11institutetext: Sapienza University of Rome, Physics Department, Piazzale
Aldo Moro 5, 00185 Rome, Italy 22institutetext: University of British
Columbia, Department of Physics and Astronomy, Vancouver, BC, V6T 1Z1, Canada
33institutetext: INFN - Sezione di Roma, Piazzale Aldo Moro 5 - I-00185, Rome,
Italy 44institutetext: INAF - Osservatorio Astronomico di Cagliari, Via della
Scienza 5 - I-09047 Selargius (CA), Italy 55institutetext: INAF - Istituto di
Radioastronomia - Via P. Gobetti, 101 - I-40129 Bologna, Italy
66institutetext: Instituto de Radioastronomia y Astrofisica, UNAM, Campus
Morelia, A.P. 3-72, C.P. 58089, Mexico 77institutetext: Instituto de
Astrofisica de Canarias, C/Via Lactea s/n, E-38205 La Laguna, Tenerife, Spain
88institutetext: Departamento de Astrofsica, Universidad de La Laguna (ULL),
E-38206 La Laguna, Tenerife, Spain 99institutetext: Infrared Processing
Analysis Center, California Institute of Technology, Pasadena, CA 91125, USA
99email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
# Study of the thermal and nonthermal emission components in M31: the Sardinia
Radio Telescope view at 6.6 GHz111Table LABEL:Tab:SRT_Sources_catalogue and
the 6.6 GHz SRT map are available in electronic form at the CDS via anonymous
ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/

S. Fatigoni 1122 F. Radiconi 1133 E.S. Battistelli 113344 M. Murgia 44 E.
Carretti 55 P. Castangia 44 R. Concu 44 P. de Bernardis 1133 J. Fritz 66 R.
Genova-Santos 7788 F. Govoni 44 F. Guidi 7788 L. Lamagna 1133 S. Masi 1133 A.
Melis 44 R. Paladini 99 F.M. Perez-Toledo 88 F. Piacentini 1133 S. Poppi 44 R.
Rebolo 7788 J.A. Rubino-Martin 7788 G. Surcis 44 A. Tarchi 44 V. Vacca 44

(Received 27 November 2020 / Accepted 20 April 2021)

###### Abstract

Context. The Andromeda galaxy is the best-known large galaxy besides our own
Milky Way. Several images and studies exist at all wavelengths from radio to
hard X-ray. Nevertheless, only a few observations are available in the
microwave range where its average radio emission reaches the minimum.

Aims. In this paper, we want to study the radio morphology of the galaxy,
decouple thermal from nonthermal emission, and extract the star formation
rate. We also aim to derive a complete catalog of radio sources for the mapped
patch of sky.

Methods. We observed the Andromeda galaxy with the Sardinia Radio Telescope at
6.6 GHz with very high sensitivity and angular resolution, and an
unprecedented sky coverage.

Results. Using new 6.6 GHz data and Effelsberg radio telescope ancillary data,
we confirm that, globally, the spectral index is $\sim 0.7-0.8$, while in the
star forming regions it decreases to $\sim 0.5$. By disentangling (gas)
thermal and nonthermal emission, we find that at 6.6 GHz, thermal emission
follows the distribution of HII regions around the ring. Nonthermal emission
within the ring appears smoother and more uniform than thermal emission
because of diffusion of the cosmic ray electrons away from their birthplaces.
This causes the magnetic fields to appear almost constant in intensity.
Furthermore, we calculated a map of the star formation rate based on the map
of thermal emission. Integrating within a radius of $R_{max}=15$ kpc, we
obtained a total star formation rate of $0.19\pm 0.01$ M⊙/yr in agreement with
previous results in the literature. Finally, we correlated our radio data with
infrared images of the Andromeda galaxy. We find an unexpectedly high
correlation between nonthermal and mid-infrared data in the central region,
with a correlation parameter $r=0.93$. Finally, by computing the logarithmic
$24_{\mu m}/21_{cm}$ ratio $q_{24\mu m}$, we find a decreasing trend with
increasing galactocentric distance and an increasing dispersion. The
logarithmic far-infrared-to-radio ratio is found to be $2.41\pm 0.04$.

###### Key Words.:

Galaxies: Individual: Messier 31; Radio continuum: galaxies, ISM; Radiation
mechanisms: nonthermal, thermal

##  1 Introduction

The Andromeda galaxy (Messier 31, M31) is the largest galaxy of the Local
Group, which includes the Milky Way, the Triangulum galaxy (M33), and several
other smaller galaxies including the satellite Messier 32 (M32) and Messier
110 (M110). Messier 31 is a large disk galaxy at a distance of 780 kpc from
Earth (Stanek & Garnavich, 1998), and has a proper redshift of z=-0.001 (de
Vaucouleurs et al., 1991), where the negative value shows that it is moving
towards the Milky Way.

The galaxy presents a large central bulge. The bulge is defined as the central
region that has an extension of $5^{\prime}$ in the optical band,
corresponding to a proper size of $1.1\,$kpc (Morton et al., 1977). The bulge
contains dilute ionized gas and a few dense star clusters. V- and I-band
Hubble Space Telescope (HST) Planetary Camera images show that the central
region seems to have two separate concentrations of light that are about 0.5”
apart (Lauer et al., 1993). One of these probably contains a dense central
object, presumably a black hole of mass $M_{BH}\simeq 10^{6}$ M⊙, while the
other might be a star cluster. There is a ring circling the bulge at a radius
of about 10 kpc, and most of the young disk stars lie in this ring or just
outside it (Haas et al., 1998; Gordon et al., 2006; Ford et al., 2013). Going
to greater distances from the center we find the HI (Braun et al., 2009) layer
that flares out to become thicker. The M31 galaxy is the brightest of the
Messier objects, with an apparent magnitude (V) of 3.4 (Gil de Paz et al.,
2007) and an apparent extension of approximately $1\times 3$ deg2 (Nilson,
1973) due to the high inclination with respect to the line of sight, which is
assumed to be of $None$ (Gordon et al., 2006) considering that $None$ means an
edge-on orientation.

Observations by the infrared Spitzer Space Telescope revealed that M31
contains about $\simeq 10^{12}$ stars (Peñarrubia et al., 2014), which is at
least twice the number of stars contained in the Milky Way, and the total mass
of the galaxy is estimated to be $M_{M31}=1.5\times 10^{12}$ M⊙ (Barmby et
al., 2006). High-resolution (FWHM=$9"$) Westerbork Synthesis Radio Telescope
(WSRT) observations of the 21 cm line by Braun et al. (2009) were used to
calculate the total neutral gas mass, which turned out to be in the range
$4.4$ to $5.5\times 10^{9}$ M⊙ (Fritz et al., 2012). CO observations by Nieten
et al. (2006) within a field of $0.5^{\circ}\times 1.8^{\circ}$ led to an
estimate of the mass of molecular gas of the order of $2.63\times 10^{8}$ M⊙
(Fritz et al., 2012). While the former observations do not include the whole
galaxy disk, they most certainly encompass the majority of the molecular gas
(about 95% of the total molecular mass, according to Nieten et al. 2006).

Because of its size and distance in the sky, M31 is an excellent laboratory
with which to further study and perhaps confirm the physical mechanisms and
properties that have so far only been studied in detail in our own Galaxy. It
also provides information about the general features of a typical disk galaxy.
The M31 galaxy has been studied at several wavelengths: Fermi LAT was used to
detect gamma-ray emission in the energy range 200 MeV - 20 GeV (Abdo et al.,
2010). NuSTAR permitted the study of hard X-ray emission allowing X-ray
binaries and neutron stars/black holes to be resolved (Wik et al., 2016). XMM,
ROSAT, and CHANDRA were used to study its X-ray emission (Stiele et al.,
2010). Ultraviolet (UV) measurements were used to study interstellar
extinction (Clayton et al., 2015) as well as the circumgalactic medium,
finding a very extended ionized medium which may overlap with a similar one
originating from our Milky Way (Lehner et al., 2020). Optical and near-
infrared (NIR) observations were used to investigate the stellar composition
(Sick et al., 2014). The IR and submillimeter (submm) emission of M31 were
studied by Fritz et al. (2012) and the HELGA collaboration. These latter
authors used data obtained from the Herschel satellite to characterize the
dust emission of M31, and to construct maps of the dust surface density, the
dust-to-gas ratio, the starlight heating intensity, and the abundance of
polycyclic aromatic hydrocarbons (PAHs).

The radio and microwave emission of M31 have been observed at high angular
resolution (FWHM $<5^{\prime}$) in the last three decades with the Effelsberg
radio telescope (Beck et al., 2020; Berkhuijsen et al., 2003, 1983), the
Westerbork Synthesis Radio Telescope (WSRT, Braun et al., 2009), and the Very
Large Array (VLA, Condon et al., 1998). Combined Effelsberg and VLA
observations at 1.46 GHz and Effelsberg data at 4.85 GHz indicate a global
nonthermal spectral index of $\alpha_{s}=1.0\pm 0.1$ (Berkhuijsen et al.,
2003), which is consistent with the value found by Battistelli et al. (2019),
who studied the overall flux density emission from radio to IR wavelengths.
Berkhuijsen et al. (2003), assuming a nonthermal spectral index equal to 1,
derived (gas-) thermal and nonthermal maps at 4.85 GHz, finding that the
thermal emission is stronger in the northern part of M31, whereas the
nonthermal emission is nearly homogeneous over the whole ring. This
homogeneity suggests that recent star formation does not lead to a local
increase in the number of relativistic electrons and/or magnetic field
strength. Strong thermal emission was also found within the central region of
the galaxy. Beck et al. (2020) used three new deep radio surveys of M31 made
with the Effelsberg radio telescope at $2.645\,GHz$, $4.85\,GHz,$ and
$8.35\,GHz$ to perform a detailed study of the spectral index maps as well as
thermal and nonthermal emission scale lengths. Beck et al. (2020), using nine
frequencies, found a total spectral index of $0.71\pm 0.02$ and a synchrotron
spectral index of $0.81\pm 0.03$; they also found that the synchrotron
emission ring is wider than the thermal emission one.

In Battistelli et al. (2019) we showed that the total integrated emission at
6.6 GHz from M31, integrated over an elliptical region of $91.5^{\prime}$ and
$59.5^{\prime}$ semi-axes, is at the level of $1.20\pm 0.06$ Jy. Comparing
different ancillary data allowed us to calculate the emission budget; we found
strong and highly significant evidence for anomalous microwave emission (AME)
at the level of 1.45${}^{+0.17}_{-0.19}$ Jy at the peaking frequency of
$\simeq$25 GHz. By decomposing the spectrum into known emission mechanisms,
such as synchrotron, free-free, thermal dust, and AME, we found that the
overall emission from M31, at frequencies below 10 GHz, is dominated by
synchrotron emission with a spectral index of 1.10${}^{+0.10}_{-0.08}$, with
subdominant free-free emission. At frequencies of $\sim$10 GHz, AME has a
similar intensity to that of synchrotron and free-free emission, overtaking
these components at frequencies between 20 GHz and 50 GHz, whereas thermal
dust emission dominates the emission budget at frequencies above 60 GHz, as
expected. For the purpose of this paper, we refer to the free-free emission as
thermal emission, and to the synchrotron emission as nonthermal emission,
assuming any other emission mechanism to be negligible at 6.6 GHz. K-band
observations with improved angular resolution (e.g., $0.9^{\prime}$ with the
Sardinia Radio Telescope at 22 GHz) will be key to disentangling AME models,
and to studying AME over an entire galaxy outside of our own with high angular
resolution (Battistelli et al. (2012), Radiconi et al. in preparation).

