2.1. Rapid ASKAP Continuum Survey

RACS-low was observed between 2019 April and 2020 June, as we described in Reference McConnellPaper I. RACS-low tiled the observable sky into 903 overlapping fields spanning declinations $-{90}^{\circ}<\delta<+{41}^{\circ}$ , each receiving 15 min of integration time. During beam-forming, RACS-low configured ASKAP’s 36 beams^{Footnote b} into the square_6x6 configuration, with a pitch of 1.05 $^{\circ}$ separating each beam centre for each field. Each field is uniquely identified by its field name. Further, each field also has corresponding target and calibration scheduling-block identifier codes (SBID, see Reference McConnellPaper I). We note that the majority of RACS-low was observed in ASKAP’s ‘multi-field’ mode, whereby multiple fields or calibrators are observed in a single scheduling block. The ASKAP Observatory now prefers a single SBID per field. In future RACS releases, therefore, an SBID will also uniquely identify an observation.

In Reference HalePaper II, we defined two subsets of RACS-low sources. Sources with Galactic latitude $|b|<{5}^{\circ}$ were classed as ‘Galactic’ with the remainder classed as ‘non-Galactic’. This split was motivated by the imaging performance of RACS-low. The snapshot imaging of RACS provides a sensitivity to a maximum angular scale of 25 $^{\prime}$ –50 $^{\prime}$ over most of the sky. RACS is therefore suited to imaging of compact sources, with a loss of sensitivity and image fidelity along the Galactic plane. Source-finding and characterisation of highly extended emission is also a unique challenge unto itself that we leave for future work. By a simple count from the catalogue, we have 704 ‘non-Galactic’ fields and 113 ‘Galactic’ fields lying along the Galactic plane. We note that some fields fall into both categories, as some portions may lie within $|b|<{5}^{\circ}$ .

Here we select 30 representative RACS-low fields for analysis in linear polarisation. In our selection of these fields we weigh several factors. First, we choose a contiguous subset of the 704 non-Galactic fields. Given the significant computational requirements for processing the entire RACS-low survey in full polarisation, we present our processing methodology here along with a catalogue produced from a small subset of the full RACS-low survey. Further, we desire a subset at an intermediate declination, allowing for coverage between, and comparison with, previous large-area surveys in both hemispheres; namely Reference Taylor, Stil and SunstrumNVSS and Reference SchnitzelerS-PASS/ATCA. We also select a region of interest within the Galaxy with a large angular extent, allowing the quality of wide-area ASKAP observations to be highlighted. Using the same methods that are presented here, we will produce and publish an all-sky compact source polarisation catalogue in a subsequent release.

For our first data release, we have selected fields towards the Spica Nebula; a Galactic H ii region ionised by the nearby star Spica / $\alpha$ Vir (Reynolds Reference Reynolds1985). We list our RACS-low fields in Table 1, and show their distribution on the sky in Fig. 1. We will provide detailed analysis of these data, and the magneto-ionic properties of the Spica Nebula, in a forthcoming paper.

Table 1. RACS fields selected for SPICE-RACS DR1. The columns (from left to right) are: the RACS-low field name, the field centre J2000 right ascension (RA), the field centre J2000 declination (Dec.), the field centre Galactic longitude (GLON), the field centre Galactic latitude (GLAT), the date of the observation, the integration time $T_{\text{int}}$ , the scheduling block identifier (SBID) of the bandpass calibration, the SBID of the target observation, the SBID of the beam-forming observation, and the number of Gaussian components in total intensity from Reference HalePaper II ^a ( $N_{\text{Gauss}}$ ).

^a Here we have only used the observations as listed above. The de-duplication process in Reference HalePaper II, however, results in some components as being listed in adjacent fields in the final catalogue. In our SPICE-RACS-DR1 catalogue we modify the tile_id and other metadata columns so only the 30 fields listed in the table above will appear. In the RACS-low catalogue the adjacent fields are: RACS_1351-06A, RACS_1328-12A, RACS_1237-12A, RACS_1213-25A, RACS_1335-25A, RACS_1424-18A, RACS_1240-25A, RACS_1331-18A, RACS_1302+00A, RACS_1303-12A, RACS_1351+00A, RACS_1326+00A, RACS_1213-18A, RACS_1212-06A, RACS_1353-12A, RACS_1416+00A, RACS_1302-06A, RACS_1305-18A, RACS_1326-06A, RACS_1239-18A, RACS_1416-06A, RACS_1212+00A, RACS_1307-25A, RACS_1237-06A, RACS_1237+00A, RACS_1357-18A, RACS_1418-12A, RACS_1429-25A, RACS_1402-25A, and RACS_1212-12A.

Figure 1. Sky coverage of SPICE-RACS DR1. We show the tiling of fields for the entirety of RACS-low in light green, and the 30 fields selected for this data release in dark green. In the inset panel we show the Stokes I rms noise (from Reference McConnellPaper I) in the region surrounding the Spica nebula. In white contours, we show emission from the nebula itself in H $\alpha$ from WHAM (Haffner et al. Reference Haffner2003), and we show the position of the Spica star with a white star.

3.1. Calibration

Our primary calibration procedures, including beam-forming, phase, and gain calibration, are all detailed in Reference McConnellPaper I. Here our starting data are the calibrated visibilities as described in Reference McConnellPaper I. each with a spectral resolution of 1 MHz across the 288 MHz bandwidth. For the purpose of full spectro-polarimetric analysis, we apply the following additional calibration steps.

3.1.2. Off-axis leakage

After the correction of the on-axis leakage, images will still suffer from instrumental polarisation. This off-axis leakage is direction-dependent and will increase in magnitude with separation from the pointing centre of each beam. The sky is dominated primarily by emission in Stokes I, and therefore leakage from I into Q, U, and V will be the dominant instrumental error.

The ASKAP dishes feature a unique mount design that allows for a third ‘roll’ axis, in addition to the altitude and azimuth axes (Hotan et al. Reference Hotan2021). This allows the reference frame of the antenna to remain fixed with respect to the sky, and therefore the parallactic angle remains fixed in time throughout a given track. Field sources, therefore, remain at fixed points in each beam throughout a given observation.

To characterise the wide-field leakage itself, we make use of the field sources to probe the primary beam behaviour, under the assumption that most are intrinsically unpolarised. This technique is also being developed for the full POSSUM survey (Anderson et al. in preparation), and draws on previous work such as Farnsworth, Rudnick, & Brown (Reference Farnsworth, Rudnick and Brown2011) Lenc et al. (Reference Lenc, Murphy, Lynch, Kaplan and Zhang2018). We begin by selecting high signal-to-noise ( $\geq$ $100\sigma$ , full-band) total intensity components. We then extract Stokes I, Q, and U spectra following the same procedure that we outline below in Sections 3.3 and 3.4.1. For each component, we fit a power-law polynomial to Stokes I of the same functional form described in Section 3.4.1, and a third-order polynomial to the fractional $q=Q/I_{\text{model}}$ and $u=U/I_{\text{model}}$ . Doing this initial fitting is required to suppress the effects of large noise spikes. In addition to residual RFI, there are further two likely sources of these spikes. First, the sensitivity per 1 MHz channel is relatively low (by a factor of $1/\sqrt{288}$ compared to the full band). Even with pure Gaussian noise, large value spikes are possible but less common. Second, since ASKAPsoft images each channel independently, each channel only receives a relatively shallow clean. This leave can leave residual sidelobes, along with other artefacts, which can appear as spikes in a given spectrum.

For each beam at each 1 MHz channel we fit a Zernike polynomial (Zernike Reference Zernike1934) as function of separation from the beam centre in the instrument frame (see Appendix A). We use the Zernike polynomials implemented in GalSim (Rowe et al. Reference Rowe2015), who follow the ‘Noll index’ ( $j_{\text{max}}$ ) convention (Noll Reference Noll1976). For our surface fitting we select $j_{\text{max}}=10$ , corresponding to the third radial order and meridional frequency of the Zernike polynomials. For all our model-fitting we use the least-squares approach, implemented in NumPy’s linalg.lstsq routine (Harris et al. Reference Harris2020).

ASKAP determines beamformer weights using a maximum sensitivity algorithm, taking the Sun as a reference source (Hotan et al. Reference Hotan2021). The beam-forming system reuses the beam-forming weights for a period of several months, using the on-dish-calibration (ODC) system to correct for any instrumental drifts that may have occurred since the beam-forming observation. We can therefore expect the primary beams, including the wide-field leakage response, to potentially change significantly after each major beam-forming weight update. We explore and quantify these changes in greater depth in the RACS-mid description paper (Duchesne et al. Reference Duchesne2023, Paper IV).

For the 30 RACS-low fields we have selected, there were four independent sets of beam weights used in the observations (see Table 1). We choose to fit frequency dependent, two-dimensional leakage surfaces independently to components from each set of beam-forming weights for each beam. In doing so, we sacrifice the number of components available to characterise the beam, but gain greater accuracy in our derived surface. For three of the four weights, we were able to obtain robust fits to the leakage with around $\sim$ 200 components per beam on average. There was only a single field using beams formed with SB8669, giving us an average of $\sim$ 20 components per beam. For this field we reduce the order of Zernike polynomial to $j_{\text{max}}=6$ to avoid over-fitting.

We find that fitted surfaces are very sensitive to outlying points, which presumably include some truly polarised components. This becomes particularly problematic when noise spikes produce very high fractional polarisation in a given channel. We therefore exclude components with q or u greater than 100%. We do this both before fitting our third-order polynomials along the frequency axis, and again before fitting a Zernike polynomial surface. We provide further detailed analysis of the fitted and residual leakage in Appendix A.

Having characterised the leakage patterns in Q and U, we perform a correction for the effects of wide-field leakage in the image domain during linear mosaicking of each beam. We produce corrected images of Stokes Q and U ( $\{Q,U\}_{\text{cor}}$ ) using the Stokes I images and the leakage maps ( $\{Q,U\}_{\text{leakage}}$ ) via

(1) \begin{equation} \{Q,U\}_{\text{cor}} = \{Q,U\} - I \times \{Q,U\}_{\text{leakage}},\end{equation}

as implemented in the YandaSoft^{Footnote d} tool linmos. The linear mosaicking process itself also suppresses the instrumental leakage. After mosaicking beams together in a field, subtracting the leakage model, and mosaicking adjacent fields into the full ‘patch’, we conclude that instrumental polarisation in our images is of the order 1% across the centre of a given field, but increases to a few percent from about 3 $^{\circ}$ separation from a field’s centre (see Fig. A.2).

