The relationship between the turnover frequency and photo-ionisation in radio sources

As stated above, only one source, J0002+2550 (WISEA J000239.39+255035.1), had sufficient rest-frame UV data to determine the ionising photon rate.⁵⁵5Of the remaining eight, only four had any rest-frame IR–UV photometry and are shown in Appendix C.

This is shown in Fig. 2, from which we obtain an ionising rate of $Q_{\text{H{\sc\,i}}}=9.3\times 10^{54}$ s^-1. This is well below the $\approx 3\times 10^{56}$ s^-1 limit at which all of the gas in the host galaxy is believed to be ionised (Curran & Whiting, 2012) and well within the regime where H i 21-cm absorption has been detected (Curran et al., 2008, 2019).

Thus, for the one source for which we could obtain the photometry, the absence of a turnover in the radio SED is consistent with a relatively low ionisation fraction. We do note, however, that this has an unusually steep spectral index in the UV band ($\alpha_{\rm UV}=-10.37$) and whether this applies to the other eight sources in the sample we cannot comment. At such high redshifts we would expect a steepening of the spectral index due to intervening hydrogen, specifically the Lyman-$\alpha$ forest (e.g. Janknecht et al. 2002; Barger & Cowie 2010) and Lyman Limits systems (e.g. Mo et al. 2010). In Fig. 2, $\lambda=1216$ Å occurs immediately after the second highest frequency point and so the above ionising photon rate should be treated as a lower limit (see Sect. 3.2).

In order to investigate the relation between the ionising photon rate and turnover frequency, we turned to a large sample, which is optically selected in order to increase the likelihood of yielding the necessary rest-frame UV photometry.

We use the sample of Curran et al. (2021), which comprises the first 100 000 QSOs of the SDSS Data Release 12 (DR12, Alam et al. 2015) with accurate redshifts. From these we shortlisted the sources which had at least one flux measurement below 10 GHz, giving 3429 radio sources. As stated above, we required at least three flux measurements to fit the radio data, which gave 1005 sources of which 499 could be fit. Of these, the ionising photon rate could be obtained for 416 source, of which 257 sources which were fit by a power law with 159 exhibiting a turnover. Since we are primarily interested in the turnover frequency, we used the polynomial results in the following analysis.

In order to supplement the radio data, we added photometry compiled in surveys with the GaLactic and Extragalactic All-sky Murchison Widefield Array (GLEAM), specifically:

1. 1.

   The 1483 sources with $72\leq\nu_{\rm TO}\leq 1400$ Mz (Callingham et al., 2017).

2. 2.

   The 373 sources of Slob et al. (2022), of which 36 exhibit $\nu_{\rm TO}\sim 150$ Mz.

3. 3.

   The 24 high redshift ($4.9\leq z\leq 6.6$) radio bright sources of Gloudemans et al. (2022).

Of the first two samples, spectroscopic redshifts were available for 214 and 49 sources, respectively. Comparing these with the SDSS sample gave just a single match, [HB89] 1252+119 (GLEAM J125438+114103) at $z=0.872$.⁶⁶6Note that there are two other coordinate matches, [HB89] 1307+121 (GLEAM  J130934+115425) and [HB89] 1442+101 (GLEAM J144516+095835), although the redshifts of the SDSS and GLEAM sources are very different – $z=2.6006$, cf. 0.34286 and $z=3.5203$, cf. 0.25578, respectively. The lowest observed rest-frame frequency is 141 MHz, compared to 676 MHz for the NED only data. For the low frequency plus NED photometry, the polynomial fit gives $\nu_{\text{TO}}=10^{9.36\pm 0.03}$ Hz = $2.29_{-0.07}^{+0.23}$ GHz and the GPS fit 251 MHz (Fig. 3), whereas the NED only data gives $\nu_{\text{TO}}=10^{8.40\pm 1.70}$ Hz = $252_{-247}^{+12400}$ MHz and the GPS fit 603 MHz⁷⁷7Which, being lower than the minimum observed frequency, would be re-fitted with a power law.. This is actually closer to the $\nu_{\text{TO}}=608$ MHz of Callingham et al. (2017)⁸⁸8325 MHz in the observed frame (J. Callingham, private communication). which uses the low frequency data.