This paper is organized as follows. The characteristics of the observations,
the scanning strategy, and the data analysis pipeline of C-band data are
described in Sect. 2. In Sect. 3 we describe all the ancillary maps used in
this paper and the recipes that were adopted to make their geometry and
angular resolution uniform. The removal of compact sources is presented in
Sect. 4. In the same section we also compare the normalized differential count
of compact sources with theoretical predictions. In Sect. 5 we carry out an
analysis of the morphology of M31 by investigating the galaxy spectral index
map. Thermal and nonthermal maps at 6.6 GHz are presented in Sect. 6, while in
Sect. 7, starting from the thermal map, we derive a map of the star formation
rate (SFR), and compare this to what was previously found in the literature.
In Sects. 8 and 9 we analyze the correlation between radio-continuum and IR
data, studying how thermal and nonthermal emissions are related to IR
emission. We also compute the parameter $q_{FIR}$ and compare the result with
the expected value. Finally, in Sect. 10 we summarize our results.

##  2 Sardinia Radio Telescope observations of M31

###  2.1 Sardinia Radio Telescope

The Sardinia Radio Telescope (SRT 222http://www.srt.inaf.it/) is an Italian
facility for radio astronomy run by the Istituto Nazionale di Astrofisica,
INAF, which was formally inaugurated in 2013 after completion of technical
commissioning (Bolli et al., 2015). The scientific commissioning of the
telescope was carried out in the period 2012-2015 (Murgia et al., 2016;
Prandoni et al., 2017). At the beginning of 2016 the first call for single-
dish early science programs (ESPs) was issued, and the observations started on
February 1, 2016. The SRT observations of M31 at 6.6 GHz presented in this
work are part of these ESPs. The SRT is placed 35 km north of Cagliari (Lat:
$None$ 29’34”N - Long: $None$ 14’42”E) on the island of Sardinia at 600 m
above sea level. The optical system is based on a quasi-Gregorian dish
antenna, with a primary mirror of 64 m in diameter, and a secondary mirror of
9 m, which is shaped in such a way as to minimize the standing wave bouncing
between the two reflectors. Three additional mirrors increase the number of
focal positions. A key feature of the SRT is its active surface, composed of a
total of 1116 electromechanical actuators able to correct deformations induced
by gravity on the primary surface. The SRT has an angular resolution of
2.9′(FWHM) at a frequency of 6.6 GHz.

The suite of backends currently available on site includes SARDARA (SArdinia
Roach2-based Digital Architecture for Radio Astronomy; Melis et al., 2018).
The SARDARA backend is a wide-band digital backend based on
ROACH2333https://casper.ssl.berkeley.edu/wiki/ROACH2 technology, and can
divide the signal in the given bandwidth into up to 16384 channels for
spectropolarimetric observations. We used this backend for our observations.

###  2.2 C-band observation

The main goal of the observations presented in this paper is to map the whole
M31 galaxy in the C-Band down to a sensitivity of 0.4 mJy/beam in order to
create a high-sensitivity/high-angular-resolution map of M31 microwave
emission covering its full extension, limited by the confusion noise at this
frequency and angular resolution (Bondi et al., 2003; Condon, 1974). We set
the Local Oscillator at 5.9 GHz and we use the SARDARA configuration with 1.5
GHz bandwidth sampled with 16384 channels of 91.6 kHz in width. The data cube
will be analyzed in greater detail in a search for spectral line emission from
various transitions (Tarchi et al., in prep.). We set the Focus Selector
filter to select the frequency range from 6.0 to 7.2 GHz, which defines our
observing bandwidth. The central frequency of our observations is therefore
6.6 GHz. Traditionally, C-band is centered at about 5 GHz, while the band
around 6.6 GHz is known as the C-high band. For simplicity, in this paper we
refer to the 6.6 GHz data as C-band data.

We decided to cover the galaxy with a rectangle of 7.4 deg2 centered at (RA;
Dec) = (0h 42m 48s; +41∘ 16′ 48′′). To the best of our knowledge, this is the
biggest map ever produced at this frequency and angular resolution with a
single-dish radio telescope.

The map size was determined by the minimum contour that includes the galaxy,
to which we added a Map Edge of 0.25 deg in each of the two scanning
directions. The minimum contour was estimated by comparing the two maps
provided by Herschel/SPIRE at 250 $\mu$m (Fritz et al., 2012) and Planck at
857 GHz (Planck Collaboration et al., 2014); which are supposed to trace the
galaxy in its maximum extension. _Planck_ and _Herchel_ $5\sigma$ contours are
presented in the right panel of Fig. 3. The aim of adding an extra padding is
to have a sky region that is completely free from M31 emission and that we can
use to estimate the background and decouple this from the signal coming from
the galaxy. We adopted an on-the-fly map-scanning strategy, which means
orthogonal subscans along right ascension (RA) and declination (Dec) spaced by
54 arcsec (FWHM$/3$). We used a scanning speed of 6′/s. Covering the 7.4 deg2
of our map required 209 RA subscans and 161 Dec subscans. Table 1 reports the
main details of the observations.

Telescope | SRT (64m)
---|---
Receiver | C-band (5.7-7.7 GHz)
Backend | SARDARA
Spectroscopy | 16384 channels
Backend bandwidth | 1.5 GHz
Channel spacing | 91.6 kHz
Observed frequency range | 6.0-7.25 MHz
Beam size | 2.9′@ 6.6 GHz
Polarization | Full Stokes
Scanning strategy | Orthogonal raster subscans
Scanning speed | 6′/s
Sample rate | 25 spectra/s
$<T_{sys}>$ | 37 K
Map size |  $2.4\,$ deg $\times\,3.1\,$ deg $\simeq 7.4\,$deg2
Total number of scans | 22 RA + 22 Dec
Total observation time | 64h
Reached sensitivity | 0.43 mJy/beam
Table 1: Main characteristics of the C-band SRT observations on M31.

Before starting the observations, a set of daily preliminary operations were
performed. These include the configuration of the backend, setting of the
amplifier attenuation factors (to ensure linearity), secondary mirror
positioning (by focusing a bright radio source), and spectroscopic system
acquisition and pointing checks. The calibrators were observed during each
session, when M31 was over the antenna critical elevation of 85 deg, the SRT
being equipped with an Alt-azimuthal mount. The observations were completed in
$64$ h, divided in 6 days, during which we carried out $22$ complete scans of
the galaxy.

###  2.3 Data acquisition

Once collected, the data were analyzed using the Single-dish Spectral-
polarimetry Software (SCUBE, see Murgia et al. 2016), which was created
specifically for treating data acquired with the SRT. SCUBE is a package
dedicated to the reduction and analysis of single-dish data and its
combination with interferometric observations. SCUBE is written in C++ and is
developed and maintained by M. Murgia and F. Govoni; it allowed us to import
the data, extract, analyze, and flag the spectra, do the baseline subtraction,
perform calibration, and generate the final maps with different methods.

As a first step in the data reduction process we removed the backend ”birdies”
(dead channels occurring in SARDARA data) and the well-known persistent radio-
frequency interference (RFI) that affect specific spectral windows at the SRT.
These omnipresent signals were removed a priori by applying the same flag
table to all observing days. All the remaining sporadic RFI signals (e.g.
satellites or other transient ambience disturbances) were removed later in the
analysis with automated flagging methods. Overall, the amount of data erased
due to RFI accounts for about 30% of the total.

The baseline subtraction was carried out differently for the calibrators and
for the M31 scans. To estimate the baseline for calibrators we made a linear
fit over $10\%$ at the beginning and at the end of each subscan, and then
removed this linear trend from the data.

To create a baseline model for the M31 scans we used a mask to cover the part
of the map that we knew could possibly contain either galactic emission from
M31 itself or any bright background point source. The remaining part of the
field of view was considered as ”cold sky” and was used to perform a linear
fit to the baseline subscan by subscan (see Sect. 2.5).

###  2.4 Calibration

Data calibration included bandpass and flux density calibration. These were
carried out at the beginning of the map extraction process and performed
channel by channel. The bandpass calibration was necessary to set the
instrument responsivity for every channel. For this purpose we used specific
calibrators, like J0542+4951 (3C147), with stable and known spectra.

We applied the gain-elevation curve correction to account for the gain
variation with elevation due to the gravitational stress changes in the
telescope structure. We do not compensate for the atmosphere opacity however,
considering that the correction is negligible at these frequencies.

The counts-to-Jansky conversion factor was derived by means of a two-
dimensional Gaussian fit to the Stokes L and R images of the standard flux
density calibrators 3C147, J0137+3309 (3C48), J1331+3030 (3C286), and
J1411+5212 (3C295). We adopted the flux density scale of Perley & Butler
(2013). The systematic uncertainty is assumed to be $5\%$ of the calibrator
flux density.

All calibrator observations were made during the interruption of the M31 scans
when the galaxy was too highly elevated on the sky or at the end of the scans.
Data were acquired during four cross scans on each of the calibration sources.
The cross scans were made at a velocity of $1$′/s and each subscan was
$20\arcmin$ long.

###  2.5 Baseline subtraction

Before stacking the subscans to form the maps, we performed a subscan baseline
removal in order to set all the emission coming from blank fields to zero. To
this purpose we applied a mask. Our mask map was created starting from three
different maps of the M31 region: the NRAO VLA Sky Survey (NVSS) map at 1.4
GHz (Condon et al., 1998) which contains point sources, the Spitzer MIPS map
at 12.5 THz (Gordon et al., 2006), and the Planck 850 GHz map (Planck
Collaboration et al., 2020) that traces the extension of the galaxy with
thermal dust emission. We renormalized and combined the set of maps creating a
mask from which the baseline is calculated. The mask, combined with the 0.25
deg $MapEdge$, was finally used to estimate the background level by fitting a
linear model to the emission in the nonmasked region.

At this point, in order to generate the final map, we performed a new flagging
of the data to remove isolated spikes or short-time RFIs in an automated
fashion. Following this approach, we were able to flag all the data that
deviate by more than $3\sigma$, where $\sigma$ is a combination between the
single subscan rms and the sky model noise according to:

$\sigma^{2}=\left(rms_{sub-scan}^{2}+\left(mod\\_errsys\cdot
model\right)^{2}\right).$ (1)

The parameter $mod\\_errsys$ was chosen to be 0.5. This method was iterated
until convergence was reached: three iterations for each set of data were
needed. At this point, it was possible to create a new set of images where, as
expected, the $rms$ level was lower with respect to the previous images.

