text:
\section{Introduction}
\label{sec:intro}

The large-scale structure of the Universe is driven by the gravitational
collapse of dark matter haloes and the subsequent merging of these into
progressively larger structures (filaments, sheets, groups, and clusters)
building the ``cosmic web''. Galaxies then form within dark matter haloes by
radiative cooling of baryons \citep{White1978}. The efficiency by which baryons
are converted into stars is highly dependent on the mass of the dark matter
halo: the stellar-mass to halo-mass relation peaks at $M_{\rm halo} \approx
10^{12} \mathrm{M_{\odot}}$, at which mass around 20 per cent of baryons are
converted into stars \citep[e.g.][]{Behroozi2010,Moster2010,Wechsler2018}.

At halo masses below this peak, the heating of gas by ultraviolet radiation from
both the host galaxy and cosmic background, and the ejection of gas by
supernova-driven winds \citep[e.g.][]{Benson2003,Baugh2006}, combine to lower
the SF efficiency in galaxies. At halo masses above $10^{12}
\mathrm{M_{\odot}}$, the overall SF efficiency of haloes is reduced from SF
being terminated, or quenched, in a subset of the galaxy population.
\citet{Peng2010} studied the fraction of passive, red galaxies in the local
Universe as a function of both mass and environment and argued that there were
two distinct mechanisms for this quenching of SF, which operates independently:
`environment quenching', which applies mainly to satellite galaxies, and `mass
quenching' of the most massive galaxies. Subsequent work has broadly confirmed
these trends \citep{Peng2012,Gabor2015,Zu2016,Cochrane2018}. Nevertheless, the
physical drivers of these quenching mechanisms remain widely debated, in
particular the extent to which mass quenching is driven by internal galactic
processes such as active galactic nuclei (AGN) as compared to the role of the
dark matter halo.

Mass quenching has frequently been attributed to the effects of AGN because AGN
activity is observed to occur in the high-mass galaxies in which SF needs to be
quenched \citep[e.g.][]{Heckman2014} and because it is well established
\citep[e.g.][]{Silk1998,Fabian1999,King2003,Fabian2012} that winds and outflows
driven by quasar activity can terminate SF in a manner that also gives rise to
the observed correlation between the mass (or velocity dispersion) of a galaxy
bulge and the mass of the central black hole (e.g.\
\citealt{Magorrian1998,Gebhardt2000}; see review by \citealt{Kormendy2013}).
However, mass quenching could also be closely linked to halo properties owing to
the strong correlation between stellar mass and halo mass. As noted by
\citet{Bower2006}, when gas falls into a dark matter halo of mass below a few
$\times 10^{11} \mathrm{M_{\odot}}$, the timescale for it to cool is shorter
than its infall time, and so it arrives at the central galaxy cold and is able
to be efficiently converted into stars. At higher halo masses, however, the
infalling gas suffers a virial shock and its cooling time exceeds the dynamical
time, leading to a hydrostatic halo of hot (X-ray emitting) gas that is built up
\citep[e.g.][]{Birnboim2003,Keres2005}. This transition in the nature of
accreted gas leads to a natural explanation for the decrease in SF efficiency in
high-mass haloes.

Regardless of the initial quenching mechanism, the hot gas in massive haloes
radiates and cools and, especially in rich, undisturbed environments, might be
expected to form a cooling flow \citep[see][]{Fabian1994}, which would result in
high levels of gas deposition and star formation (SF) in the central galaxy. A
source of heating is required to offset the cooling and prevent this. It is now
widely accepted that radio AGN are responsible for this: radio-AGN activity
(either current or recently terminated) is seen in essentially all of the
central galaxies of cool-core clusters \citep{Burns1990,Dunn2006,Best2007} and
these AGN are able to deposit the jet energy directly and efficiently into the
intracluster medium by inflating bubbles/cavities \citep[see reviews
by][]{McNamara2007,Fabian2012}. Estimates of the radio-AGN heating rate in these
clusters \citep[as determined from the cavity enthalpy coupled with a buoyancy
timescale; e.g.][]{Birzan2004,Cavagnolo2010} show that these are well matched to
the radiative cooling rates of the hot gas \citep[e.g.][]{McNamara2007}. This
forms a natural feedback cycle, whereby the hot gas offers both a source of fuel
for the radio AGN through cold chaotic accretion \citep[e.g.][]{Gaspari2013} and
a confining medium for the radio source to expand against and deposit its energy
into \citep[see review by][]{Heckman2014}.

Radio AGN feedback is likely to be important on galactic scales as well and
massive galaxies are also located in hot hydrostatic envelopes
\citep[e.g.][]{Croston2007, Mingo2011, Mingo2012}. It was the inclusion of this
`jet-mode' (or `maintenance-mode') feedback from radio AGN into semi-analytic
models of galaxy formation that allowed \citet{Croton2006} and \citet{Bower2006}
to naturally explain the shape of the galaxy luminosity function and the bimodal
nature of the galaxy population. Modern hydrodynamical simulations, including a
feedback term (associated with radio AGN) that suppresses gas cooling in hot
haloes, are also successful at reproducing the local observed trends in galaxy
properties \citep[e.g.][]{Gabor2015}. These results provide support for
so-called halo-quenching models for the switch off of SF in massive galaxies.

Observationally, considerable advances in our understanding of the importance of
radio AGN have been made over the last one to two decades; a major driver of
this has been the availability of uniform, wide-area spectroscopic surveys such
as the Two-degree Field Galaxy Redshift Survey \citep[2dFGRS;][]{Colless2001}
and the Sloan Digital Sky Survey \citep[SDSS;][]{York2000,Stoughton2002},
coupled with wide-area radio surveys, especially the National Radio Astronomy
Observatory (NRAO) Very Large Array (VLA) Sky Survey \citep[NVSS;][]{Condon1998}
and the Faint Images of the Radio Sky at Twenty centimetres survey
\citep[FIRST;][]{Becker1995}. Cross-matching of radio and optical surveys
\citep[e.g.][]{Sadler2002,Best2005a,Mauch2007,Best2012} has allowed detailed
statistical studies of the prevalence and properties of radio-AGN activity.

\citet{Best2005b} showed that the fraction of massive galaxies that host
radio-AGN activity is a very strong function of stellar mass ($f_{\rm rad}
\propto M_*^{2.5}$) or black hole mass ($f_{\rm rad} \propto
M_{\mathrm{BH}}^{1.6}$), reaching as high as $\approx 30$ per cent at the
highest stellar masses, to the radio luminosity limit of their analysis
($L_{\mathrm{1.4\,GHz}} \approx 10^{23}$W\,Hz$^{-1}$); see also \cite{Brown2011}
for a deeper study of a much smaller sample. \citet{Best2006} and
\citet{Best2007} built upon this to estimate the time-averaged heating rate
associated with this radio-AGN activity, assuming that all massive galaxies go
through recurrent radio-AGN outbursts and that the observed radio-AGN prevalence
could be used as a measure of the duty cycle of the AGN activity. They found
that this heating rate exceeded the heating rate that is necessary to
counterbalance the typical radiative cooling losses of the hot gas, and that
radio-AGN heating is therefore able to explain the old, red, and dead nature of
massive galaxies in hot haloes.

Although this broad understanding is in place, several open questions remain
about the detailed process of radio-AGN feedback. One such question concerns the
duty cycle of radio-AGN activity. \citet{Kauffmann2009} and \citet{Best2012}
studied the distribution of Eddington-scaled accretion ratios at low Eddington
ratios and found that, down to their limit of around $L/L_{\rm Edd} = 10^{-3}$,
these follow roughly a power law with increasing numbers of low Eddington ratio
sources \citep[see also][]{Heckman2014}. Deeper radio data are needed to probe
lower radio luminosities, and hence track the full distribution of accretion
rates, over wide enough sky areas to build significant samples. Such data would
also allow a much larger fraction of lower stellar mass sources to be detected,
enabling investigation of the relationships between duty cycle, radio luminosity
distribution, and stellar mass.

The LOFAR Two-Metre Sky Survey \citep[LoTSS; ][]{Shimwell2017} offers a new
opportunity to advance these studies and address some of these questions. This
large-area 150\,MHz survey with the LOw Frequency ARray \citep[LOFAR;
][]{vanHaarlem2013} reaches more than an order of magnitude deeper than the
FIRST survey for sources of typical spectral index, and is even more
advantageous for steep spectrum sources (older electron populations).
Furthermore, it also has high sensitivity to extended radio structure, thus
avoiding the need for the complicated combination of FIRST and NVSS that
\citet{Best2005a} needed to adopt. This paper cross-matches data from the first
data release \citep[DR1; ][]{Shimwell2018} of LoTSS with the main galaxy sample
of the SDSS to provide new insights into the prevalence, duty cycle, and impact
of radio-AGN activity in the local Universe.

The layout of the paper is as follows. Section~\ref{sec:sample_data} describes
the sample of galaxies to be studied and the cross-matching between the radio
and optical catalogues. Section~\ref{sec:radio_loud} then outlines the procedure
to filter the radio-AGN subset of these sources from the bulk population of
star-forming galaxies (SFGs). The radio spectral index properties of these
sources, and the resultant local radio luminosity function at 150\,MHz are
discussed in Sections~\ref{sec:spectral_index} and~\ref{sec:lum_func},
respectively. Section~\ref{sec:fractions} then considers the prevalence of radio
AGN as a function of both stellar mass and black hole mass, comparing the
results to previous studies, and using the large sample to break the degeneracy
between these two parameters. Section~\ref{sec:accretion_rates} examines the
distribution of Eddington-scaled accretion rates of the most massive galaxies.
Finally, the results of the paper are summarised in
Section~\ref{sec:conclusions}, and their implications are discussed. Throughout
the study, a cosmology with $\mathrm{\Omega}_{\mathrm{\Lambda0}} = 0.7$,
$\mathrm{\Omega}_{\mathrm{m0}} = 0.3$, and $\mathrm{H}_{\mathrm{0}} = 70$ km
s$^{-1}$ Mpc$^{-1}$ is assumed.

\section{The sample and data}
\label{sec:sample_data}

The LoTSS \citep[][]{Shimwell2017} is a high-resolution survey that will cover
the full northern hemisphere at frequencies ranging from 120 to 168
MHz\footnote{Although the central frequency of the LoTSS band is 144\,MHz, the
bandwidth is large and the sensitivity-weighted mean frequency depends on
position within the mosaic owing to the frequency-dependent primary beam size.
To avoid an impression of undue precision, we use 150\,MHz throughout the paper
to refer to the LoTSS frequency.}. The LoTSS DR1 \citep[][]{Shimwell2018} covers
424 square degrees centred in the Hobby-Eberly Telescope Dark Energy Experiment
\citep[HETDEX; ][]{Hill2008} Spring Field region (right ascension 10h45m00s to
15h30m00s and declination 45$^\circ$00$\arcmin$00$\arcsec$ to
57$^\circ$00$\arcmin$00$\arcsec$) and contains over 300,000 sources with a
signal of at least five times the noise level. It is composed of 58 overlapping
pointings with 8 hours of observation in each one. The median rms noise is
$\approx$ 71\,$\mathrm{\mu}$Jy~beam$^{-1}$ and 95 per cent of the area in the
release has an rms noise level below 150\,$\mathrm{\mu}$Jy~beam$^{-1}$. The
angular resolution is 6 arcseconds and the positional accuracy is better than
0.2 arcseconds for high signal-to-noise sources; the positional accuracy
increases to $\approx$ 0.5 arcseconds for the faintest sources with a flux
density of less than 0.6 mJy. Detailed information about the LoTSS DR1 is
presented in \citet{Shimwell2018}. The radio sources were associated with their
optical and mid-infrared counterparts in the Panoramic Survey Response and Rapid
Response System \citep[Pan-STARRS;][]{Kaiser2002, Kaiser2010} and the Wide-field
Infrared Survey Explorer \citep[WISE;][]{Wright2010} surveys (with a
completeness and reliability at the $\approx 99$ per cent level) using a
combination of statistical likelihood ratio techniques and visual classification
methods \citep{Williams2018}. Finally, photometric redshifts were derived and
rest-frame magnitudes were obtained for the sources \citep[][although those data
are not used in the current paper]{Duncan2018}.

