cut (in metres) used for each SBID is provided in the RACS database and as an additional item in the FITS header under the MINUV keyword. These SBIDs still feature the largest number of residual ghost sources and general artefacts as they are typically in the Galactic Plane, and we recommend users exercise caution when inspecting the images if interested in real extended sources.

2.3.3. Peeling

A significant source of artefacts in the initial RACS-mid imaging is contribution from bright off-axis sources. Both large, extended radio sources as well as compact sources radiate sidelobes and additional direction-dependent artefacts through the imaged field of view when they fall within one of the sidelobes of the primary beam. The first primary beam sidelobe is $\sim2$ deg from the beam centre at 1367.5 MHz. In extreme cases, sidelobes or artefacts from such sources can cause the deconvolution to diverge rendering a single beam image unusable. To mitigate this issue, we opt for a ‘peeling’ and subtraction approach for particularly problematic sources. This by-eye selection typically includes sources $\gtrsim 10$ Jy at 1367.5 MHz. This ‘peeling’ largely follows the definition and process described by Noordam (Reference Noordam, Oschmann and Jacobus2004, see also Smirnov 2011b), though we use a combination of (1) direct visibility subtraction, (2) true directional peeling, and (3) temporary mainlobe subtraction prior to peeling, depending on signal-to-noise ratio (SNR) and complexity of the offending source. The modes are generally used together—directional subtraction follows a round of peeling to remove residual emission due to, e.g., a difference in the directional gain amplitudes with the original data.

Direct visibility subtraction. For direct subtraction, the visibilities are phase-rotated to the direction of the bright source to be removed, and the source is imaged using the widefield imager WSClean (Offringa et al., Reference Offringa, McKinley and Hurley-Walker2014; Offringa & Smirnov, Reference Offringa and Smirnov2017)Footnote ¹¹. A mask is created to exclude the sky outside a circular aperture enclosing the source, and a CLEAN component model is derived from that masked image, and from it corresponding model visibilities M. Using the Jones matrix formalism of the radio interferometer measurement equation (Hamaker et al., Reference Hamaker, Bregman and Sault1996; Smirnov, 2011a), the modified visibilities are then computed as

(1) \begin{equation} V_{pq}^\prime = V_{pq} - M_{pq} \, ,\end{equation}

for correlated visibilities formed by antennas p and q. The resultant visibility data, $V^\prime$ , is then phase-rotated back to the original direction. This subtraction procedure is always run if the source is outside of the specified FoV for a given SBID. There is no benefit in subtracting within the main lobe FoV as this is functionally similar to deconvolving the source during normal imaging and would result in a missing source in the image.

Directional peeling. With a sufficient SNR for the off-axis source, a round of gain calibration on the derived CLEAN model can be reliably performed. The CLEAN model, M, is then subtracted after applying the inverse of the derived gains, G. Thus, the source-subtracted visibilities, $V^\prime$ , are

(2) \begin{equation} V_{pq}^\prime = V_{pq} - G_{p}^{-1} M_{pq} (G_{q}^{H})^{-1} \, ,\end{equation}

where the superscript H is the Hermitian transpose. This is a form of direction-dependent calibration, but is only applied to the source model and not the data itself. As this involves solving for gain solutions, a sufficiently high SNR is required to determine reliable gains for all antennas whether or not the mainlobe model has been subtracted. The choice of cut-off SNR is dependent on source structure and can typically be lower for point sources. As the default mode of gain calibration here is to solve for both phases and amplitudes (which tend to produce the best results), an additional directional subtraction round is always run afterwards.

Mainlobe subtraction. If the SNR of the source is not sufficient for good solutions during directional peeling, but direct subtraction by itself is not sufficient to remove all unwanted artefacts, then subtracting the field model within the mainlobe of the primary beam prior to peeling can help. In this process, we start with a temporary directional subtraction of the bright off-axis source followed by shallow imaging of the mainlobe. The field model within the mainlobe is subtracted, and the bright off-axis source is returned to the data for gain calibration. The field model is then also returned to the data once gain solutions are derived, and the bright off-axis source is peeled as per usual.

Table 3 lists all SBIDs within which sources have been peeled and/or subtracted. Generally, only sources off the Galactic Plane are included as the Galactic Plane poses additional imaging challenges (see Section 2.3.2). Sources are chosen after a full round of imaging to visually identify which sources cause problems. For each SBID with a problematic source, all 36 beams are independently re-processed and the source is only peeled or subtracted if it is greater than 1.2 deg away from the beam centre of a given beam. Because we do not peel the source from all beams, artefacts persist for some beams; though they are now localised to within $\sim 1.2$ deg of the source. Figure 6 shows the peeling result for SB20734 that contains Virgo A. The dot-dash, grey circle indicates the 1.2 deg radius outside of which peeling is performed, and the dashed, green circle indicates the aperture within which Virgo A is modelled. In the figure, we also show a zoom-in of a region directly to the north of Virgo A, highlighting the improvements. A python-based pipelineFootnote ¹² for peeling is used as a manual intermediate step in the ASKAPSoft pipeline prior to self-calibration once a problematic source has been identified, requiring its location and approximate size.

Table 3. List of peeled sources, aperture within which they are eventually subtracted (see main text) and SBIDs they are subtracted out of for beams where they are $\geq 1.2$ deg from the beam centre.

^aOnly the inner lobes and core are included as the large-scale outer lobes are largely undetected in the RACS-mid data (see Section 2.3.2 for some discussion of the angular scale sensitivity).

^bBoth Orion A and B are peeled/subtracted in the same tiles in that order, as they have an angular separation of $\sim 3.8$ deg

Figure 6. SB20734 containing Virgo A before peeling (left) and after peeling (right). A zoom-in of the region above Virgo A before and after peeling is shown in the bottom row, with a solid, black box indicating its location in the top panels. The dot-dash, grey circle has a 1.2 deg radius: beams with centres within this radius are excluded from peeling. The dashed, green circle indicates the radius within which Virgo A is modelled. Black crosses indicate the beam centres. The median rms noise, $\sigma_{\text{rms}}$ , in the top panels is quoted for the full tile excluding the 1.2 deg circle containing Virgo A (which is unchanged after peeling). In the bottom panels $\sigma_{\text{rms}}$ is quoted for the zoomed-in region only.