###  2.6 Map making

In the creation of our final total intensity map, we combined the scans by
taking an average of all of them in order to increase the signal-to-noise
ratio. A direct stacking would be heavily affected by the scanning noise
present in the individual on-the-fly maps, which takes the form of the typical
pattern of ”stripes” oriented along the scanning direction. These stripes can
be due to imperfect RFI removal or short-term fluctuations in the atmosphere
or in the receiver gain. Sometimes these features are not totally removed
during the flagging and the baseline subtraction process, and so they persist
in the final map. These features are easier to identify if we analyze the map
in the perpendicular direction with respect to the direction of the corrupted
subscan (Emerson & Graeve, 1988). To this purpose, SCUBE implements both
Fourier and wavelet map-making methods. In the Fourier transform plane, the
scanning noise terms from any single scan appear only on a narrow band passing
through the origin, with a width inversely proportional to the characteristic
scale lengths of the baseline drifts in the original scan. The final map is
created by incorporating a second scan in an orthogonal direction so that once
the second scan is also transformed in the Fourier plane, the second map only
intersects the error terms of the first map at the origin. The knowledge of
the approximate scale length of the baseline drift of each map, and so the
width of each band error in the Fourier plane, enables an optimally weighted
combination of Fourier terms of each data set to be made (Emerson & Graeve,
1988). In the wavelet space, the logic is the same, except that the unwanted
scanning noise is isolated and filtered out using a convenient merging of the
stationary wavelet transform (SWT) coefficients (Murgia et al., 2016). One
advantage of the SWT method compared to the Fourier method is that there is no
need to determine an optimal width for the weights of the noise bands. While
the noise bands partially overlap at the center of the Fourier plane, they are
completely separated in the wavelet domain. Indeed, the horizontal and
vertical detail coefficients of the two orthogonal scans can be opportunely
mixed while the diagonal detail coefficients and the approximation
coefficients are averaged.

We verified that both the Fourier and SWT de-stripping techniques give
consistent results. However, we noted that the SWT performed slightly better
than the Fourier method and therefore we opted the former.

As a last step in the map-making process we corrected the final map for the
residual base level. To this purpose, we masked the emission from M31 and
applied the Papoulis-Gerchberg Algorithm (PGA, Papoulis, 1975) to model and
interpolate the surrounding base level over the excluded data. SCUBE provides
an application of the PGA that allows the user to reconstruct the missing part
of an image by imposing a low pass filter in the spectral domain. The signal
extrapolation is carried out by iterating alternately between time (spatial)
and spectral domains until convergence is reached. To this end, we masked out
an elliptical region of 91′.5 and 59′.5 semi-major and semi-minor axis,
respectively (PA=-52$\deg$), centered on the innermost regions, and we
reconstructed the missing signal by imposing that the reconstructed map
retains no power below a spatial scale corresponding to the ellipse major
axis. This is achieved by a convolution with a Gaussian tapering in the
spectral domain. The reconstructed base-level image is then subtracted from
the final map, which allows the removal of residual large-scale features left
over after baseline removal and map-making, and large-scale features in
foreground and background emission, bringing the map back to a zero-level
outside the galaxy, as already shown in Battistelli et al. (2019). Figure 1
shows our final map of M31 at 6.6 GHz.

###  2.7 Map statistics

In order to provide an estimate of the sensitivity level reached in our map,
we masked the emission from point-like radio sources. Details on how compact
sources are removed from the map are reported in Sect. 4. In our source-
subtracted map, considering our pixel size of $0.9\arcmin$, and the SRT beam
in the C-band, we derived the $rms$ fluctuations from two $0.4\,$deg2
uncontaminated regions. We repeated the same analysis on the SWT map before
and after applying the flagging. We found consistent (although slightly
different) results in the two maps: we obtained a rms mean value of 0.44
mJy/beam before flagging and 0.43 mJy/beam after flagging. Therefore, from
this first analysis, we have a final value for the rms of 0.43 mJy/beam. This
value is fully compatible with the expected value from confusion noise,
assuming a spectral index of 0.7: $confnoise$ = 0.44 mJy/beam (Condon, 1974).

|
---|---
Figure 1: Left: Final M31 images after averaging over the whole bandwidth at
6.6 GHz. Right: Final M31 image after subtraction of point sources. Both the
maps are characterized by an angular resolution of $2.9^{\prime}$. The typical
beam size is indicated with the black circle in the bottom-left corner of each
panel.

##  3 Ancillary data

In this section we describe the ancillary maps used throughout this work. The
ancillary maps are described in two different sections, one relative to radio
and microwave wavelengths and the other to IR observations.

###  3.1 Radio data

For radio wavelengths, we used three maps acquired with the Effelsberg Radio
Telescope444https://www.mpifr-bonn.mpg.de/en/effelsberg,555 https://www.mpifr-
bonn.mpg.de/3265873/m31, at 1.46 GHz, 2.7 GHz, and 4.85 GHz. With its
100-metre primary mirror diameter, Effelsberg reaches an angular resolution of
$5.0\arcmin$ (Beck & Grave, 1982) and $2.8\arcmin$ (Berkhuijsen et al., 2003)
at 2.7 GHz and 4.85 GHz, respectively. The 1.46 GHz map angular resolution of
$0.75\arcmin$ (Beck et al., 1998) was achieved combining Effelsberg data with
VLA data. Moreover, we used the SRT map at 1.385 GHz (Melis et al., 2018),
with an angular resolution of $13.9^{\prime}$, to correlate radio and IR data.

###  3.2 Infrared data

Infrared data were used to characterize stellar and dust emission and explore
the presence of possible correlations with the radio continuum emission (see
Sect. 8). For this purpose, we used the Spitzer Space
Telescope666http://www.spitzer.caltech.edu/ maps at 24 $\mu$m and 3.6 $\mu$m.
These two maps were chosen for correlation purposes because the 24 $\mu$m
(Gordon et al., 2006) map is a SFR tracer, while the 3.6 $\mu$m (Barmby et
al., 2006) is a tracer of older stars. The angular resolution of the Spitzer
24 $\mu$m map is $0.1\arcmin$, while the 3.6 $\mu$m map is characterized by a
resolution of $0.04\arcmin$.

In order to estimate the radio/IR ratio, $q_{FIR}$, following the convention
by Helou et al. (1985), we also used the IRAS 60 $\mu m$ map (Xu & Helou,
1996) and the Herschel 100 $\mu$m map (Fritz et al., 2012). The angular
resolutions of the two maps are $4.0^{\prime}$ and $0.21^{\prime}$,
respectively.

###  3.3 Ancillary data treatment

Each one of the aforementioned maps is characterized by a given angular
resolution and geometry (i.e., pixelization on the sky). In order to compare
different maps, it was necessary to bring all of them to the same coordinate
system and angular resolution.

We used the $Montage$ $6.0$ software777http://montage.ipac.caltech.edu/ to
modify the coordinate system of the maps and convert it to the SRT C-band
pixelization. Once all the maps were in the same World Coordinate System (WCS,
Mink, 1997) reference frame, they were convolved to the same angular
resolution.

Depending on the purpose, three different final angular resolutions were used:
$14^{\prime}$, $5.0^{\prime}$, and $3^{\prime}$.

  * •

For the sole purpose of estimating the $q_{FIR}$ parameter, the Herschel 100
$\mu$m map and the IRAS 60 $\mu$m map were convolved to the common resolution
of $14^{\prime}$.

  * •

Convolution of all the maps to the resolution of the Effelsberg 2.7 GHz map
($5.0^{\prime}$) was carried out to disentangle thermal and nonthermal
emission (see Sect. 6) and to correlate these two emission mechanisms to IR
data (see Sect. 8).

  * •

We convolved the two Spitzer maps, the 1.4 GHz Effelsberg, and our SRT C-band
image to $3^{\prime}$ to extract a spectral index map (see Sect. 5).

##  4 Compact sources

A large number of sources can be seen in the $2.4\times 3.1$ deg2 C-band map
shown in the left panel of Fig. 1. In order to calculate a proper estimate of
the intrinsic brightness of M31, it is necessary to identify and remove all
the signals coming from these sources. To this end, we generated a point-
source catalog of M31 at 6.6 GHz. Aside from point-source subtraction, we used
this catalog to verify the theoretical prediction on the number of sources we
expect to find with a certain flux density in a given area, according to a
specific model.

We started by extracting a point-source catalog from the NRAO VLA Sky Survey
at $1.4$ GHz (Condon et al., 1998) which has an angular resolution of 45″ and
a rms of 0.5 mJy/beam. We relied on the relatively high angular resolution and
sensitivity of the NVSS to obtain a good prior for the coordinates of the
points sources in the direction of the M31 field. This is particularly helpful
in the subtraction of those point sources seen in projection against (or
embedded in) the extended emission of the galaxy, as we show in Figure 2 where
the SRT C-band contours are added to the NVSS image. Indeed, the finer NVSS
resolution also allowed us to resolve nearby sources not easily discernible
from our single-dish resolution. We note that we excluded the M31 central
region extended emission from the NVSS image in the point-source subtraction
process.

Figure 2:  NVSS image at 1.4 GHz with the SRT contours at 6.6 GHz. Levels
start at 2 mJy/beam and increase by a factor of two. The inset shows a detail
of the M31 ring. The extended emission of the central region (grey area) has
been excluded from the point-source search.

We select in the NVSS map all the point sources with a peak flux density of
$F>5$ mJy at 1.4 GHz. By fitting these point sources with a 2D Gaussian, we
compiled a catalog that contains the peak flux densities and the coordinates
for a large number of sources both inside and in the immediate surroundings of
the M31 field.

In order to account for nonzero offset over which the source may lie, we added
to the Gaussian model a plane with an offset and a tilt as free parameters.
This made the fit more accurate for sources with a significant local base
level, in particular those seen in projection over the M31 emission. As a
further step, the NVSS catalog coordinates were used as input parameters in
the CATS888www.sao.ru/cats/ (Verkhodanov et al., 1997) database in order to
extract the radio flux density information available in the literature about
these sources at other frequencies in the radio band. We then extracted the
flux densities of the point sources present in the 6.6 GHz SRT image using as
a prior the coordinates of the NVSS catalog. By fitting the same 2D Gaussian +
base level model over each one of the NVSS coordinates pairs we obtained the
peak flux density at 6.6 GHz. The fit was performed only if the SRT peak in
the NVSS position is $>3\sigma$, where $\sigma$ is the noise of the map, or
1.6 mJy/beam/pixel for the SRT C-band map. Out of the all the sources present
in the initial NVSS catalog, we detected 93 counterparts at 6.6 GHz inside the
field of the C-band SRT map broadly delimited by the black contours in Fig. 2.

Finally, we merged the CATS catalog with the SRT C-band catalog leading to a
final list of point sources containing both the source positions and all the
spectral information available in the literature for these objects. The radio
spectra of all these compact sources were modeled with a modified power:

$log(F)=-\alpha\cdot log\left(\frac{\nu}{\nu_{ref}}\right)+log(A)+k\cdot
e^{-log\left(\frac{\nu}{\nu_{ref}}\right)},$ (2)

where $k$ is a curvature term, $A$ is the amplitude at the reference frequency
$\nu_{ref}$, and $\alpha$ is the spectral index of the unmodified power law.
The model converges to the usual power law with index $\alpha$ in the limit
$k\rightarrow 0$. We adopted for reference frequency the fixed value
$\nu_{ref}=1$ GHz. The observed radio spectra were fitted with the modified
power law model when at least three data points were available, otherwise we
fixed the curvature to $k=0$, resorting to an ordinary power law.