The galaxy sample selected for this study is based on the spectroscopic data
from the seventh data release of the SDSS  \citep[SDSS DR7;][]{Abazajian2009}.
There are 34709 galaxies from the SDSS Main Galaxy Sample \citep[broadly
complete between magnitudes 14.5 and 17.77 in r band;][]{Strauss2002} that
overlap with the region covered by the LoTSS DR1. An upper redshift limit of $z
= 0.3$ is applied, which reflects the approximate limit of main galaxy sample
spectroscopy ($n_{z > 0.3} = 47$). A lower redshift limit of $z=0.01$ is also
applied, as below this the SDSS fibres probe such a small region of the galaxy
that derived galaxy parameters become unreliable ($n_{z < 0.01} = 413$).
Galaxies with a non-zero $z$ warning flag were discarded as their redshifts
could be unreliable ($n_{\mathrm{z\,warning}} = 34$). The catalogue contains
duplicated observations for some of the objects and a single observation of each
object was selected to avoid multiple counting on the statistics
($n_{\mathrm{duplicates}} = 711$).  With all of these constraints, a sample of
33504 galaxies was obtained in the LoTSS DR1 area.

All the SDSS sources are bright in the optical band and have clear corresponding
PanSTARRS detections. Therefore, the radio counterparts of SDSS galaxies were
found by cross-matching the positions of SDSS galaxies with the PanSTARRS
counterparts of the LoTSS sources. LoTSS sources present accurate
PanSTARRS-LoTSS cross identifications even for extended sources due to the
efforts shown in \citet{Williams2018}. The positions of the SDSS sources were
cross-matched with the optical (PanSTARRS) positions using a search radius of 2
arcseconds. This radius is sufficiently large to obtain essentially all the
genuine counterparts with no significant contamination by random matches. In any
case, the results are largely insensitive to this choice of cross-match radius
if it is within the range of 1 to 2 arcseconds since the PanSTARRS to SDSS
astrometric agreement is typically accurate to fractions of an arcsecond. A
LoTSS counterpart was found for 10615 of the SDSS sample (32 per cent). The
LoTSS data release provides mosaics with the rms noise level for each position,
which also allows the flux density limit at 150 MHz to be determined for the
SDSS galaxies without a LoTSS cross-match. This limit was set at five times the
rms noise level, which corresponds to the approximate catalogue limit for LoTSS
\citep{Shimwell2018}. There are 43 sources for which the measured LoTSS total
flux density is slightly below five times the rms noise. For completeness these
sources were removed from the catalogue and considered as non-detections.

The MPA-JHU\footnote{https://wwwmpa.mpa-garching.mpg.de/SDSS/DR7/} value-added
catalogue \citep{Brinchmann2004arxiv} offers improved and additional information
to the SDSS data. In particular, the nebular emission lines were measured after
the subtraction of the contribution of the stellar populations
\citep{Tremonti2004} and are used for the selection of AGN in the next section.
We also use the stellar mass of the galaxies presented in the catalogue, as
derived by \citet{Kauffmann2003a}. Additional parameters used in this paper are
the 4000\AA\ break strength ($D_{\rm 4000}$), and the velocity dispersion
($\sigma$). The latter can be used to estimate the mass of the central black
hole: the relation adopted in this paper is that of \citet{McConnell2013}, i.e.
$\log_{10}(M_{\mathrm{BH}}/\mathrm{M_{\odot}}) = 8.32 + 5.64
\log_{10}[\sigma/(200 \mathrm{km\,s^{-1}})]$. \citet{Best2012} cross-matched the
SDSS DR7 spectroscopic catalogue against the NVSS and FIRST radio surveys to
derive the 1.4\,GHz properties of these sources; data from their catalogues are
also used in the analysis.

\section{Selection of radio AGN}
\label{sec:radio_loud}

A significant challenge in deep radio surveys such as LoTSS is to separate radio
AGN from SFGs, for which the radio emission is associated with the SF in the
galaxy, but there may or may not also be a radio-quiet AGN present; radio AGN
are defined in this work as galaxies for which their radio emission is dominated
by a jet originating from the central supermassive black hole.  Star-forming
galaxies emit at radio wavelengths primarily due to synchrotron emission from
shocks associated with supernovae, and hence their radio luminosity correlates
broadly with the SFR (e.g.\ \citealt{Condon1992}), although there are
indications that the relationship can also depend on other parameters such as
galaxy mass (e.g.\ \citealt{Gurkan2018,Read2018}). Star-forming  galaxies
dominate the radio source counts at flux densities $S_{\rm 150\,MHz} \le 1\,$mJy
\citep{Wilman2008,deZotti2010,Williams2016,CalistroRivera2017} and cannot be
ignored at higher flux densities.

A common method of separating SFGs from AGN is through their optical
emission-line properties, in particular using emission line diagnostic diagrams
\citep{Baldwin1981,Kauffmann2003,Kewley2006}. One shortcoming of this method is
that it cannot be used for objects without clear emission line detections.
Although for radio-selected samples, these sources are mostly weak-lined radio
AGN \citep[e.g.][]{Sadler2002}. An additional complication is the radio-quiet
AGN. The origin of the radio emission in these sources remains widely debated,
with both SF and weak radio jets contributing, and the dominant mechanism
varying between sources \citep[e.g.][and references therein]{White2017}. In the
local Universe, regardless of the presence or not of weak jets, radio-quiet AGN
are frequently associated with SFGs (e.g.\ \citealt{Kauffmann2003}; see also
discussion in \citealt{Heckman2014}) and, at the depth of surveys like LoTSS,
such SF may lead to the galaxy being detected in the radio. The emission line
fluxes of such sources can be driven by the hard ionising spectrum of the AGN,
leading to an AGN classification by optical line diagnostics, but such
SF-dominated sources should not be included in samples of radio AGN.

An alternative approach to identify radio AGN is to compare the radio and
far-infrared luminosities of sources \citep[e.g.][]{Machalski1999, Sabater2012,
Mingo2016}. This technique makes use of the relatively tight far-infrared radio
correlation (FIRC) of SFGs \citep[e.g.][]{Yun2001, Smith2014}: objects with a
radio excess above that predicted from the FIRC can be identified as radio AGN.
This is a powerful method (albeit still  limited by the intrinsic scatter in the
FIRC), but is generally not applicable because of the absence of sufficiently
deep far-IR data.

\citet{Best2005b} proposed an alternative method for identification of radio
AGN, using the `$D_{\rm 4000}$ versus $L_{\rm rad}$/$M_*$' plane. The parameter
$D_{\rm 4000}$ is the strength of the 4000\AA\ break in the galaxy spectrum, and
$L_{\rm rad}$/$M_*$ is the ratio of radio luminosity to stellar mass.
\citeauthor{Best2005b} showed that for a wide range of different SF histories,
SFGs occupy largely the same region of this plane, since both $D_{\rm 4000}$ and
$L_{\rm rad}$/$M_*$ depend broadly on the specific SFR of the galaxy.
\citeauthor{Best2005b} used these theoretical tracks to propose a diagnostic
division line between radio AGN and SFGs whereby the radio-quiet AGN were
classified together with the SFGs.

\citet{Kauffmann2008} considered an alternative diagnostic, making use of the
ratio of radio luminosity to H$\alpha$ luminosity. In this work, the latter is
used as a proxy for SFR, following the same principles as the separation via the
FIRC. Using this separation they also proposed a modification to the division
line on the $D_{\rm 4000}$ versus $L_{\rm rad}$/$M_*$ diagnostic of
\citet{Best2005b}. \citet{Best2012} then further built upon this work by
devising a scheme to separate radio AGN from SFGs using a combination of the
revised $D_{\rm 4000}$ versus $L_{\rm rad}$/$M_*$ diagnostic, the emission line
diagnostic diagram, and the radio luminosity to H$\alpha$ luminosity relation.

In this paper, the identification of the radio-AGN subsample builds upon the
work of \citet{Best2012}, but with some significant modifications. First, the
division lines are recast into 150\,MHz luminosities (as detailed in the next
subsection) owing to the observing frequency of LoTSS. Second, a fourth
diagnostic test is also included in the analysis, made feasible by the
availability of WISE data.  Star-forming galaxies separate from the typical
hosts of radio AGN in their WISE colours, particularly in W2--W3 \citep[4.6 to
12 micron colour;][]{Yan2013}: \citet{Herpich2016} use a cut at W2--W3 = 2.5 (in
Vega magnitudes; W2--W3 $\approx 0.7$ in AB magnitudes) to separate galaxies
with and without SF, although it is also clear from their plots that this
separation is far from clean. Third, we make use of the analysis by
\citet{Gurkan2018} of the LOFAR data in the Herschel-ATLAS North Galactic Plane
(hereafter H-ATLAS) field \citep{Hardcastle2016}. These LOFAR data are similar
to those of LoTSS and have the same SDSS and WISE properties available. But
\citet{Gurkan2018} have done detailed spectral energy distribution fitting,
incorporating the available Herschel data in a self-consistent manner using
\textsc{MAGPHYS} (\citealt{daCunha2008}; see also \citealt{Smith2012}) to derive
accurate SFRs and allow more precise separation of SFGs from radio AGN based on
the radio luminosity to SFR relation. This complementary sample therefore
provides an opportunity to test and optimise the calibration of the diagnostic
division lines, and the resultant combined classification scheme, as outlined
below.

\subsection{Adopted individual diagnostics for population separation}
\label{sec:sfagndiags}

Figure~\ref{fig:sfagn} shows the four classification methods along with the
classification lines adopted. Sources are colour-coded by their final adopted
classification (see Section~\ref{sec:final_class}). The details of the
classification in each of the four diagnostics is now discussed in turn.