2.4.1. The effect of PAF beam-forming

Over the course of processing data for RACS-mid, we found that the primary beam response measured from holography was not constant in time. Significant changes appear to occur after the digital beam-former weights are re-measured. This process of measuring the parent beam-former weights is performed with a cadence of a few weeks to a few months. Hotan et al. (Reference Hotan, Bunton and Chippendale2021, see also Hotan et al. Reference Hotan, Bunton and Harvey-Smith2014; McConnell et al. Reference McConnell, Allison and Bannister2016) describes the digital beam-forming procedure and how the beam-former weights are also updated using the on-dish calibration (ODC) system to prevent degradation of the beams between parent weight measurements (see also van Cappellen et al., Reference van Cappellen, Oosterloo and Verheijen2022; Dénes et al., Reference Dénes, Hess and Adams2022, for a description of the beam-forming for the PAFs of Apertif on the Westerbork Synthesis Radio Telescope). The derivation of beam-forming weights uses a method to maximise SNR when pointing at the Sun. Changes to solar features may cause the resulting digital beam response to shift and/or change shape if the centroid of the solar emission is not constant between the beam-forming observations. RACS-mid was observed over 9 disjoint sets of dates, each with a different set of parent beam weights (hereinafter we refer to these periods as BWT-1 to BWT-9), and for a majority of the observations no appropriate matching holographic measurement of the primary beam is available. Application of mismatched primary beam responses can result in brightness scale errors up to a factor of 2 at the beam edges for all beams. While the digital beam-former weights are updated more frequently with the ODC system, we do not see significant brightness scale discrepancies between these ODC updates.

For BWTs with applicable holographic measurements, the holographic Stokes I primary beam measurements are used (five periods), but these constitute only $\sim 4\%$ of the total survey. Most of the observations were taken during BWT-1, with $\sim 90\%$ of the total observations. Table 4 summarises the BWTs and indicates the range of SBIDs (and associated observation dates) applicable and whether matching holographic primary beam measurements are available.

2.4.2. Measurement of the Stokes I primary beam response

In lieu of post-mosaicking corrections (cf. Paper I), we opt to measure the primary beam response per beam for BWT-1–3 and BWT-5 which do not have appropriate holographic measurements. A similar, non-holographic approach has been adopted by Kutkin et al. (Reference Kutkin, Oosterloo and Morganti2022) for Apertif post-imaging primary beam corrections. We use in-field sources extracted from apparent brightness images of the $\sim 40\,000$ individual beams and compare these to NVSS measurements where available. Beams from tiles that lie outside of the NVSS coverage are excluded. We also exclude beams from SBIDs in the Galactic Plane and those with peeled sources due to the increase in artefacts in those fields. We use the aegean source-finder with a detection threshold of $10\sigma$ to create per-beam source lists—this yields of order 50–200 sources per beam per SBID prior to cross-matching. We cross-match these individual single-beam source-lists to the NVSSFootnote ¹³, including only compactFootnote ¹⁴ and isolatedFootnote ¹⁵ sources. The final per-beam cross-matched source-lists contain $\sim 50$ –100 sources per SBID. We assume a beam attenuation of the form,

(3) \begin{equation} A^{b}_{\text{attenuation}} = \frac{S_{b}\left(1400/1367\right)^\alpha}{S_{\text{NVSS}}}, \,\end{equation}

for sources with apparent flux density $S_{b}$ in RACS-mid beam, b. We also assume a nominal $\alpha = -0.7$ to scale NVSS measurements to 1 367.5 MHz, though the final attenuation patterns are normalised after modelling.

The measured $A^{b}_{\text{attenuation}}$ is median-binned in tile (l,m) coordinates, stacking all SBIDs for a given BWT. Bin sizes range to $2.1 \times 2.1$ arcmin $^2$ for BWT-1 to $6.9 \times 6.9$ arcmin $^2$ for BWT-2, BWT-3, and BWT-5. The larger bin size used for the later BWTs is to account for more sparsely sampled beams. We fit 2-D models to the binned measurements of $A^{b}_{\text{attenuation}}$ as a function of (l,m) using standard least-squares methods. While generic 2-D polynomial and elliptical Gaussian models are tested we find these do not represent the attenuation patterns for all beams. Instead we find Zernike polynomial models (Zernike, Reference Zernike1934)Footnote ¹⁶ fit the primary beam main lobe patterns well. Zernike polynomials have been used for modelling holographic primary beam measurements from the VLA (e.g. Iheanetu et al., Reference Iheanetu, Girard and Smirnov2019; Sekhar et al., Reference Sekhar, Jagannathan, Kirk, Bhatnagar and Taylor2022) and MeerKAT (e.g. Asad et al., Reference Asad, Girard and de Villiers2021; Sekhar et al., Reference Sekhar, Jagannathan, Kirk, Bhatnagar and Taylor2022). A brief comparison of some alternate beam attenuation models are shown in Appendix A.

Each binned $A^{b}_{\text{attenuation}}$ dataset is fit with a Zernike polynomial of reasonably high Noll index (Noll, Reference Noll1976). We use the Akaike information criterion (AIC; Akaike, Reference Akaike1974) to select the appropriate Noll index for each beam. The choice of Noll index varies per beam and per BWT, increasing slightly for BWT-1 with larger bins, and range from 38–99 for BWT-1, 32–40 for BWT-2, 38–58 for BWT-3, and 22–41 for BWT-5. The different BWTs and beams have significant differences in the density of sources in the sidelobes, accounting for some of the variation seen in the selected Noll indices. Consequently, the sidelobes of the primary beam are generally poorly modelled and are clipped in the final beam models. For comparison, Sekhar et al. (Reference Sekhar, Jagannathan, Kirk, Bhatnagar and Taylor2022) use a Noll index of 66 to model both VLA and MeerKAT beams, though in that case the sidelobes are well modelled with their holographic measurements. Attenuation patterns for each beam are additionally clipped below 12% which reflects the clip used during mosaicking. The individual beam images are incorporated into the FITS file format used by the observatory to store the holographic primary beam measurement and are used by the ASKAPSoft mosaicking software.

Figure 7 shows the binned, measured and Zernike model Stokes I response for beam 35 from BWT-1, as well as the binned, (and regridded) measured, Zernike model, and holographic model response for beam 35 from BWT-4. The BWT-1 beam 35 measured data use a $2.1 \times 2.1$ arcmin $^{2}$ bin size, and the BWT-4 beam 35 measured data use a $10.4 \times 10.4$ arcmin $^{2}$ bin size. For display purposes the BWT-4 data are regridded to the same bin size as the BWT-1 data, including interpolation. The ratios between the measured and model responses are also shown to highlight the offset in brightness scale that would be introduced when using the holographic model from a different BWT. The main difference we see between BWTs is a shift in peak position of the beam, but there is also a small deviation in the shape that is more difficult to account for in simply shifting the holographic model beam positions. A small additional offset is observed between the BWT-5 holographic model and the BWT-5 Zernike model which results in a 10–20% variation in brightness scaling towards the beam edges. Further examples of beams 15 and 35 for BWT1–3&5 are shown in Figure 8, highlighting the measured attenuation pattern per source, the resulting model at each source’s location, and residuals after application of the model to the measured sources. Figure 9 shows the Stokes I beam models for all beams of BWT-1 to highlight the variation in the beam shapes across the full footprint.