---
Figure 3: Fit of the spectrum of the source 4C +39.03 detected in the SRT map.
The green open point represents the SRT flux density measurement at 6.6 GHz.
The filled red dots represent the measurements taken from the literature. The
values of the ”local” spectral indexes at 0.15, 1.4, and 6.6 GHz are also
reported. In the right panel we also outline the Planck 857 GHz and the
Herschel SPIRE $250\,\mu$m $5\sigma$ contours in dot-dashed cyan and yellow,
respectively.

For significantly curved spectra, only the spectral index between a very close
couple of frequencies is meaningful. Nonetheless, a ”local” spectral index
value at a given frequency, $\alpha_{\nu}$, can be easily computed
differentiating Eq. 2 with respect to $log\left(\frac{\nu}{\nu_{ref}}\right)$:

$\alpha_{\nu}=\alpha-k\cdot
log\left(\frac{\nu}{\nu_{ref}}\right)e^{-log\left(\frac{\nu}{\nu_{ref}}\right)}.$
(3)

This can be viewed as the spectral index between an infinitesimally close
couple of frequencies at $\nu$.

An example of a spectral fitting of one of the sources is shown in Fig. 3. In
general, we observe very good agreement between the SRT flux densities and the
measurements from the literature. Moreover, in Fig. 3 we show that in the case
of this bright source there is also evidence that the SRT in-band spectral
index measured at four distinct frequencies between 6.0 and 7.2 GHz is
consistent with the trend of both the model fit and the measurements at nearby
frequencies taken from the literature.

The image of the SRT point-source model at 6.6 GHz is shown in Fig. 4 at an
angular resolution of 2.9′. We recall that the SRT at C-band point-source
model is a subset of the larger initial catalog based on the NVSS which
comprises a larger field in the sky. Indeed, by using the NVSS-based
coordinates and the best-fit parameters of the modified power law in Eq. 2, it
is in principle possible to generate a point-source model image at any desired
frequency and angular resolution. Moreover, this procedure was followed to
remove the point-source contribution from the global M31 spectral energy
distribution (SED) in Battistelli et al. (2019).

A catalog of all the compact radio sources in the SRT map and the best-fit
parameters are reported in Table LABEL:Tab:SRT_Sources_catalogue, while in
Fig. 31 we show all the 93 fits of the spectra. Table
LABEL:Tab:SRT_Sources_catalogue also reports the spectral indexes measured
between 1.4 and 6.6 GHz, $\alpha_{1.4GHz}^{6.6\,GHz}$. The data at 1.4 GHz
refer to the NVSS catalog.

Figure 4: The SRT C-band point-source model.

###  4.1 Spectral index distribution

After obtaining the spectral index $\alpha_{1.4\,GHz}^{6.6\,GHz}$ for each
source detected in C-band, it is possible to study the spectral index
distribution, which is shown in Fig. 5. The average spectral index value is
0.72, while the median value is 0.81. The spectral index bin with most
occurrences is $0.85-0.95$.

As expected, in most cases, the spectral index exhibits a positive value. Only
2 out of the 93 sources exhibit a negative spectral index, or equivalently an
inverted spectrum. Approximately 74 % of the sources exhibit a spectral index
of $\alpha>0.5,$ indicating that their emission is most likely optically thin
nonthermal synchrotron emission. This population is typically composed of
background sources and/or supernova remnants (SNRs) in the M31 field, as has
been shown by Galvin & Filipovic (2014) and Lee & Lee (2014).

Figure 5: Spectral index distribution between 1.4 and 6.6 GHz for the 93
detected point sources in the field of M31.

###  4.2 Differential source counts

Given the C-band flux densities for each of the 93 sources, the differential
source counts were computed by binning the 0.007-0.45 Jy flux density range
according to the values reported in Table 2. It is possible to compare the
source counts with a theoretical model: here we use the latest version of the
4.8 GHz model999http://w1.ira.inaf.it/rstools/srccnt/ by Bonato et al. (2017).
As our measurements were performed at $6.6\,GHz$, we extrapolated them to 4.8
GHz flux density for each source, according to:

$F_{4.8\,GHz}=F_{6.6\,GHz}\left(\frac{6.6}{4.8}\right)^{\alpha_{1.4\,GHz}^{6.6\,GHz}},$
(4)

where $\alpha_{1.4\,GHz}^{6.6\,GHz}$ is the source spectral index; see Table
LABEL:Tab:SRT_Sources_catalogue.

The choice of 0.007 Jy as the lower value to study the differential source
counts can be justified assuming a Gaussian distribution, an rms value of
$\sim 1.6$ mJy/pixel within the C-band map extrapolated to 4.8 GHz, and a
3$\sigma$ peak cut-off. We find that the sources with flux density of 0.007 Jy
are detected with a probability of greater than $95\%$. It is important to
stress that a non-negligible fraction ($\sim 19\%$) of the detected sources
have extrapolated flux densities at 4.8 GHz lower than 0.007 Jy; we do not
make use of these.

Before comparing the measurements extrapolated to 4.8 GHz with the source-
count model, the source counts $n(F)$ derived from the SRT flux densities were
fitted with a power law $n(F)=A\left(\frac{F}{Jy}\right)^{-\gamma}$, where
both the number density of sources per flux density interval at 1 Jy $A$ and
the spectral index $\gamma$ are free parameters. The result of the fit in
logarithmic scale is:

$\frac{dN}{dF}=\left(300\pm 80\right)\left(\frac{F}{Jy}\right)^{-1.98\pm
0.07}Jy^{-1}sr^{-1}.$ (5)

The differential source counts and the best-fit curve are shown in Fig. 6. The
horizontal error bars represent the bin widths, while the vertical ones
correspond to the Poisson error associated with the measurements.

Range (Jy) | Count |  $Log_{10}$(Diff. Count) | Theor. count
---|---|---|---
0.007 - 0.01 | 22 | 6.54 | 21
0.01 - 0.0135 | 16 | 6.33 | 14
0.0135 - 0.02 | 15 | 6.03 | 14
0.02 - 0.05 | 14 | 5.34 | 18
0.05 - 0.15 | 5 | 4.37 | 7
0.015 - 0.45 | 3 | 3.67 | 2
Table 2: Number of sources detected in C-band map in a given flux density bin.
In the last column we show the number of expected sources in each bin from the
model we are considering. Figure 6: Point-source differential source counts
and the fitted power law in double log scale. The error bars parallel to the
flux density axis indicate the bin widths.

At this point we can compare our source sample with the model from Bonato et
al. (2017). This theoretical model returns the predicted Euclidean normalized
differential radio source counts per steradian for a discrete series of flux
densities. The Euclidean normalized differential counts is defined as
$F^{2.5}\frac{dN}{dF}$, where $F$ is in Jy. In Fig. 7, our source sample is
compared with the 4.8 GHz model taking into account the $\sim 7$ deg2 observed
area. The data are in good agreement with the model for the low-,
intermediate-, and high-density values. The tension in the last point in Fig.
7 can be solved by removing from the counts the BLAZAR $B3-0035+413$ that at
4.8 GHz has an extrapolated flux density of $\sim 0.44Jy$.

Integrating the model from 0.007 to 0.45 Jy, we find that the expected total
number of sources is $\sim 76$, while in the same range we detect 75 sources.
This appears to confirm the agreement between the detected and expected point
sources.

Figure 7: Normalized differential point-source counts compared with the Bonato
et al. (2017) model at 4.8 GHz.

###  4.3 The SRT C-band sample of supernova remnants

With the previously compiled catalog of compact sources detected in our map,
we can perform a correlation between SNRs and SNR candidates in the field of
M31. For this purpose, we used the Lee & Lee (2014) SNR and SNR candidate
catalog, which includes 156 sources, obtained with H$\alpha$ and [Sii] images
of M31. Using a $1.45^{\prime}$ search radius we cross-matched the SNRs and
SNR candidates catalog with our sample of point sources. The choice of a
search radius of $1.45^{\prime}$ is motivated by the size of the
FWHM${}_{SRT}=2.9^{\prime}$. We found then that source numbers 241, 359, 622,
626, and 645 have at least one corresponding source in the SNR catalog. In
particular, for the SRT source number 645 we have four corresponding sources
within $1.45^{\prime}$, three corresponding sources for the SRT source number
622, and 2 for number 241. The other two SRT sources have only one
corresponding source each. In Fig. 8 we show all the 156 SNRs and SNR
candidates (white contours) and the sources that have a corresponding source
within a radius of $1.45^{\prime}$ (cyan circles).

Figure 8: The SRT C-band map with compact sources. The 156 SNRs and SNR
candidates from (Lee & Lee, 2014) are overplotted with white contours. The SRT
sources that have a corresponding source within a radius of 1.45’ are shown
with cyan circles.

In Table 3 we report the list of sources with a match in the Lee & Lee (2014)
catalog, together with the ID in that same catalog, the distance from the
center, and the spectral index between 1.4 and 6.6 GHz.

SRT source n. | Lee SNR n. | Distance(′) | $\alpha_{1.4\,GHz}^{6.6\,GHz}$
---|---|---|---
241 | 50 | 0.87 | 0.42
359 | 17 | 0.93 | 0.49
622 | 156 | 0.96 | 0.16
626 | 144 | 0.78 | 0.31
645 | 97 | 0.09 | 0.01
Table 3: The SRT C-band sources with a correspondence in the Lee et al. (2014)
catalogue.

We note that the sources with a corresponding source in the SNR catalog are
mostly characterized by a relatively flat spectrum. This is a bias most likely
introduced by the achieved sensitivity at 6.6 GHz with SRT: if we assume two
sources with the same 1 GHz amplitude, it is much more likely to detect the
source with a flat spectrum than the source with a steeper one.

##  5 M31 morphology

Studies of M31 have revealed a normal disk galaxy with spiral arms wound up
clockwise. The spiral arms are separated from each other by at least 4 kpc.
The spiral pattern is somewhat distorted by the interactions of the galaxy
with M32 and M110 (Gordon et al., 2006). The Spitzer Space Telescope (SST)
revealed two spiral arms emerging from a central bar. The arms have a
segmented structure and continue beyond the galaxy’s ring mentioned above. The
ring shape of M31 was discovered in the 1970s (Pooley, 1969; Berkhuijsen &
Wielebinski, 1974). Comparing the radio emission at 6.6 GHz and the location
of the 3691 Hii regions presented in Azimlu et al. (2011) we find a strong
correlation in the ring, as one can see in Fig. 9. The Hii level contours
denote 1, 3, or 6 Hii regions inside each SRT C-band pixel, and so they are
not flux density contours but are simply an indication of the positions of the
Hii regions.

Figure 9: The SRT C-band map of M31 with selected Hii region contours. We note
the remarkable agreement between the distribution of the Hii regions and the
location of the 10 kpc ring and the 15 kpc structure.