The upper left panel of Figure~\ref{fig:sfagn} shows $D_{4000}$ {\it versus}
$L_{\rm 150\,MHz}$/$M_*$, based on that of \citet{Best2005b}. The upper dotted
line shows the revised division proposed by \citet{Kauffmann2008} and adopted by
\citet{Best2012}; the radio luminosity is converted to 150\,MHz from 1.4\,GHz by
assuming a spectral index of $\alpha = 0.7$ (for $S_\nu \propto \nu^{-\alpha}$;
this spectral index is the canonical value from \citealt{Condon2002}, with a
similar typical value found for LOFAR sources by \citealt{CalistroRivera2017}).
However, applying these classifications to the H-ATLAS data indicated that this
conservative cut left many genuine radio AGN within the star-forming region and
led to significant numbers of misclassifications in the final combined
classifications. In contrast, \citep[as noted by][]{Kauffmann2008} the original
cut of \cite{Best2005b} led to some SFGs being misclassified as radio AGN and,
especially at the lower radio luminosities probed by LoTSS, also led to
misclassification of some radio AGN with large values of $D_{4000}$ as SFGs.
Therefore, a second diagnostic line was introduced that tracked the original
\citet{Best2005b} line until $D_{4000} = 1.7$ and then continued horizontally
(lower dotted line on the upper left panel of Figure~\ref{fig:sfagn}); this form
was chosen to maximise agreement with the more sophisticated \citet{Gurkan2018}
classifications of the (much smaller) H-ATLAS sample. Galaxies above the upper
dotted line were classified as radio AGN via this diagnostic, galaxies below the
lower dotted line were classified as SFGs, and galaxies between the two lines
were deemed intermediate. Only a handful of galaxies could not be classified on
this diagnostic because of an absence of a reliable mass measurement.
Table~\ref{tab:sfagn2} shows the number of objects classified into each category
by each diagnostic. It also shows the resultant overall classification
statistics of these galaxies (see Section~\ref{sec:final_class}) and can thus be
used to judge the importance of each diagnostic in the overall classification.

\begin{table}
\centering
\caption{Numbers of sources classified in each category by each different
classification method. `AGN' and `SF' are (radio) AGN and star-forming
classifications, respectively. `Intermediate' are the intermediate
classifications for the $D_{4000}$ versus $L_{\rm 150\,MHz}$/$M_*$ or
$L_{\mathrm{H}\alpha}$ versus $L_{\rm rad}$ diagnotics; `unclassified' sources
lack data for classification on that diagnostic. The bracketed number on the
second row for each diagnostic shows the number of these sources that end up
with an overall classification of `radio AGN' (with the rest classified as
star-forming, which may include radio-quiet AGN). This can therefore be used to
gauge the importance of each diagnostic in the overall classification. For
example, the majority of objects classified as `AGN' by the BPT diagnostic are
ultimately deemed to be radio-quiet AGN so this classification is not given much
weight, although classification as `SF' by the same diagnostic is almost
invariably secure.}
\label{tab:sfagn2}
\begin{tabular}{ccccc}
  \hline
  Diagnostic & \multicolumn{4}{c}{Number classified in that category} \\
  method     & \multicolumn{4}{c}{(number with overall class of radio AGN)}\\
  & AGN & SF & Intermediate & Unclassified \\
\hline
\hline
$D_{4000}$ vs.    & 703 & 8392 & 1510 & 10 \\
$L_{\rm 150\,MHz}$/$M_*$     & (703) & (97) & (1318) & (3) \\
\hline
BPT                   & 2865 & 5012 & 0 & 2738 \\
                      & (544) & (10) & (0) & (1567) \\
\hline
$L_{\mathrm{H}\alpha}$ vs. & 659 & 8622 & 1146 & 188 \\
$L_{\rm 150\,MHz}$       & (659) & (532) & (753) & (177) \\
\hline
WISE                  & 1759 & 8039 & 0 & 781 \\
W2-W3                 & (1426) & (105) & (0) & (590) \\
\hline
\end{tabular}
\end{table}

The upper right panel of Figure~\ref{fig:sfagn} shows the [O{\sc iii}]/H$\beta$
versus [N{\sc ii}]/H$\alpha$ line diagnostic diagram that is widely used to
separate AGN and SFGs owing to the relative strength of the four emission lines
involved and the broad independence of the line ratios on dust extinction. The
division proposed by \citet{Kauffmann2003} is adopted: AGN and SFGs are divided
at log([O{\sc iii}]/H$\beta$) $=$ 1.3 + 0.61 / (log([N{\sc ii}]/H$\alpha$) $-$
0.05). Seventy-four per cent of the radio source sample are classifiable by this
method, while the rest lack detections for at least one of the lines. This is a
significantly higher fraction of classifiable objects than the $\sim 30$ per
cent found by \citet{Best2012}, largely because the deeper radio survey contains
a far higher fraction of (stronger lined) SFGs. As discussed above (and evident
on Figure~\ref{fig:sfagn}) objects selected as AGN by this diagnostic may be
significantly contaminated by radio-quiet AGN, and so for these objects this
diagnostic diagram is not given much weight in the final classification (cf.
Table~\ref{tab:sfagn2}~and~\ref{tab:sfagn}). However, the absence of AGN
signatures provides a more useful diagnostic: essentially all $\sim 5000$
sources ($=47$ per cent of the sample) that fall in the SFG region of this
diagnostic plot have an overall classification of SFG (see
Table~\ref{tab:sfagn2}).

The $L_{\mathrm{H}\alpha}$ versus $L_{\rm 150\,MHz}$ diagnostic is shown in the
lower left panel of Figure~\ref{fig:sfagn}. \citet{Best2012} used just a single
conservative cut on this diagnostic. However, as with the $D_{\rm 4000}$ versus
$L_{\rm 150\,MHz}$/$M_*$ diagnostic, comparison with the H-ATLAS data indicates
that any single separation line leads to significant numbers of
misclassifications. Therefore, again, two separation lines are adopted: $\log
(L_{\mathrm{H}\alpha} / \mathrm{L_{\odot}}) = \log (L_{\rm 150\,MHz} / {\rm
W\,Hz}^{-1}) - 16.9$ and $\log (L_{\mathrm{H}\alpha} / \mathrm{L_{\odot}}) =
\log (L_{\rm 150\,MHz} / {\rm W\,Hz}^{-1}) - 16.1$. Galaxies with measured
H$\alpha$ luminosity, or an upper limit on this, below the lower line are
classified as radio AGN on this diagnostic. Those which have measured H$\alpha$
luminosities, or upper limits, between the two lines are deemed intermediate,
and those with a measured H$\alpha$ luminosity above the upper line are
classified as SFGs. We found 1.7 per cent of sources have upper limits on
$L_{\mathrm{H}\alpha}$ above the upper line and are left unclassified.

Finally, the lower right panel of Figure~\ref{fig:sfagn} shows a plot of W1--W2
versus W2--W3 mid-infrared WISE colours. The sources are classified on this plot
according to a simple division at W2--W3 = 0.8 (AB). This value was again
optimised based on comparison with the H-ATLAS sample and is very similar to the
division adopted by \citet{Herpich2016}. We found 7 per cent of sources did not
possess a W2--W3 colour and so were unclassified. This diagnostic is somewhat
crude, and was largely only used where the other diagnostics produced
intermediate or contradictory classifications.

\subsection{Combination of diagnostics and final classification}
\label{sec:final_class}

For each individual classification method, sources may be classified as a radio
AGN, classified as having their radio emission associated with SF, or be
unclassified. For two of the four diagnostics, an intermediate classification is
also possible. This leads to 144 different possible combinations of
classifications, where in some cases the classifications may disagree.
\citet{Best2012} discussed how to combine these individual classifications in
order to obtain a single overall classification. Their classifications are
broadly adopted here, but need to be expanded to incorporate the new
intermediate classes and the WISE diagnostics. In this work, the approach taken
was to apply these diagnostics to the equivalent data for the H-ATLAS sample
from \citet{Gurkan2018}, and compare against  the more sophisticated
classifications available for that sample. Specifically, for each possible
combination of the four diagnostics, the H-ATLAS sources with that combination
were identified, and the majority classification of that sample (SF or
radio AGN) was adopted as the overall classification for that combination. In
most cases this outcome was very clear with the H-ATLAS sources having (almost)
uniformly the same classification; only for a few of the (less common)
combinations, where the different diagnostic methods suggest different
classifications, was the H-ATLAS sample also more heterogeneous in its
classifications, suggesting (as expected) more uncertainty in the  overall
classification. Based on this comparison with H-ATLAS, the overall potential
contamination from misclassification is estimated to be $\la 3$ per cent.

Table~\ref{tab:sfagn} shows the final classifications adopted and the number of
sources in each category; to save space, classifications with fewer than five
sources are not listed separately, but the overall contribution of these is
indicated at the bottom of the table. In total, 2121 sources are classified as
radio AGN, and 8494 sources are classified as having their radio emission
associated with SF\footnote{These AGN/SF classifications are used in
\citet{Hardcastle2018} to help to establish criteria to define larger AGN and SF
galaxy samples from the wider LoTSS sample without SDSS spectroscopy.}.

\begin{table}
\centering
\caption{Number of sources and overall classification for different combinations
of the four classification methods (with at least five sources).  `AGN' are
sources classified as (radio) AGN; `SF' means that the radio emission is
identified to be primarily powered by star formation (but a radio-quiet AGN may
also be present); `Int' are intermediate classifications for the $D_{4000}$
versus $L_{\rm 150\,MHz}$/$M_*$ or $L_{\mathrm{H}\alpha}$ versus $L_{\rm rad}$
diagnotics; `Unc' (unclassified) sources lack data for classification on that
diagnostic.}
  \label{tab:sfagn}
\begin{tabular}{cccccc}
\hline
 $D_{4000}$ vs. & BPT & $L_{\mathrm{H}\alpha}$ vs. & WISE & No. of &
Overall\\
 $L_{\rm 150\,MHz}$/$M_*$ & & $L_{\rm 150\,MHz}$ & W2--W3 & sources &
class  \\
\hline
\hline
Unc  & SF   & SF   & SF   &    5 & SF   \\
SF   & Unc  & Unc  & SF   &   11 & SF  \\
SF   & Unc  & Unc  & AGN  &    5 & AGN  \\
SF   & Unc  & SF   & Unc  &    8 & SF  \\
SF   & Unc  & SF   & SF   &  892 & SF  \\
SF   & Unc  & SF   & AGN  &  102 & SF  \\
SF   & Unc  & AGN  & Unc  &    5 & AGN  \\
SF   & Unc  & AGN  & SF   &   36 & AGN  \\
SF   & Unc  & Int  & Unc  &    8 & SF  \\
SF   & Unc  & Int  & SF   &   83 & SF  \\
SF   & Unc  & Int  & AGN  &   21 & AGN  \\
SF   & SF   & SF   & Unc  &  100 & SF  \\
SF   & SF   & SF   & SF   & 4791 & SF  \\
SF   & SF   & SF   & AGN  &   25 & SF  \\
SF   & SF   & Int  & Unc  &    5 & SF  \\
SF   & SF   & Int  & SF   &   63 & SF  \\
SF   & AGN  & SF   & Unc  &   29 & SF  \\
SF   & AGN  & SF   & SF   & 1758 & SF  \\
SF   & AGN  & SF   & AGN  &  240 & SF  \\
SF   & AGN  & Int  & Unc  &   10 & AGN  \\
SF   & AGN  & Int  & SF   &  179 & SF  \\
SF   & AGN  & Int  & AGN  &   14 & AGN  \\
AGN  & Unc  & Unc  & Unc  &    8 & AGN  \\
AGN  & Unc  & AGN  & Unc  &  211 & AGN  \\
AGN  & Unc  & AGN  & AGN  &  251 & AGN  \\
AGN  & Unc  & Int  & Unc  &   65 & AGN  \\
AGN  & Unc  & Int  & AGN  &   70 & AGN  \\
AGN  & AGN  & AGN  & Unc  &    8 & AGN  \\
AGN  & AGN  & AGN  & SF   &    5 & AGN  \\
AGN  & AGN  & AGN  & AGN  &   43 & AGN  \\
AGN  & AGN  & Int  & Unc  &    5 & AGN  \\
AGN  & AGN  & Int  & SF   &    5 & AGN  \\
AGN  & AGN  & Int  & AGN  &   20 & AGN  \\
Int  & Unc  & Unc  & Unc  &   67 & AGN  \\
Int  & Unc  & Unc  & AGN  &   92 & AGN  \\
Int  & Unc  & SF   & Unc  &   38 & SF  \\
Int  & Unc  & SF   & SF   &   28 & SF  \\
Int  & Unc  & SF   & AGN  &  194 & AGN  \\
Int  & Unc  & AGN  & Unc  &   45 & AGN  \\
Int  & Unc  & AGN  & AGN  &   43 & AGN  \\
Int  & Unc  & Int  & Unc  &  131 & AGN  \\
Int  & Unc  & Int  & SF   &   49 & AGN  \\
Int  & Unc  & Int  & AGN  &  256 & AGN  \\
Int  & SF   & SF   & SF   &    7 & SF  \\
Int  & SF   & SF   & AGN  &    8 & AGN  \\
Int  & AGN  & SF   & Unc  &   17 & AGN  \\
Int  & AGN  & SF   & SF   &   64 & SF  \\
Int  & AGN  & SF   & AGN  &  312 & AGN  \\
Int  & AGN  & Int  & Unc  &   15 & AGN  \\
Int  & AGN  & Int  & SF   &   50 & SF  \\
Int  & AGN  & Int  & AGN  &   86 & AGN  \\
\hline
\multicolumn{4}{c}{Other combinations} & 24 & AGN \\
\multicolumn{4}{c}{Other combinations} &  8 &  SF \\
\hline
\multicolumn{4}{c}{Total} & 2121 & AGN \\
\multicolumn{4}{c}{Total} & 8494 &  SF \\
\hline
\end{tabular}
\end{table}