Figure 7. Measured and model $A_{\text{attenuation}}$ of beam 35 for BWT-1 and BWT-4. Diagonal panels show, from top to bottom, (1) the binned, measured attenuation from BWT-1, (2) the best-fit Zernike polynomial model for BWT-1, (3) the binned, measured attenuation from BWT-4 (after regridding and interpolation), (4) the holographic measurements from SB28507 used for BWT-4, and (5) the best-fits Zernike polynomial model for BWT-4. Plots underneath the diagonals are the ratio difference between the various patterns (as top model / bottom model). All patterns are clipped at 0.12 to reflect the cutoff used during mosaicking.

Figure 8. Central beam 15 [(i), (iii), (v), (vii)] and corner beam 35 [(ii), (iv), (vi), (viii)] Stokes I $A^b_{\text{attenuation}}$ modelling for BWT-1 (top row), BWT-2 (second row), BWT-3 (third row), and BWT-5 (bottom row). Left panels. Measured attenuation pattern showing individual sources. Centre. The fitted Zernike model at the location of the individual sources. Right. Ratio of the measured and model attenuation patterns representing residuals. The colourmap for $A^b_{\text{attenuation}}$ is clipped at 0.12, corresponding to the blue sources.

Figure 9. Stokes I model beams for BWT-1 for all beams in the footprint. Beams are clipped at 12% attenuation and are arranged to match the footprint (Figure 1).

We do not have the SNR to accurately measure the spectral-dependence of the beams. As shown in Figure 2, the beam-averaged FWHM as measured by holography varies from 1.30 to 1.18 degrees at the low- and high-frequency ends of the (unflagged) band. As the Zernike beam models are determined from the MFS apparent brightness images, any residual frequency dependence is implicitly captured by the Zernike models and the 0 $^{\text{th}}$ -order Taylor maps after primary beam correction will not have residual frequency-dependent spectral effects. Mosaicking for SBIDs that use these in-field measured beams does not provide a frequency-dependent correction for the 1 $^{\text{st}}$ -order Taylor maps that are created during imaging. These 1 $^{\text{st}}$ -order Taylor maps will only be correct at the centre of each beam. Mosaic images are trimmed to remove additional primary beam sidelobe components present above 12% for the SBIDs weighted using holographic measurements.

2.4.3. Measurement of the Stokes V widefield leakage

To characterise the widefield leakage from Stokes I into V for all BWTs, we follow a similar method to our characterisation of the total intensity response described above. This method is also being used for leakage characterisation of Stokes I into Q and U by Thomson et al. (in prep) for SPICE-RACS , and has been used for widefield leakage correction of the Westerbork Synthesis Radio Telescope (Farnsworth et al., Reference Farnsworth, Rudnick and Brown2011), the MWA (Lenc et al., Reference Lenc, Anderson and Barry2017), and other ASKAP data (e.g. POSSUM; West et al., in prep; Gaensler et al., in prep). We begin using the same total intensity source catalogues used in Section 2.4.2, and extract the corresponding uncorrected Stokes V flux density at the peak total intensity pixel location from each beam image. The number of detectable circularly polarised sources is expected to be low (e.g. Lenc et al.,Reference Lenc, Murphy, Lynch, Kaplan and Zhang2018; Callingham et al., Reference Callingham, Shimwell and Vedantham2023) and we assume the selected sources are unpolarised with no preferred handedness (as suggested by previous Stokes V surveys, e.g. Rayner et al., Reference Rayner, Norris and Sault2000; Lenc et al., Reference Lenc, Murphy, Lynch, Kaplan and Zhang2018; Callingham et al., Reference Callingham, Shimwell and Vedantham2023). As above, we model the 2-D widefield leakage surface using a Zernike polynomial using least-squares for each BWT. In contrast to our total intensity modelling, we cut out components that are $<100\sigma_{\text{rms},I}$ or are separated from the beam centre by more than $1^\circ$ . In addition to this sample cut, model-fitting here is also performed on individual sources rather than binned data in contrast to the approach taken with the Stokes I beam modelling. As we do not normalise the leakage surface, any residual position-independent leakage introduced or left-over from on-axis corrections using PKS B1934 $-$ 638 are also included in these widefield models.

Initially inspecting the distribution of V/I, we find some points show spuriously high fractional circular polarisation, despite our cuts. In Figure 10 we show the standard deviation of V/I as a function of SBID. We see that there are some observations with high variance, including some entire BWT. For our purposes, we initially excluded SBIDs with a V/I standard deviation greater than the $84^{\text{th}}$ percentile of the entire set of observations, however, due to small number of remaining SBIDs for most BWTs we opt to relax this to the $99.7^{\text{th}}$ percentile for BWT-3–9. After excluding outlying sources, we fit Zernike polynomials up to a maximum Noll index of 10, and select the best model according to the AIC. Finally, we regrid and interpolate our fitted models to exactly match the corresponding holography images produced for the Stokes I beams.

Figure 10. V/I across all SBIDs for beam 0 prior to mosaicking and leakage correction (i) and for the full tiles after mosaicking and applying leakage correction (ii). Top panels. $|V/I|$ for all sources with $S_I > 100\sigma_{\text{rms},I}$ . Bottom panels. The standard deviation of $V/I$ for each observation. Each BWT (advancing from left to right; see Table 4) is shaded a different colour.

Figure 11 shows the leakage surface for beam 35 for BWT-1 and BWT-4 and compares to the BWT-4 holographic measurement of the same beam as in Figure 7 for the Stokes I response. In Figure 11, the data are binned as in the Stokes I case, though we do not bin the data for fitting. The residual differences between the measured and model leakage surfaces are shown. There is an offset between the holographic model and Zernike models, though we do not use the holographic model for widefield leakage correction. We show additional examples of our fitted surfaces for beam 15 and 35 for BWT-1–3 in Figure 12, again showing the residual difference between the measured $V/I$ and the fitted leakage surfaces. Finally, model Stokes V beams for BWT-1 are shown in Figure 13 for all beams in the footprint. The partition between negative and positive leakage in the Zernike polynomial surfaces generally resemble those found for other instruments (e.g. the VLA and MeerKAT; Sekhar et al., Reference Sekhar, Jagannathan, Kirk, Bhatnagar and Taylor2022), taking into account all beams being offset from the optical axis due to their arrangement in the closepack36 PAF footprint. Some beams (e.g. 12/23 and 13/22, see Figure 13) have non-standard shapes, though their symmetry in the PAF footprint suggests this is an accurate representation of the leakage for these beams. While PAF-based Stokes V beam modelling is not available in the literature for comparison, the leakage patterns vary between the PAF beams which is seen in leakage maps for Stokes Q and U for ASKAP (Thomson et al., in prep) and in Stokes Q for Apertif (Dénes et al., Reference Dénes, Hess and Adams2022).