The ring structure is believed to be the result of the interaction with M32
over 200 million years ago. The M32 galaxy likely passed through the disk of
M31, which left M32 stripped of more than half of its mass (D’Souza & Bell,
2018). The nucleus of M31 is home to a dense, compact star cluster. A total of
35 black holes have been detected in M31, of which 7 are within 0.3 kpc of its
center (Barnard et al., 2013). These black holes formed by gravitational
collapse of massive stars and have a mass of between approximately five and
ten times that of the Sun.

Radio and microwave observations revealed emission from the central region and
only from the external ring, while the central bar is not detected. The
central bar, which is detected in all the IR images, is also not visible in
the ancillary Effelsberg radio maps. As one can clearly see in Fig. 9, the 10
kpc ring and the 15 kpc structure, noted in IR maps, (Haas et al., 1998;
Gordon et al., 2006) are well correlated with the location of the Hii regions
in the SRT map. The 15 kpc structure is unfortunately at the edge of the
Effelsberg maps, and in the following section, when these maps are used, it is
not possible to disentangle thermal and nonthermal emission within these
external regions.

The M32 and NGC205 galaxies, which fall inside the SRT coverage, are not
detected because of their low radio emission. Brown et al. (2011) found that,
at 1.4 GHz, M32 and NGC205 have respectively a flux density of $0.7\pm 0.5$
mJy and $0.1\pm 0.5$ mJy. Also, if a flat radio spectral index is assumed for
these galaxies (most favorable scenario), their flux densities are compatible
with the SRT confusion noise and therefore cannot be detected.

In the following, we investigate the M31 radio features by creating a spectral
index map, and by studying the spectral index trend within the central region
and the entire galaxy.

###  5.1 Spectral index map

We start by building a spectral index map across the whole M31. In general, a
flat spectral index is expected in regions whose emission is dominated by a
thermal component, while a steeper spectrum is an indication of a substantial
presence of nonthermal emission.

The spectral index value has been calculated on a pixel-by-pixel basis by
taking into account only the pixels with flux densities greater than $3\sigma$
in all considered maps, where $\sigma$ is the map sensitivity expressed in Jy
beam-1. Indeed, pixels below this threshold are those where the spectral index
becomes very uncertain because of the presence of faint radio emission and
radio noise. The spectral index map was created using Effelsberg maps at
$1.46$ GHz and $4.85\,$GHz and the SRT map at 6.6 GHz. We used the maps
projected and then convolved to $3^{\prime}$ as explained in Sect. 3.

The spectral index value and its $1\sigma$ uncertainty were computed pixel by
pixel by fitting the relation:

$f(\nu)=A\left(\frac{\nu}{1\,GHz}\right)^{-\alpha},$ (6)

where both $A$ and $\alpha$ are free parameters. To perform the fitting, we
used a Python implementation of Goodman and Wear’s Markov chain Monte Carlo
(MCMC) Ensemble sampler (Foreman-Mackey et al., 2013). The same MCMC was used
in the following sections, to fit the emission in the central region and that
in the ring, and to disentangle thermal and nonthermal emission maps. The
spectral index map and its $1\sigma$ uncertainty are shown in Fig. 10.

|
---|---
Figure 10: Left panel: Spectral index map extracted from 1.46 and 4.85 GHz
Effelsberg maps and our SRT 6.6 GHz map. Right panel: Spectral index 1$\sigma$
noise map. The typical beam size (FWHM$=2.9^{\prime}$) is indicated with the
black circle in the bottom-left corner of each panel.

Globally, the spectral index map shows values of $\alpha\simeq 0.7$ (light
blue and light green pixels). In the northern and southern ring and in the
central region, we can also find small patches (violet and blue pixels) not
located at the edge of the map where the SRT and Effelsberg S/N is lower, with
a spectral index around $\alpha=0.4-0.5$. These are the regions where thermal
free-free emission is expected to be dominant and are likely to be Hii
regions, which are mainly located in these parts of the galaxy, as shown in
Azimlu et al. (2011). In some inner and outer ring regions we find that the
spectral index is larger than 0.9. This means that these regions are
characterized mostly by synchrotron emission, with the steep slope indicating
that an old population of electrons is responsible for the nonthermal
emission.

Spectral index maps of M31 have been built before (Berkhuijsen et al., 2003;
Beck et al., 2020). We find excellent agreement between our spectral index map
shown in Fig. 10 and the one extracted by Berkhuijsen et al. (2003) , which is
not surprising given the fact that, with the exception of the SRT C-band map,
to extract the spectral index we used the same set of maps, at the same
angular resolution. The spectral index values found by Beck et al. (2020)
using Effelsberg maps at 1.465 and 8.350 GHz are generally smaller than ours
by 0.2-0.3. We can ascribe this discrepancy to the difference in total flux
density between the SRT 6.6 GHz map and the new Effelsberg 8.35 GHz map. More
specifically, the latter shows a flux density that is $\sim 15$% higher than
the former. We identified some factors that could account for this
discrepancy:

  * •

The point sources in the two maps have been subtracted with different methods.
Residual point sources in one map that have been removed from the other can
affect the flux difference in both directions.

  * •

Maps have been made with different telescopes, the data have been analyzed
independently, and different software has been used.

  * •

The total area covered by the two maps is different; this can affect the
background estimate and the baseline subtraction.

Moreover, Battistelli et al. (2019) showed that AME is the dominant emission
mechanism in M31 in the frequency range 10-60 GHz. This means that, at 8.35
GHz, AME must make up $10-15$% of the total integrated flux density, causing
the spectral trend to drift away from a simple power law. Therefore, AME can
account for part of the difference between the two spectral index maps.

In the following sections, we explain how we used our spectral index map to
choose regions in which to perform a detailed study of the decomposition
between thermal and nonthermal emissions.

###  5.2 Spectral index trend

Hereafter, we evaluate the spectral index gradient across the galaxy and, in
doing so, we separately analyze the properties of the central region. For this
analysis we used the Effelsberg map at 2.7 GHz and all the other radio maps
projected and convolved to $5^{\prime}$ as explained in Sect. 3. We fitted a
simple power law with two free parameters using the integrated flux densities
in each given region instead of single pixel values, following the same
procedure for the extraction of the spectral index map.

For the galaxy central region, we evaluated the spectral index within four
concentric rings centered at $(Ra,Dec)=(10.70,41.28)$ deg (see Table 4 for
more details). For each ring, the total flux density was evaluated as:

$F=\left(\bar{S}_{in}-\bar{S}_{ext}\right)\Omega_{in},$ (7)

where $\bar{S}_{in}$ is the brightness inside the integration region,
$\bar{S}_{ext}$ is the background brightness and $\Omega_{in}$ is the solid
angle subtended by the region. The thermal brightness uncertainty was computed
as explained in Rubiño-Martín et al. (2012) and then combined with the
calibration error101010The calibration errors are considered to be $5\%$ for
all the maps..

Ring n. | Galactoc. | Spectral | $\sigma$
---|---|---|---
| Distance(kpc) | Index | S.Index
1 | 0-0.34 | 0.62 | 0.14
2 | 0.34-1.02 | 0.63 | 0.10
3 | 1.02-1.70 | 0.68 | 0.09
4 | 1.70-2.38 | 0.77 | 0.09

Table 4: Spectral index values in the central region as a function of
galactocentric distance.

The spectral index as a function of distance from the galaxy center and the
four rings where it was calculated are shown in Fig. 11. For each ring we used
a position angle of $-52$ deg in the $(Ra,Dec)$ system and an eccentricity
equal to 0.26. The spectral index within the central region is found to be
approximately constant within the uncertainties. We further investigate the
central region emission in Sect. 6.1 by decomposing thermal and nonthermal
emission.

Figure 11: Upper panel: Four elliptical rings used to compute the spectral
index trend with the distance from the center of M31. Lower panel: Spectral
index average radial profile calculated within the rings shown above.

We also repeated the same analysis for the whole galaxy: in this case we used
seven elliptical rings, centered at $(Ra,Dec)=(10.70,41.28)$ deg, with the
same position angle and eccentricity as those used for the central region. The
details of the seven regions and the spectral index values are reported in
Table 5.

Ring n. | Galactoc. | Spectral | $\sigma$
---|---|---|---
| Distance(kpc) | Index | S.Index
1 | 0-1.13 | 0.66 | 0.05
2 | 1.13-3.40 | 0.90 | 0.05
3 | 3.40-5.66 | 1.05 | 0.04
4 | 5.66-7.92 | 0.94 | 0.04
5 | 7.92-10.18 | 0.83 | 0.04
6 | 10.18-12.45 | 0.87 | 0.05
7 | 12.45-14.14 | 1.01 | 0.05

Table 5: Spectral index values as a function of the galactocentric distance
for the whole galaxy.

The seven elliptical rings and the spectral index as a function of the
distance from the galaxy center are shown in Fig. 12. The spectral index shows
a clear positive gradient in the innermost 6 kpc (first three rings). In the
outermost rings, the gradient changes sign twice, at $\sim 6$ (decreasing) and
at $\sim 10$ kpc (increasing again), even though the values are compatible
with a flat trend between $\sim 8$ kpc and $\sim 12$ kpc. Overall the 10 kpc
ring presents smaller values with respect to the peak reached within the third
ring. The average value of $\sim 0.9$ found at distances greater than $\sim 4$
kpc strongly hints at synchrotron emission being the predominant mechanism at
these frequencies. The point at highest galactocentric distance corresponds to
the edge of the ring: as expected, because of the aging of electrons, we found
a steeper spectral index value. The same spectral index trend and similar
values have been found by Berkhuijsen et al. (2003).

Figure 12: Upper panel: Seven elliptical rings used to compute the spectral
index trend with the distance from the galaxy center. Lower panel: Spectral
index average radial profile calculated within the rings shown above.

##  6 Thermal versus nonthermal emission

In this section we report a full decomposition of the M31 emission into
thermal and nonthermal components. We start by analyzing the central region
and the full 10 kpc ring individually, and then the northern and the southern
halves of the 10 kpc ring individually as well (Fig. 13). It is interesting to
conduct a separate analysis on the latter two regions because they appear to
have different properties in the spectral index map.

Figure 13: Two regions we selected as the north and south ring for
disentangling the total thermal and nonthermal emission.

For this decomposition, we used the maps reported in Table 6, all projected to
the SRT C-band map coordinate system and convolved to a resolution of
$5^{\prime}$. For all the regions mentioned above, the flux densities were
fitted with a synchrotron+free-free model with three free parameters: the
free-free and synchrotron amplitudes at 1 GHz, and the synchrotron spectral
index.

$F_{sync}(\nu)=A_{sync}\nu^{-\alpha_{sync}},$ (8)

$F_{f-f}(\nu)=A_{f-f}T_{e}^{-0.5}g_{ff}.$ (9)

We used Eqs. 8 and 9 to describe synchrotron and free-free emissions,
respectively. The electron temperature $T_{e}$ in Eq. 9 has been fixed at
$8000\,K$ throughout the paper. Fixing the electron temperature is not
fundamental in the decomposition between thermal and nonthermal emission
because the free-free amplitude is left as a fit parameter, but it is
important in the SFR extraction. In the latter case, we fix $T_{e}=8000\,K$ as
we expect the 10 kpc ring to host most of the M31 star forming regions (Ford
et al., 2013) and we know that in our Galaxy the electron temperature at
around a galactocentric distance of 10 kpc is $\sim 8000\,K$ (Paladini et al.,
2004).