We also performed full photometry searches on the low frequency samples. Of the sources with at least three radio photometry measurements, 260 have spectroscopic redshifts, of which 24 were best fit by a power law and 201 with a turnover.⁹⁹9Although the remaining 35 appeared to be amply sampled, some gave inverted spectra and the bunching of data at similar frequencies in others prevented a fit. Just 55 of these had sufficient rest-frame UV photometry to obtain the ionising photon rate and adding these to the SDSS sources with a measured $Q_{\text{H{\sc\,i}}}$, gave 263 first order fit sources and 208 second order fit.

In Fig. 4 we show the radio and UV luminosity distributions of the combined SDSS plus low frequency sample.

The UV and radio luminosities are strongly correlated, with a Kendall-tau test giving a probability of $p(\tau)=9.20\times 10^{-13}$ of this arising by chance, which is significant at $Z(\tau)=7.14\sigma$, assuming Gaussian statistics. Also, it is apparent that the 1st order fit sources are generally more luminous at $\lambda=21$ cm than those with a 2nd order fit.

This could be due to the flagging of 2nd order fits as 1st order if the fitted turnover frequency is below the minimum observed, which will have the effect of boosting the luminosity if the putative turnover occurs at $\nu_{\text{TO}}<1.4$ GHz. This may be evident in Fig. 5, where

the 2nd order fit sources are, on average, more fully sampled, thus making it possible that some of the 1st order fit sources may exhibit a turnover if sampled to a similar degree. The median radio bandwidths of the low frequency observations is $\Delta\nu\sim 750$ MHz (Callingham et al., 2017; Slob et al., 2022) and so we remove our most extreme outliers (at $\Delta\nu\stackrel{{\scriptstyle<}}{{{}_{\sim}}}1$ GHz) from the sample in the following analysis.

As discussed in Sec. 2.2.1, the higher the redshift the more we expect the rest-frame UV continuum to be absorbed by intervening hydrogen.

In the top panel of Fig. 6, we can see a clear Malmquist bias, where the volume probed at low redshift is insufficient to detect the rare high luminosity sources, while at high redshift only the high luminosity sources are detected. We therefore use the spectral index to quantify a correction for the intervening hydrogen. In Fig. 7 we show all of the SDSS sources for which we could obtain an ionising photon rate,

from which we can see a steepening in the spectral index with redshift. A linear fit gives $\alpha_{\rm UV}=-0.16z-1.20$, so, on the basis that the intrinsic spectral index is $\alpha_{\rm UV}=-1.20$ at $z\sim 0$, we correct each measured value of $\alpha_{\rm UV}$ by subtracting $-0.16z$ (Fig. 8).¹⁰¹⁰10$\alpha_{\rm UV}=-1.20$ at $z\sim 0$ is similar to values in the literature, e.g. $\alpha_{\rm UV}=-1.4$ at $z<1.44$ (Shull et al., 2012) and $\alpha_{\rm UV}=-1.7$ at $\left<z\right>=1$ (Telfer et al., 2002).

Although not valid for an individual source, using this to correct the UV spectral index to obtain the intrinsic value provides a statistical correction for the redshift, and thus a better estimate of the ionising photon rate before the absorption of the UV continuum by intervening gas.¹¹¹¹11There is also the issue of dust suppressing the shorter wavelengths in the rest-frame, leading to underestimates in $Q_{\text{H{\sc\,i}}}$. However, intrinsic dust will also provide shielding for the gas countering the degree of ionisation somewhat.

From the corrected spectral indices we recalculate the ionising photon rates, according to Equ. 3. Showing these in Fig. 9,

we see little difference between the 1st and 2nd order fit sources. A $t$-test gives a probability of $p(t)=0.952$ of the mean $Q_{\text{H{\sc\,i}}}$ values being the same and a Kolmogorov-Smirnov test gives $p(KS)=0.032$ of the populations being drawn from the same sample.¹²¹²12For the uncorrected ionising photon rates these are $p(t)=0.992$ and $p(KS)=0.045$.

In Fig. 10 we show the turnover frequency versus the ionising photon rate,

from which we see a correlation between $\nu_{\rm TO}$ and $Q_{\text{H{\sc\,i}}}$, which has a probability $p(\tau)\leq 9.10\times 10^{-6}$ of arising by chance. Given this correlation, we now assume that the sources which exhibit no turnover have not been searched to sufficiently low frequencies. We include these sources by assigning the lowest observed rest-frame frequency as the upper limit to this (e.g. Fanti et al. 1990). We also apply this to the 2nd order fit sources where the turnover frequency yielded by the fit is lower than the lowest observed rest-frame frequency (Fig. 11).