3.1.3. Ionospheric Faraday rotation

We perform a correction for ionospheric Faraday rotation in the image domain using a time-integrated correction. We make use of the FRion^{Footnote e} (Van Eck in prep.) framework to both derive and apply the ionosphere correction. Following Van Eck (Reference Van Eck2021), our observed complex polarisation ( $P_{\text{meas}}(\lambda^2)$ ) relates to the polarisation from the sky ( $P_{\text{sky}}(\lambda^2)$ ) by

(2) \begin{equation} P_{\text{meas}}(\lambda^2) = P_{\text{sky}}(\lambda^2) \Theta(\lambda^2),\end{equation}

where $\Theta$ is the integrated effect of the ionosphere, $\lambda$ is the observed wavelength, and $P=Q+iU$ . We can then compute $\Theta$ as

(3) \begin{equation} \Theta(\lambda^2) = \frac{1}{t_s - t_f} \int_{t_s}^{t_f} e^{2i\lambda^2\phi_{\text{ion}}(t)} dt,\end{equation}

where $t_s$ and $t_f$ are the starting and ending times of an observation, and $\phi_{\text{ion}}(t)$ is the ionospheric Faraday rotation as a function of time t. Note that the amplitude ( $|\Theta|$ ) and the phase ( $\arg\Theta$ ) of $\Theta$ (in the range $-\pi/2<\arg\Theta<+\pi/2$ ) can be interpreted as the depolarisation and change in polarisation angle, respectively, due to the ionosphere.

FRion derives the time-dependent Faraday rotation using RMExtract (Mevius Reference Mevius2018) before computing and applying the integrated correction. For our observations, FRion and RMExtract derive 5 time samples of the ionosphere. Other than stochastic variations, we find no large trends in the extracted ionospheric Faraday rotation as a function of time for our RACS observations. We show the distribution of the ionospheric RM in Fig. 2, along with the distribution of each field’s altitude and azimuth angle, with respect to ASKAP. For all our 30 fields, the magnitude of the ionospheric Faraday rotation was less than 1 rad m $^{-2}$ , with a median value of $-0.73^{+0.10}_{-0.07}$ rad m $^{-2}$ . Within each day of observing the standard deviation of the ionospheric Faraday rotation is 0.016(0.002) rad m $^{-2}$ . We also see that the distribution of observing altitude and azimuth is relatively narrow. We can conclude that the small fluctuations ionospheric RM in SPICE-RACS-DR1 driven primarily by variations in the ionospheric total electron column and we are mostly looking along the same local geomagnetic field.

Applying a time-integrated correction comes with a few caveats. First, in the case of a highly disturbed ionosphere over the period of integration, the value of $|\Theta(\lambda^2)|$ can go to 0, resulting in catastrophic depolarisation. Even in less dire cases, however, uncertainties in $\Theta(\lambda^2)$ will be propagated into our estimated $P_{\text{sky}}(\lambda^2)$ , and will also amplify the errors in $P_{\text{meas}}(\lambda^2)$ . Each of these effects become worse as $|\Theta(\lambda^2)|\rightarrow0$ , but are negligible as $|\Theta(\lambda^2)|\rightarrow1$ . In Fig. 3 we show the distribution of the absolute value and angle of $\Theta(\lambda^2)$ as function of frequency across our observations. We find that $0.9999929<|\Theta(\lambda^2)|\leq1$ for all our fields. This is due to both our short (15 min) integration time, as well as a relatively stable ionosphere during our observations. A time-integrated ionospheric Faraday rotation correction is well suited to short, RACS-style, observations.

Figure 2. Impact of the ionosphere in SPICE-RACS-DR1. In the top panel we show the probably distribution function (PDF) of rotation measure (RM) over time, in the middle we show the PDF of the telescope altitude angle, and in the lower panel we show the PDF of the telescope azimuth angle.

Figure 3. Integrated effect of ionospheric Faraday rotation as a function of frequency ( $\Theta$ ). We can interpret the absolute value $|\Theta|$ the depolarisation from the ionosphere, and the phase $\arg(\Theta)$ as the change in polarisation angle.

3.2. Imaging

For each field we perform imaging of the calibrated visibility data using the ASKAPsoft software applications and pipeline (Guzman et al. Reference Guzman2019) on the Pawsey Supercomputing Research Centre,^{Footnote f} producing an image cube for each of ASKAP’s 36 beams in Stokes I, Q, and U. At all stages, the pipeline treats each beam as an independent observation of the sky. Each cube is channelised at 1 MHz spectral resolution, spanning 744–1032 MHz in 288 channels.

Imaging and deconvolution are handled by the cimager application. We outline the key parameters that we specify for cimager in Table 2. Our approach is similar to that described in Reference McConnellPaper I. As we are producing image cubes, however, we make some different choices in imaging parameters. We produce images $\sim$ ${2.8}^{\circ}$ in size, with a pixel grid allowing 10 samples (2.5 $^{\prime\prime}$ pixels) across the final point-spread function (PSF, 25 $^{\prime\prime}$ full width at half maximum). Wide-field effects are corrected using the w-projection algorithm (Cornwell, Golap, & Bhatnagar Reference Cornwell, Golap and Bhatnagar2008). We use the BasisfunctionMFS algorithm for multi-scale deconvolution, allowing 3 major cycles with a target residual threshold of 0.06 mJy PSF $^{-1}$ .

Table 2. Key imaging parameters used in ASKAPsoft.

RACS-low was observed with the third axis rotated to maintain a constant feed rotation of –45 $^{\circ}$ from the North celestial pole in the plane of the sky. We enable the parallactic angle correction, with a rotation of +90 $^{\circ}$ , in cimager which produces correct polarisation angles in the frame of the sky. For future observations (with SBID $\geq 16000$ ), parallactic angle correction is applied to the visibilities on ingest from the telescope.

For spectro-polarimetric analysis, we require a uniform resolution across all of our image planes and along the frequency axis. As described in Reference McConnellPaper I, the combination of wide-field and snapshot imaging results in variation in the PSF from beam-to-beam as a function of position on the sky. In Reference HalePaper II, a common angular resolution of 25 $^{\prime\prime}$ was selected to maximise uniform sky coverage. To satisfy our resolution requirement we first utilise the Wiener filter pre-conditioning of ASKAPsoft with robust weighting (Briggs Reference Briggs1995) $-0.5$ to ensure our lowest frequency channels can meet our required PSF size. Similar to Reference HalePaper II, we then convolve each restored image plane to a common, circular 25 $^{\prime\prime}$ using a Gaussian kernel. From here we break from ‘standard’ ASKAP processing, and the use of the ASKAP pipeline.

3.3. Cutout procedure

The image cubes for each Stokes parameter, each with 2881 MHz channels, require large amounts of disk space to store. In contrast to previous treatments of ASKAP images (e.g. Anderson et al. Reference Anderson2021), we do not produce mosaics of each observed field. The snapshot imaging of RACS provides a sensitivity to a maximum angular scale of 25 $^{\prime}$ –50 $^{\prime}$ over most of the sky. Reference HalePaper II finds that $\sim$ 40% of RACS sources are unresolved at 25 $^{\prime}$ . As such, images from RACS contain a large portion of empty sky. We take advantage of this fact, and produce cutouts around each source for Stokes I, Q, and U, for each beam cube in which they appear across all fields in total intensity. We use the criterion that a source must be within 1 $^{\circ}$ of a beam centre to be associated with that beam. We define the upper (lower) cutout boundary for a given source as the maximum (minimum) in RA/Dec of all Gaussian components comprising that source in the total-intensity catalogue, offset by the major axis of that component. We further pad the size of each cutout by three times the size of the PSF (75 $^{\prime\prime}$ ). In this way, each cutout contains all the components we require for our later catalogue construction, and enough data to estimate the local rms noise.

The trade-off we make here is significantly reduced data size in exchange for many more data products to manage. We track these data products, as well as their metadata, in a mongoDB^{Footnote g} database. Within this database, we create four ‘collections’^{Footnote h}:

• beams: One ‘document’^{Footnote i} per source. Lists which RACS-low field, and beams within that field, a given source appears in.

• fields: One ‘document’ per field. Contains meta-data on each field as described in Reference McConnellPaper I. Built directly from the RACS database.^{Footnote j}

• islands: One ‘document’ per source. Contains data and meta-data on a given RACS-low source. Initially populated by the source catalogue from Reference HalePaper II.

• components: One ‘document’ per Gaussian component. Contains data and meta-data on a given RACS-low Gaussian component. Initially populated by the Gaussian catalogue from Reference HalePaper II.

We now apply our image-based corrections for ionospheric Faraday rotation, as described in Section 3.1. Using the YandaSoft tool linmos, we simultaneously correct for the primary beam and wide-field leakage, and mosaic the images of each source across different beams within a field. At this stage we have a single cutout, corrected for the primary beam and wide-field leakage, for each source within each field. We cutout and then mosaic, equivalent to mosaicking first and then cutting out, in order to make it more efficient.

Finally, we can mosaic adjacent fields together to correct for the loss of sensitivity towards the edge of a field. We search for all sources that appear in more than one field. Due to the RACS-low tiling scheme, a source can appear in four fields at most. Having found all repeated sources, we again use linmos to perform a weighted co-addition of the images. This now provides us with a single image cube of each source in Stokes I, Q, and U, corrected for the primary beam, wide-field leakage, and ionospheric Faraday rotation across our 30 selected fields.

3.4. Determination of rotation measure

3.4.1. Extraction of spectra

In this first data release, we provide a polarised component catalogue, complementing the total intensity Gaussian catalogue described in Reference HalePaper II. As such, we leave both direct source-finding on the polarisation images as well as further treatment of the image cubes (e.g. 3D RM-synthesis) to a future release.