A summary of the flux densities we used as data points for the fit, divided by
region, is reported in Table 6, where the reported flux density uncertainties
are a combination of statistical and calibration errors. A summary of the fit
results is reported in Table 7.

Map | Frequency(GHz) | Central Region [Jy] | 10 kpc Ring [Jy] | North Ring [Jy] | South Ring [Jy]
---|---|---|---|---|---
VLA+EFF21 | 1.46 | $0.252\pm 0.020$ | $3.47\pm 0.17$ | $1.858\pm 0.093$ | $1.526\pm 0.077$
EFF11 | 2.7 | $0.184\pm 0.010$ | $2.04\pm 0.10$ | $1.130\pm 0.057$ | $0.861\pm 0.043$
EFF6 | 4.85 | $0.113\pm 0.006$ | $1.20\pm 0.06$ | $0.671\pm 0.034$ | $0.506\pm 0.025$
SRT | 6.6 | $0.087\pm 0.005$ | $0.90\pm 0.05$ | $0.493\pm 0.025$ | $0.388\pm 0.020$
Table 6: Flux densities used as data points for the fit. The flux density
uncertainties are a combination of statistical and calibration errors.

Finally, in Sect. 6.5 we carry out a detailed separation into components,
pixel by pixel, for the whole galaxy and produce thermal and nonthermal
emission maps.

###  6.1 Central region

In order to evaluate the total flux density in the central area, we used a
circular region centered at $(Ra,Dec)=(10.70,41.28)$ deg with a radius of
$6^{\prime}$.

In Fig. 14 we show the data points and the thermal, nonthermal, and total
emission fit models. The final $\chi^{2}$ is 4.0. According to the fit, at 1
GHz, the thermal emission within the central region is about $\sim 8\%$ of the
total emission, while extrapolating the flux density to the SRT C-band
frequency we conclude that the total thermal emission in the central region,
at 6.6 GHz, is $\sim 26\%$ of the total emission.

Figure 14: Fit to the flux densities of the central region (see Table 7). The
fit is characterized by $\chi^{2}=4.0$. The error bars in the plot correspond
to 2$\sigma$.

###  6.2 10 kpc ring

For the 10 kpc ring, the flux density was extracted inside a region bounded by
two ellipses, one with a major semi-axis of $25^{\prime}$ and another with
major semi-axis of $65^{\prime}$, both with the same eccentricity of 0.26 and
the same position angle of $-52$ deg in the $(Ra,Dec)$ coordinate system.

In Fig. 15 we plot the data points and the fit model that yields a total
$\chi^{2}$ of 0.6. The synchrotron spectral index is found to be compatible
with the value found by Berkhuijsen et al. (2003), $\alpha_{sync}=1.0\pm 0.1,$
and with previous studies of the M31 SED.

Figure 15: Fit to the flux densities of the ring (see Table 7). The fit is
characterized by $\chi^{2}=0.6$. The error bars in the plot correspond to
2$\sigma$.

We find that, at 1 GHz, the thermal emission within the ring is about $7\%$ of
the total emission. At 6.6 GHz, the whole ring emission is dominated by
nonthermal emission, contributing to $\sim 70\%$ of the total emission budget.
The $\sim 30\%$ thermal fraction is similar to the value of 26% that we
extracted in Sect. 6.1 for the central region, indicating that the total
contribution of the thermal emission at 6.6 GHz in not negligible.

###  6.3 Northern ring region

In the following, we present a repetition of the previous analysis, this time
focusing on the northern half of the 10 kpc ring shown by the green region in
Fig. 13. For this purpose, we used the same ellipses as in Sect. 6.2. Data
points and the fit model are reported in Fig. 16. The fit is characterized by
a $\chi^{2}=1.5$.

Figure 16: Fit to the flux densities of the northern half of the 10kpc ring
(see Table 7). The fit is characterized by $\chi^{2}=1.5$. The error bars in
the plot correspond to 2$\sigma$.

We note that in the upper ring the free-free emission at 1 GHz is only a
fractional part of the total emission. Moreover, we have only a marginal
detection for the free-free emission: the 1 GHz free-free amplitude is $\sim
1.5\sigma$, which is compatible with zero emission.

The thermal flux density, extrapolated to 6.6 GHz is about one-third of the
synchrotron emission, while at 1 GHz it is 16 times weaker.

###  6.4 Southern ring region

We repeated here the analysis we carried out in Sect. 6.3, this time focusing
on the southern half of the 10 kpc ring, the region delimitated by yellow
dash-dotted lines in Fig. 13. Again, for this purpose, we used the same
ellipses as in Sect. 6.2.

From an inspection of the spectral index map, we expect to find a synchrotron
spectral index which is higher than what we found for the northern region.
Also, looking at the distribution of Hii regions within the ring (Fig. 18), we
expect a total thermal emission fraction at 6.6 GHz which is compatible with
what we extracted for the northern ring. Figure 17 shows the best-fit curve:
the fit model is characterized by $\chi^{2}=0.3$.

Figure 17: Fit to the flux densities of the southern half of the 10kpc ring
(see Table 7). The fit is characterized by $\chi^{2}=0.3$. The error bars in
the plot correspond to 2$\sigma$.

As expected, the synchrotron spectral index value found for the southern
region is steeper than the one we found for the northern region. However, due
to the high uncertainties, the two synchrotron spectral indices turn out to be
in agreement.

|
---|---
Figure 18: Left panel: the thermal (free-free) emission map at 6.6 GHz with
Hii contours. Contour levels indicate 2, 3, 6, 9, and 12 Hii regions inside
the C-band pixel. Right panel: Same as left panel, but for the synchrotron
component (nonthermal) with SNR contours. Contour levels show 1, 2, and 3 SNRs
inside the C-band pixel. The resolution of both thermal and nonthermal maps is
$5^{\prime}$.

We note from the extrapolated flux densities at $6.6\,$ GHz that within the
southern ring the nonthermal emission is essentially twice the thermal
emission. The integrated thermal emission in the two half ring regions is of
the same order of magnitude, as predicted, while nonthermal emission was found
to be higher in the northern half of the ring. The thermal emission fraction
at $6.6\,$GHz in the southern half was found to be $38\%$, while in the
northern half it makes up $26\%$ of the total budget.

Fit results
---
Fit parameters | Central region | 10 kpc ring | North reg. | South reg.
$\alpha_{sync}$ | $0.83^{+0.19}_{+0.10}$ | $1.06^{+0.18}_{-0.12}$ | $1.01^{+0.17}_{-0.10}$ | $1.13^{+0.23}_{-0.16}$
$A_{sync}^{1GHz}$ [Jy] | $0.322^{+0.026}_{-0.027}$ | $4.84^{+0.32}_{-0.31}$ | $2.58^{+0.16}_{-0.16}$ | $2.13^{+0.15}_{-0.14}$
$A_{ff}^{1GHz}$ [Jy] | $0.029^{+0.029}_{-0.021}$ | $0.31^{+0.24}_{-0.21}$ | $0.16^{+0.14}_{-0.11}$ | $0.18^{+0.11}_{-0.11}$
$F_{tot}^{6.6GHz}$ [Jy] | $0.091\pm 0.028$ | $0.92\pm 0.27$ | $0.51\pm 0.14$ | $0.40\pm 0.13$
$F_{ff}^{6.6GHz}$ [Jy] | $0.024\pm 0.020$ | $0.26\pm 0.18$ | $0.13\pm 0.10$ | $0.15\pm 0.09$
$F_{sync}^{6.6GHz}$ [Jy] | $0.067\pm 0.019$ | $0.66\pm 0.19$ | $0.39\pm 0.10$ | $0.25\pm 0.9$
$\chi^{2}$ | $4.0$ | $0.7$ | $1.5$ | $0.3$
Table 7: Summary table of thermal and nonthermal emission decomposition done
in the central region, the 10 kpc ring and in the NE and SW halves of the
ring. In the first three rows, fit results are reported while in rows 4, 5,
and 6 the total, thermal, and nonthermal emission at $6.6\,$ GHz are shown,
respectively. In the last row we report the goodness of the fit for 1 degree
of freedom.

###  6.5 Thermal and nonthermal maps

As anticipated by the spectral index map, we found a typical synchrotron
spectral index ($0.5\leq\alpha\leq 0.8$) in the central region, and a steeper
spectrum with $\alpha\geq 0.8$ for the 10 kpc ring. We can deduce from this
that the galaxy presents sources of active electron injection in the central
region, while the ring mainly hosts old electrons.

By using the free-free and synchrotron emission models in Eqs. 9 and 8 to
reproduce the emission at the four different considered frequencies, we
derived the thermal and the nonthermal contributions to the total flux density
at a resolution of $5^{\prime}$. We decided to leave the synchrotron and free-
free emission amplitude at 1 GHz as free parameters, while the synchrotron
spectral index has been fixed to 1.06 in the ring and to 0.83 in the central
region. Fixing the latter improves sensitivity on synchrotron and free-free
amplitude values, removing the intrinsic degeneration between free-free and
synchrotron emission models. Following this procedure, we were able to build
spatially resolved emission maps for the thermal and nonthermal emission,
which we present in Fig. 18.

The thermal map shown in the left panel of Fig. 18 shows a concentrated
thermal emission in the southern and central part of the ring and diffuse
emission in the northern ring as expected given the distribution of Hii
regions (e.g., Azimlu et al., 2011). The regions of high thermal emission in
the ring match the ones identified in previously published M31 thermal maps
(Berkhuijsen et al., 2003), and we generally find good agreement in the whole
ring and in the central region. Tabatabaei et al. (2013) separated free-free
and synchrotron emission in M31 using a template for the free-free emission
from H$\alpha$ emission line. Despite their different frequency, angular
resolution, and decomposition technique, we noted good agreement in the
location of the brightest thermal spots.

The value of the thermal peak emission in the ring is $\sim 19$ mJy/beam,
whereas the nonthermal intensity only reaches $\sim 10$ mJy/beam. The thermal
peak is located in the central part of the ring, $(Ra;Dec)=(11.17;41.35)$ deg.
In the southern ring, the thermal peak is $\sim 16$ mJy/beam, while nonthermal
emission is about one-third of that. High thermal emission is also found
within the central region, in agreement with the results presented in
Berkhuijsen et al. (2003). On the other hand, the H$\alpha$ map extracted by
Devereux et al. (1994) shows a bright bulge. These results seem to suggest
that the M31 bulge is also a star-forming region, despite no evidence of star
formation in the central region being found by Ford et al. (2013). In the
following section, we investigate the SFR in more detail, starting from the
given thermal map. Globally, the average free-free noise values resulting from
the pixel-by-pixel MCMC fit were found to be in the range $0.5-2\,$ mJy/beam.
We stress the perfect agreement between our thermal map and the H$\alpha$ map
by Devereux et al. (1994) convolved to $5^{\prime}$; see Fig. 19.

Figure 19: Thermal map ($FWHM=5^{\prime}$) with H$\alpha$ contours by Devereux
et al. (1994).