\begin{figure*}
  \begin{tabular}{cc}
    \psfig{file=sfcut_d4000_pm.pdf,width=8.8cm,clip=} &
    \psfig{file=sfcut_bpt2.pdf,width=8.8cm,clip=} \\
    \psfig{file=sfcut_ha_lrad.pdf,width=8.8cm,clip=} &
    \psfig{file=sfcut_wisecut.pdf,width=8.8cm,clip=} \\
  \end{tabular}
  \caption{Location of the LoTSS sources on the four diagnostic plots used to
separate the radio AGN from those galaxies where the radio emission is powered
by SF. {\it Upper left:} The `$D_{4000}$ versus $L_{\rm 150\,MHz}$/$M_*$'
method, developed by \citet{Best2005a}. Upper right: The [O{\sc iii}]/H$\beta$
versus [N{\sc ii}]/H$\alpha$ emission line ratio diagnostic diagram of
\citet{Baldwin1981}. Lower left: The $L_{\mathrm{H}\alpha}$ versus $L_{\rm
150\,MHz}$ relation. Lower right: This WISE W1--W2 versus W2--W3 colour-colour
diagnostic. In all plots, the dotted lines indicate the division(s) used for
that classification method (see Section~\ref{sec:sfagndiags} for further
details); in the cases of the upper left and lower left plots, there are two
division lines; sources between the two lines are deemed to be `intermediate' by
that classification method. The different symbols reflect the final combined
classification of each source: radio AGN are plotted as red diamonds and SFGs as
black crosses. In the lower left plot, arrows indicate upper limits to the
H$\alpha$ luminosity. }
  \label{fig:sfagn}
\end{figure*}

\section{Spectral indices of local radio AGN}
\label{sec:spectral_index}

It is well established that the low-luminosity radio AGN have typically much
smaller physical sizes that their more luminous counterparts. \citet{Best2005a}
found that the vast majority of radio sources in their SDSS-NVSS-FIRST
cross-match are unresolved at the $\approx 5 \arcsec$ resolution of the FIRST
survey \citep[see also][]{Baldi2010}. Higher resolution radio studies have
confirmed these sources to be AGN, but have shown that they either remain
unresolved down to sub-arcsecond (sub-kpc) resolution or present weak jets on
scales of at most a few kiloparsec \citep{Baldi2015}. The core radio
morphologies of these sources and their host galaxy properties largely resemble
those of the more powerful extended radio sources, but they lack the extended
emission; the explanation for this is still a matter of debate \citep[e.g.][and
references therein]{Baldi2018}. For our radio-AGN sample, 28 per cent of the
sources show multiple components and 15 per cent can be modelled by a single
Gaussian but are resolved according to the criteria shown by
\citet{Shimwell2018}. The remaining 57 per cent of the sources are unresolved
single components, although this figure raises quickly with decreasing flux
density.

A characteristic property of these radio AGN that can offer clues as to their
nature is the radio spectral index: compact radio cores and hotspots display
much flatter spectra than extended radio components. \citet{deGasperin2018} have
measured the spectral index properties of radio sources from wide-area sky
surveys and have found evidence that these flattened towards lower radio flux
densities; similar flattening at lower flux densities has been seen in deeper
studies of small sky areas \citep[e.g.][]{Prandoni2006} and is often ascribed to
lower luminosity AGN. However, the spectral indices of these local very
low-luminosity AGN have only been investigated for small samples
\citep{deGasperin2011}, indicating a mix of flat and steep spectrum sources.

Of the 2121 LoTSS-selected radio AGN, 496 of these sources are in common with
the NVSS-FIRST sample of \citet{Best2012}, and therefore have available 1.4\,GHz
flux densities. The cross-matching process of \citet[][see also
\citealt{Best2005a}]{Best2012} uses an approach that combines NVSS and FIRST
data, such that the matched sources benefit from the angular resolution of FIRST
(which is similar to that of LoTSS) but also the sensitivity to extended
structure offered by NVSS (which LoTSS also possesses due to the dense LOFAR
core). This combination thus minimises any possible biases in spectral indices
originating from a mismatch in resolution/sensitivity of different surveys. The
relation between LoTSS luminosity and stellar mass, including which galaxies are
also detected in NVSS/FIRST, is shown in Fig.~\ref{fig:mlum}. In this figure it
is clear how LoTSS probes a population of galaxies that remained undetected in
NVSS/FIRST.

\begin{figure}
   \centering
\includegraphics[width=\linewidth]{lum-stellar_mass.pdf}
   \caption{Distribution of LoTSS luminosities with respect to the stellar
masses. The SF galaxies detected only in LoTSS are plotted with different
symbols (circles) than those that are also detected in NVSS/FIRST (stars). Radio
AGN are indicated with three-spoked asterisks in red for galaxies detected only
in LoTSS and in olive for galaxies detected in both LoTSS and NVSS/FIRST.}
   \label{fig:mlum}
\end{figure}

The distribution of 150\,MHz to 1.4\,GHz spectral indices for these radio AGN is
shown in Fig.~\ref{fig:alpha}, as a function of 1.4\,GHz flux density; the LoTSS
data are sufficiently deeper than the NVSS/FIRST data that the LoTSS limit
provides essentially no constraint on this plot. At high flux densities ($S_{\rm
1.4\,GHz} > 20$\,mJy) the median spectral index is $0.63$, which is very much in
line with the canonical value of 0.7 \citep{Condon2002}. The scatter of the
spectral indices at the lower flux densities are mainly caused by the
uncertainty in the measurements but a possible general trend towards flat or
even inverted spectrum sources was also explored.

The median spectral index decreases towards 0.4 at the lowest flux densities,
and the faintest radio sources display a wide range of spectral indices,
including some flat spectrum sources. If correct this would be consistent with
the suggestion of \citet{Whittam2017}, using higher frequency source counts, of
an increasing core fraction towards lower flux densities. However, it must be
noted that at the lowest flux densities the distribution may be skewed by
selection biases in the FIRST-NVSS sample as the 3\,mJy flux density limit of
that sample (set by the NVSS limits) is approached, boosting the number of
apparently flatter spectrum sources. These selection biases were investigated
using a Monte Carlo simulation. The distribution of spectral indices for sources
with $S_{\rm 1.4\,GHz} > 20$\,mJy was fitted using a Gaussian. Then, for each
source with a flux density measured at 150\,MHz, 10000 random spectral indices
were drawn from the Gaussian and the corresponding $S_{\rm 1.4\,GHz}$ was
calculated. If this was below the NVSS detection limit (3\,mJy) then that
iteration of that source was discarded (as it would not have been within the
subsample of sources with measured spectral indices); otherwise it was retained.
The retained values were then used to compute the mean spectral index as a
function of 1.4\,GHz flux density. The results of the simulation are plotted in
Fig.~\ref{fig:alpha}. It is clear that the trend of the spectral index to
decrease at the lower flux densities is consistent with being driven entirely by
the biases arising from the combination of statistical flux errors and selection
limits. That would indicate that the bulk of the compact radio AGN are not
simply flat spectrum radio cores with an absence of extended emission, but
rather that they are broadly scaled-down versions of the more luminous extended
sources \citep[see also discussion in][]{Baldi2018}. This conclusion fits in
with the view of the recurrent nature of radio-AGN activity, discussed in
Section~\ref{sec:accretion_rates}.

\begin{figure}
   \centering
\includegraphics[width=\linewidth]{alpha2.pdf}
   \caption{Spectral index distribution of the radio-AGN galaxies with both
LoTSS and 1.4\,GHz measurements, as a function of 1.4\,GHz flux density.
Individual sources are shown as blue dots and the medians of different bins in
1.4\,GHz flux density are shown as orange crosses. The vertical lines correspond
to the standard deviation on each bin. The median spectral index at flux
densities $S_{\rm 1.4\,GHz} > 20$\,mJy is 0.63. The weak decrease in median
spectral index at lower flux densities may be driven by selection biases in the
1.4\,GHz sample, as indicated by the black dashed line that simulates the effect
of this (see text for details).}
   \label{fig:alpha}
\end{figure}

The typical spectral index found in Figure~\ref{fig:alpha} is also needed to
provide a reliable conversion factor for comparison of the LoTSS results with
previous measurements at higher frequencies.  For the remainder of the paper
(and for consistency with the value adopted for the AGN/SF separation) the
canonical spectral index for radio AGN of 0.7 is adopted, which is close to the
observed median. This spectral index corresponds to the following conversion in
luminosities: $\log_{10}(L_{\mathrm{1.4\,GHz}}) \approx \log_{10}(L_{\rm
150\,MHz})-0.68$.

\section{Local 150\,MHz luminosity function}
\label{sec:lum_func}

The local 150\,MHz radio luminosity functions for both SFGs and radio AGN were
determined using the standard technique, $\rho = \sum_i 1/V_i$
\citep{Schmidt1968,Condon1989}, where $V_i$ is the volume within which source
$i$ could be detected. The calculation of $V_i$ for each source requires careful
consideration of both the radio and optical redshift limits, in particular
taking into account the variable flux limit of the LoTSS survey as a function of
sky location \citep{Shimwell2018}.

For each source, $V_i$ was calculated as follows. First, the redshift range $z <
0.3$ was divided into narrow redshift slices (in practice, 3000 slices of width
$\Delta z = 0.0001$ were used). The radio luminosity of the source in question
was then used to calculate the flux density that the source would have at the
mid-point of each redshift slice, assuming a spectral index of 0.7. The LoTSS
rms maps were used to calculate the sky area over which a source of that flux
density could be detected above the 5$\sigma$ limit \citep[see][]{Shimwell2018},
and hence the volume available to that source within the redshift slice.
Finally, the optical magnitude limits of the SDSS main galaxy selection ($14.5 <
r < 17.77$) were used to determine upper and lower redshift limits at which the
source could be included in the SDSS sample (if these magnitude limits lay
outside the sample selection limits of $z_{\rm min}=0.01$ and $z_{\rm max} =
0.3$ then the latter were used instead). The available volume in all redshift
slices between those two limits was then summed. The radio luminosity functions
were then derived by summing $1/V_i$ across all sources, and Poisson statistics
were used to estimate the uncertainties.  It should be emphasised that at some
luminosities the formal statistical uncertainties are small and are likely to be
underestimates, and systematic errors such as AGN/SF classification dominated
the error budget.  No correction is made for incompleteness in the LoTSS survey,
but as shown by \citet{Shimwell2018} this is small above the 5$\sigma$ threshold
(at least for point sources, which the low-luminosity radio AGN are likely to
be). It may possibly contribute to the slight downturn in the faintest
luminosity bin.