Here we use our prior knowledge from the total intensity catalogue, and extract single-pixel spectra in Stokes I, Q, and U from the location of peak total intensity. It is important to note that since we do not use source-finding, which can decompose a larger island of emission into overlapping Gaussians, our pencil-beam extractions may contain contributions from multiple components which are listed in the Reference HalePaper II. catalogue. In lieu of complete component fitting and separation, we provide a flag if given component is blended with another (see Section 3.5.1) and additional columns to allow users to easily examine the relative flux of components that are blended with a component of interest (see Section 3.5).

We also extract a ‘noise’ spectrum for each component; estimating the rms noise ( $\sigma_{QU}$ ) from each source cutout cube per channel. We define an annulus with an origin at the centre of the image, an inner radius of 10 pixels ( $={25}^{\prime\prime}=1\,$ PSF) and an outer radius of either 31 pixels ( $={77.5}^{\prime\prime}=3.1\,$ PSF) or the largest radius allowable within the image if the cutout is less than 62 pixels across. For the ensemble of pixels lying within this annulus, we compute the median absolute deviation from the median (MADFM) of this ensemble for each channel and Stokes parameter. The MADFM is more robust to outlying values than the direct standard deviation. We then correct the MADFM value by a factor of $1/0.6745$ which allows it to approximate the value of the standard deviation. This estimator of the noise is potentially sensitive to a component falling inside of our annulus, which would result in an over-estimation. Such a scenario could, for example, arise in the case of a cutout around a double radio source. We perform a brief exploration of this effect in Appendix B, and we find that even in the case of a 1 Jy beam $^{-1}$ component in the annulus the error on our noise estimation remains below 50%.

Before performing further analysis we apply flagging to the 1D spectra. We apply this to remove the effects of narrow-band radio-frequency interference (RFI) and channels for which the clean process diverged. For each Stokes parameter we apply iterative sigma-clipping to a threshold of $5\sigma$ about the median, as measured by the robust MADFM. We mask any channel flagged across all Stokes parameters. Across all of the 105912 components, the average number of channels in our spectra after flagging is $270.0^{+8.0}_{-46}$ out of 288.

To mitigate the effect of the Stokes I spectral index, we perform RM-synthesis (Burn Reference Burn1966; Brentjens & de Bruyn Reference Brentjens and de Bruyn2005) on the fractional Stokes parameters ( $q=Q/I$ , $u=U/I$ ). We fit a power-law model as function of frequency ( $\nu$ ) to each extracted Stokes I spectrum of the form:

(4) \begin{equation} I(\nu_c) = A \nu_c^{\alpha}, \end{equation}

(5) \begin{equation} \nu_c = \frac{\nu}{\nu_0},\end{equation}

where $\nu_0$ is the reference frequency, and all other terms are our model parameters. The reference frequency $\nu_0$ is determined independently for each spectrum as the mean frequency after flagging out bad channels. We fit using the SciPy’s optimize.curve_fit routine, which also provides the uncertainties on each fitted parameter. We iterate through the number of fitted spectral terms; first fitting just A, and then $A,\alpha$ . We select the ‘best’ model as the one with minimum Akaike information criterion (AIC, Akaike Reference Akaike1974), so long as there is not a simpler model within an AIC value of 2 (following Kass & Raftery Reference Kass and Raftery1995). We also assess the quality of the fit, and apply a flag if we determine that the model is unreliable. We detail these quality checks in Section 3.5.1.

3.4.2. Rotation measure synthesis

We use RM-synthesis (Burn Reference Burn1966; Brentjens & de Bruyn Reference Brentjens and de Bruyn2005) in combination with RM-clean (Heald, Braun, & Edmonds Reference Heald, Braun and Edmonds2009) to quantify the Faraday rotation of our spectra, implemented in RM-Tools ^{Footnote k} (Purcell et al. Reference Purcell, Van Eck, West, Sun and Gaensler2020). For a recent review, and in-depth derivation, of Faraday rotation in astrophysical contexts we refer the reader to Ferrière, West, & Jaffe (Reference Ferrière, West and Jaffe2021).

In brief, as linearly polarised emission propagates through a magneto-ionised medium, the polarisation angle ( $\psi$ ) will change in proportion to both the ‘Faraday depth’ ( $\phi$ ) and the wavelength squared. Faraday depth is a physical quantity, defined at every point in the interstellar medium along the LOS (Ferrière Reference Ferrière2016)

(6) \begin{equation} \phi(r) = \mathcal{C} \int_0^r n_e(r^{\prime}) B_\parallel (r^{\prime}) dr^{\prime},\end{equation}

where r is the distance from the observer to the source along the LOS, $n_e$ is the thermal electron density, $B_\parallel$ is the magnetic field projected along the LOS (taken to be positive towards the observer), and $\mathcal{C}\equiv\frac{e^3}{2\pi m_e^2 c^4}\approx0.812$ for $B_\parallel$ in $\unicode{x03BC}$ G, $n_e$ in cm $^{-3}$ , and r in pc.

In the most simplistic case, where a single source of polarised emission is behind a uniform rotating volume along the LOS, the Faraday depth will collapse to a single value referred to as the rotation measure (RM). In this case the change in $\psi$ is linearly proportional to $\lambda^2$ , where RM is the constant of proportionality:

(7) \begin{equation} \psi = \frac{1}{2}\arctan\left(\frac{U}{Q}\right) \end{equation}

(8) \begin{equation} \Delta\psi = \phi\lambda^2 = \text{RM}\lambda^2.\end{equation}

The RM is a observational quantity which, if the simplistic case above holds, we can relate to the physical ISM via (Ferrière et al. Reference Ferrière, West and Jaffe2021)

(9) \begin{equation} \text{RM} = \mathcal{C} \int_0^d n_e B_\parallel ds,\end{equation}

where d is the distance from the source to the observer.

RM-synthesis is a Fourier-transform-like process, with many analogies to aperture synthesis in one dimension. The result of RM-synthesis is a spectrum of complex polarised emission ( $pI=P=Q+iU=Le^{2i\psi}$ , where L is the linearly polarised intensity and $\psi$ the polarisation angle) as a function of Faraday depth ( $F(\phi)$ , Burn Reference Burn1966):

(10) \begin{equation} F(\phi) = \frac{1}{\pi} \int_{-\infty}^{+\infty}p(\lambda^2) e^{-2i\phi(\lambda^2 - \lambda^2_0)} d\lambda^2,\end{equation}

where $\lambda^2_0$ is the reference wavelength-squared to which all polarisation vectors are de-rotated. Following Brentjens & de Bruyn (Reference Brentjens and de Bruyn2005) $\lambda_0^2$ is often set to be the weighted mean of the observed $\lambda^2$ range, but it can be set to an arbitrary real value such that $\lambda^2>0$ with consequences in the conjugate domain, as per the Fourier shift theorem. For our purposes, we choose to set $\lambda^2_0$ as the weighted average of the observed band.

Here we refer to $F(\phi)$ as the ‘Faraday spectrum’. RM-Tools computes the direct, discrete transform of the complex polarisation. We can take the RM of this emission to be the point of maximum polarised intensity ( $L=\sqrt{Q^2+U^2}$ ) in this spectrum:

(11) \begin{equation} \text{RM} \equiv \text{argmax}{|F(\phi)|}.\end{equation}

In RM-synthesis, we weight by the inverse of the measured variance per channel to maximise sensitivity. Noise bias in the polarised intensity is corrected by RM-Tools for all sources with polarised signal-to-noise-ratio (SNR) greater than 5 following George, Stil, & Keller (Reference George, Stil and Keller2012):

(12) \begin{equation} L_{\text{corr}} = \sqrt{L^2 - 2.3\sigma_{QU}^2}.\end{equation}

Observationally, we sample a discrete and finite range in $\lambda^2$ , causing the Faraday spectrum to be convolved with a transfer function referred to as the RM spread function (RMSF). The properties of the RMSF set the observational limits of the Faraday spectrum. We list these properties in Table 3. Since we deconvolve our Faraday spectrum, we replace the observed RMSF with a smooth Gaussian function. RM-Tools selects the width of the Gaussian function by fitting to the amplitude of main lobe of the computed RMSF. The key properties, namely the resolution (FWHM) in Faraday depth ( $\delta\phi$ ), the maximum Faraday depth ( $\phi_{\text{max}}$ ), and the maximum Faraday depth scale ( $\phi_{\text{max-scale}}$ ), are given by (Brentjens & de Bruyn Reference Brentjens and de Bruyn2005; Dickey et al. Reference Dickey2019):

(13) \begin{equation} \delta\phi \approx \frac{3.8}{\Delta\lambda^2}, \end{equation}

(14) \begin{equation} \phi_{\text{max}} \approx \frac{\sqrt{3}}{\delta\lambda^2}, \end{equation}

(15) \begin{equation} \phi_{\text{max-scale}} \approx \frac{\pi}{\lambda^2_{\text{min}}},\end{equation}

where $\lambda^2_{\text{min}}$ is the smallest observed wavelength-squared, $\Delta\lambda^2$ is the total span in wavelength-squared $\Delta\lambda^2=\lambda^2_{\text{max}} - \lambda^2_{\text{min}}$ , and $\delta\lambda^2$ is the width of each $\lambda^2$ channel (RM-tools takes the median channel width). Cotton & Rudnick (submitted) argue that $\phi_{\text{max-scale}}$ over-estimates the maximum Faraday depth scale. They provide the quantity

(16) \begin{equation} \mathcal{W}_{\text{max}} = 0.67 \left( \frac{1}{\lambda^2_{\text{min}}} - \frac{1}{\lambda^2_{\text{max}}} \right),\end{equation}

which provides the Faraday depth scale at which the power over the entire band drops by a factor of 2. We list all these key values in Table 3, and we show the ideal RMSF in Fig. 4. Here we choose to use 100 samples across the main lobe of the RMSF, which provides us improved precision for RM-clean, and allow RM-Tools to automatically select a maximum Faraday depth based on the available channels for a given spectrum.

Table 3. Observational properties of SPICE-RACS DR1.