The nonthermal emission map is shown in the right panel of Fig. 18. This map
is much smoother compared to the thermal one, especially within the ring,
confirming the results at several wavelengths and angular resolutions obtained
by Berkhuijsen et al. (2003); Tabatabaei et al. (2013); Beck et al. (2020).
The nonthermal emission amplitude varies in the ring between $\sim 5$ mJy/beam
and $\sim 19$ mJy/beam, while the thermal map shows values even two orders of
magnitude lower than the peak in the central ring. This nonthermal uniformity
can be explained with the spread of the relativistic electrons from their
birth regions into the whole ring along the magnetic field lines, as shown in
Berkhuijsen et al. (2003). Further details about the magnetic fields and the
polarized emission in M31 based in the SRT C-band data will be presented in
Murgia et al. (in prep.). The average nonthermal noise was found to be in the
range $0.5-1\,$ mJy/beam.

As sources of free-free and synchrotron emission, it is important to correlate
the thermal and nonthermal maps with the position of the Hii regions and SNR
or SNRs candidates, respectively. In the left panel of Fig. 18, the Hii region
contours are overlayed on the thermal map at 6.6 GHz. We used contours
relative to 2, 3, 6, 9, and 12 Hii regions inside the C-band pixel. We note
the good agreement between the brightest thermal regions and the areas where
more than two Hii regions are located. The only exception is the central
region of M31. Here we find a non-negligible thermal emission but no Hii
region counterpart.

In the right panel of Fig. 18, the SNR region contours are overlayed on the
nonthermal map at 6.6 GHz. We find excellent agreement between the location of
the SNRs and nonthermal emission at 6.6 GHz. Contour levels are 1, 2, and 3
SNRs inside the C-band pixel.

##  7 Star formation rate in M31

Here, we use the thermal map derived in Sect. 6 to compute the total SFR of
M31. In addition, we also generate a SFR map for the galaxy.
The SFR and thermal emission are related by a linear model. The thermal flux
density at a given frequency $\nu$ is proportional to the number, $N_{c}$, of
Lyman continuum photons (Lyc) absorbed by the gas in the Hii regions. The
relation between $N_{c}$ and the thermal flux density is reported below
(Mezger et al., 1974):

$N_{c}[s^{-1}]=4.761\cdot
10^{48}\alpha(\nu,T_{e})^{-1}\nu[GHz]^{0.1}T_{e}^{-0.45}S_{th}[Jy]D[kpc]^{2},$
(10)

where $S_{th}$ is the thermal flux density at frequency $\nu$, and we assumed
the M31 distance to be $D=780$ kpc. For the electron temperature we used
$T_{e}=8000$ K while $\alpha(\nu,T_{e})$ is a slow varying function, tabulated
in Mezger & Henderson (1967). We considered $\alpha(\nu,T_{e})=1$ at radio
wavelengths. This gives Nc = 3.11 $\times$ 1052 s-1. Given $N_{c}$, the SFR
can be found using a simple relation. In fact, the ratio of SFR to Lyman
continuum photon rate is given by Kennicutt et al. (1994) and Kennicutt
(1998):

$SFR\left[M_{sun}\cdot yr^{-1}\right]=7.5\cdot 10^{-54}\cdot
Nc\;\;\;\left[\mbox{photons}\cdot\mbox{s}^{-1}\right].$ (11)

The left hand side of Eq. 11 is the continuous SFR required to maintain a
steady-state population of ionizing stars that produces an observed ionizing
photon rate $N_{c}$.

Using the conversion factor from $N_{c}$ to SFR from Eq. 11 and given the
thermal map extracted in Sect. 6, we created a full SFR map. The output map is
reported in Fig. 20. From this map we extracted a total star formation rate of
$SFR_{radio-freefree}=0.19\pm 0.01$ M⊙ yr-1, which is consistent with the
findings of Ford et al. (2013) and Xu $\&$ Helou (1996). This total SFR value
was found by integrating the SFR map in Fig. 20 in all the available pixels
(i.e., in all the pixels with $S/N>3$), which corresponds to an integration
radius of $R_{max}\sim 15\,$ kpc. Here, and for the following integrated SFR
values in this section, the reported error is a sum of the SFR map statistical
uncertainty and the calibration error.

Figure 20: SFR (left) and SFR noise (right) maps extracted from radio and
free-free emission. The radio–free-free map used as a starting point is the
one described in Sect. 6.

The obvious advantage of using a radio tracer of the recent star formation
activity is that it is almost unaffected by dust extinction, unlike other
classical tracers at optical and UV wavelengths (e.g., the H$\alpha$
luminosity and the near- or far-UV luminosity). The main issue with methods
that use ionization of hydrogen, and the related free-free emission, to trace
star formation from massive stars is the direct absorption of Lyman continuum
photons by dust. In this case, the ionizing photons are removed altogether and
are no longer available to ionize hydrogen. Thus, no emission from free-free
continuum emission will result (Kennicutt & Evans, 2012).

The actual impact of Lyc photon absorption by dust has been notoriously
difficult to establish because of the absence of a “ground truth” (or
reference) with which to compare measurements. Models are therefore needed,
and these show that the level of Lyc absorption depends on the assumptions of
the geometry of the nebulae (Dopita et al. 2003, and references therein). In
Tabatabaei & Berkhuijsen (2010), a lower limit was put on the M31 ring SFR in
the radial range $6<R<17$ kpc, after dust attenuation correction. This
resulted in $SFR_{H\alpha}[6<R<17$ kpc$]=0.27$ M⊙ yr-1. We extracted the total
SFR in a similar region, finding $SFR_{radio-freefree}[6<R<15$ kpc$]=0.16\pm
0.01$ M⊙ yr-1. In Kang et al. (2009), the authors extracted the SFR of young
star forming regions (¡10 Myr) from UV observations as a function of the
galactocentric distance within a $R_{max}=27$ kpc; their Fig. 13 suggests that
the SFR in the region $6<R<15$ kpc corresponds to $\sim 47\%$ of the total,
while $31\%$ can be found in the region $15<R<17$ kpc. Using these estimates,
we extrapolated the total SFR from free-free emission within the annular
region $6<R<17$ kpc from our data, finding $SFR_{radio-freefree}[6<R<17$
kpc$]=0.27\pm 0.01$ M⊙yr-1, which is in very good agreement with the value
found in Tabatabaei & Berkhuijsen (2010).

In the following we compare the SFR map derived from SRT data with the one
from Ford et al. (2013) (F13 hereafter). The latter was obtained by combining
far-UV (FUV) and 24 $\mu$m (mid-infrared (MIR)) data, which accounts for both
optically visible and dust obscured star formation. Furthermore, the
contribution to the emission in the FUV and to the heating of dust emitting in
the MIR coming from the old stellar populations was carefully taken into
account. This is an issue in M31 in particular, given the very luminous and
dense bulge. As we discuss later, this is an issue for the SFR calculated from
the free-free emission as well. The two maps are shown in Fig. 21.

Figure 21: Two SFR maps extracted from two different tracers: the free-free-
based SFR map (top) with a resolution of $5^{\prime}$, and the Ford-Hires
UV+MIR SFR map (bottom) convolved to $5^{\prime}$ angular resolution.

In order to be able to compare the two maps, we converted the UV SFR map,
originally in units of SFR surface density, into units of M⊙ yr-1 ,
multiplying the value of each pixel by the projected pixel area in kpc2. With
respect to the UV+MIR SFR map, the SRT map shows a less smooth star formation
distribution, reaching higher peaks. This is due to the different star
formation tracers used to extract the two maps: the UV and 24 $\mu$m are more
uniformly distributed than the HII regions responsible for free-free emission.
Moreover, in the case of free-free emission, the gas is ionized by stars that
are younger than the stars responsible for the heating in the UV and MIR
emission. Therefore, while the SRT map represents the present SFR, the F13
data show the SFR for older stars. Another morphological difference that
immediately emerges from the comparison of the two maps is the star formation
in the central region. The map extracted from radio–free-free emission shows a
peak of SFR in the central area of the galaxy, while the F13 map shows almost
no star formation in the central region. The total SFR integrated limited to
the central area is $Centre\\_SFR_{radio-freefree}=(12.8\pm 0.7)\cdot 10^{-3}$
M⊙ yr-1 for the SRT map. We also extracted the same quantity from the F13 map,
and found $Centre\\_SFR_{UV}=(3.5\pm 0.2)\cdot 10^{-3}$ M⊙ yr-1. This value
has been extracted from both maps integrating on an elliptical region of
ellipticity $e=0.26$ at the center of the SRT map, with the same position
angle as the galaxy, namely $\theta=-52\deg,$ and a semi-major axis of
$10^{\prime}$ (2.3 kpc). No noise cut was applied to the two maps for this
purpose.

The excess of thermal emission we find in the central region is not due to
star formation activity. The center of M31 has been known to host ionized gas
for at least four decades now, the presence of which has been proven by
several works (e.g., Deharveng et al., 1982; Jacoby et al., 1985; Devereux et
al., 1994). Deharveng et al. (1982) suggested that horizontal-branch stars
could be a likely source of ionizing photons as the detected UV emission quite
closely follows the optical one, the latter surely being dominated by old
stellar populations. Line flux density ratio diagnostics, such as
[Nii]/H$\alpha$ (found to be $\sim 3$ on average, see Rubin & Ford, 1971),
also support an origin different from star formation. Using HST images, King
et al. (1992) noted that, within the bulge, there are no bright point-like
sources that could be identified as O stars, and Devereux et al. (1994)
suggest that the peculiar filamentary morphology displayed by the H$\alpha$
emission within the bulge is unlikely attributable to star forming regions.
The presence of these filaments indicates the existence of shocks, which
represent a possible origin of ionized gas. Results supporting this view were
obtained by analyzing IR emission, using both SED fitting (see Smith et al.,
2012; Viaene et al., 2014) and radiative transfer models (Viaene et al.,
2017). All such studies were in agreement in attributing both the dust heating
and the UV luminosity to the old stellar populations hosted within the bulge.
The presence of this kind of stars in the central region is confirmed by the
high correlation we find in the center between our thermal map at 6.6 GHz and
the Spitzer map at 3.6 $\mu$m, as discussed later in Sect. 8.1. Taking into
account this set of considerations, in the following we mask the central
region and do not include it in the SFR investigation.

In order to quantify the level of correlation between the SFR map generated
from radio and free-free emission and the same map extracted from UV emission,
we created a T-T plot. For this purpose, all the pixels with a S/N of lower
than 3 in the radio–free-free thermal map have been excluded from the
analysis. The plot is reported in Fig. 22. The two maps show good correlation
in the ring region, with a correlation coefficient of $r_{ring}=0.75$. As
expected (see Smith et al., 2012; Ford et al., 2013; Viaene et al., 2014), the
ring is the region with the strongest star formation activity.

Figure 22:  T-T plot between the radio–free-free SFR map and the F13 map with
the central area masked. The red contours on the top two maps highlight the
regions with $S/N>3$ in our SFR map (top-left panel), and therefore the pixels
that have been considered in this analysis. The plot shows a good correlation
between the two maps in the ring region with a correlation parameter of
$r_{ring}=0.75$.