Figure 11. A comparison of the measured and modelled leakage of Stokes I into V ( $V/I$ ) for beam 35 in BWT-1 and BWT-4. Diagonal panels show, from top to bottom, (1) the binned, measured leakage from BWT-1, (2) the fitted Zernike polynomial for BWT-1, (3) the binned, measured leakage from BWT-4, (4) the leakage model derived from holographic measurements, and (5) the fitted Zernike polynomial for BWT-4. Plots underneath the diagonals are the residual differences (as top $-$ bottom) and all patterns are clipped with reference to (1).

Figure 12. Central beam 15 [(i), (iii), (v), (vii)] and corner beam 35 [(ii), (iv), (vi), (viii)] Stokes V leakage modelling results for BWT-1 (top row), BWT-2 (second row), BWT-3 (third row), and the subset SB21616–SB21710 from BWT-1 (bottom row). Le_ panels. Measured V/I leakage pattern with individual sources. Centre. The fitted Zernike model at the location of the individual sources. Right. Residual leakage patterns. Only sources within 1 deg of the beamcentre are included. Note (ii) is similar to the top three panels of Figure 11.

Figure 13. Stokes V model beams ( $V/I \times A_{\text{attenuation}}$ ) for BWT-1 for all beams in the footprint. Beams are clipped at 12% Stokes I attenuation and are arranged to match the footprint (Figure 1).

The SBID subset SB21616–SB21710 showed spuriously high residual leakage after correction compared to the remainder of BWT-1. This subset contains 91 SBIDs corresponding to a single calibrator, SB21637. We create models for this subset separate from BWT-1 for both the Stokes I response and the $V/I$ leakage. We find that some beams in this subset have notably different leakage patterns to the remainder of BWT-1. Figure 12(Vii) shows beam 15 for the SB21616–SB21710 and 12(viii) shows beam 35. While beam 15 in this subset resembles beam 15 from the full BWT-1 subset, beam 35 differs significantly. Other beams, including 12 and 23, were also found to have spuriously high ( $|V/I| >0.03$ ) leakage sources when using the full BWT-1 leakage correction. For the SB21616–SB21710 subset, we use leakage models derived from those SBIDs only. We find no substantial difference in the Stokes I response in the SB21616–SB21710 subset and continue to use the full BWT-1 Stokes I model for these SBIDs. It is not clear what caused the change in leakage characteristics for select beams within this SBID subset.

The residual per-SBID $|V/I|$ leakage after mosaicking and application of the leakage surface is also shown in Figure 10(ii) to highlight the reduction of leakage.

2.5.1. Stokes I

The number of sources with Stokes I flux densities $>10\sigma_{\text{rms},I}$ per adjacent beam cross-match ranges from $\sim 25\,000$ to $\sim 53\,000$ , with corner and edge beams featuring fewer sources due to fewer adjacent beams. Figure 14 shows the ratio of Stokes I flux density measurements of sources in adjacent beams as a function of distance from the beam centres. The plot shows the cross-match results for each beam separately, arranged to match the PAF footprint (see Figure 1). As each beam is cross-matched to each adjacent beam, the individual beam results are not independent. We calculate the median flux density ratio in bins separately for the Zernike-based models (white circles, with $\sim 24\,000$ to $\sim 50\,000$ sources per beam) and the holography-based models (red squares, with $\sim 1\,100$ to $\sim 2\,400$ sources per beam). The overall median ratio is $1.00_{-0.11}^{+0.12}$ . Generally there is good agreement in adjacent beams with two main exceptions: roll-off in the holography-derived models beyond $\sim 0.8$ deg from the beam centre of order $\sim 10$ % and offsets in beams 5 and 30. The corner beams 5 and 30 feature median ratios of $0.95_{-0.08}^{+0.10}$ and $1.07_{-0.10}^{+0.10}$ for the Zernike subset ( $0.90_{-0.09}^{+0.09}$ and $1.12_{-0.12}^{+0.13}$ , for the holography subset), respectively.

Figure 14. 2-D histogram of Stokes I flux density ratios of sources detected across adjacent beams as a function distance from the reference beam centre. Binned median flux density ratios for the Zernike models (white circles) and holography models (red squares) are shown, along with $16^{\text{th}}$ and $84^{\text{th}}$ percentiles for the corresponding bins. The per-beam plots are arranged to match the footprint (Figure 1). The colour scale is linear in the reported range.

2.5.2. Stokes V

A similar process is repeated for Stokes V, though we first look at unpolarised sources in adjacent beams after application of the widefield leakage corrections and primary beam correction. For this, we use sources detected at $>10\sigma_{\text{rms},I}$ in both beams, but restrict to sources with $|S_V| < 3\sigma_{\text{rms},V}$ in the reference beam. The adjacent beam Stokes V measurement is not restricted. This results in a similar number of sources per beam as in the Stokes I comparison in the previous section. Figure 15 shows the unpolarised sources in the 36 beams with the difference in Stokes V measurements between adjacent beams. In Figure 15 we also show the binned medians for the selected sources, along with the $16^{\text{th}}$ and $84^{\text{th}}$ percentiles. The median $3\sigma_{\text{rms},V}$ for the relevant beam is shown in Figure 15, with the median $\sigma_{\text{rms},V}$ per beam ranging from 154 to 200 $\mu$ Jy PSF $^{-1}$ . The overall median difference in Stokes V measurements from sources in adjacent beams is $0_{-503}^{+505}$ $\mu$ Jy PSF $^{-1}$ and shows an increase at small and large separations. This is consistent with elevated noise towards the edges of the adjacent and reference beams, respectively.

Figure 15. Difference in Stokes V measurements between adjacent for unpolarised ( $S_V < 3\sigma_{\text{rms},V}$ in the reference beam) Stokes I sources. The white circles are medians in bins, and the white, solid lines are the corresponding $16^{\text{th}}$ and $84^{\text{th}}$ percentiles. The dashed, red lines indicate the median $3\sigma_{\text{rms},V}$ . The per-beam plots are arranged to match the footprint (Figure 1). The colour scale is linear in the reported range.