It is important to note that here our Faraday resolution ( $\delta\phi$ ) is less than maximum Faraday depth scale ( $\phi_{\text{max-scale}}$ , $\mathcal{W}_{\text{max}}$ ). As such our observations lose sensitivity to polarised emission from Faraday depth structures with characteristic widths greater than these scales (see discussions in e.g. Schnitzeler & Lee Reference Schnitzeler and Lee2015; Van Eck et al. Reference Van Eck2017; Dickey et al. Reference Dickey2019). The true total polarised emission from such Faraday thick emission cannot be recovered directly from our Faraday spectra, but instead must be carefully modelled (e.g. Van Eck et al. Reference Van Eck2017; Thomson et al. Reference Thomson2019). We aim to lift this constraint in the future through the inclusion of higher frequency RACS observations.

Figure 4. Ideal, uniformly weighted, RM spread function (RMSF).

3.4.4. Faraday complexity

We wish to draw a distinction between two sets of classifications that are used in the literature: Faraday ‘thick’ vs ‘thin’ and Faraday ‘complex’ vs ‘simple’. The terms ‘thin’ and ‘thick’ in Faraday space are analogous to ‘point’ and ‘extended’ source classifications in the image domain, respectively. Classifying in this way is therefore reliant on both the Faraday resolution ( $\delta\phi$ ) and maximum scale ( $\phi_{\text{max-scale}}$ , $\mathcal{W}_{\text{max}}$ ). The simplest Faraday spectrum occurs in a ‘Faraday screen’ scenario, where a polarised source illuminates a foreground Faraday rotating medium (Brentjens & de Bruyn Reference Brentjens and de Bruyn2005). In such a scenario, the fractional q and u spectra will be strictly sinusoidal as a function of $\lambda^2$ . Identically, this will correspond to a delta-function in the Faraday spectrum. ‘Faraday complexity’ (i.e. classified as ‘simple’ or ‘complex’), as summarised in Alger et al. (Reference Alger2021), is a deviation from this simplistic case. Determining whether a spectrum is Faraday thick or thin requires modelling and careful analysis of the spectrum (e.g. Van Eck et al. Reference Van Eck2017; O’Sullivan et al. Reference O’Sullivan, Lenc, Anderson, Gaensler and Murphy2018; Ma et al. Reference Ma2019; Thomson et al. Reference Thomson2019). For the purposes of our catalogue, we provide metrics to classify spectra as simple or complex, and leave further investigation of the spectral properties to future work.

We quantify the Faraday complexity in two ways. First, since we perform RM-clean, we can compute the second moment of the model components ( $m_{2,CC}$ , Anderson et al. Reference Anderson, Gaensler, Feain and Franzen2015). Following Dickey et al. (Reference Dickey2019), the zeroth, first, second, and simplified second moments of a Faraday spectrum are:

(17) \begin{equation} M_0 = \sum_{i=1}^n F(\phi) d\phi, \end{equation}

(18) \begin{equation} M_1 = \frac{\sum_{i=1}^n F(\phi) \phi}{\sum_{i=1}^n F(\phi)}, \end{equation}

(19) \begin{equation} M_2 = \frac{\sum_{i=1}^n F(\phi) (\phi - M_1)^2}{\sum_{i=1}^n F(\phi)}, \end{equation}

(20) \begin{equation} m_2 = \sqrt{M_2} \end{equation}

respectively. We compute $m_{2,CC}$ by substituting the polarised intensity of the RM-clean components for $F(\phi)$ in Equation (19). Finally, we can produce a normalised complexity metric by dividing the second moment by the width of the RMSF ( $\delta\phi$ ).

The second complexity parameter we provide is the $\sigma_{\text{add}}$ quantity provided by RM-Tools. A detailed description and investigation of this parameter is provided in Purcell & West (Reference Purcell and West2017) and will be published in Van Eck et al. (in preparation). In brief, $\sigma_{\text{add}}$ is computed by first subtracting a best-fit Faraday screen model from the spectrum. The distribution of residual spectrum is then normalised, and a standard normal distribution is fit. $\sigma_{\text{add}}$ is then computed for Stokes $q=Q/I$ and $u=U/I$ as the additional scatter beyond the standard normal distribution. Here we report $\sigma_{\text{add}}$ as:

(21) \begin{equation} \sigma_{\text{add}} = \sqrt{\sigma_{\text{add},q}^2 + \sigma_{\text{add},u}^2}.\end{equation}

We also compute the uncertainty on $\sigma_{\text{add}}$ using a Monte-Carlo approach. RM-Tools reports the 16 $^{\text{th}}$ and 84 $^{\text{th}}$ percentile errors on $\sigma_{\text{add},q}$ and $\sigma_{\text{add},u}$ . We take the probability distribution of these values to be log-normal, and draw a thousand random samples from these distributions. We then compute $\sigma_{\text{add}}$ for each sample and report the median value and error from the resulting distribution.

Finally, it is important to note that these complexity metrics are agnostic to the source of the complexity they are quantifying. In addition the distinctions between ‘thick’ and ‘thin’ that we describe above, these metrics are also unable to determine whether the complexity is intrinsic to the source or is instrumental. Instrumentally, there are many potential sources of spectral complexity including image artefacts, residual RFI, bandpass errors, and so on. Here we make no post hoc correction of the complexity metrics to try to nullify these effects. Instead, we prefer to correct such issues at their source and note that residual effects may still be present in our data.

3.5. The catalogue

Here we describe our Gaussian component catalogue. We describe the flags we use within this catalogue and our recommended subsets in Sections 3.5.1 and 3.5.2, respectively. We have used the RM-Table ^{Footnote m} (Van Eck et al. Reference Van Eck2023) format to construct our catalogue, which provides a minimum set of columns to include. We present the first two rows of our catalogue in Table 4. Under this format, we reproduce some columns from the Reference HalePaper II total total intensity Gaussian catalogue. Where this is the case, we indicate the corresponding column name in the total intensity catalogue. Throughout we use quote position and polarisation angles in Celestial coordinates, measured North towards East following the IAU convention (IAU 1973).

Table 4. The first two rows of the SPICE-RACS DR1 catalogue. We have transposed the table for readability. We define all column names in Section 3.5.

We provide the following RM-Table standard columns:

• ra—Right ascension of the component. RA in Reference HalePaper II.

• dec—Declination of the component. Dec in Reference HalePaper II.

• l—Galactic longitude of the component.

• b—Galactic latitude of the component.

• pos_err—Positional uncertainty of the component. Taken as the largest of either the error in RA or Dec. (see below).

• rm—Rotation measure.

• rm_err—Error in RM.

• rm_width—Width in Faraday depth. Taken as the second moment of the RM-clean components ( $m_{2,CC}$ )

• complex_flag—Faraday complexity flag. See Section 3.5.1 below.

• rm_method—RM determination method. RM Synthesis— Fractional polarisation for all components.

• ionosphere—Ionospheric correction method, FRion for all components.

• stokesI—Stokes I flux density at the reference frequency from the fitted model.

• stokesI_err—Error in the Stokes I flux density. Noise in Reference HalePaper II.

• spectral_index—Stokes I spectral index. Taken as the fitted term $\alpha$ in Equation (4).

• spectral_index_err—Error in the Stokes I spectral index.

• reffreq_I—Reference frequency for Stokes I.

• polint - Polarised intensity at the reference frequency.

• polint_err—Error in polarised intensity.

• pol_bias—Polarisation bias correction method. ‘2012- PASA…29..214G’ for all components, referring to George et al. (Reference George, Stil and Keller2012).

• flux_type—Spectrum extraction method. ‘Peak’ for all components as we extract spectra from the peak in Stokes I.

• aperture—Integration aperture. 0 $^{\circ}$ for all components as we extract single-pixel spectra.

• fracpol—Fractional linear polarisation (in the range 0–1).

• polangle—Electric vector polarisation angle.

• polangle_err—Error in the polarisation angle.

• reffreq_pol—Reference frequency for polarisation.

• stokesQ—Stokes Q flux density.

• stokesU—Stokes U flux density.

• derot_polangle—De-rotated polarisation angle.

• derot_polangle_err—Error in the de-rotated polarisation angle.

• beam_maj—PSF major axis.

• beam_min—PSF minor axis.

• beam_pa—PSF position angle.

• reffreq_beam—Reference frequency for the PSF.

• minfreq—Lowest frequency contributing to the RM.

• maxfreq—Highest frequency contributing to the RM.

• channelwidth—Median channel width of the spectrum.

• Nchan—Number of channels in the spectrum.

• rmsf_fwhm—Full-width at half maximum of the RMSF.

• noise_chan—Median noise per-channel in Stokes Q and U.

• telescope—Name of telescope. ‘ASKAP’ for all components.

• int_time—Integration time of the observation.

• epoch—Median epoch of the observation.

• leakage—Instrumental leakage estimate at the position of the component.

• beamdist—Angular separation from the centre of nearest tile. Separation_Tile_Centre in Reference HalePaper II.

• catalog—Name of catalogue. ‘SPICE-RACS-DR1’ for all components.

• cat_id—Gaussian component ID in catalogue. Gaussian_ID in Reference HalePaper II.

• complex_test—Faraday complexity metric. sigma_add OR Second moment for all components.

• dataref—Data reference. Our CASDA collection, https://data.csiro.au/collection/csiro:58508 for all components.

We provide the following additional columns, which are outside of the RM-Table standard:

• ra_err—Error in Right Ascension. E_RA in Reference HalePaper II.

• dec_err—Error in Declination. E_Dec in Reference HalePaper II.

• total_I_flux—Total flux density in Stokes I of the component. Total_Flux_Gaussian in Reference HalePaper II.

• total_I_flux_err—Error in total flux density in Stokes I. E_Total_Flux_Gaussian in Reference HalePaper II.

• peak_I_flux—Peak flux density of the component in Stokes I. Peak_Flux in Reference HalePaper II.

• peak_I_flux_err—Error in peak flux density in Stokes I. E_Peak_Flux in Reference HalePaper II.

• maj—Major axis of the fitted Gaussian in in Stokes I. Maj in Reference HalePaper II.

• maj_err—Error in the major axis of the Gaussian fit. E_Maj in Reference HalePaper II.

• min—Minor axis of Gaussian fit. Min in Reference HalePaper II.

• min_err—Error in the minor axis of the Gaussian fit. E_Min in Reference HalePaper II.

• pa—Position angle of the Gaussian fit. PA in Reference HalePaper II.