The derived radio luminosity functions of AGN and SF galaxies are provided
separately in Table~\ref{tab:lumfunc} and shown in Figure~\ref{fig:lumfunc}.
For comparison, the local 1.4\,GHz luminosity functions of radio AGN and SF
galaxies are overlaid, converted to 150\,MHz by adjusting the break luminosity
using the spectral index of 0.7. The plotted line for the radio AGN is the
parameterisation of \citet{Heckman2014}, which averages over a wide selection of
previous 1.4 GHz radio luminosity function determinations
\citep{Machalski2000,Sadler2002,Best2005a,Mauch2007,Best2012}, while that of SF
galaxies uses the parameterisation of \citet{Mauch2007}. As can be seen, the
agreement between the 1.4\,GHz and 150\,MHz radio luminosity functions is good
\citep[as is that with earlier 150\,MHz luminosity functions in the H-ATLAS
field by][]{Hardcastle2016}. The slight offset in the SF galaxies can either be
explained by the use of a different spectral index (the agreement above the
break is almost perfect if $\alpha=0.6$ is used instead), or be caused by
slightly different redshift distributions of the two samples, given the strong
cosmological evolution of this population. These results give confidence in the
robustness of the separation of SF galaxies from AGN, which is important for the
analysis in the subsequent sections of the paper.

The luminosity function of radio AGN continues to increase with the same
power-law slope down to the lowest flux densities probed by LoTSS. As argued by
\citet{Mauch2007}, this cannot continue indefinitely without the integrated
space density of radio AGN exceeding the space density of massive galaxies which
are believed to host these objects. \citet{Mauch2007} calculated a limit of
$L_{\rm 1.4\,GHz} \approx 10^{19.5}$\,W\,Hz$^{-1}$ below which the radio
luminosity function must turn down if their hosts are all brighter than $L^*$
ellipticals, where $L^*$ is the break of the optical luminosity function, and
\citet{Cattaneo2009b} derived a similar value assuming that the host galaxies
require a black hole more massive than $10^6\,\mathrm{M_{\odot}}$. This limit
corresponds to around $10^{20.2}$\,W\,Hz$^{-1}$ at 150\,MHz, less than an order
of magnitude below the faintest luminosities probed by LoTSS. This issue is
investigated in more detail in the following sections, by breaking down the
radio-AGN prevalence by both (stellar or black hole) mass and radio luminosity
separately.

\begin{figure}
   \centering
\includegraphics[width=\linewidth]{lumfuncs_lotss_v2.pdf}
   \caption{ Local 150\,MHz luminosity functions of AGN (red circles) and SF
galaxies (blue diamonds) separately. Also shown for comparison (black dashed
lines) are the local 1.4\,GHz luminosity function for radio AGN \citep[using the
parameterisation of][]{Heckman2014} and SF galaxies \citep[using the
parameterisation of][]{Mauch2007}, converted to 150\,MHz by scaling the break
frequency with a spectral index of 0.7.}
   \label{fig:lumfunc}
\end{figure}

\begin{table}
\caption{Local 150\,MHz luminosity functions of AGN and SF galaxies separately.
The columns show the range of 150\,MHz radio luminosities considered in each
bin, the total number of radio AGN in that bin, the derived space density of
these (in units of number per log$_{10} L$ per Mpc$^3$), the total number of
SFGs and their space density. Uncertainties are statistical Poissonian
uncertainties only.}
  \label{tab:lumfunc}
\begin{tabular}{crcrc}
\hline
&
\multicolumn{2}{c}{Radio AGN} &
\multicolumn{2}{c}{Star-forming galaxies} \\
${\rm log}_{10} L_{\rm 150\,MHz}$ &
N & ${\rm log}_{10} \rho$ &
N & ${\rm log}_{10} \rho$ \\
{\footnotesize(W Hz$^{-1}$)} &
& {\footnotesize (log$_{10}L)^{-1}$Mpc$^{-3}$} &
& {\footnotesize (log$_{10}L)^{-1}$Mpc$^{-3}$} \\
\hline
\hline
 21.10--21.40 &  12 & $-3.75^{+0.12}_{-0.17}$ &  120 & $-2.81^{+0.04}_{-0.04}$
\\
 21.40--21.70 &  40 & $-3.69^{+0.06}_{-0.07}$ &  353 & $-2.72^{+0.02}_{-0.02}$
\\
 21.70--22.00 & 105 & $-3.73^{+0.04}_{-0.04}$ &  811 & $-2.69^{+0.01}_{-0.01}$
\\
 22.00--22.30 & 171 & $-3.93^{+0.03}_{-0.03}$ & 1459 & $-2.73^{+0.01}_{-0.01}$
\\
 22.30--22.60 & 268 & $-4.03^{+0.03}_{-0.03}$ & 1935 & $-2.86^{+0.01}_{-0.01}$
\\
 22.60--22.90 & 332 & $-4.13^{+0.03}_{-0.03}$ & 1896 & $-3.10^{+0.01}_{-0.01}$
\\
 22.90--23.20 & 276 & $-4.38^{+0.03}_{-0.03}$ & 1114 & $-3.52^{+0.01}_{-0.01}$
\\
 23.20--23.50 & 251 & $-4.47^{+0.03}_{-0.03}$ &  487 & $-4.03^{+0.02}_{-0.02}$
\\
 23.50--23.80 & 151 & $-4.65^{+0.05}_{-0.06}$ &  165 & $-4.54^{+0.04}_{-0.05}$
\\
 23.80--24.10 & 118 & $-4.78^{+0.05}_{-0.06}$ &   38 & $-5.37^{+0.07}_{-0.09}$
\\
 24.10--24.40 &  81 & $-5.01^{+0.05}_{-0.06}$ &   14 & $-5.76^{+0.10}_{-0.14}$
\\
 24.40--24.70 &  83 & $-5.06^{+0.05}_{-0.05}$ &     &                     \\
 24.70--25.00 &  75 & $-5.12^{+0.05}_{-0.06}$ &     &                     \\
 25.00--25.30 &  48 & $-5.37^{+0.06}_{-0.08}$ &     &                     \\
 25.30--25.60 &  34 & $-5.55^{+0.07}_{-0.09}$ &     &                     \\
 25.60--25.90 &  19 & $-5.92^{+0.09}_{-0.11}$ &     &                     \\
 25.90--26.20 &  14 & $-6.07^{+0.10}_{-0.14}$ &     &                     \\
 26.20--26.50 &   6 & $-6.47^{+0.15}_{-0.23}$ &     &                     \\
 26.50--26.80 &   3 & $-6.55^{+0.20}_{-0.41}$ &     &                     \\
 \hline
\end{tabular}
\end{table}

\section{Fraction of radio AGN}
\label{sec:fractions}

\begin{figure*}
   \centering
\raisebox{0.2cm}{\includegraphics[width=0.43\linewidth]
{radio-loud_fraction-stellar_mass.pdf}}
\includegraphics[width=0.55\linewidth]
{radio-loud_fraction-lum_lim-stellar_mass-comparison.pdf}
   \caption{Left panel: Fraction of galaxies that host a radio AGN above a given
luminosity limit, as a function of  stellar mass, for multiple radio luminosity
limits (shown in different colours). Right panel: The fraction of galaxies
hosting radio AGN brighter than a given radio luminosity, separated by their
stellar mass. The various stellar mass ranges are shown in different colours.
The solid lines represent the results of this (150\,MHz) study and dotted lines
show the results from \citet{Best2005b} at 1.4\,GHz (converted assuming a
spectral index of 0.7). The error bars correspond to a confidence interval
equivalent to one sigma. The fractions of radio AGN are shown only up to
$L_{\mathrm{150\,MHz}} \approx 10^{25}\,\mathrm{W\,Hz^{-1}}$ because the number
of galaxies above this limit is relatively low and render very large error bars.
The highest mass bin would reach a value compatible with 100 per cent within the
error at $L_{\mathrm{150 MHz}} \leq 10^{21.7}\, \mathrm{W\,Hz^{-1}}$ but those
points are not shown owing to the large error bars.}
   \label{fig:fractions_stellar}
\end{figure*}

It is well-established that there is a strong increase in the prevalence of
radio-AGN activity with the mass of the galaxy \citep[e.g.][]{Best2005b,
Mauch2007, Sabater2013}. The left panel of Fig.~\ref{fig:fractions_stellar}
shows the fraction of the SDSS galaxies that host a radio AGN above a given
luminosity limit, as a function of the stellar mass of the galaxy; this is shown
for different cut-offs on the radio luminosity, from $L_{\mathrm{150\,MHz}} \geq
10^{21}\,\mathrm{W\,Hz^{-1}}$ to $L_{\mathrm{150\,MHz}} \geq
10^{25}\,\mathrm{W\,Hz^{-1}}$ in increments of 0.5 dex. For each limit, the
fraction is computed by considering only the galaxies that could be detected
above the given luminosity limit, with a flux density of five times the rms
noise level. The error bars were computed using the Agresti-Coull binomial
proportion confidence interval \citep{Agresti1998} with the confidence interval
covering 68 per cent of the probability (roughly equivalent to 1 sigma in a
normal distribution).  The results show the expected increase in the prevalence
of radio AGN with stellar mass, as seen in previous studies at high
luminosities, but are able to extend this down to lower radio luminosities.
Remarkably, for limits of $L_{\mathrm{150\,MHz}} \geq
10^{21}\,\mathrm{W\,Hz^{-1}}$, the fraction of galaxies hosting radio AGN
reaches 100 per cent for stellar masses above $10^{11}$\,M$_{\odot}$. At limits
below or equal to $L_{\mathrm{150\,MHz}} \geq 10^{22}\,\mathrm{W\,Hz^{-1}}$, the
100 per cent fraction is reached for masses above $10^{12}$\,M$_{\odot}$. This
is consistent with the results of \citet{Brown2011}, who studied a much smaller
sample of nearby very massive galaxies and also found near ubiquity of radio
emission from either AGN or SF activity.

The right panel of Fig.~\ref{fig:fractions_stellar} shows the same results in an
alternative way. The fraction of galaxies hosting radio AGN is presented with
respect to the luminosity limit for multiple strata of stellar mass. For
comparison, this figure also shows the results of \citet{Best2005b} at higher
luminosities at 1.4\,GHz, converted using a spectral index of 0.7. The results
agree in the overlapping luminosities, but the current analysis extends almost
two orders of magnitude lower in terms of radio luminosity. The new data discard
the flattening towards lower luminosity limits that was hinted at in
\citet{Best2005b} and \citet{Mauch2007}, instead indicating that the prevalence
reaches 100 per cent by the lowest luminosities, at least for the higher mass
ranges.

\begin{figure*}
   \centering
\raisebox{0.2cm}{\includegraphics[width=0.43\linewidth]
{radio-loud_fraction-bh_mass.pdf}}
\includegraphics[width=0.55\linewidth]
{radio-loud_fraction-lum_lim-bh_mass-comparison.pdf}
   \caption{Left panel: Fraction of galaxies that host a radio AGN above a given
radio luminosity, as a function of the black hole mass, for various radio
luminosity limits (shown in different colours). Right panel: The fraction of
galaxies hosting radio AGN brighter than a given radio luminosity, separated by
their black hole mass. Colours, lines, and errors bars as in
Figure~\ref{fig:fractions_stellar}. In the left panel, the line corresponding to
$L_{\mathrm{150\,MHz}} \leq 10^{21}\,\mathrm{W\,Hz^{-1}}$ is omitted from the
plot owing to its large error bars.}
   \label{fig:fractions_bh}
\end{figure*}