For circularly polarised sources, there are a total of 18 adjacent beam detections with $|S_V| > 10\sigma_{\text{rms},V}$ in each beam. Figure 16 shows the absolute ratio of Stokes V measurements from adjacent beams, coloured by beam number. Sources with the same sign in adjacent beams are indicated with circles, and sources with opposing signs are indicated with diamonds. The median ratio over all beams is $1.02_{-0.09}^{+0.17}$ . Two of the detections show opposing signs, which correspond to a single source (NVSS J205111 $+$ 081859) detected across beams 9 and 16 at $10.2\sigma_{\text{rms},V}$ and $28.3\sigma_{\text{rms},V}$ , respectively, in SB21634. A single detection of PKS B1340 $-$ 373 in beam 23 from SB21680 shows $|S_V/S_{V,\text{adjacent}}| \approx 2$ , corresponding to a residual leakage in the final mosaicked tile of $|V/I|\approx 0.015$ . Both SB21634 and SB21680 are part of the subset noted in Section 2.4.3 that show different leakage patterns for some beams. This suggests the residual leakage in these SBIDs may differ from the remainder of the survey despite the separate leakage models used.

3.3.1. Absolute flux density scale

As an initial check of the absolute flux density scale, we follow Paper I by comparing RACS-mid flux density measurements of a selection of bright calibrator sources modelled by Perley & Butler (Reference Perley and Butler2017). We also check the flux density of PKS B1934 $-$ 638 by comparison to the model reported by Reynolds (Reference Reynolds1994). For this purpose, we integrate the flux density above $2\sigma_{\text{rms}}$ directly from the maps within apertures that cover twice the angular extent reported by Perley & Butler (Reference Perley and Butler2017)—assuming 0.1 arcsec extent for PKS B1934 $-$ 638—convolved with the Gaussian restoring PSF of the images they are measured from. Figure 26 shows the RACS-mid flux density measurements against the model flux density. All source measurements lie within $\sim 6\%$ of model values except for extended sources. Typically the more extended the source is, the less agreement between measurement and model. This is particularly significant for Fornax A and Taurus A (enclosed by black diamonds in Figure 26), though in these cases the additional 75 m (u,v) cut reduces the measurement in RACS-mid further (see Section 2.3.2 and Section 3.5).

Figure 16. Absolute Stokes V flux density ratios for sources detected in adjacent beams, as a function of angular separation from the reference beam centre. The sources are coloured by reference beam number. The black, solid line indicates a ratio of 1, the black, dashed line is the median ratio, and the black, dot-dash lines indicate the 16 $^{\text{th}}$ and 84 $^{\text{th}}$ percentiles.

3.3.2. The brightness scale across the sky

New source-lists are generated by selavy after the tile-dependent brightness scaling described in Section 2.6. These per-SBID source-lists are used in standard pipeline processing for per-SBID validation and are also used for full-survey validation focused on flux densities/brightness scaling and astrometry (Section 3.6). Source-finding is performed to a limit of $5\sigma$ in the individual tile images. These source lists are cross-matched to various catalogues: the NVSS; the Sydney University Molonglo Sky Survey (SUMSS; Bock et al., Reference Bock, Large and Sadler1999; Mauch et al., Reference Mauch, Murphy and Buttery2003); the TIFRFootnote ¹⁷ GMRTFootnote ¹⁸ Sky Survey (alternate data release 1; Intema et al., Reference Intema, Jagannathan, Mooley and Frail2017); and RACS-low; and sources in the third realisation of the International Celestial Reference Frame (ICRF3; Charlot et al., Reference Charlot, Jacobs and Gordon2020). Despite similarity in frequency to FIRST, we do not use FIRST for flux density comparisons as FIRST was observed over different frequency ranges for different parts of the survey due to the upgrade to the VLA (Helfand et al., Reference Helfand, White and Becker2015). Comparisons with NVSS, SUMSS, and RACS-low are useful in determining the quality (or consistency) of the final brightness scaling, particularly as NVSS and SUMSS catalogues have already been used extensively in deriving the primary beam response for a majority of the observations (Section 2.4) and general brightness scaling per tile (Section 2.6).

Figure 27(i) shows the histogram of spectral indices derived from comparisons with SUMSS, RACS-low, and TGSS (excluding the Galactic Plane where source spectra may not reflect extragalactic source properties). The median spectral indices across the survey region are found to be $\alpha_{\text{TGSS}}^{\text{RACS-mid}} = -0.69_{-0.22}^{+0.25}$ , $\alpha_{\text{SUMSS}}^{\text{RACS-mid}} = -0.81_{-0.42}^{+0.50}$ , and $\alpha_{\text{RACS-low}}^{\text{RACS-mid}} = -0.88_{-0.28}^{+0.39}$ , with uncertainties derived from the 16 $^{\text{th}}$ and 84 $^{\text{th}}$ percentiles of each distribution. For SUMSS, this is consistent with the spectral index used for SBID-dependent scaling earlier. For RACS-low, this is consistent to the spectral index between RACS-low and NVSS reported in Paper II. The TGSS comparison results in a spectral index representative of flatter spectra and is also consistent with similar comparisons in Paper II for RACS-low. Similarly, the median spectral index when matching to TGSS is the same median in the NVSS–TGSS spectral index catalogue constructed by Gasperin et al. (2018): $\alpha_{\text{TGSS}}^{\text{NVSS}} = -0.70_{-0.21}^{+0.26}$ . Direct comparisons of the flux densities for NVSS, SUMSS, and RACS-low over the full survey (excluding the Galactic Plane) are shown in Figure 27(ii). These comparisons assume $\alpha=-0.82$ for NVSS and SUMSS (as in Section 2.6), and $\alpha=-0.88$ for RACS-low. As expected, due to SBID-dependent scaling, NVSS and SUMSS comparisons show a median ratio of 1: $S_{\text{RACS-mid}}/S_{\text{NVSS}} = 1.00^{+0.20}_{-0.14}$ , $S_{\text{RACS-mid}}/S_{\text{SUMSS}} = 1.00^{+0.27}_{-0.18}$ , and $S_{\text{RACS-mid}}/S_{\text{RACS-low}} = 1.00^{+0.18}_{-0.11}$ , with uncertainties estimated as for $\alpha$ .

Figure 17. Time-dependence of the flux density scale across the survey with comparison to NVSS (red, filled circles) and SUMSS (blue, open squares). Top. Logarithmic flux density ratios for comparisons with NVSS and SUMSS, with no frequency scaling. Middle. SBIDs with sources in the central 2 deg of the tile within the declination range $-40 \leq \delta_{\text{J2000}} \leq -30$ with both NVSS and SUMSS cross-matches. Bottom. All SBIDs in the top panel but after scaling the source flux densities assuming a power law with index $\alpha = -0.82$ . The black, dashed line in all panels corresponds to flux densities ratios of unity. The coloured lines correspond to medians for each survey comparison.