• pa_err—Error in the position angle of the Gaussian fit. E_PA in Reference HalePaper II.

• dc_maj—Major axis of the deconvolved Gaussian fit. Maj_DC in Reference HalePaper II.

• dc_maj_err—Error in the major axis of the deconvolved Gaussian fit. E_Maj_DC in Reference HalePaper II.

• dc_min—Minor axis of deconvolved Gaussian fit

• dc_min_err—Error in minor axis of deconvolved Gaussian fit. E_Min_DC in Reference HalePaper II.

• dc_pa—Position angle of deconvolved Gaussian fit. Min_DC in Reference HalePaper II.

• dc_pa_err—Error in position angle of deconvolved Gaussian fit. E_Min_DC in Reference HalePaper II.

• N_Gaus—Number of Gaussians associated with the source. N_Gaus in Reference HalePaper II.

• source_id—Total intensity component ID. Source_ID in Reference HalePaper II.

• tile_id—RACS tile ID. Tile_ID in Reference HalePaper II.

• sbid—ASKAP scheduling block ID. SBID in Reference HalePaper II.

• start_time—Observation start time. Obs_Start_Time in Reference HalePaper II.

• separation_tile_centre—Angular separation from the nearest tile centre, an alias for beamdist above. Separation_Tile_Centre in Reference HalePaper II.

• s_code—Source complexity classification (see Reference HalePaper II. S_Code in Reference HalePaper II.

• fdf_noise_th—Theoretical noise in the Faraday spectrum.

• refwave_sq_pol—Reference wavelength squared used for polarisation. Corresponds to reffreq_pol above.

• snr_polint—SNR in polarised intensity.

• stokesI_model_coef—A comma-separated list of the Stokes I model coefficients corresponding to Equation (4). This format is compatible with the fitting routines in RM-Tools.

• fdf_noise_mad—Noise in the Faraday spectrum derived using the median absolute deviation.

• fdf_noise_rms—Noise in the Faraday spectrum derived using the standard deviation.

• sigma_add— $\sigma_{\text{add}}$ complexity metric (see Equation (21)).

• sigma_add_err—Error in $\sigma_{\text{add}}$ complexity metric.

Further, we add the following Boolean flag columns (see Section 3.5.1 below):

• snr_flag—Signal-to-noise flag.

• leakage_flag—Leakage flag.

• channel_flag—Number of channels flag.

• stokesI_fit_flag_is_negative—Stokes I model array contains negative values.

• stokesI_fit_flag_is_close_to_zero—Stokes I model array is close to 0.

• stokesI_fit_flag_is_not_finite—Stokes I model array contains non-numerical values.

• stokesI_fit_flag_is_not_normal—Stokes I model residuals are not normally distributed.

• stokesI_fit_flag—Overall Stokes I model fit flag.

• complex_sigma_add_flag— $\sigma_{\text{add}}$ complexity flag.

• complex_M2_CC_flag—Second moment complexity flag.

• local_rm_flag—RM is an outlier compared to nearby components.

Finally, our work has not yet provided the following columns which are part of RM-Table. As such, these all take on their default or blank values:

• rm_width_err—Error in the RM width.

• Ncomp—Number of RM components.

• fracpol_err—Error in fractional polarisation.

• stokesQ_err—Error in Stokes Q flux density.

• stokesU_err—Error in Stokes U flux density.

• stokesV—Stokes V flux density.

• stokesV_err—Error in Stokes V.

• interval—Interval of time between observations.

• type—Source classification.

• notes—Notes.

4.1. Spectral noise

We begin our assessment of our catalogue by inspecting the properties of the noise as a function of frequency, Stokes parameter, and direction on the sky. In Fig. 5 we show the distribution of our estimated rms noise per channel, per Stokes parameter across all components in our region. In Stokes I, we find an average rms of $5.7^{+2.9}_{-1.8}$ mJy PSF $^{-1}$ channel $^{-1}$ , which corresponds to a band-averaged value of $340^{+170}_{-110}$ $\unicode{x03BC}$ Jy PSF $^{-1}$ (assuming noise scaling with the square-root of the number of channels), with a $90^{\text{th}}$ percentile value of 580 $\unicode{x03BC}$ Jy PSF $^{-1}$ . This is marginally higher than values reported in Reference McConnellPaper I and Reference HalePaper II for multi-frequency synthesis (MFS) images, where we found a median and $90^{\text{th}}$ -percentiles rms noise of 250 and 330 $\unicode{x03BC}$ Jy PSF $^{-1}$ , respectively. We can attribute this, in part, to local rms variations due to bright sources and declination effects, our larger final PSF (25 $^{\prime\prime}$ , achieved by convolution with a Gaussian kernel) and to our shallower deconvolution threshold. The latter is necessitated by the lower signal-to-noise in each channel image of our frequency cubes against the full-band MFS images. We also note that some observations that do not appear in Reference McConnellPaper I are included in Reference HalePaper II catalogue due to their PSF. Some of these fields have a higher rms noise than ones in Reference McConnellPaper I, but could be convolved to the common 25 $^{\prime\prime}$ resolution.

Figure 5. Measured rms noise in each Stokes parameter across all observed fields. (a) Noise as a function of frequency. We show the median noise with a solid line, and the $\pm1\sigma$ range as a shaded region. (b) The cumulative distribution function (CDF) of estimated band-averaged noise for each Stokes parameter. In Stokes Q and U we are approaching the theoretical noise limit, whereas in Stokes I the noise by a factor of 3–4 higher. We attribute this to the higher level of artefacts and sidelobes in the Stokes I images.

Looking to Stokes Q and U, we find that the rms noise is $1.33^{+0.42}_{-0.27}$ and $1.26^{+0.42}_{-0.27}$ mJy PSF $^{-1}$ channel $^{-1}$ , respectively. Converting to an estimated band-averaged value, we find a median noise of $78^{+25}_{-16}$ and $74^{+25}_{-16}$ $\unicode{x03BC}$ Jy PSF $^{-1}$ for Q and U, respectively. After performing RM-synthesis the median noise in linear polarisation is $80^{+28}_{-16}$ $\unicode{x03BC}$ Jy PSF $^{-1}$ . RM-Tools measures this noise value by taking the MAD of the Faraday spectrum, excluding the main peak. As such, Faraday complexity can induce an increased measured ‘noise’ into this computation. Overall, these values are close to the expected rms values for naturally weighted images. Even though we use $-0.5$ robust image weighting, this is applied per-channel. As such, computing an average along the frequency axis after this channel imaging should approach the naturally weighted noise.

In Fig. 5a we can see that the rms noise in Stokes Q and U has a weak local maximum around $\sim$ 900 MHz. We find no such feature in Stokes I, even after removing the apparent spectral dependence. We are able to attribute this feature to observations made earlier in the RACS-low observing campaign. As outlined in Table 1, our subset of fields is comprised roughly of two epochs; which can be selected with SBID greater or lesser than 9000. We find that the early set of observations (SBID $<9000$ ) exhibit this local maximum, whereas the Q,U noise in later observations (SBID $>9000$ ) are approximately flat as function of frequency. We therefore attribute this feature to lower quality bandpass solutions during early RACS-low observations.

Finally, we can turn our attention to the spatial distribution of the rms noise in I and L across the observed fields, which we show in Fig. 6. In both maps we see artefacts around the brightest sources in the field. In particular, the worst noise appears in the North-West portion of the field, surrounding 3C273 (source_id: RACS_1237+00A_3595), which has an integrated Stokes I flux density of $\sim$ 64 Jy in RACS-low. Looking throughout the rest of the field we can see greater variance in the total intensity noise, indicating the impact of sources to the rms noise. By contrast, in the polarised intensity noise the square sensitivity pattern of the beam configuration is clearly apparent. This is to be expected as RACS-low was conducted with a non-interleaved square_6x6 footprint, and a wide (1.05 $^{\circ}$ ) beam pitch. We conclude that, away from the brightest sources, we are approaching the sensitivity limit in Stokes Q and U, but are artefact-limited in Stokes I. We note that since we are close to the thermal noise limit in Stokes Q and U our noise measurements become sensitive to residual calibration errors, such as the feature we describe above.

Figure 6. Spatial distribution of rms noise in (a) Stokes I and; (b) linearly polarised intensity (L). White stars indicate the position of components with a Stokes I flux density $>$ 3 Jy PSF $^{-1}$ . Our linear mosaicking of adjacent beams and fields (weighted by inverse-variance) produces a spatial pattern in the resulting noise. This effect is particularly noticeable in the $\sigma_L$ distribution, where the boundaries of our 30 square fields are apparent.

Users of our catalogue should remain aware that the noise in our catalogue is not spatially uniform. Without appropriately weighting for uncertainties, this distribution can cause spurious correlations to appear when assessing bulk properties and statistics within our catalogue. In addition to the flag columns we provide, users should also make appropriate use of the uncertainty columns when undertaking statistical analysis with our catalogue.

4.2. Linear polarisation and rotation measure

We now inspect the polarisation properties of SPICE-RACS components after applying our internal flags to select our recommended basic subsets (see Section 3.5.2). We summarise the basic statistics of these subsets in Table 5, following the approximate format of Adebahr et al. (Reference Adebahr2022) (their Table 5) for later comparison. Of the 105912 total components we obtain reliable fits (goodI) to the Stokes I spectrum for 24680 ( $\sim$ 23%) components, we detect 9092 ( $\sim$ 8%) components in linear polarisation (goodL), and 5818 ( $\sim$ $5.5\%$ ) components have a reliable RM (goodRM). Given our observed area of $\sim$ 1300 deg $^{2}$ , our average RM density is $\sim$ 4 deg $^{-2}$ (the goodRM subset) with $\sim$ 6.5 deg $^{-2}$ for polarised sources above $5\sigma$ (the goodL subset).

Table 5. Summary of polarisation statistics for components in SPICE-RACS (following the format of Adebahr et al. Reference Adebahr2022). Each row corresponds to the subsets in Section 3.5.1 (goodI, goodL, and goodRM) and I denotes all the components detected in Stokes I. We define columns as follows: ‘N’ represents a number count of a given subset, ‘F’ a fraction, and ‘ $L/I$ ’ the average linear polarisation fraction. The subscripts S and E denote the subset of sources that are unresolved and resolved, respectively. The error ranges given represent the 16 $^{\text{th}}$ and 84 $^{\text{th}}$ percentiles of the population distribution.

