Interplanetary Scintillation studies with the Murchison Wide-field Array III: Comparison of source counts and densities for radio sources and their sub-arcsecond components at 162 MHz

We derive the source counts from the $\sim 900\,$sq.deg. MWA field presented in Chhetri et al. (2018), made using a single 5-minute observation centred at 162 MHz. This field contains 2550 objects at a signal-to-noise (S/N) ratio $\geq$5, which we shall refer to as the “Total Population".

We determined the magnitude of the interplanetary scintillation (IPS) exhibited by our sources by measuring the standard deviation of all pixels in images produced at a half-second cadence to produce a “variability image”. Further details of the variability image are discussed in Morgan et al. (2018) and Chhetri et al. (2018) (hereafter, Papers I and II respectively). Following the convention established in Paper I, we term the continuum image (i.e. the mean flux density image) the “standard image".

In Chhetri et al. (2018), we derived the normalised scintillation indices (NSIs) for all sources in the field. The NSI gives the fraction of flux density that is concentrated on sub-arcsec scales. The relevant angular scale probed by IPS, the Fresnel scale, is $\sim$ 0.3 arcsecond for our field at 162 MHz. By imposing the criterion NSI$\geq$0.90 we select compact sources whose flux is almost entirely confined to $<$ 0.3 arcsecond angular scales. Of the 247 “Sources with compact structure" that exhibit IPS with a S/N$\geq 5$ in the variability image, there are 93 sources (the “Point source population”) that meet this NSI criterion. Note that in contrast to Paper II we only consider those sources for which there is a detection of a point source (aside from the validation of our method in Section 2.2.1). The upper limits on NSI used in Paper II are only relevant when comparisons are being drawn between compact and non-compact sources, which is not relevant to the the work presented here. This also allows us to use the full sample of compact sources, rather than restricting ourselves to a high S/N subsample.

There are three inherent complications to the analysis of the source counts in this dataset: (i) there is a large gradation in sensitivity across the field due to the attenuation of the primary beam; (ii) the scintillation index (and therefore point source sensitivity) are also a function of solar elongation and; (iii) derivation of the compact source counts relies on the interpretation of the statistics of the standard deviation of the flux density, rather than the mean flux density. Point (i) is discussed directly below, while discussion of points (ii) and (iii), being IPS-specific, are deferred to Section 2.3.

In paper II we presented a catalogue of compact sources. The compact flux densities of the sources were not included in the catalogue explicitly. These flux densities are obtained from the primary beam corrected and absolute flux density corrected variability image, with appropriate background correction discussed in section 3.2.1 of Paper II.

The counts of sources with compact structure presented in Fig. 4 follow the overall population above $\sim$3 Jy, within the range of uncertainties, then appear Euclidean down to our sensitivity limit of 0.4 Jy. Our results of these counts can be compared to the work by Artyukh & Tyul’bashev (1996). They conducted a similar survey of radio sources exhibiting interplanetary scintillation at 102 MHz. The IPS survey was made using the Large Phased Array (LPA) of the Lebedev Institute of Physics in Pushchino. The LPA has a very large collecting area (30 000 m² at the zenith) and hence very high instantaneous sensitivity to IPS but has much lower angular resolution than the MWA, so is seriously influenced by source confusion. To overcome this limitation they used the 7C radio source survey (McGilchrist et al., 1990) to identify the scintillating sources and to obtain a value for the total (non-scintillating) flux density. Despite this the scintillating sources are themselves still affected by confusion in the Pushchino survey and source identifications are often ambiguous. Our MWA observations with $>$2550 beam areas per scintillating source have negligible confusion and the total intensity can be un-ambiguously determined from the same IPS observations. Our result of counts of compact detections are in excellent agreement with their Figure 1 above a flux density of 0.5 Jy, which is their stated completeness limit. We even see the slight (but probably not significant) excess in the range 0.5 to 1.0 Jy. However, we do not observe the sharp drop in the differential counts of scintillating sources below 0.6 Jy reported in their paper. While we do not have sufficient sensitivity to probe counts at the faintest level they reach, we suggest that they have not been able to fully correct for completeness and confusion at these faint levels.

Wilman et al. (2008) have constructed a simulation of 20°$\times$20°of extragalactic radio continuum sky, as part of the SKA Simulated Skies (S³) project²²2http://s-cubed.physics.ox.ac.uk/. The output of the simulation consists of various “Galaxy” types (FRI, FRII, Quasars and star-forming galaxies) made up of sets of “components” (cores, lobes, hotspots and disks) appropriate to each galaxy type. The flux density of each component at a variety of frequencies including 151 MHz ($S_{151}$) is provided, as well as each component’s morphology (major axis $a$, minor axis $b$ and position angle). We derived the counts of compact objects predicted by this simulation as follows. First, all components for all galaxies with total flux density greater than 100 mJy were selected. The compact flux density of each component (as would be measured by IPS) was estimated to be

where the Fresnel scale, $\theta_{F}$, is assumed to be 0.3″(see Section 4.1 of Paper II for further details). Next, each group of components that lay within $\theta_{F}$ of each other were blended into a single component by summing their IPS flux densities (as is appropriate for coherent combination of IPS variability). Then, each group of components that lay within 3′(the inteferometric resolution of the MWA) were blended into a single component by summing their IPS flux densities in quadrature (as is appropriate for incoherent combination of IPS variability). The resulting sources with a flux density > 100 mJy were preserved. The difference in frequencies (their 151 MHz vs. our 162 MHz) is assumed to be negligible.