Figure 28 shows the median brightness scaling over the whole sky with comparison to the NVSS, SUMSS with the second epoch Molonglo Galactic Plane Survey (MGPS-2; Murphy et al., Reference Murphy, Mauch and Green2007), and RACS-low. While there is general agreement across the sky (by construction due to the use of NVSS and SUMSS in SBID-dependent brightness scaling), variation exists, particularly in the Galactic Plane (see Section 3.3.3). The comparison with RACS-low [Figure 28(iii)] also shows the same position-dependent variation with respect to NVSS and SUMSS revealed in Paper I and Paper II. Likely the positional variation between RACS-mid and RACS-low is a result of uncorrected per-observation brightness scale variations reported in Paper I.

3.3.3. The brightness scale in the Galactic Plane

In the Galactic Plane ( $|b| \lesssim 5^\circ$ ), RACS-mid generally reports a higher flux density than expected compared to NVSS [Figure 28(i)], with $S_{\text{RACS-mid}}/S_{\text{NVSS,GP}} = 1.04_{-0.16}^{+0.28}$ . A similar increase, though smaller in magnitude, is seen in comparison to RACS-low [Figure 28(iii)] with $S_{\text{RACS-mid}}/S_{\text{RACS-low}} = 1.01_{-0.18}^{+0.33}$ . While the Galactic Plane tiles are scaled in brightness according to the elevation-dependent fit shown in Figure 18, a separate time-dependent effect remains uncorrected, as described in Section 2.6. With reference to the NVSS, the regions around Cygnus A and between $l \gtrsim 350^\circ$ and $l \lesssim 55^\circ$ show an increase in brightness of order 15–25% which is beyond what is seen in Figure 17 for non-Galactic Plane SBIDs. Artefacts from the bright, extended sources in the Galactic Plane (and Cygnus A) may also influence the flux density measurements where they overlap with real sources.

Figure 18. Median tile flux density ratios between the NVSS (red, filled circles) and SUMSS (blue, open squares) after scaling in frequency assuming $\alpha = -0.82$ as a function of elevation. The solid black line shows the 5 $^{\text{th}}$ -order polynomial fit to the flux density ratios.

In addition to the main all-sky catalogues noted in Section 3.3.2, we also cross-match the RACS-mid source lists with a collection of smaller-area Galactic Plane surveys: MGPS-2, THORFootnote ¹⁹ (Beuther et al., Reference Beuther, Bihr and Rugel2016; Wang et al., Reference Wang, Beuther and Rugel2020), MAGPISFootnote ²⁰ (White et al., Reference White, Becker and Helfand2005; Helfand et al., Reference Helfand, Becker, White, Fallon and Tuttle2006), and the CGPSFootnote ²¹ (Taylor et al., Reference Taylor, Gibson and Peracaula2003) where overlap exists with RACS-mid. The MGPS-2 comparison is shown alongside SUMSS in Figure 28(ii) and does not show the same increase in brightness seen with either RACS-low or NVSS comparisons, with a marginally lower median $S_{\text{RACS-mid}}/S_{\text{MGPS-2}} = 0.98_{-0.21}^{+0.29}$ . Figure 29 shows the binned flux density ratios for sources within $|b|\leq 5^\circ$ from the Galactic Plane surveys as a function of l along with NVSS and RACS-low. We note the overall median flux density ratios as $S_{\text{RACS-mid}}/S_{\text{THOR}} = 1.26_{-0.24}^{+0.30}$ (5231 sources), $S_{\text{RACS-mid}}/S_{\text{MAGPIS}} = 0.88_{-0.15}^{+0.20}$ (1481 sources), and $S_{\text{RACS-mid}}/S_{\text{CGPS}} = 1.02_{-0.20}^{+0.32}$ (14246 sources).

As only compact sources are used for comparison, it is unlikely RACS-mid recovers a larger fraction of flux density for the extended radio sources in the Galactic Plane. RACS-low showed a similar increase in flux density with respect to NVSS in the Galactic Plane, and it is clear from comparison with smaller Galactic Plane surveys there is large variation within this complex region of the sky.

3.3.4. Accuracy of the primary beam corrections

To confirm the accuracy and consistency of the primary beam corrections after mosaicking, we also show the median flux density ratios as a function of position over the PAF footprint for the tile, binned in (l,m). Figure 30 shows the comparison for NVSS [30(i)], SUMSS [30(ii)], and RACS-low [30(iii)] for the full RACS-mid survey (where overlaps exists with the relevant comparison survey). We find residual errors on the edges of the tiles due to errors in the individual beams which are reduced in the centre of the tile. Additional errors are seen, particularly for southern fields and those in BWT-4 using holographic primary beam correction. In Figure 31 we show the same median tile flux density scale split into three subsets: BWT-1–3&5 cross-matched to NVSS [31(i)] and SUMSS [31(ii)], both using exclusively Zernike model primary beam responses, and BWT-4&6–9 cross-matched to NVSS [31(iii)] using the holography approach. This highlights that the edge effect seen in Figure 30(ii) is present for both the Zernike and holography subsets, though the residual flux density ratio is flipped N-S. The error appears to increase away from the location where the primary beam correction is defined. As the Zernike models are defined with reference to the NVSS in the range $-40^\circ < \delta_{\text{J2000}} < +48^\circ$ , they would be most applicable to the celestial equator. For holography, PKS B0407 $-$ 658 is used as the target source and $\delta_{\text{J2000}} \approx -65^\circ$ is the reference location. Due to the limited holography availability for RACS-mid, future investigation with RACS data is reserved for work with RACS-high and the second epoch of RACS-low, both of which have been observed alongside corresponding holographic measurements. In a future paper, when cataloguing, we will be linearly mosaicking nearby tiles as in Paper II which reduces this error for most tiles. With the improved data quality and consistency, RACS-mid and other recent ASKAP imaging is becoming sensitive to higher order effects, such as antenna pointing errors, which are yet to be fully analysed and corrected.

Figure 19. Example RACS-mid image.

Figure 20. A comparison of RACS-mid (top left) with RACS-low (bottom left), and RACS-mid and RACS-low convolved to a 25 arcsec resolution (top right and bottom right, respectively), featuring a radio galaxy of $\sim 6$ arcmin angular extent associated with ESO 509 $-$ G108. The size of the PSF for each image is shown in the bottom left as an ellipse. Note the colour scale varies between the images to reflect the change in angular resolution.