We can compare our recovered source densities to an ‘expected’ value by extrapolating from Rudnick & Owen (Reference Rudnick and Owen2014b,Reference Rudnick and Owena). Rudnick & Owen (Reference Rudnick and Owen2014b) provide a power-law scaling relation with polarised source density $\propto \sigma_L^{-0.6}$ . Assuming that beam and wavelength-dependent depolarisation are negligible to first order, we find an expected spatial density of 9(4) deg $^{-2}$ and 7(3) deg $^{-2}$ at $5\sigma$ and $8\sigma$ , respectively, for SPICE-RACS. As we detail in Appendix A, our residual widefield leakage is $\geq$ 1% across our survey area, whereas Rudnick & Owen (Reference Rudnick and Owen2014b) report residual leakage of $\sim$ $0.4\%$ . If we are able to further improve our residual leakage in future releases, we can therefore expect to approach these values.

In Fig. 7 we show the distribution of the (bias-corrected) linearly polarised flux density against the total flux density, split by our basic subsets. The distribution of reliable polarised sources bears overall resemblance to previous large radio polarimetric surveys (e.g. Hales et al. Reference Hales, Norris, Gaensler and Middelberg2014; Anderson et al. Reference Anderson2021; O’Sullivan et al. Reference O’Sullivan2023). Notably, however, due to our residual leakage we have no components with $\leq$ 1% fractional polarisation in our goodL or goodRM subsets. Looking now at Fig. 8, we show the distribution of the absolute RM value against the polarised SNR for the same subsets. At SNRs less than $8\sigma$ we can see a large spike in high absolute value RMs, which are certainly spurious. This gives us confidence in our recommended thresholds for the goodRM basic subset. However, we also see that the bulk population of RMs, which we can take to be real, appear to extend below the $8\sigma$ threshold. We conclude that there are therefore many more scientifically useful RMs below the $8\sigma$ $L/\sigma_L$ level that can be extracted with appropriate and careful weighting of the uncertainties and systematics.

Figure 7. Linearly polarised flux density (L) against Stokes I flux density for SPICE-RACS DR1. Each coloured/shaped marker corresponds to our basic subsets as defined in Section 3.5.2: The blue circles represent components for which we have a reliable fit to the Stokes I spectrum, but the linear polarisation may be spurious, the orange squares represent components that have a reliable linear polarisation detection but a potentially unreliable RM, and the green triangles represent components with a reliable RM. The dashed black lines show contours of constant fractional polarisation, and the grey shaded region is the area of $>$ 100% fractional polarisation. Where the scatter points become over-dense we show the density of points as a 2D histogram. The contour levels of the histogram are at the $2^{\text{nd}}$ , 16 $^{\text{th}}$ , 50 $^{\text{th}}$ , 84 $^{\text{th}}$ , and 98 $^{\text{th}}$ percentiles.

Figure 8. The absolute value of the rotation measure ( $|\text{RM}|$ ) as a function of the linearly polarised signal-to-noise ratio ( $L/\sigma_L$ ). Each coloured/shaped marker corresponds to our basic subsets as defined in Section 3.5.2 as per Fig. 7. The dashed and dotted vertical lines represent the $8\sigma$ and $5\sigma$ levels, respectively. We note that to be included in the goodRM and goodL subsets a component must have $L/\sigma_L$ $\geq8$ and $\geq5$ , respectively (see Section 3.5.2 for further details). Where the scatter points become over-dense we show the density of points as a 2D histogram. The contour levels of the histogram are at the $2^{\text{nd}}$ , 16 $^{\text{th}}$ , 50 $^{\text{th}}$ , 84 $^{\text{th}}$ , and 98 $^{\text{th}}$ percentiles.

Using the classification from Reference HalePaper II (see their Section 5.2.1 and Equation (1)), we further subdivide our catalogue into unresolved and extended components. For our DR1 subset, we find the same fraction of components ( $\sim$ 40%) are unresolved in total intensity at 25 $^{\prime\prime}$ as in the all-sky catalogue. For components detected in linear polarisation (both goodL and goodRM), we find $\sim$ 20% are unresolved. Looking at the fractional polarisation across our subsets, we find the median linear polarisation is around $\sim$ 3% for both goodL and goodRM and for unresolved and extended components. In this comparison we note, however, that our linear polarisation analysis uses pencil-beam spectral extraction compared to the source-finding used in Reference HalePaper II.

We show the spatial distribution of RMs from the goodRM subset in Fig. 9. As previously mentioned, we will provide a more detailed description and analysis of the structure present in these RMs in a forthcoming paper. We show the distribution of both absolute RM and RM in Fig. 10 and with and without the local_rm_flag. Before applying the local_rm_flag, the RMs span from –273(2) to 362(4) rad m $^{-2}$ . The median absolute RM is about 9 rad m $^{-2}$ with a standard deviation of 13 rad m $^{-2}$ . Within this set there are only 3 components with an absolute RM greater than 100 rad m $^{-2}$ . Applying the local_rm_flag, we find 81 components are identified as being outliers with respect to their local RM value. After excluding these values, the overall RM distribution remains mostly unchanged (as shown in Fig. 10), with most of flagged RMs having slightly higher absolute RM ( $>$ 10 rad m $^{-2}$ ) than the total distribution. We leave investigation of the properties, and potential sources, of these outlying RMs to future work.

Figure 9. Spatial distribution of rotation measures (RM) across the sky from (a) SPICS-RACS-DR1 (goodRM subset, see Section 3.5.2), (b) Reference SchnitzelerS-PASS/ATCA, and (c) Reference Taylor, Stil and SunstrumNVSS sing nearest-neighbour interpolation.

Figure 10. The cumulative distribution (CDF) of (a) absolute RM (log-scale) and (b) RM with and without applying the local_rm_flag. Here we select RMs from the goodRM subset (see Section 3.5.2).

We now assess our RMs by crossmatching against historical polarisation catalogues. We crossmatch against the Van Eck et al. (Reference Van Eck2023) consolidated catalogue (v1.1.0) using the STILTS (Taylor Reference Taylor, Gabriel, Arviset, Ponz and Enrique2006) tmatch2 routine with a maximum sky separation of 60 $^{\prime\prime}$ (the beam-width of Reference Taylor, Stil and SunstrumNVSS, which comprises the majority of the catalogue). From our total 105912 components, we find 1553 matches in the consolidated catalogue; 1080 of these have reliable RMs in SPICE-RACS-DR1 with remaining 473 being flagged. In Table 6 we show which flags were applied to the set of 473 flagged components and which original survey they were matched with. Across each of the matched surveys the majority of components were flagged for being below our fractional polarisation leakage threshold (leakage_flag). After stokesI_fit_flag, the most common flag is snr_flag. In future releases, we plan to improve both our leakage threshold and our signal-to-noise near bright components which will help to reduce the number of true polarised components flagged in SPICE-RACS.

Table 6. The 1553 SPICE-RACS-DR1 components that crossmatch with historical surveys. For each survey we give the number and percentage (in parentheses) of components flagged by a given criterion (see Section 3.5.1). In the final column we give the counts of unflagged components.

Looking now to the crossmatched components with reliable SPICE-RACS RMs (the goodRM subset), we compare the RMs from all surveys with at least two matched components in Fig. 11. The included surveys are Reference Taylor, Stil and SunstrumNVSS, Reference SchnitzelerS-PASS/ATCA, Farnes, Gaensler, & Carretti (Reference Farnes, Gaensler and Carretti2014), Tabara & Inoue (Reference Tabara and Inoue1980), POGS-II (Riseley et al. Reference Riseley2020), and POGS-I (Riseley et al. Reference Riseley2018). Out of these surveys, Reference Taylor, Stil and SunstrumNVSS, Reference SchnitzelerS-PASS/ATCA, and Farnes et al. (Reference Farnes, Gaensler and Carretti2014) each have $\geq$ 10 matched components. For each of these larger surveys, 90% of the matched RMs are within 2.6, 9.2, and 13 standard deviations, respectively. We note that Farnes et al. (Reference Farnes, Gaensler and Carretti2014) derived their broadband results from high-frequency data; they estimate $\sim$ 5 GHz as the approximate reference frequency. Farnes et al. (Reference Farnes, Gaensler and Carretti2014) also find a similar scatter as we see in Fig. 11 when they compare their RMs to Reference Taylor, Stil and SunstrumNVSS (see their Fig. 6). We conclude that the large discrepancy between SPICE-RACS and Farnes et al. (Reference Farnes, Gaensler and Carretti2014) is due to the properties of the latter catalogue, and not spurious results from SPICE-RACS DR1.

Figure 11. Rotation measures of SPICE-RACS-DR1 against other surveys. The SPICE-RACS RMs are drawn from the goodRM subset (see Section 3.5.2). We show points that have a RM difference ( $\Delta\text{RM}$ ) of less than $5\sigma$ in green, and the remaining outlier points in black. Due to the high number of matched components, we show the inlying points from Reference Taylor, Stil and SunstrumNVSS as a density plot. The contour levels of the 2D histogram are at the $2^{\text{nd}}$ , 16 $^{\text{th}}$ , 50 $^{\text{th}}$ , 84 $^{\text{th}}$ , and 98 $^{\text{th}}$ percentiles. We note that the Farnes et al. (Reference Farnes, Gaensler and Carretti2014) RMs have a reference frequency of $\sim$ 5 GHz, and also showed similar scatter when compared to the Reference Taylor, Stil and SunstrumNVSS RMs.