Fig.~\ref{fig:fractions_bh} considers the prevalence of radio-AGN activity in
relation to the mass of the black hole; it is similar to
Fig.~\ref{fig:fractions_stellar} but for black hole masses instead of stellar
masses. In the left panel of Fig.~\ref{fig:fractions_bh} an increase in the
fraction of radio AGN is seen with respect to the black hole mass. As with
previous 1.4\,GHz studies, this has a shallower slope than found for stellar
mass. Furthermore, in this case the slope appears to flatten at the highest
black hole masses for each luminosity bin and the maximum fraction reaches at
most 30 to 50 per cent. The right panel of Fig.~\ref{fig:fractions_bh} compares
the results against the 1.4\,GHz data. For this comparison, the data of
\citet{Best2005b} were re-analysed using new black hole masses derived using the
velocity dispersion to black hole mass relation adopted in this paper (see
Sect.~\ref{sec:sample_data}; this is different from the relation that
\citeauthor{Best2005b} originally used). Once again, the agreement is excellent,
and in this case the flattening persists and the fraction never reaches 100 per
cent even for the highest black hole masses.

\begin{figure*}
   \centering
\includegraphics[width=0.45\linewidth]
{rl_fraction-bh_mass-mass_bins-llim_22_5.pdf}
\includegraphics[width=0.45\linewidth]
{rl_fraction-mass-bh_mass_bins-llim_22_5.pdf}
   \caption{Left panel: Fraction of galaxies that host a radio AGN with
$L_{\mathrm{150\,MHz}} \geq 10^{22.5}\,\mathrm{W\,Hz^{-1}}$, as a function of
the black hole mass, for multiple stellar mass bins (shown in different
colours). Right panel: The fraction of galaxies hosting radio-AGN with
$L_{\mathrm{150\,MHz}} \geq 10^{22.5}\, \mathrm{W\,Hz^{-1}}$, as a function of
the stellar mass, separated by their black hole mass (shown in different
colours).}
   \label{fig:mass_mass_bh}
\end{figure*}

The stronger dependence of the radio-AGN fraction on stellar mass than on black
hole mass found in Figs.~\ref{fig:fractions_stellar}~and~\ref{fig:fractions_bh}
suggests that stellar mass is a more important factor in triggering radio-AGN
activity. However, there is a strong correlation between black hole mass and
stellar mass \citep[e.g.][]{Reines2015} that must be considered before reaching
this conclusion. There is also a consideration that the black hole masses have
larger uncertainties on their measurements because of the underlying scatter in
the relationship between black hole mass and velocity dispersion from which they
are estimated. In order to disentangle the effect of black hole mass and stellar
mass, the fraction of radio AGN with respect to the black hole mass was computed
in  bins of different stellar masses (left panel of Fig.~\ref{fig:mass_mass_bh})
and the fraction with respect to the stellar mass was computed in bins of
different black hole masses (right panel of Fig.~\ref{fig:mass_mass_bh}) for
galaxies with $L_{\mathrm{150\,MHz}} \geq 10^{22.5}\,\mathrm{W\,Hz^{-1}}$. This
threshold in flux density maximises the number of galaxies available for the
stratified study, which minimises the size of the error bars and scatter of the
radio-AGN fractions, but consistent results are found for other radio luminosity
limits. The error bars and fractions are computed as in
Figs.~\ref{fig:fractions_stellar}~and~\ref{fig:fractions_bh}. From
Fig.~\ref{fig:mass_mass_bh} it is clear that the fraction of radio AGN is mainly
driven by the stellar mass. The black hole mass also has some residual effect
but minimal in comparison with the effect of the stellar mass; indeed it is
plausible that the remaining trend with black hole mass is simply a result of a
residual correlation between black hole and stellar mass within the narrow
stellar mass bin. The dominance of the stellar mass dependence points at an
external trigger, rather than the properties of the black hole itself being the
key driver. It is consistent with the properties of the gas that fuels the AGN
being closely linked to the stellar mass of the galaxy
\citep[e.g.][]{Sabater2013}, perhaps through the strong relationship between
stellar mass and halo mass.

It is apparent in the left hand panels of both Fig.~\ref{fig:fractions_stellar}
and Fig.~\ref{fig:fractions_bh} that there is a visible reduction in the slope
of the $f_{\rm rad}$ versus $M_*$ relation from higher to lower luminosity
limits. \citet{Janssen2012} split the radio-AGN population into high- and
low-excitation sources, which correspond broadly to those fuelled at high
Eddington rates by the accretion of cold gas in SFGs, and those fuelled at low
rates by the accretion of gas from cooling hot haloes (e.g.\
\citealt{Hardcastle2007}; see discussion in \citealt{Heckman2014}).
\citeauthor{Janssen2012} showed that the high-excitation population of radio AGN
presents a more moderate dependency with the mass than low-excitation radio-AGN,
and are more dominant at lower masses. The flattening in
Fig.~\ref{fig:fractions_stellar} may well be related to this, as the lower radio
luminosities typically probe lower stellar masses.

\section{Distribution of Eddington-scaled accretion rates}
\label{sec:accretion_rates}

The distribution of Eddington-scaled accretion rates of AGN has been previously
studied by \citet{Kauffmann2009} and \citet{Best2012}.
\citeauthor{Kauffmann2009} examined the accretion rates of emission-line
selected AGN, deriving radiative AGN luminosities by scaling from the [O{\sc
iii}]~5007 emission line. They found that red (quenched) galaxies followed a
power-law distribution of Eddington-scaled accretion rates, rising towards lower
accretion rates down to their lower observable limit of $L/L_{\mathrm{Edd}} \sim
10^{-3}$. \citeauthor{Best2012} studied radio-selected AGN and derived
Eddington-scaled luminosities by summing the radiative luminosity (scaled from
[O{\sc iii}]~5007) and the jet mechanical luminosity (from the radio luminosity;
see below). They also found that the distribution rose strongly down to
$L/L_{\mathrm{Edd}} \sim 10^{-3}$ for the bulk of the radio population and that
this was dominated by the jet mechanical luminosity except for the small
population of strong emission line radio sources, which mostly occurred at high
Eddington ratios \citep[see also discussion in][]{Heckman2014}.

With the deeper LoTSS DR1 data it is possible to reach much lower limits in the
Eddington ratio distribution. In this paper, this is only considered for the
high stellar mass population (masses higher than $10^{11}$\,M$_{\odot}$).
Consideration of just this population has two advantages: First, at these masses
it was shown in Sec.~\ref{sec:fractions} that essentially all galaxies host a
radio AGN, so the full Eddington rate distribution should be recoverable;
second, at these masses the sample is expected to be dominated by the jet-mode
AGN \citep[e.g.][]{Janssen2012}, meaning that a homogeneous population of
sources is probed and that the Eddington ratio can be estimated using just the
jet mechanical power.

The mechanical power of the radio jets in a given radio source would ideally be
inferred directly from the properties of the radio source, by comparison with
radio source evolution models \citep[see, for example,][]{Hardcastle2018}.
However, such inference requires knowledge of the environments of the radio
sources, which is not available for the current sample. Furthermore, radio
source evolution models are generally most accurate for \citet{Fanaroff1974}
Class II (FRII) sources, whereas at the low radio luminosities probed in this
work most of the sources are either of FRI class or are compact. A common
alternative approach for estimating the jet mechanical energy is therefore to
scale directly from the radio luminosity. Such a conversion does not capture the
intrinsic physical variations for individual sources; the radio luminosity of a
radio source evolves throughout its lifetime even if the jet power remains
constant\citep[ e.g.][]{Kaiser1997}. But for population-based analyses, such as
that in this paper, the use of an average conversion factor can provide a
suitably accurate approximation.

The jet mechanical power to radio luminosity conversion is commonly estimated
from the cavities inflated by radio sources in the surrounding intergalactic or
intracluster medium, as observed in X-rays. The total mechanical energy supplied
by the radio jets is typically assumed to be 4$pV$, where $p$ is the pressure of
the surrounding medium, $V$ the volume of the cavity, and the factor 4 comes
from summing the work performed to inflate the cavities ($pV$) and the enthalpy
of the relativistic plasma in the radio lobes (3$pV$). When combined with an
estimate of the source age, for which the buoyancy timescale of the cavities is
typically used \citep[e.g.][]{Churazov2001}, this allows the mechanical power of
the jet to be estimated. This is found to correlate with the observed radio
luminosity \citep{Birzan2004,Rafferty2006,Birzan2008,Cavagnolo2010}, although
with significant scatter, as expected from the discussion above.

An alternative jet mechanical power to radio luminosity conversion was presented
by \citet{Willott1999} based on the synchrotron properties of the radio source.
For this, assumptions need to be made about the composition of the radio jet
plasma \citep[see also][]{Croston2018}, and the cut-offs of the electron energy
distribution. These are similar to the assumptions required for direct jet power
inference for individual sources.

\citet{Heckman2014} showed that the scaling relations determined from
cavity-based methods and those of \citeauthor{Willott1999} provide broadly
consistent estimates of the jet mechanical powers, at least at the
moderate-to-high radio luminosities that dominate the energetic output of the
AGN; this gives confidence that the assumptions made in each method are
understood and broadly justified. They proposed the following
population-averaged conversion between radio luminosity and jet
mechanical power:$$P_{\rm mech,cav} = 2.8 \times 10^{37} \left(\frac{L_{\rm
1.4\,GHz}}{10^{25}\,{\rm W\,Hz}^{-1}}\right)^{0.68}\,\mathrm{W}.$$

\noindent This expression is adopted in this work, modified to 150\,MHz
assuming a spectral index of 0.7.

Using this, the distribution of Eddington-scaled accretion ratios for galaxies
with masses between $10^{11}$\,M$_{\odot}$ and $10^{12}$\,M$_{\odot}$ is shown
in Fig.~\ref{fig:mech_over_ed}.  For each bin in
$L_{\mathrm{mech}}/L_{\mathrm{Edd}}$, this is calculated by considering the
number of galaxies with detections in this range of accretion rates as compared
to all the galaxies that could be detectable to that Eddington limit (with a
detection above five times the rms noise level). The errors correspond to the 95
per cent confidence interval using the Agresti-Coull binomial proportion
confidence intervals. The accretion rate distribution increases from higher to
lower accretion rates as previously determined, with this continuing down to
$L_{\mathrm{mech}}/L_{\mathrm{Edd}} \approx 10^{-5}$ when the distribution
flattens. A remaining small proportion of the galaxies have
$L_{\mathrm{mech}}/L_{\mathrm{Edd}}$ below $10^{-5.75}$, indicated by the orange
point on Fig.~\ref{fig:mech_over_ed} (which distributes these over the next four
bins), indicating that the distribution must fall below
$L_{\mathrm{mech}}/L_{\mathrm{Edd}} \sim 10^{-6}$.

\begin{figure}
   \centering
\includegraphics[width=0.86\linewidth]
{mech_over_ed.pdf}
   \caption{Distribution of Eddington-scaled accretion rate for radio AGN (solid
blue line). The error bars correspond to the 95 per cent confidence interval.
The cumulative distribution obtained from the Kaplan-Meier estimator (solid
green line) running from higher towards lower accretion rates is shown. The
orange cross represents the remaining fraction of galaxies spread over the
following 4 bins. The distribution flattens and peaks at
$L_{\mathrm{mech}}/L_{\mathrm{Edd}} \approx 10^{-5}$. A Kaplan-Meier estimator
fit finds the median of the distribution at $L_{\mathrm{mech}}/L_{\mathrm{Edd}}
= 10^{-4.98}$. We note that the conversion between radio luminosities and
mechanical power is uncertain at low luminosities
\citep[e.g.][]{Hardcastle2018}.}
   \label{fig:mech_over_ed}
\end{figure}

The distribution of Eddington-scaled accretion rates was also checked using
survival analysis. The Kaplan-Meier estimator \citep{Kaplan1958} was computed
using the \texttt{Lifelines} package \citep{python_lifelines} in
\texttt{Python}. Left-censoring was used, with detections marked as events and
non-detections as limits using the $L_{\mathrm{mech}}/L_{\mathrm{Edd}}$
corresponding to five times the rms noise limit at the position. The results
fully agree with the binned method shown before. The median of the distribution
is found at $L_{\mathrm{mech}}/L_{\mathrm{Edd}} = 10^{-4.98}$; the mean of the
distribution is at $L_{\mathrm{mech}}/L_{\mathrm{Edd}} = 10^{-3.53}$.