Figure 21. A radio galaxy associated with 2MASS J10340385 $+$ 1840490 as seen in the RACS-mid (right), NVSS (centre), and FIRST (right) images. Note the colour scale varies between the three images due to the difference in angular resolution.

Figure 22. PSF variation between tiles across the full survey area in equatorial coordinates. A single major axis (left panel) and minor axis (right panel) is associated with each tile. Note each panel has different colour scaling to highlight the variation in major and minor axes independently.

As a check of the internal consistency of the primary beam models after application during linear mosaicking, we compare the flux densities of sources detected in two tiles (in overlapping regions between tile mosaics). Sources are selected by cross-matching individual source-lists and considering sources with cross-matches within half their reported size. We restrict this to compact ( $S_{\text{int}} / S_{\text{peak}} < 1.2$ ) sources only. This process is similar to the per-beam comparisons shown in Section 2.5.1 except cross-matches occur between observations and we only consider sources once. Figure 32 shows a 2-D histogram of the integrated flux density ratios ( $S_{\text{int,1}} / S_{\text{int,2}}$ ) for $100\sigma_{\text{rms}}$ sources present in two tiles, plotted as a function of the sum of the source separations from their respective tile centres. We also show medians along with 16 $^{\text{th}}$ and 84 $^{\text{th}}$ percentiles in $\sim 8$ arcmin bins. While some features appear in the histogram, these are due to larger densities of sources at specific separation distances and the reported summary statistics do not vary as a function of distance from the tile centre in these overlap regions. The overall median ratio is $0.99_{-0.07}^{+0.07}$ .

3.3.5. Brightness scale uncertainty

We find that despite tile edge effects, the uncertainty in the brightness scale for each BWT, as a function of declination, and internally is reasonably consistent. Therefore to derive a brightness scale uncertainty for RACS-mid, we focus on the bulk NVSS-derived flux density ratios for $100\sigma$ sources to reduce contribution from frequency differences and low-SNR measurement errors. As an estimate of this uncertainty, we use the maximum between the 16 $^{\text{th}}$ and $84^{\text{th}}$ percentile of the $S_{\text{RACS-mid}} / S_{\text{NVSS}}$ ratio to avoid being overly conservative within the fairly well-modelled tiles (within the central $6\,\text{deg} \times 5\,\text{deg}$ ). This yields a brightness scale uncertainty of 6%, consistent with scatter in measurements of bright calibrator sources (Figure 26). The exceptions to this are the edges of the PAF footprint (within $\sim 0.5$ deg of the footprint boundary). In this exterior region, we follow Paper I and estimate the uncertainty by cross-matching compactFootnote ²² sources that are detected in adjacent tiles outside of the central $6\,\text{deg} \times 5\,\text{deg}$ . This provides a median integrated flux density ratio of $1.00_{-0.12}^{+0.13}$ . Adding the tile exterior uncertainty in quadrature with the NVSS-based uncertainty we estimate a brightness scale uncertainty, $\xi_{\text{RACS-mid}}$ , of

(5) \begin{equation} \xi_{\text{RACS-mid}}(l,m) = \begin{cases} 0.06 S_{\text{RACS-mid}}, & \text{if}~(|l| < 3^\circ, |m| < 2.5^\circ) \\[5pt] 0.14 S_{\text{RACS-mid}}, & \text{otherwise} \end{cases}\end{equation}

as a function of (l,m) across a given tile for all SBIDs. These externally-estimated uncertainties are consistent with the internal accuracy of the primary beam discussed in the previous section and Section 2.5.1.

3.4.1. Stokes V residual leakage

To assess the residual leakage of Stokes I in to V, we extract residual leakage using the selavy Stokes I source-lists. In the Stokes V images we take the absolute brightest pixel within a 9-pixel box around the Stokes I position, avoiding sources near beam edges which may be reduced in size in either the Stokes I or V images. Residual leakage tables for each SBID are stored in the RACS database as an additional source list for use in data validation. Figure 33 shows the mean $|V/I|$ for sources with $S_I >500\sigma_{\text{rms},I}$ in HEALPix bins across the sky.

Figure 23. The median and maximum PSF major and minor axes in declination, binned to approximately match tile separation.

Figure 34 shows the residual $|V/I|$ leakage as a function of distance from the tile centre and declination for the same sample of sources. In Figure 34 we also show the binned median $|V/I|$ split between the SB21616–SB21710 subset, which is corrected using a single-day leakage model (see Section 2.4.3) and the remaining SBIDs. The median $|V/I|$ is $0.0014_{-0.0007}^{+0.0012}$ for the main subset and $0.0017_{-0.0009}^{+0.0022}$ for the SB21616–SB21710 subset. For the main subset, we estimate the residual leakage from $3\sigma$ of the $|V/I|$ distribution: $|V/I| \lesssim 0.009$ . Figure 34 shows higher residual leakage in the SB21616–SB21710 subset, and we estimate the residual leakage for this subset separately as $|V/I| \lesssim 0.024$ . These are shown on Figure 34, along with the binned $3\sigma$ as a function of tile centre separation and declination. The higher residual leakage found for the SB21616–SB21710 subset explains the spurious sources noted Section 2.5.2, including the sign-changing source NVSS J205111 $+$ 081859 which has $|V/I| \approx 0.016$ in the mosaic of SB21634.

3.4.2. Comparison of bright Stokes V sources with the literature

Sources with high fractional circular polarisation are generally associated with stars, including pulsars (e.g. Lenc et al., Reference Lenc, Murphy, Lynch, Kaplan and Zhang2018; Pritchard et al., Reference Pritchard, Murphy and Zic2021). We create a subset of the Stokes V sources from aforementioned leakage tables by extracting sources detected at both $10\sigma_{\text{rms,}I}$ and $10\sigma_{\text{rms,}V}$ . From these 1 011 sources, we inspect the top twenty with the highest fractional polarisation. We find that five correspond to either extended/double radio sources with mismatched peak Stokes V peak flux density measurements that are normally removed in isolated/compact source cuts. The remaining fifteen are associated with stars (and pulsars) or are otherwise related to stellar systems. This includes the magnetic chemically peculiar star CU Vir detected with $V/I \sim 0.66$ , which is known to have pulsed circularly polarised radio emission (e.g. Lo et al., Reference Lo, Bray and Hobbs2012) and has been previously detected by RACS-low (Pritchard et al., Reference Pritchard, Murphy and Zic2021).