In Fig. 12 we compare linear polarisation from Reference Taylor, Stil and SunstrumNVSS components with those that cross-match in our catalogue. Since Reference Taylor, Stil and SunstrumNVSS was derived from observations at $\sim$ 1.4 GHz we expect that most of our lower-frequency observations should be depolarised with respect to the higher frequency data (Sokoloff et al. Reference Sokoloff1998). If we assume that components are depolarising due to a random, external dispersion in RM described by $\sigma_{\text{RM}}$ (see e.g. Sokoloff et al. Reference Sokoloff1998; O’Sullivan et al. Reference O’Sullivan, Lenc, Anderson, Gaensler and Murphy2018, Equations (41) and (2), respectively), then the fractional polarisation is given by

(22) \begin{equation} \frac{L}{I}(\lambda) \propto e^{-2\sigma^2_{\text{RM}}\lambda^4}.\end{equation}

We note that this $\sigma_{\text{RM}}$ is referring to a dispersion within a given spectrum, and should not be confused with RM dispersion between sources (e.g. Schnitzeler Reference Schnitzeler2010). Given observations at two reference wavelengths ( $\lambda_1$ , $\lambda_2$ ), $\sigma_{\text{RM}}$ can be estimated as

(23) \begin{equation} \sigma_{\text{RM}} = \sqrt{ \frac{\ln{ \left[\frac{(L/I)_1}{(L/I)_2}\right]}}{2\left(\lambda_2^4 - \lambda_1^4\right)}}.\end{equation}

For our set of components which match with Reference Taylor, Stil and SunstrumNVSS, we find average polarisation fractions of $3.4^{+3.2}_{-1.7}$ and $7.3^{+4.7}_{-3.2}$ for SPICE-RACS and Reference Taylor, Stil and SunstrumNVSS, respectively. Converting to $\sigma_{\text{RM}}$ using Equation (23) we find an average estimated value of $5.8^{+1.5}_{-1.6}$ rad m $^{-2}$ . For comparison, O’Sullivan et al. (Reference O’Sullivan, Lenc, Anderson, Gaensler and Murphy2018) reported a value of 14.1(1.8) rad m $^{-2}$ from QU-fitting spectra between 1 to 2 GHz. Given our lower frequency band, we expect to probe an overall sample of more Faraday simple sources than O’Sullivan et al. (Reference O’Sullivan, Lenc, Anderson, Gaensler and Murphy2018). We derive this expectation from the fact that complex polarised sources are known to depolarise as a function $\lambda^2$ (Burn Reference Burn1966; Tribble Reference Tribble1991; Sokoloff et al. Reference Sokoloff1998). To first order, we can therefore expect to find a higher average fractional polarisation at higher frequency and vice versa. Therefore, after taking into account both spectral index and observational sensitivity, sources detected at low frequencies are almost all Faraday simple (e.g. O’Sullivan et al. Reference O’Sullivan2023). Conversely, higher frequency observations (e.g. Anderson et al. Reference Anderson, Gaensler, Feain and Franzen2015; O’Sullivan et al. Reference O’Sullivan, Lenc, Anderson, Gaensler and Murphy2018) are able to detect the population of Faraday complex sources.

Figure 12. The fractional linear polarisation ( $L/I$ ) of SPICE-RACS components cross-matched with components from Reference Taylor, Stil and SunstrumNVSS. In (a) we show the scatter points as a 2D density histogram where the data are over-dense. The contour levels of the 2D histogram are at the $2^{\text{nd}}$ , 16 $^{\text{th}}$ , 50 $^{\text{th}}$ , 84 $^{\text{th}}$ , and 98 $^{\text{th}}$ percentiles. Here the background colour scale is the estimated $\sigma_{\text{RM}}$ (see Equation (23)) for a given fractional polarisation ratio. In (b) we bin collections of matched components in a 2D histogram and compute the median complexity metric $\lt\sigma_{\text{add}}\gt$ within each bin.

There is also a small subset of components for which the SPICE-RACS fractional polarisation is greater than Reference Taylor, Stil and SunstrumNVSS. Inspecting the Faraday complexity metrics, we find that $\sigma_{\text{add}}$ is $3.1^{+4.2}_{-1.7}$ where SPICE-RACS has a higher fractional polarisation than Reference Taylor, Stil and SunstrumNVSS, compared to $1.64^{+1.3}_{-0.51}$ for the remainder. Reference Taylor, Stil and SunstrumNVSS suffers from bandwidth depolarisation for $|\text{RM}|>100$ rad m $^{-2}$ (Taylor et al. Reference Taylor, Stil and Sunstrum2009; Ma et al. Reference Ma2019), however almost all of our detected RMs are below this threshold. Further, we find no correlation between absolute RM and complexity, nor these outlying components. We can therefore attribute the subset of sources with higher fractional polarisation to repolarisation effects (e.g. Anderson et al. Reference Anderson, Gaensler, Feain and Franzen2015). Further investigation of depolarisation/repolarisation requires deeper analysis (e.g. QU-fitting), which we leave for future work.

From the matched RMs, we identify outlying points as those beyond $5\sigma$ in the difference in RM ( $\Delta\text{RM}$ ). We investigate two potential components that can explain the outlying points. First, since each of these surveys were conducted at different frequencies and with different bandwidths, components which are Faraday complex can produce different measured RMs in the different surveys. We provide a detailed assessment of our complexity metrics below. Secondly, as we found above, polarisation leakage remains a significant systematic effect in our observations. Crucially, if we have underestimated the amount of instrumental leakage in the position of a given component, our measured RM will be contaminated. In the case of leakage from Stokes I into Q and U, the smooth Stokes I spectrum will impart a false RM signal at 0 rad m $^{-2}$ .

In Fig. 13 we show the distributions of both our complexity metrics, $\sigma_{\text{add}}$ and $m_{2,\text{CC}}/\delta\phi$ , and our leakage threshold for both inlying and outlying components. We find that outlying components are statistically more complex. For the inlying set $\sigma_{\text{add}}=1.67^{+1.7}_{-0.51}$ and $m_{2,\text{CC}}/\delta\phi=0.14^{+0.32}_{-0.13}$ , whereas for the outlying set $\sigma_{\text{add}}=3.7^{+4.0}_{-1.9}$ and $m_{2,\text{CC}}/\delta\phi=0.72^{+3.9}_{-0.45}$ . Looking at our leakage threshold distributions, we find that the upper percentile portion of the outlier set tends to be higher than the inlier set; with the 97.6 $^{\text{th}}$ percentile being 3.4% and 2.3%, respectively. Stated another way, if an outlying component is towards the edge of a field, with a higher leakage threshold, the outlier is more likely to be closer to the edge of the field with respect to an inlying component. Finally, we confirm that complexity and the leakage threshold are not correlated for inlying components, which we show in Fig. 14. For outlying components there is not a general correlation, however almost all of the outliers that have a high leakage value are also classed as complex. We discuss potential sources of this in Section 4.3.

Figure 13. Distributions of inlying (green) and outlying (black) RMs (as shown in Fig. 11) as a function of (a) normalised Faraday complexity and (b) leakage threshold.

Figure 14. Scatter between our $\sigma_{\text{add}}$ complexity metric and leakage threshold for inlying (green) and outlying (black) points (as shown in Fig. 11). Due to the density of inlying points we show the data as a 2D density histogram.

Finally, we can assess our RM density and compare with historical surveys. We construct three coarse HEALPix^{Footnote n} grids with $N_{\text{side}}$ parameters 32, 16, and 8, corresponding to pixel resolutions of $\sim$ ${110}^{\prime}$ , $\sim$ ${220}^{\prime}$ , and $\sim$ ${440}^{\prime}$ , respectively. We then count the density of RMs from SPICE-RACS-DR1 (goodRM subset), Reference Taylor, Stil and SunstrumNVSS, and Reference SchnitzelerS-PASS/ATCA on the $N_{\text{side}}$ 32, 16, 8 grids, respectively. We show these density maps in Fig. 15. Our SPICE-RACS RM grid covers a total sky area of 1306 deg $^{2}$ , with an RM density of $4.2^{+2.7}_{-2.1}$ deg $^{-2}$ . The lowest RM densities occur on the edge of our processed region, where our sensitivity decreases and leakage increases, as well as in the area affected by artefacts around 3C273. For comparison, we find RM densities of $1.1^{+0.4}_{-0.4}$ deg $^{-2}$ and $0.3^{+0.2}_{-0.1}$ deg $^{-2}$ for Reference Taylor, Stil and SunstrumNVSS and Reference SchnitzelerS-PASS/ATCA respectively.

Figure 15. RM density of SPICE-RACS compared to other large area surveys. Here we show the sky using an orthographic projection in equatorial coordinates, centred on the SPICE-RACS DR1 area. We use HEALPix grids with $N_{\text{side}}$ of 32, 16, and 8 for panels (a), (b), (c), respectively.

4.3. Faraday complexity

As we describe in Sections 3.4.4 and 3.5, we provide two normalised complexity metrics, $\sigma_{\text{add}}$ and $m_{2,\text{CC}}/\delta\phi$ . Having investigated the $\sigma_{\text{add}}$ parameter, and its reported error $\delta\sigma_{\text{add}}$ , we find that the metric is susceptible to two failure cases; each applying when $\sigma_{\text{add}}/\delta\sigma_{\text{add}}$ is small. First, if a single channel (or small subset of channels) are affected by a noise or RFI spike, the resulting distribution (which is computed internally within RM-Tools) of $\sigma_{\text{add},Q}$ or $\sigma_{\text{add},U}$ is heavily skewed to larger values. This causes an over-reported value in the $\delta\sigma_{\text{add}}$ . Further, since we compute $\sigma_{\text{add}}$ as the quadrature sum of $\sigma_{\text{add},Q}$ and $\sigma_{\text{add},U}$ (Equation (21)), themselves each being positive definite values, we will have a Ricean-like bias at low signal-to-noise. We therefore only consider $\sigma_{\text{add}}$ a reliable metric where $\sigma_{\text{add}}/\delta\sigma_{\text{add}}>10$ (where complex_sigma_add_flag is True). Where we do classify $\sigma_{\text{add}}$ as reliable, we indeed find that increasing values of $\sigma_{\text{add}}$ correspond to greater complexity in the spectra (see Appendix C).

In Fig. 16 we show the distributions and correlation between these two metrics for all SPICE-RACS components with reliable RMs. Of the 5818 components in the goodRM subset, we find 695 components that are classified as complex. We see that the $\sigma_{\text{add}}$ metric is more sensitive than $m_{2,\text{CC}}/\delta\phi$ , with $\sigma_{\text{add}}$ flagging 692 components as complex and $m_{2,\text{CC}}/\delta\phi$ flagging 178. Further, $m_{2,\text{CC}}/\delta\phi$ only flags 3 components that were not flagged by $\sigma_{\text{add}}$ ; whereas $\sigma_{\text{add}}$ flags 517 components that were not flagged by $m_{2,\text{CC}}/\delta\phi$ .