If all the galaxies belong to the same population the relative fractions can be
directly linked to the time spent by the AGN at each accretion rate (duty
cycle). It is interesting to note, therefore, that although the distribution
peaks (i.e.\ galaxies spend most of their time) at relatively low accretion
rates, the bulk of the time-averaged energy output for these galaxies is
produced during the relatively short times corresponding to the higher
Eddington-scaled accretion rates. The Kaplan-Meier distribution can be weighted
by the energy output corresponding to each $L_{\mathrm{mech}}/L_{\mathrm{Edd}}$
and, integrating along this distribution, it is possible to find the average
energy output corresponding to a given $L_{\mathrm{mech}}/L_{\mathrm{Edd}}$. In
this way, it is found that 50 per cent of the energy output is emitted when
$L_{\mathrm{mech}}/L_{\mathrm{Edd}} \ge -2.5$ ; the other 50 per cent is emitted
at accretion rates below this value. Using the original distribution, the
average time spent by a radio AGN at $L_{\mathrm{mech}}/L_{\mathrm{Edd}} \geq
-2.5$ is only $\approx$ 1.7 per cent of the total. Hence, 50 per cent of the
energy is released during the $\le 2$ per cent of the time spent at the highest
accretion rates.

Considering that the distribution covers the full range of accretion rates, the
total time-averaged mechanical energy output for high-mass radio AGN can be
estimated. To do that, the mean accretion rate was computed from the mean
Eddington-scaled value using the typical mass for the black holes in the
selected stellar mass range: the median black hole mass of
$10^{8.3}\,\mathrm{M_{\odot}}$ corresponds to $L_{\mathrm{Edd}} =
10^{39.5}\,\mathrm{W}$, and this $L_{\mathrm{mech}}$ is expected to have an
average value of $\approx 10^{36}\,\mathrm{W}$ ($10^{43}\,\mathrm{erg\,s^{-1}}$)
through the lifetime and population of radio AGN. This result is broadly in line
with the findings of \citet{Best2006}

It is interesting to compare this average AGN heating rate against the cooling
radiation losses of the X-ray haloes surrounding these galaxies.
\citet{Kim2015}, and references therein, consider scaling relations between
X-ray luminosity and K-band luminosity. Considering the median stellar mass of
the galaxies in the mass range selected above ($10^{11.21}\,
\mathrm{M_{\odot}}$) and assuming that for these old ellipticals the K-band
luminosity can be used as a proxy of the stellar mass (with a mass-to-light
ratio of order unity), this AGN heating rate is significantly above the X-ray
luminosity expected for individual ellipticals. Instead, it is more similar to
the values found by \citeauthor{Kim2015} for the central galaxies of groups and
clusters, and of the same order of magnitude of that associated with the
inflation of bubbles/cavities in clusters \citep[e.g.][]{McNamara2007,
Diehl2008}. This is consistent with recurrent radio-AGN activity in these
high-mass galaxies providing sufficient energy to control cooling in their
surrounding groups and clusters and providing the feedback that maintains the
host galaxies as quenched, as expected \citep[see also][]{Hardcastle2018}.

\section{Summary and conclusions}
\label{sec:conclusions}

The LoTSS DR1 and the SDSS main spectral galaxy sample have been combined to
study the prevalence of radio AGN in the local Universe ($0.01 \leq z \leq
0.3$). There are 33504 SDSS galaxies in the LoTSS area and 32 per cent of these
are detected by LoTSS. The main results of this study are as follows:

\begin{itemize}
\item The method to classify and separate radio AGN from SFGs has been adapted
to the 150 MHz frequency of LoTSS and improved by both the inclusion of WISE
colours and the calibration of diagnostic lines against the well-studied sample
of \citet{Gurkan2018}. A total of 2121 radio AGN were identified.

\item The distribution of 150\,MHz to 1.4\,GHz spectral indices was obtained. A
median spectral index of 0.63, in line with the canonical value of 0.7, is found
at high flux densities ($S_{\rm 1.4\,GHz} > 20$\,mJy). The median spectral index
decreases at lower flux densities, but simulations indicate that this may be
entirely driven by selection effects whereby steeper spectrum sources are missed
in the shallower high-frequency data sets.

\item The local radio luminosity function at 150\,MHz has been derived for radio
AGN and SFGs separately down to lower luminosities than has previously been
possible. The radio-AGN luminosity function agrees well with previous
determinations of the luminosity function at 1.4\,GHz, which gives confidence in
the robustness of the AGN/SF separation method.

\item The prevalence of radio AGN, as selected at 150\,MHz, has been studied as
a function of the stellar and black hole masses. As previously seen at high
luminosity at higher frequencies, the fraction of galaxies hosting radio AGN
rises strongly with mass. This relation is seen to be flatter at lower
luminosities and lower stellar masses.

\item Stellar mass is a stronger driver of the fraction of AGN than the black
hole mass. Indeed, once the correlation between black hole and stellar mass is
accounted for, the radio-AGN fraction still rises strongly with stellar mass (at
fixed black hole mass) but shows little or no dependence on black hole mass (at
fixed stellar mass). This is indicative of the stellar mass being better linked
to the properties of the gas that fuels the AGN and drives the activity.

\item Remarkably, 100 per cent of galaxies with  masses higher than
$10^{11}\,\mathrm{M_{\odot}}$ host radio AGN when limits below or equal to
$L_{\mathrm{150\,MHz}} \geq 10^{21}\,\mathrm{W\,Hz^{-1}}$ are considered. This
suggests that the most massive galaxies are always switched on, even if it is at
relatively low radio powers.

\item The full distribution of Eddington-scaled accretion rates was derived for
the most massive galaxies (stellar masses between $10^{11}$\,M$_{\odot}$ and
$10^{12}$\,M$_{\odot}$). This is shown to rise with decreasing Eddington
fraction down to $L_{\mathrm{mech}}/L_{\mathrm{Edd}} \sim 10^{-5}$ and to drop
sharply below $L_{\mathrm{mech}}/L_{\mathrm{Edd}} \sim 10^{-6}$. However, the
bulk of the energy output is produced during the relatively short times that AGN
are powering jets close to their maximum powers.
\end{itemize}

The use of LoTSS has permitted the exploration of a wider parameter space than
previous studies. The results are very much in line with previous work, but the
extension to lower luminosities has greatly extended the scope of that work. The
discovery that the most massive galaxies are always on as a radio source at the
luminosity levels that LoTSS reaches and that stellar mass appears to be a more
important driver of activity than black hole mass, both indicate that the
radio-AGN activity is controlled by the fuel supply to the radio AGN, which
connects more closely to the stellar mass. This fits the popular picture whereby
this fuel is connected to cooling of hot gas within the dark matter halo, and
the radio-AGN activity is instrumental in maintaining the host galaxies as old,
red, and dead.

Considering that the DR1 of LoTSS covers only 2 per cent of the final LoTSS
area, an improvement of at least one order of magnitude in the volume of LoTSS
data can be expected in the near future (next two to three years). These larger
samples will allow a more detailed exploration of additional aspects of the
triggering of radio AGN. In particular, it will be very interesting to directly
test the picture above by separating the radio-AGN luminosity distribution by
additional parameters such as environment \citep[see][]{Sabater2013,Sabater2015}
and mass to separate the roles of these two critical parameters. Furthermore,
the forthcoming availability of extensive spectroscopic data for LoTSS sources
out to higher redshifts, through the WEAVE-LOFAR survey \citep[][]{Smith2016},
will allow cosmic evolution to be examined. These topics will constitute the
focus of future detailed studies.

\begin{acknowledgements}
We would like to thank the anonymous referee for useful comments that
improved this paper.

This paper is based (in part) on data obtained with the International LOFAR
Telescope (ILT) under project codes LC2\_038 and LC3\_008. LOFAR
\citep{vanHaarlem2013} is the LOw Frequency ARray designed and constructed by
ASTRON. It has observing, data processing, and data storage facilities in
several countries, which are owned by various parties (each with their own
funding sources) and are collectively operated by the ILT foundation
under
a joint scientific policy. The ILT resources have benefited from the following
recent major funding sources: CNRS-INSU, Observatoire de Paris and Université
d'Orléans, France; BMBF, MIWF-NRW, MPG, Germany; Science Foundation Ireland
(SFI), Department of Business, Enterprise and Innovation (DBEI), Ireland; NWO,
The Netherlands; The Science and Technology Facilities Council, UK.

Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan
Foundation, the Participating Institutions, the National Science Foundation, the
U.S. Department of Energy, the National Aeronautics and Space Administration,
the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education
Funding Council for England. The SDSS Web Site is http://www.sdss.org/.
The SDSS is managed by the Astrophysical Research Consortium for the
Participating Institutions. The Participating Institutions are the American
Museum of Natural History, Astrophysical Institute Potsdam, University of Basel,
University of Cambridge, Case Western Reserve University, University of Chicago,
Drexel University, Fermilab, the Institute for Advanced Study, the Japan
Participation Group, Johns Hopkins University, the Joint Institute for Nuclear
Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the
Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos
National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the
Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio
State University, University of Pittsburgh, University of Portsmouth, Princeton
University, the United States Naval Observatory, and the University of
Washington.

This work was carried out in part on the Dutch national e-infrastructure with
the support of SURF Cooperative through e-infra grants 160022 \& 160152.

PNB, JS, and RKC are grateful for support from the UK Science and Technology
Facilities Council (STFC) via grant ST/M001229/1 and ST/R000972/1.

KJD and AS acknowledge support from the European Research Council under the
European Unions Seventh Framework Programme (FP/2007-2013)/ERC Advanced Grant
NEWCLUSTERS-321271

MJH and WLW acknowledge support from the UK Science and Technology Facilities
Council
ST/M001008/1.

JHC acknowledges support from the STFC under grants ST/R00109X/1 and
ST/R000794/1.

APM would like to acknowledge the support from the NWO/DOME/IBM programme "Big
Bang Big Data: Innovating ICT as a Driver For Astronomy", project \#628.002.001.

GG acknowledges the CSIRO OCE Postdoctoral Fellowship.

F.d.G. is supported by the VENI research programme with project number 1808,
which is financed by the Netherlands Organisation for Scientific Research (NWO).

LKM acknowledges the support of the Oxford Hintze Centre for Astrophysical
Surveys, which is funded through generous support from the Hintze Family
Charitable Foundation. This publication arises from research partly funded by
the John Fell Oxford University Press (OUP) Research Fund.

IP acknowledges support from INAF under PRIN SKA/CTA 'FORECaST'

SM acknowledges funding through the Irish Research Council New Foundations
scheme and the Irish Research Council Postgraduate Scholarship scheme.

This research made use of \textsc{Astropy}, a community-developed core Python
package for
Astronomy (Astropy Collaboration, \citeyear{astropy}, \citeyear{astropy2});
\textsc{Ipython}
\citep{IPython}; \textsc{matplotlib} \citep{matplotlib}; \textsc{numpy}
\citep{numpy}; \textsc{pandas} \citep{pandas}; \textsc{scipy}
\citep{scipy}, \textsc{TOPCAT} \citep{TOPCAT}, and \textsc{KERN} suite
\citep{Kern}.