We can now compare the resulting source counts of this simulation against our counts of sources with compact structure (see Fig. 4). As noted by Wilman et al. (2008) the (S³) simulations should be used with caution outside the range of parameters which are constrained by their current (i.e. 2008) knowledge but our IPS observation can be used to test aspects of the model using new information not available in 2008 and as appropriate can be used to improve the simulation model. In our flux density range we need only consider the radio loud AGN components of the model which includes the cores, lobes and hot spots. Our data shows a remarkable agreement in the density of sources with compact structure and reasonable agreement for the slope of the source counts for these sources. Inspection of the simulated component list at 162 MHz shows that our counts of all sources with compact structure is dominated by hotspots so we can explore this aspect of the model. The agreement in density confirms the use of old hot spot data from Jenkins & McEllin (1977) which sets the fraction of the total flux density in the hot spots. The fact that the slope of the simulated counts for scintillating components is the same as the counts for total flux density is no surprise since Wilman et al. (2008) assume that hot spots have the same luminosity function and spectral shape as the lobes. The actual agreement with our observed slope validates this quite reasonable assumption.

For the reasons given above the source counts of all 247 objects showing scintillation presented in the previous section do not lend themselves to direct cosmological interpretation. We now consider the more readily interpretable counts of point sources which are presented in Fig. 5.

Readhead & Longair (1975) selected quasars from the 3C catalogue with flux density > 2 Jy and used IPS to identify sources with fractional flux from compact components $\geq$ 0.40. They interpreted the resulting source counts to be evidence for strong evolution of quasars. Our sky coverage is much smaller than the total 3C catalogue, limiting our numbers at the high flux density end. By contrast, we probe an order of magnitude lower in flux density and restrict the fractional flux for compact components to $\geq$ 0.90, selecting only sources smaller than 0.3 arcseconds.

The counts obtained for our point source population, shown in Fig. 5, show a Euclidean profile (ie flat in this normalized plot) across the entire flux density range that we are sensitive to, over almost two orders of magnitude. The GLEAM source counts deviate from Euclidean below $\sim$ 1 Jy. A difference in the two distributions would be consistent with the understanding that a) the extended radio galaxies and the compact point source populations (eg. AGNs) have different lifespans so they must evolve differently, b) that their luminosity functions must be different, or c) a combination of both.

Also plotted in Fig. 5 are the source counts from the S³ simulations. The NSI for each simulated source was determined by counting the flux density of all components that lay within an MWA beam of a compact source, allowing the ratio of compact vs. total flux density (i.e. the NSI) to be computed. Looking at those that would be detected by the MWA as point sources, we see that, in contrast to the compact component source counts, the simulation is not able to reproduce our observed point source counts. With such a complex simulation, determining which input parameters might be adjusted in order to produce output more consistent with our results is not straightforward, especially with only one realisation of the simulation which covers less than a steradian, and so we do not attempt this here. However we note that the combination of compact morphologies from IPS, along with detailed spectral information from GLEAM will provide strong constraints for testing such models.

One important result from paper II was that $\sim$ 40% of the compact source population was composed of peaked spectrum objects, and interestingly the S³ simulations do define 19/34 galaxies that we classify as compact as ‘GPS’ AGN. Callingham et al. (2017) identified over 1400 peaked spectrum sources in the GLEAM survey, including sources with peaks outside the GLEAM frequencies. Peaked spectrum sources (eg. the gigahertz peaked-spectrum (GPS) sources) show morphologies similar to FR II sources but are confined to <10 kpc in linear size, and are generally considered to be younger stages of radio galaxies (eg. O’Dea, 1998). Using their catalogue, we found 34 peaked-spectrum sources in our point source population. One source (GLEAM J013243-165444) that is identified as a peaked spectrum object was found to be a blazar in paper II, and not a peaked spectrum source in the traditional sense. We removed this source from our list, leaving 33 peaked spectrum sources.

The remaining 60 sources in our point source population show power-law spectra with spectral indices on a continuum, with 18 of them showing flat-spectrum ($\alpha_{GLEAM}\,\geq$ -0.5) and the remainder steep ($\alpha_{GLEAM}\,<$ -0.5) as seen in Fig. 6, and Table 2. Note that there is no obvious bimodality in the distribution of spectral indices; the classification is purely conventional. The flat-spectrum sub-population of point sources are the beamed compact cores of radio galaxies. Only one source (the above mentioned GLEAM J013243-165444) shows Fermi gamma-ray association. Here, we note that the flat- or steep-spectrum hot-spots on radio galaxies only form a minor part (<2%, based on work by Readhead & Hewish (1976)) of this population as they mostly show NSI lower than 0.9 by virtue of their being embedded in extended radio lobes or having two or more compact components with their IPS flux adding incoherently to lower their NSI.