Figure 16. A comparison of the normalised Faraday complexity metrics described in Section 3.4.4. Here we only show points for which $\sigma_{\text{add}}/\delta\sigma_{\text{add}}>10$ , which is why the $m_{2,CC}$ CDF does not reach 0. The scatter plot shows the correlation between these metrics, with points coloured by the components’ polarised signal-to-noise ratio ( $L/\sigma_L$ ). Black points indicate components that have been flagged as having RMs which are outlying from the local ensemble of SPICE-RACS RMs. For each metric, we also show the distribution of the metric values for both inlying and outlying components. We note that the $m_{2,CC}$ values (vertical axis) appear quantised due to the discretisation and numerical precision of the Faraday depth axis and the placement of clean components.

For components that are classed as complex, we see a good correlation between the two metrics. We check whether the components that were flagged as complex were also flagged as outlying from their local ensemble RMs, but we find no significant deviation from the distribution of inlying points. We do, however, find that the complexity correlates with polarised signal-to-noise ( $L/\sigma_L$ ). To better understand this correlation we directly inspect the spectra themselves.

In Fig. 17 we show the Stokes I spectra as a function of frequency, the L, Q, U spectra as a function of $\lambda^2$ , and the clean, dirty, and model Faraday spectra. Here we show four representative spectra nearest to the 50 $^{\text{th}}$ , 84 $^{\text{th}}$ , 97.7 $^{\text{th}}$ , and 99.9 $^{\text{th}}$ percentile in $L/\sigma_L$ . Looking at the brightest Stokes I spectrum we can see both a power-law structure as well as a sinusoidal ripple across the band.

Figure 17. Example spectra of four sources from SPICE-RACS nearest to the 99.9 $^{\text{th}}$ , 97.7 $^{\text{th}}$ , 84 $^{\text{th}}$ , and 50 $^{\text{th}}$ percentile in polarised signal-to-noise from top to bottom. In the left column we show the Stokes I data and our fitted power-law model. In the middle column we show the fractional Stokes Q and U, and polarised intensity L. Here the scatter points are the observed data divided by the Stokes I model, and the solid lines is the RM-clean model transformed to $\lambda^2$ space. In the right-hand column we show the dirty, clean, and model Faraday dispersion functions (FDFs) in polarised intensity.

ASKAP has a known ripple across its band with a period of $\sim$ 25 MHz, which has been attributed to a standing wave effect between the dish surface and the receiver (Sault Reference Sault2015). We will leave a deep investigation of characterising and mitigating this effect to future work. We do note however, that the period of the ripple may vary in concert with other observational parameters such as elevation. Further, the ripple in our data is seen after the application of the bandpass calibration. Therefore, residual errors from the bandpass are also superimposed. For the purposes of this investigation we will only consider a constant 25 MHz ripple.

We consider the two ways a 25 MHz ripple may present itself in our observations. As described in Sault (Reference Sault2015), the standing wave ripple presents itself in the instrumental gains. For a given component which emits in all Stokes parameters, this ripple can present itself in the data multiplicatively with the true emission from the sky. Secondly, in case of a bright Stokes I component, leakage of the ripple in Stokes I can be introduced additively to the other Stokes parameters.

We will now consider how the two cases that can introduce the standing wave ripple into Stokes Q and U may affect our Faraday spectra. First, it is important to consider that the ripple is periodic in frequency, and will therefore distribute across a broad range of Faraday depths. In Fig. 18 we show the noiseless Faraday spectrum, with $\text{RM}=0$ , in the presence of both a gain ripple and leakage ripple. In each case we have generated a unit polarised signal with the amplitude of the ripple also being unity. In the case of a gain ripple, the RMSF is effectively given higher sidelobes. These sidelobes translate in Faraday depth with the true Faraday depth of the component. In the case of leakage, the response has no dependence on the Faraday depth of the source; the true spectrum is superposed with the response from the ripple. For both the leakage ripple and $\text{RM}=0$ gain ripple case, we find the leakage response spans 302–764 rad m $^{-2}$ in $|\phi|$ at the 50% level. Lastly, we find that the leakage has a marginally higher amplitude in the Faraday spectrum than a gain ripple. For a ripple with unit amplitude, the maximum amplitude of the Faraday spectrum response is 24% and 15% for the leakage and gain ripples, respectively. These responses decrease linearly with the ripple amplitude. In conclusion, we urge caution in the interpretation of high signal-to-noise spectra. Whilst they are not included in our catalogue, care should be taken in analysis of secondary components in our clean Faraday spectra. In particular, secondary RM components found in the range $300<|\phi|<800$ rad m $^{-2}$ should be treated with a reasonable level of scepticism.

Figure 18. The effect of the 25 MHz standing-wave ripple on an ideal Faraday depth spectrum. (a) The Faraday spectrum for an ideal, noiseless component (black) in the presence of a leakage ripple (solid green) and a gain ripple (dashed green). (b) The maximum peak of the leakage ripple in the Faraday spectrum relative to a unitary polarised signal as a function of the ripple amplitude.

4.4. Spectral indices

Using our fitted models to the Stokes I spectra we are able to assess the distribution of spectral indices. Here we are interested in the spectra of all components, regardless of their polarisation. We identify components with a reliable fit in our catalogue by indexing where both the stokesI_fit_flag and channel_flag are false (see Section 3.5.1). Of the 105912 components we analyse, we successfully fit 24680 spectra in Stokes I. Since we image each 1 MHz independently, this number is in line with our expectations. If we assume that the rms noise increases in proportion to the square root of the number of channels, a $5\sigma$ detection using the full band is only $0.3\sigma$ in a single channel; with only $\sim$ 10% of our sample of components having $5\sigma$ per channel. Further, we find that goodI subset is indeed drawn from the high SNR Stokes I components.

Of the goodI subset, 18021 components have a reliable fit to the spectral index, with the remaining 6675 spectra only being fit with a flat intensity model. We make three notes of caution in interpreting our spectral indices. First, they are derived from, at most, 288 MHz of bandwidth and are therefore not as robust as broad-band spectral indices. Secondly, our applied holographic primary beams may differ slightly from the true primary beams. If the error in our holographic beams is chromatic, this will also affect our spectral indices. Finally, since we both apply a constant 100 m uv-cut to the visibilities and extract spectra from the peak pixel, our spectral indices may be unreliable for extended sources.

We find an error-weighted average spectral index of $-0.8\pm0.4$ (here the errors represent the error-weighted ensemble standard deviation). We show the distribution of our fitted spectral indices, as a function of flux density, in Fig. 19. Above about 1 Jy PSF $^{-1}$ we have relatively few components, making bulk statistics unreliable. Below 1 Jy PSF $^{-1}$ , however, we see that most bright components are in line with the bulk average index of $\sim-0.8$ , but tend to flatten with decreasing flux density to $-0.4\pm0.9$ . Our bulk spectral indices, and trend to flatten with decreasing flux density, are consistent with previous spectral index measurements at $\sim$ 1 GHz (e.g. de Gasperin, Intema, & Frail Reference de Gasperin, Intema and Frail2018); giving us confidence in our in-band results.

Figure 19. The spectral index of Stokes I spectra against flux density from our fitted spectra (taken from the peak pixel). In green we show the measurements from each component, including errors in flux density and spectral index. In orange we show the components which have a flat spectral index fit; for all these components we take the spectral index to be $\alpha=0$ . Where the points are over-dense, we show the distribution as a density plot. The contour levels of the 2D histogram are at the $2^{\text{nd}}$ , 16 $^{\text{th}}$ , 50 $^{\text{th}}$ , 84 $^{\text{th}}$ , and 98 $^{\text{th}}$ percentiles. In black we show the error-weighted mean spectral index in flux density bins, along with the error-weighted standard deviation.

5.1. PolSpectra columns

The PolSpectra standard defines the following mandatory columns:

• source_number

  Simple source identifier (incrementing integer number).

• ra

  As per Section 3.5.

• dec

  As per Section 3.5.

• l

  As per Section 3.5.

• b

  As per Section 3.5.

• freq

  Frequency array in Hz.

• stokesI

  Stokes I flux density array in Jy/PSF.

• stokesI_error

  Errors on Stokes I flux density array in Jy/PSF.

• stokesQ

  Stokes Q flux density array in Jy/PSF.

• stokesQ_error

  Errors on Stokes Q flux density array in Jy/PSF.

• stokesU

  Stokes Q flux density array in Jy/PSF.

• stokesU_error

  Errors on Stokes U flux density array in Jy/PSF.

• beam_maj

  As per Section 3.5.

• beam_min

  As per Section 3.5.

• beam_pa

  As per Section 3.5.

• Nchan

  As per Section 3.5.

In addition, we also provide the following columns:

• cat_id

  As per Section 3.5.

• telescope

  As per Section 3.5.

• epoch

  As per Section 3.5.

• integration_time

  As per Section 3.5.

• leakage

  As per Section 3.5.

• channel_width

  As per channelwidth in Section 3.5.

• flux_type

  As per Section 3.5.

• faraday_depth

  The Faraday depth array in rad m $^{-2}$ .

• faraday_depth_long

  Double-length Faraday depth array (in rad m $^{-2}$ ), matching the RMSF for use in RM-clean

• FDF_Q_dirty

  The real part of the dirty Faraday spectrum array in

  Jy/PSF/RMSF.

• FDF_U_dirty

  The imaginary part of the dirty Faraday spectrum array in Jy/PSF/RMSF.

• FDF_Q_clean

  The real part of the clean Faraday spectrum array in Jy/PSF/RMSF.

• FDF_U_clean

  The imaginary part of the clean Faraday spectrum array in Jy/PSF/RMSF.

• FDF_Q_model

  The real part of the clean model of the Faraday spectrum array in Jy/PSF/RMSF.

• FDF_U_model

  The imaginary part of the clean model of the Faraday spectrum array in Jy/PSF/RMSF.

• RMSF_Q

  The real part of the RMSF array in normalised units.

• RMSF_U

  The imaginary part of the RMSF array in normalised units.