In order to incorporate the limits, we use the Astronomy SURVival Analysis (asurv) package (Isobe et al., 1986) which adds these as censored data points, allowing a generalised non-parametric Kendall-tau test. Including these weakens the correlation ($p(\tau)\leq 2.3\times 10^{-4}$)¹³¹³13For the uncorrected ionising photon rates this is $p(\tau)=1.63\times 10^{-4}$ ($3.77\sigma$)., which may suggest, if $\nu_{\text{TO}}\propto Q_{\text{H{\sc\,i}}}$, that the 1st order fit sources are not dominated by those which have not been searched to low enough frequencies to detect the turnover. However, given the lower frequency ranges of the 1st order fit sources (Sect. 2.3.1), the detection of a turnover if the sampling were ample cannot be ruled out. Hence, it is also possible that the inclusion of these sources only acts to introduce noise.

While we have found a strong correlation between the turnover frequency and the ionising photon rate, one should be aware of:

1. 1.

   For the one SDSS source which did match the low frequency catalogues, the lowest rest-frame frequency yielded by NED was 676 MHz, compared to 141 MHz from the low frequency data (Sect. 2.2.2). Thus, until observed and included in NED, allowing automated photometry scraping, it is reasonable to assume that many of the SDSS (1st order fit) sample have not been observed to sufficiently low frequencies.

2. 2.

   The 1st order fit sources generally have sparser sampling across the radio band (Sect. 2.3.1). So, until the power-law sources are sampled as comprehensively as those exhibiting a turnover, we cannot be certain that a potential turnover does not occur within the observed range.

3. 3.

   As seen in Fig. 10, there is a large variation in the uncertainties in the turnover frequencies. Two examples, at either end of the range, are shown in Fig. 12.

   

   

   In a large enough sample we expect as may overestimates as underestimates, so that the biases introduced by these would be largely averaged out. Ideally, however, these should all be small and so, again, more comprehensive sampling of the radio band is required.

4. 4.

   From the upper limits in Fig. 11, we see that the lowest observed frequency is correlated with the ionising photon rate, which is most likely driven by the redshift, where sources are observed to the same minimum frequency in a given survey, confirmed in Fig. 13.

   

   This is also evident in Gloudemans et al. (2023), where the high redshifts ($z=5.6-6.6$) cause the low observed frequencies ($\stackrel{{\scriptstyle>}}{{{}_{\sim}}}0.1$ GHz) to be insensitive to turnover frequencies below $\sim 1$ GHz.

From Equ. 8, the ionising photon rate required to completely ionise a neutral plasma ($n_{\rm p}=n_{\rm e}$) of constant density within a Strömgren sphere of radius $r_{\text{str}}$ is

For a range of ionising photon rates we use this to obtain the electron density, which we insert into Equ. 5, giving the absorption coefficient $\kappa_{\nu}$. For a constant electron density through a path $r_{\text{str}}$, $\tau_{\nu}=\kappa_{\nu}r_{\text{str}}$ which, via Equ. 6, gives the variation in flux density with frequency (Fig. 14).

From this, it is clear that for a given density, temperature and radius, the turnover frequency is correlated with the ionising photon rate, as is seen for the observational data (Sect. 2.4.1).

A more physically realistic model, especially on galactic scales, is an exponential decrease in gas density with distance from the nucleus of the galaxy (e.g. Baldwin 1982; Begeman et al. 1991). O’Dea & Baum (1997) use a distribution of the form $n=n_{0}r^{-a}$, where $0\stackrel{{\scriptstyle<}}{{{}_{\sim}}}a\stackrel{{\scriptstyle<}}{{{}_{\sim}}}3$. However, this gives $n=0$ at $r=0$ and an infinite column density for $N_{\text{H{\sc\,i}}}=\int_{0}^{\infty}ndr$. We therefore use the exponential sphere model, $n=n_{0}e^{-r/R}$, which gives $N_{\text{H{\sc\,i}}}=\int_{0}^{\infty}ndr=n_{0}R$.

Using the Milky Way’s $N_{\text{H{\sc\,i}}}=3.3\times 10^{22}$ $\hbox{{\rm cm}}^{-2}$ and breaking the $n_{0}R$ degeneracy via the gas mass (Curran & Whiting, 2012)

where $F=20$ is the Galactic flare factor (Kalberla & Kerp, 2009), gives $M_{\text{gas}}=5.3\times 10^{9}$ M_⊙, $n_{0}=1.88$ $\hbox{{\rm cm}}^{-3}$ and $R=5.66$ kpc for the Milky Way (Fig. 15).