\end{acknowledgements}

\bibliographystyle{aa}

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text:
Home World-News Macron promises a French 'renaissance' at inauguration – Irish Times
Macron promises a French 'renaissance' at inauguration – Irish Times
The newly elected French president, Emmanuel Macron, promised to restore France's battered self-confidence in a brief inaugural address at the Élysée Palace on Sunday.
"I will convince our compatriots that the power of France is not in decline, but that we are on the threshold of an extraordinary renaissance," he said.
The world's eyes are on France, Mr Macron said repeatedly. Alluding to the Brexit vote, Donald Trump and Marine Le Pen, the extreme right-wing leader he defeated on May 7th, he said, "Everywhere, people were asking if the French would fall back on an illusory past . . . turn their back on the Enlightenment, or if . . . they were going to embrace the future and give themselves a new élan."
Mr Macron promised to "refound" and "relaunch" the EU.
"We will need a more efficient, more democratic, more political Europe, " he said. "I shall work for that."
At age 39, Mr Macron is France's youngest president. "The confidence that the French have shown me fills me with immense energy," he said. "I'll be at work from this evening." He is expected to name his prime minister on Monday, and will travel to Berlin to see German chancellor Angela Merkel on Monday afternoon.
Mr Macron wants his French "renaissance" to combine liberal, free market economics with preservation of the welfare state.
'Liberating and protecting'
"Liberating and protecting, that is how our future will be made," he said later at Paris town hall.
Though Mr Macron often laments political bickering and the paralysis of "the system", he refrains from casting individual blame. He paid homage to his last seven predecessors.
Gen Charles de Gaulle, who founded the Fifth Republic in 1958, "gave France back her role in the concert of nations," Mr Macron said. Georges Pompidou made France an industrial power. Valéry Giscard d'Estaing modernised society. François Mitterrand reconciled French and European dreams. Jacques Chirac "knew how to say No to the warmongers" when George W Bush invaded Iraq. Nicolas Sarkozy devoted great energy to resolving the 2008 financial crisis.
That left François Hollande, who had made Mr Macron a presidential advisor, then minister for the economy. On Sunday, all talk of betrayal was forgotten. Mr Macron said the 2015 Paris accord on climate change made Mr Hollande a "forerunner" and said he "protected the French in a world struck by terrorism".
Thanks to Emmanuel Macron, Mr Hollande will not go down in history as the president who turned France over to the Front National. Mr Hollande waited on the steps of the Élysée as Mr Macron walked up the red carpet for his inauguration. Their eyes locked, and Mr Hollande broke into a smile.
Transmitted codes
During an hour-long private meeting, Mr Hollande is believed to have transmitted the codes for France's force de frappe, and briefed his successor on the war in Syria.
Mr Macron walked Mr Hollande to his waiting car, and joined in applause for the departing president. Like François Mitterrand in 1995, Mr Hollande drove to the socialist party headquarters, where he claimed to have left the country in better condition than when he took office.
There was a sense of a generational putsch, of power having skipped politicians in their 40s and 50s. Mr Macron referred repeatedly to future generations. "We shall fight not only for this generation, but for generations to come . . . We must build the world that our young people deserve."
Though he is not yet 40, Mr Macron has more experience of the Élysée and EU summits than Mr Hollande had when he took office. The new president learned from Mr Sarkozy, the "hyper-president" who lost his bid for re-election, and from Mr Hollande, the would-be "normal president" who could not even stand for re-election.
French president Emmanuel Macron kisses his wife Brigitte Trogneux during the handover ceremony in Paris on Sunday. Photograph: Philippe Wojazer/Reuters
"I don't believe in a 'normal' president," Mr Macron said last October. "That's not what the French want. On the contrary, the concept destabilises them, makes them feel insecure."
The president of the constitutional council, Laurent Fabius, called on Mr Macron to "calm anger, heal wounds, lift doubts, trace a path and embody hope. Your success will be the success of France."
"I know that the French expect a great deal of me," Mr Macron said. "They are right . . . Nothing will weaken my determination."
During the campaign, Mr Macron's critics questioned whether such a young man was capable of leading France's armed forces. In the police and military, support ran high for Ms Le Pen. France is still under a state of emergency.
That may explain the martial feeling of much of the ceremony. In the Élysée garden, Mr Macron stood to attention before army, air force and navy contingents, while the Republican Guard band played the Marseillaise. In the afternoon, he visited wounded soldiers at the Percy military hospital. Unlike his predecessors, Mr Macron made his inaugural drive up the Champs-Élysées standing in a military command car.
After laying a wreath at the tomb of the unknown soldier, Mr Macron shook hands with the crowds around the Arc de Triomphe, where a man with links to the extreme right tried to assassinate president Jacques Chirac on Bastille Day 2002.
The walkabout on the Champs-Élysées and a second session outside Paris town hall in the afternoon must have given Mr Macron's security detail cold sweats.
"You can't listen to the security guys," Macron told staff who tried to dissuade him from confronting angry factory workers between the two rounds of the presidential election, as shown in a TF1 documentary last week. "I'll never be safe, because that's the way the country is now. You have to take the risk," Mr Macron said. "Because if you listen to the security guys, you end up like Hollande. You may be safe, but you're dead."
At his last engagement of the day, Mr Macron spoke to 800 French officials and foreign diplomats at Paris town hall. He made peace with the socialist mayor Anne Hidalgo, who opposed last year's labour law and who supported the socialist candidate Benoît Hamon against Mr Macron. The new president agreed to join Ms Hidalgo's campaign to hold the 2024 Olympics in Paris.
Paris is "the beating heart of our country", Mr Macron said, "A wounded heart, that has paid a heavy price to terrorism."
He praised Ms Hidalgo's generous response to the refugee crisis, and for making Paris "a major actor against climate change".
Implicitly recognising the hold of the far right over much of rural France, Mr Macron said it wasn't enough to recognise successful areas of the country. He had to "enable every town, every department, every region to become successful too. This reconciliation of all France will be the heart of my mission."
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Students interested in attending the second week of this year's Young Leaders Institute must submit the online application by Friday, May 17.
It is free to apply; however, a nonrefundable $50 participation fee (per week) will be due upon acceptance into the Institute. Scholarships are available for students with demonstrated need. Spots are limited, so students are encouraged to apply early to increase their chances of admission.
Throughout the week, students will learn the value of collaboration as they work in teams, researching, debating, and presenting solutions for the challenges facing our world. They will be mentored throughout the week by Asia Society staff and local experts in their respective fields. All content provided will help students form arguments when taking the side of an issue and be invaluable when presenting policy recommendations on the Institute's last day.
Over 120 students have benefited from the Young Leaders Institute in 2016, 2017, and 2018. The Institute's fourth year will include a repeat of the Middle East session and new sessions on U.S.-China relations and on energy and security, respectively. Asia Society staff will also arrange a field trip on one of the days of the week.
For questions, please contact Mei Qi, Business and Policy Programs Manager, at MQi@AsiaSociety.org or 713.496.9915.
Dr. Emran El-Badawi is program director and associate professor of Middle Eastern Studies at the University of Houston. He founded the Arab Studies program at UH and he has designed, implemented, and assessed degree programs in the Humanities and Sciences. These include degrees in Arab-Middle East Studies, Religious Studies, and interdisciplinary studies in Energy, Development and Sustainability, with a focus on U.S.-Middle East relations. El-Badawi has consulted for various industries, including government, law and oil & gas. He is also active in program development and fund raising. El-Badawi is also the founding executive director and treasurer of the International Qur'anic Studies Association, which is the world's first learned society of its kind. Part of IQSA's goal is to bridge the divide between scholars in the west and those in Muslim majority countries through international conferences.
El-Badawi has published in English as well as Arabic and has made dozens of national as well as international media contributins or appearances, including for The New York Times, Al-Jazeera, Forbes, Christian Science Monitor, and Association Relative à la Télévision Européenne (ARTE). His awards include honorable acclaim by the British-Kuwait Friendship Society Book Prize for his book on The Qur'an and the Aramaic Gospel Traditions. His current research projects include a book on 'Arab Liberalism' between secular nationalism and political Islam, as well as a project documenting the evolution of Eastern Church/Canon Law to Shariah Law. El-Badawi has a bachelor's degree in Computer Science from Rutgers University, and he received his Ph.D. with honors in Near Eastern Languages and Civilizations from the University of Chicago.
Tom Kolditz is the founding Director of the Ann & John Doerr Institute for New Leaders at Rice University—the most comprehensive, evidence-based, university-wide leader development program in the world. Serving all 6200 students at Rice, the Doerr Institute is among the largest leader development enterprises in Texas. Formerly teaching at the Yale School of Management, he was chairman of the Department of Behavioral Sciences and Leadership at West Point, the founding director of the West Point Leadership Center, co-founded the Thayer Leader Development Group, and built a leader development consultancy, Saxon Castle. Brigadier General Kolditz received the Distinguished Service Medal, the Army's highest award for service, and is a Fellow in the American Psychological Association. Dr. Kolditz has published in the Proceedings of the National Academy of Sciences and the World Economic Forum. In 2017, he received the prestigious Warren Bennis Award for Excellence in Leadership, a distinction shared by noteworthy leaders including Howard Schultz, Doris Kearns Goodwin, and Bill George. He is ranked #6 in Coaching by a UK professional ranking organization, Global Gurus. He holds a BA from Vanderbilt University, three master's degrees, and a PhD in psychology from the University of Missouri.
Vani Rao is the Diversity & Inclusion Director for Quanta Services. In this newly created role, she will partner with Quanta's global operating units to drive profitability through diverse practices and inclusive behaviors. Prior to joining the Quanta family, Vani worked for BP for 15 years with roles in upstream projects & engineering, continuous improvement, planning, capability and most recently in Diversity & Inclusion with accountability for BP America's employee resource groups. Vani received her Bachelor's degree in Psychology from Austin College and MBA from Columbia University. She is also certified in project management (Project Management Professional®) and Myers-Briggs Type Indicator. She is also a 2012 APAWLI (Asian Pacific American Women's Leadership Institute) Fellow, a signature leadership program for APA women. As a classically trained dancer, Vani indulges her lifelong love for Indian performing arts and education by serving on Asia Society's Advisory Board and by supporting charitable organizations in the US and India focused on upliftment. Most of her spare time is devoted to mentoring students and supporting the Dallas Cowboys.
Dr. Tabaar is currently working on two research projects: nuclear statecraft in hybrid regimes, and ideological strategies of political actors. He teaches courses on US foreign policy in the Persian Gulf, religion and politics in Iran, and Middle East politics. Dr. Tabaar has a BA in social sciences from the University of Tehran, an MA in sociology from the New School for Social Research, an MA in international relations from the University of Chicago, and a PhD in government from Georgetown University.
Mary Reed is a Social Studies teacher at Memorial High School. She earned her B.S. in 1975 from Oklahoma State University. After graduation, she taught United States History in Alief ISD and served as Social Studies Department Chairman. At Memorial High School she has taught AP World History and World Geography at the PreAP, GT, and academic levels. She has served as PTA president and Campus Improvement Team member, District Improvement Team member, and curriculum writer. Being raised on three continents has given Mary an appreciation of other cultures and a love of travel. She has experienced 52 different countries, and imparts her global curiosity to her students both inside the classroom and through leading international summer trips. She is a scholastic coach to several teams including: UIL Current Issues and Events, UIL Social Studies, World Affairs Council Academic WorldQuest Challenge, and Texas Quiz Bowl. She is also the sponsor of her high school chapter of Model United Nations. She has been a featured presenter in other school districts, at professional conferences, and for the World Affairs Council of Greater Houston. She recently presented "Korea: the Forgotten Victory" in Washington, D.C.
Topics may include bilateral trade negotiations, the countries' role in global financial institutions, and China's soft power in Asia.
Topics may include the future of renewable fuels, the importance of growing demand, and managing risk in potential geopolitical flashpoints.