The different spectral properties of sub-populations that form our point source population are summarised in Table 2.

In Fig. 6 we plot the low frequency SEDs of individual peaked and non-peaked spectrum sources separately. Their spectral properties and their SED inform us that the counts seen in Fig. 5 are a composite of the source counts of these two different sub-populations, and that their different evolution and luminosity function profiles pose challenge to a direct interpretation of Fig. 5.

At high frequencies, the synchrotron self-absorbed cores of AGNs show flat spectra and dominate the total radio source population, while a class of compact objects with steep spectra form a small fraction of the Total Population (Chhetri et al., 2013). These compact steep-spectrum (CSS) sources are considered to be at more evolved stages in radio galaxy evolution than the GPS sources. With the hypothesis that the steep-spectrum compact objects in our subsample are CSS objects, with their spectral peaks below frequencies covered by GLEAM, we combined them with the peaked source population to form a single sub-population. With this, the GPS/CSS objects become the single most dominant group, comprising 81% of the total low frequency sub arcsecond population, while the flat-spectrum objects form only a minority (19%).

We made counts of the different sub-populations identified above and compare them in Figure 7. We tested against the hypothesis that the counts of different sub-populations are consistent to within a constant multiplicative factor of the GLEAM counts and other counts of interest (ie. counts of sub-populations = f $\times$ GLEAM counts), where the fraction “f" is the best-fitting scaling parameter. Table 3 lists the reduced $\chi^{2}$ values and the scaling factors used for different sub-populations. The reduced $\chi^{2}$ were calculated as:

where, N is the number of flux density bins, $\mu_{1},\sigma_{1}$ are the bin values and uncertainties for the population under consideration, and $\mu_{2},\sigma_{2}$ are the bin values and uncertainties for the model.

We did not undertake a more rigorous comparison of the source count distributions using, for instance, a two-sided Kolomogorov-Smirnov test, because it is already clear from our crude comparison of the binned data using the reduced $\chi^{2}$ metric that the errors in various distributions are too large to identify any points of difference between the distributions. We expect this picture to change with better statistics achievable with our future widefield IPS survey.

Currently, an improved understanding may be gained from the counts of peaked spectrum sources made using Callingham et al. (2017) data with appropriate completeness corrections to lower flux densities. Here as well, future widefield IPS survey will provide complementary information by identifying CSS sources to complete the picture.

Knowledge of the compact source counts is also important for the design and use of long-baseline interferometers, including the SKA_LOW which will reach arcsecond resolution on its longest baselines. Moldón et al. (2015) and subsequently Jackson et al. (2016) have conducted a search for compact sources that are suitable calibrators for international baselines of the LOFAR telescope with correlated flux density of $\gtrsim$50–100 mJy at frequencies around 110–190 MHz, on scales of a few hundred milliarcseconds. They find around 1 suitable calibrator source per square degree on 200-300 km international baselines of LOFAR, which drops to 0.5 sources per square degrees on 200-600 km baselines.

Moldón et al. also address the problem of selecting likely compact calibrators a priori. At higher frequencies this is done by selecting flat-spectrum sources, which is a very well-justified approach, since these are extremely likely to be very compact (Chhetri et al., 2013). Moldón et al. note that spectral index at higher frequencies is a poor predictor of compactness at low frequencies; however they find that flat spectra at low frequencies is much more likely to select satisfactory primary calibrators. Fig. 7 clearly shows that by selecting peaked sources as well as flat-spectrum sources, the number of high-quality compact calibrators that will be selected is almost doubled in the flux density range 0.4–2 Jy. Additionally, significant numbers of calibrators at even higher flux densities will be selected. Furthermore, we showed in Paper II that all peaked sources in the Total Population were compact, so the use of peaked sources in addition to flat-spectrum sources does not compromise reliability.

The calibration of SKA_LOW is entirely dependent on having a grid of calibrators in the field of view (FoV) dense enough to provide the direction dependent calibration of multiple ionospherically limited patches. To this effect, we can provide information at two levels. The density of point sources on baselines out to about 350 km at 162MHz (< 1$\arcsec$ in size) is 0.2 per square degree at 0.4 Jy (this is approximately an order of magnitude higher than that estimated by the S³ model in Fig. 5). However if more complex sources can be modeled in the calibration strategy we can use all scintillating sources with a density of 0.7 per square degree and these will have at least some visibility on baselines out to 350  km.

The S³ model in Fig. 4 indicates that the density of compact components follows a trend similar to the overall low radio frequency source population below $\sim$ 0.4 Jy. Although these simulations cannot exactly reproduce our results at higher flux densities, in the absense of real measurements we use the simulations as a guide to extrapolate down another order of magnitude and estimate a density of compact source density of 4.6 sources/deg.² at 40 mJy, a flux density scale useful for SKA_LOW calibrators.