For the exponential sphere complete ionisation of the gas occurs when the ionising photon rate reaches (Curran & Whiting, 2012)

Using Equ. 11 to obtain $n_{0}$ for a given $R$, where the gas is completely ionised (Fig. 16),

gives the optical depth through the gas as (Equ. 5)

Integrating this gives

which, again via Equ. 6, gives the variation in flux density with frequency (Fig. 17, left). Like the constant density distribution, this exhibits an increase in the turnover frequency with ionising photon rate.

Although, unlike the constant density model, at large distances the path length is not required for the optical depth calculation, a value is required for the flux density (Equ. 6) and we have assumed that $r_{\rm em}=2.3R$ as the extent over which the emission can be detected. However, while the flux densities are not absolute, the turnover frequency is independent of the choice of $r_{\rm em}$.

For the Milky Way the gas distribution is a combination of both the constant and exponential profile (Kalberla & Kerp, 2009);

where $n_{0}=1.45$ $\hbox{{\rm cm}}^{-3}$, $R=3.15$ kpc and $r_{0}=7$ kpc (see Fig. 15). For this distribution complete ionisation of the gas occurs when the ionising photon rate is (Curran, 2024)

Again, assuming that all of the gas is ionised, we use the electron density for a given ionising photon rate (Equ. 13) to calculate the optical depth through the gas

which gives

This results in the frequency profiles shown in Fig. 17 (middle), where, again, we have assumed $r_{\rm em}=2.3R$ as the size of the emission region, although this does not affect the apparent $Q_{\text{H{\sc\,i}}}-\nu_{\rm TO}$ correlation.

Since our sample comprises radio loud AGN, it is likely that most of the host galaxies are elliptical. These can be modelled via several similar density profiles, e.g. the Jaffe profile (Jaffe, 1983), which models the distribution of light in a spherical galaxy as $n=n_{0}\,(r_{\rm s}/r)^{2}/4\,\pi\,(1+r/r_{\rm s})^{2}$, where $r_{\rm s}$ is the radius which contains half the total emitted light. However, as seen in Fig. 18 this has an asymptotic density distribution, as does the NFW profile (Navarro et al., 1996), $n=n_{0}\,(r_{\rm c}/r)/(1+r/r_{\rm c})^{2}$, where $r_{\text{ c}}$ is the radius of the halo core.

These can be avoided by employing the halo density distribution of an isothermal sphere, for example $n=n_{0}/(1+r/r_{\text{ c}})^{2}$ (Begeman et al., 1991) or equation 7 of Capelo et al. (2010),

where $G$ is the gravitational constant, $\rho_{\text{c}}$ is the critical density of the Universe $r_{\text{s}}$ is a scale radius, $\mu$ the mean molecular weight of the gas and $m_{\text{p}}$ the proton mass. We select this, the simplest model, cf. the polytropic and stellar models, to limit the number of free parameters. Using the column density of the Milky Way at $r=35$ kpc ($N_{\text{H{\sc\,i}}}=3.25\times 10^{22}$ $\hbox{{\rm cm}}^{-2}$) to constrain the characteristic density gives $\delta_{\text{c}}=2370$ for $T=10\,000$ K and $r_{\text{s}}=R$.

Given its relative simplicity, while yielding a similar density profile to that of Capelo et al. (2010), we use the profile of Begeman et al., for which complete ionisation of the gas occurs when

Proceeding as above, the optical depth through the completely ionised gas is

giving

which has a similar form to the exponential density (Equ. 12) and frequency (Fig. 17, right) profiles.

For all of the above models $\nu_{\rm T0}\propto Q_{\text{H{\sc\,i}}}^{~{}~{}0.97}$, although there is some variation in the constant of proportionality and each model therefore predicts quite different values for the turnover frequency,

which can vary by up to an order of magnitude for a given ionising rate (Table 1). Only where there is a constant density component, will we see a turnover at $\nu_{\rm TO}\sim 1$ GHz for $Q_{\text{H{\sc\,i}}}\sim 10^{56}$ s^-1 (sufficient to ionise all of the gas in a large spiral). For the exponential and spherical models it appears that $Q_{\text{H{\sc\,i}}}\stackrel{{\scriptstyle>}}{{{}_{\sim}}}10^{58}$ s^-1 is required for $\nu_{\rm TO}\sim 1$ GHz, which is consistent with the hypothesis that sources with no turnover are not subject to complete ionisation.