As many circularly polarised sources are variable, to assess the absolute flux scale and sign of the Stokes V data we also cross-match the $>10\sigma_{\text{rms},V}$ Stokes V sources with the pulsar catalogue reported by Johnston & Kerr (Reference Johnston and Kerr2018). This catalogue is reported at 1 400 MHz with measurements integrated over pulse profiles. With a 2 arcsec cross-match (with no proper motion correction), we find fourteen pulsars from the ATNF pulsar catalogueFootnote ²³, of which eleven are cross-matched with the Johnston & Kerr (Reference Johnston and Kerr2018) catalogue. We re-measure the Stokes I and V flux densities within apertures as in Section 3.3.1 instead of relying on the peak flux measurement as in the leakage tables. Figure 28 shows the RACS-mid $V/I$ compared to the Johnston & Kerr (Reference Johnston and Kerr2018) $V/I$ after a sign flip (to account for a difference in sign convention for the pulsar catalogue) coloured by their Stokes V SNR. There is agreement with the signs for each pulsar with $|V/I|$ typically slightly larger in RACS-mid. The ratio of RACS-mid fractional polarisation to the pulsar catalogue values has a median of $1.08_{-0.19}^{+0.70}$ . We also show $V/I$ for PKS B1934 $-$ 638 for reference, assuming zero circular polarisation, which is used for on-axis leakage correction prior to widefield corrections.

Figure 24. Mosaic of individual Stokes I rms noise maps across the full survey in equatorial coordinates. The region of Galactic latitude covering $|b| < 5^\circ$ is bounded by the dashed, black lines.

No sources from the validation files that satisfy $10\sigma_{\text{rms,}I}$ and $10\sigma_{\text{rms,}V}$ are cross-matched with the circularly polarised LOFAR Two-metre Sky Survey detections (V-LoTSS; Callingham et al., Reference Callingham, Shimwell and Vedantham2023) despite some overlap in the the respective survey areas. More in-depth comparisons will be made in future Stokes V cataloguing work.

Figure 25. Binned noise ( $\sigma_{\text{rms}}$ ) across tiles as a function of declination for median and minimum values of $\sigma_{\text{rms}}$ . The solid, black and dot-dash, grey lines indicate median values over the full survey, for Stokes I and V, respectively.

Figure 26. Integrated flux density of bright calibrator sources modelled by Perley & Butler (Reference Perley and Butler2017) with the inclusion of PKS B1934 $-$ 638 (Reynolds, Reference Reynolds1994) as a function of their model flux density. Sources are coloured by their angular extent, but clipped (orange) when smaller than the RACS-mid resolution. Sources in tiles with a $(u,v)<$ 75 m visibility cut are enclosed by black diamonds. Measurement and brightness scale uncertainty are plotted but are smaller than the visible marker size.

3.6.1. Beam-to-beam consistency

Variation in astrometry between individual beams of order $\lesssim 2.5$ arcsec ( $\sim 1$ pixel) was noted in Paper I. The cause of scatter between adjacent beams is largely due to self-calibration and will therefore affect all sources regardless of SNR. To investigate this for RACS-mid, we cross-matched compact, isolated sources in the individual apparent brightness beam images used for primary beam response modelling (Section 2.4.2). This was done for beams 35 and 29, beams 35 and 34 (top left corner of the footprint, see Figure 1), and beams 20 and 15 (centre of the footprint) for all SBIDs in the survey. Figure 36 shows $\Delta\alpha\cos\left(\delta\right)$ and $\Delta\delta$ for the cross-matched sources in beams 20 and 15, beams 35 and 29, and beam 35 and 34. We split the data into low- and high-SNR subsets, with $10 < S_I / \sigma_{\text{rms},I} < 100$ ( $\sim 22\,000$ sources) and $S_I > 100\sigma_{\text{rms},I} $ ( $\sim 2\,500$ sources) for each subset, respectively. Both the low- and high-SNR subset show similar features. Offsets in $\alpha_{\text{J2000}}$ become scattered towards low declination, in line with fewer sources per declination bin combined with a larger PSF minor axis (Figure 23). The PSF position angle is typically close to zero elsewhere aligning the minor axis closely with $\alpha_{\text{J2000}}$ . Scatter in $\Delta\delta$ increases at high and low declination are largely due to the increase in PSF major axis. In the case of $\Delta\delta$ , there is an additional bias in these low/high-declination offsets. These offsets appear independent of source SNR.

3.6.2. External accuracy

To assess the external astrometric accuracy of RACS-mid, we matched the positions of bright ( $>10\sigma$ ), isolated (no neighbours within 150 arcsec), and compact ( $0.8 < S_{\text{int}}/S_{\text{peak}} < 1.2$ ) sources to counterparts in other radio surveys. In Figure 37, we show the astrometric offsets between 1 875 matches to the ICRF3 [37(i)] with median offsets (with $1\sigma$ uncertainties) of $\Delta\alpha\cos{(\delta)} = -0.20 \pm 0.53$ arcsec, $\Delta\delta = 0.02 \pm 1.04$ arcsec; 509 487 matches to NVSS [37(ii)] with median offsets of $\Delta\alpha\cos{(\delta)} = -0.20 \pm 2.01$ arcsec, $\Delta\delta = -0.14 \pm 2.34$ arcsec; and 380 066 matches to RACS-low [37(iii)] with median offsets of $\Delta\alpha\cos{(\delta)} = 0.40 \pm 0.90$ arcsec, $\Delta\delta = -0.19 \pm 1.00$ arcsec. The scatter in the cross-matched positional offsets is generally confined to the pixel scale of RACS-mid (2 arcsec), except in the case of NVSS where the larger pixels of the NVSS images (15 arcsec) increase the offset scatter further.

Figure 38 shows the ICRF3 offsets as a function of distance from the tile centre and declination. There is no change in offsets as a function of distance from the tile centre. The high-declination (corresponding to low elevation) sources feature $\Delta\delta$ offsets with increased scatter which can be partially attributed to the larger PSF major axis. The median $\Delta\delta$ offset at high declination drops below 0, with binned-median offset of $-1.32\pm1.86$ arcsec in the highest declination bin. This is indicative of a systematic effect, but this remains within one standard deviation of the binned offsets. A similar offset is seen for low-declination sources, though with less certainty due to the smaller sky area covered and correspondingly fewer sources. The high-declination offsets are similar to what is seen in the beam-to-beam comparisons, though in the low-declination case the offsets against the ICRF3 are reversed.

As variation in astrometric precision changes per SBID and per beam, we do not offer any astrometric correction per image. There are insufficient ICRF3 sources per full mosaic to make such corrections for each individual beam image (typically 1–10 ICRF3 sources over a single tile mosaic). There are no other surveys that cover the full RACS-mid survey region at a similar (or better) resolution besides RACS-low, which has the same astrometric errors. Bulk offsets (i.e. as a function of declination) may be corrected for in full-sky catalogues as part of the next data release.