As stated in Sect. 1, the turnover frequency is anti-correlated with the projected linear size, which is clearly evident in the O’Dea & Baum (1997) data (Fig. 20).¹⁵¹⁵15$\ell$ is calculated using updated spectroscopic redshifts and cosmological parameters (Planck Collaboration et al., 2020).

On average, the turnover frequencies we derive are slightly lower than those of O’Dea & Baum, which could be due to the recent increase in the availability of low frequency data (see Appendix B). Out of the 66 sources with a measured linear size, 38 exhibit a turnover, 27 are flagged as upper limits, since the fit gives a turnover below the lowest observed frequency, and one has an inverted spectrum (FBQS J133037.6+250910). Including the limits, and excluding J1330+2509, we obtain $p(\tau)=4.56\times 10^{-14}$ for the correlation, which is similar to that using the values of O’Dea & Baum directly, where there are no limits.¹⁶¹⁶16$p(\tau)=1.58\times 10^{-13}$ ($7.38\sigma$).

If the extent of the radio emission scales with $R$, the anti-correlation between turnover frequency and size is evident from our model. Furthermore, for an ionising photon rate of $Q_{\text{H{\sc\,i}}}\sim 10^{55}$ s^-1 (J0002+2550, Fig. 19) and $R\sim\ell$, the model is in agreement with the observations (Sect. 1); that is, $\nu_{\rm TO}\sim 0.4-5$ GHz for $R\stackrel{{\scriptstyle<}}{{{}_{\sim}}}1$ kpc (HFPs/GPSs) and $\nu_{\rm TO}\stackrel{{\scriptstyle<}}{{{}_{\sim}}}0.5$ GHz for $R\stackrel{{\scriptstyle>}}{{{}_{\sim}}}10$ kpc (CSSs).

In order to generalise this, we need to break the degeneracy between $n_{0}$ and $R$ in Equ. 12. For this we use the exponential model for which we can use the fact that, for a given mass, the product $n_{0}R^{3}$ is constant (Equ. 10).¹⁷¹⁷17For the spherical model (Sect. 3.1.4), $M_{\text{gas}}=2\pi m_{\rm p}n_{0}\int_{0}^{\infty}r^{2}dr/(1+r/r_{\text{c}}^{2})=4\pi\,n_{0}\,r_{\rm c}^{3}[1+\frac{r}{r_{\rm c}}-\frac{r_{\rm c}}{r_{\rm c}+r}-2\ln(r_{\rm c}+r)+2\ln(r_{\rm c})]_{0}^{\infty}$. From a low-redshift survey of 21-cm emission from the 1000 H i brightest galaxies in the southern sky, Koribalski et al. (2004) find a range of $M_{\rm gas}\sim 10^{7}-10^{11}$ M_⊙, with a median of $M_{\rm gas}=2.9\times 10^{9}$ M_⊙. Using these masses and assuming the Galactic flare factor, for a given scale-length we determine $n_{0}$, which is then used in Equs. 12 and 6 to produce an SED (as for Fig. 19), from which we obtain the turnover frequency.¹⁸¹⁸18In Equ. 6, $r_{\rm em}$ and $D_{\rm L}$ are only required to obtain the absolute flux density.

The results are shown in Fig. 21, where we see the predictions overlap the observations of O’Dea & Baum (1997) for the range of gas masses expected (assuming $R=\ell$). We also note that the median gas mass of $M_{\rm gas}=2.9\times 10^{9}$ M_⊙ passes close to the middle of the distribution of the observed data. This is also closely aligned with the two models of O’Dea & Baum, which are based on synchrotron self-absorption models. Thus, for a gas scale-length close to the radio source size, our simple free-free model can reproduce the observed $\nu_{\rm TO}$–$\ell$ relationship.

Lastly, we note that, while we are considering the ionisation of the large-scale gas, the material causing the turnover in the radio SED may be mainly located within the central $\sim 1$ kpc (e.g. Dallacasa et al. 2000; Callingham et al. 2015; Maccagni et al. 2018). However, for the exponential and isothermal sphere models of the gas density, we see that at 100 MHz the absorption is already optically thick within this region (Fig. 22).