We also analyzed the SFR radial trend (see Fig. 23 and Table 8) by computing
the integrated SFR in annular regions of increasing distance from the center
of the galaxy. An image of the regions of integration is reported in the top
panel of Fig. 23, while Table 8 reports the radial SFR trend extracted from
our map. In this step of the analysis, we compared our radio–free-free SFR map
with the F13 SFR map as well as with the flux density extracted from the
H$\alpha$ (Devereux et al., 1994) and CO (Nieten et al., 2006) maps, these
latter two being proportional to the recent SFR. For these two maps, the flux
density values have been renormalized to the radio–free-free mean SFR value in
order to be able to visually compare them to the other two radial trends in
the plot.

---
Figure 23: Top panel: Regions of integration used to extract the integrated
SFR values reported in the bottom panel and in Table 8. Bottom panel:
Integrated SFR in annular regions of increasing galactocentric distance for
four different tracers. For the radio–free-free and UV SFR maps, the plot
illustrates the SFR value integrated in annular regions of increasing
galactocentric distance (see top panel) and renormalized by the area of the
region. For H$\alpha$ and CO maps, we report the flux density renormalized to
the mean SRT-integrated SFR value in order to be able to visually compare the
four different trends. The SFR values from the radio–free-free map are also
reported in Table 8.

Ring n. | Inner major semi axis | Outer major semi axis | SFRradio-ff
---|---|---|---
| [arcmin] | [kpc] | [arcmin] | [kpc] |  $10^{-3}$[M⊙ yr-1 kpc-2]
2 | 10 | 2.3 | 20 | 4.5 | $0.56\pm 0.03$
3 | 20 | 4.5 | 30 | 6.8 | $0.38\pm 0.02$
4 | 30 | 6.8 | 40 | 9.1 | $0.86\pm 0.04$
5 | 40 | 9.1 | 50 | 11.3 | $1.75\pm 0.09$
6 | 50 | 11.3 | 60 | 13.6 | $1.16\pm 0.06$
7 | 60 | 13.6 | 65 | 14.7 | $0.44\pm 0.02$

Table 8: Total SFR values found from the radio–free-free map by integrating in
the annular regions shown in Fig. 23. The ring number follows the same
numbering as that used in Table 5.

The radial trend is similar for all the star formation tracers, with a SFR
peak at $\leavevmode\nobreak\ 45^{\prime}$ (or equivalently $10$ kpc) from the
center of the galaxy. The SFR derived from the F13 map is compatible with the
one calculated from radio–free-free everywhere in the galaxy apart for the
second and third regions of integration. Finding a lower SFR from the
radio–free-free map in these two regions is expected. Indeed, in the F13 map
the arms of the galaxy are visible at this distance from the center of the
galaxy, while we do not detect the same structure in the SRT map.

##  8 Radio versus IR cross-correlation

A linear correlation is known to exist between the radio continuum and MIR
galactic emission (de Jong et al., 1985; Helou et al., 1985; Yun et al.,
2001). We investigate this correlation with Spitzer-MIPS $24\mu$m on SRT
total, thermal, and nonthermal maps within M31. Moreover, we used the Spitzer-
IRAC map at 3.6 $\mu$m to investigate how the presence of old stars affects
both thermal and nonthermal emission within the M31 central region.

###  8.1 The relation between radio continuum emission and warm dust

In this section, we present a study of the relation between the spatially
resolved 6.6 GHz and the warm dust emission as sampled by the map at 24
$\mu$m, observed with the MIPS instrument on board the SST. The latter map has
been transformed to the SRT map coordinate system and then convolved from its
angular resolution of $0.1^{\prime}$ to $2.9^{\prime}$.

Taking into account only pixels with $S/N>3$ we obtained the correlation shown
in Fig. 24 over the whole galaxy. The correlation parameter over the whole
galaxy is found to be $\sim 0.75$. Here and in the following, for each
correlation between radio and MIR data, we extracted the power-law exponent
(b) of the MIR (X)-radio (Y) correlation fitting $log(Y)=a+b*log(X)$. In each
case, we fitted a power law determining the ordinary least-squares bisector in
the log-log plane (Isobe et al., 1990). For the whole galaxy, we found
$b=0.75\pm 0.01$ indicating that the correlation is nonlinear.

Figure 24: Correlation between the Spitzer 24 $\mu m$ maps and the SRT 6.6
GHz. The best linear fit is indicated with the red line. The log-log slope for
the whole galaxy is $0.75\pm 0.01$. Only pixels above $3\sigma$ have been
used.

We note that the points at higher flux densities in Fig. 24 belong to the
central region of the galaxy. In order to perform a more detailed analysis, we
repeated the study of the radio–MIR correlation, separating pixels belonging
to the ring and to the central region. This was done by exploiting the same
regions as we did in Sect. 6 to disentangle thermal and nonthermal emission
within the central region and the ring. The results are shown in Fig. 25. The
correlation is significantly better in the central region (correlation
parameter of $\sim 0.95$) and consequently gets worse for pixels within the
ring (with a correlation parameter of $\sim 0.65$), even though a clear trend
is still present. We find that the power-law exponent for the ring only is
compatible with that found for the whole galaxy, while for the central region
we find $b=1.47\pm 0.04$ indicating that the relation between MIR and radio is
more than linear.

Figure 25: Upper panel: 24 $\mu$m–radio correlation for pixels within the 10
kpc star forming ring. The log-log slope for the ring is $0.73\pm 0.01$. Lower
panel: Same correlation as in the left panel but for pixels within the central
region. The log-log slope for the central region is $1.47\pm 0.04$. Only
pixels above $3\sigma$ have been used.

###  8.2 The infrared to thermal and nonthermal correlation

In Sect. 8.1 we showed that a good correlation exists between the 24 $\mu$m
and the 6.6 GHz emission. Now we analyze this correlation further by
investigating the correlation between MIR and both the thermal and nonthermal
emission at 6.6 GHz, with the aim of gaining more insight into the origin and
characteristics of the two kinds of emission. We used the thermal and
nonthermal maps extracted with the MCMC fit in Sect. 6. Considering that the
thermal and nonthermal maps have a resolution of $5^{\prime}$, the Spitzer map
has been convolved to the same resolution starting from the original
$0.1^{\prime}$ map. With the same fit we also extracted thermal and nonthermal
noise maps. These noise maps have been used to select only the pixels where
thermal or nonthermal detection is $>3\,\sigma$. The correlation factors over
the whole galaxy were found to be $\sim 0.72$ and $\sim 0.84$ , respectively,
for thermal and nonthermal emission.

Figure 26 presents a pixel-by-pixel correlation between the thermal and
nonthermal emission and the 24 $\mu$m emission, taking into account the seven
elliptical rings used in Sects. 5 and 7. In general, we note a strong
correlation in the central region (rings 1 and 2), then the correlation
decreases around the 5 kpc structure (ring 3), increases around the 10 kpc
ring (rings 4, 5, and 6) and then decreases again out to the 15 kpc structure
(ring 7).

Figure 26: Top panel: M31 thermal–MIR correlation. Bottom panel: M31
nonthermal–MIR correlation. With $\rho$ we indicate the Pearson correlation
coefficient within each elliptical ring. Only pixels above $3\sigma$ have been
used.

We find that, in general, the correlation between nonthermal and MIR is very
close to linear for all the considered regions while it is clear that the
thermal T-T plot shows two different trends for the central region and the
ring. This could suggest different emission mechanisms within the two regions.
As we did in Sect. 8.1, it can be interesting to disentangle the emission from
the ring and the central region and analyze possible changes in the
correlation parameters found above.

The correlation plots for both thermal and nonthermal emission, including only
the pixels in the ring region, are reported in Fig. 27. In this case, the
correlation parameters, of $\sim 0.78$ for nonthermal emission and $0.71$ for
thermal emission, are very close to the ones we found for the whole galaxy.
Correlations between thermal and nonthermal emission with MIR data in the M31
ring have been found before (Berkhuijsen et al., 2013). The latter authors
showed that the log-log slope of the correlation between nonthermal and MIR
depends on frequency, finding $0.61\pm 0.02$ and $0.97\pm 0.09$ using 20.5 and
6.3 cm data, respectively. At 4.5 cm, we found a log-log slope of $0.82\pm
0.02$. This value is intermediate with respect to the previous ones. We
noticed that the log-log slope between nonthermal and MIR in the ring is
strongly affected by data cut-off, and for this reason we cannot make a direct
comparison with the frequency dependence reported in Berkhuijsen et al.
(2013). For instance, taking into account only pixels above five times the
linear offset we find a log-log slope of $1.07\pm 0.10$. The power-law
exponent extracted for the thermal–MIR correlation, of namely $0.96\pm 0.02$,
is not significantly lower than the one extracted by previous authors at 20.5
cm: $1.08\pm 0.02$. The power-law exponent we found for the M31 ring between
thermal and MIR emission confirms the linear relation derived by Xu et al.
(1992) in the Large Magellanic Cloud.

Figure 27: Top panel: M31 ring thermal–MIR correlation.
Bottom panel: M31 ring nonthermal–MIR correlation. Log-log slopes for thermal
and nonthermal emission in the ring are respectively $0.96\pm 0.02$ and
$0.82\pm 0.02$. Only pixels above $3\sigma$ have been used.

The correlation between the IR thermal dust emission and nonthermal
synchrotron emission within galaxies is most likely linked to the formation of
massive stars, as explained in Harwit & Pacini (1975) and Helou et al. (1985).
In Sect. 7, we showed that the M31 ring presents active star forming activity.
The dust grains absorb UV and optical photons from young massive stars, and
re-radiate them at MIR wavelengths. The same young massive stars are the
progenitors of supernovae and supernovae remnants that can accelerate the
electrons responsible for synchrotron radiation. This mechanism therefore
provides a natural explanation to the strong correlation between nonthermal
and MIR emissions. The linear correlation between thermal radio and MIR
emissions in the M31 ring is not surprising considering that both of them are
associated with young ionizing stars.

We find a stronger correlation between the central region and the ring. The
radio–MIR correlation parameter for thermal emission within the central region
is $\sim 0.89$ and this is $\sim 0.93$ for nonthermal emission. Figure 28
shows the two radio–MIR correlation plots for the central region of M31. The
power-law exponents between thermal or nonthermal emission and MIR emission
were found to be $1.80\pm 0.06$ and $1.07\pm 0.02$, respectively.

Figure 28: Top: M31 central region thermal–MIR correlation. Bottom: M31
central region nonthermal–MIR correlation. It is evident, especially from the
bottom panel, that the scatter plots show sharp edges. This comes naturally
from the fact that the central region points are distributed over a circular
area with sharp edges. Log-log slopes for thermal and nonthermal emission in
the central region are respectively $1.80\pm 0.06$ and $1.07\pm 0.02$. Only
pixels above $3\sigma$ have been used.

Unlike the ring, the central region of M31 does not show star forming activity
(Ford et al., 2013). This means that the correlation between nonthermal and
MIR emission cannot be explained by the presence of young massive stars. To
understand the correlation with the M31 central region, we used the IRAC-
Spitzer map at 3.6 $\mu$m, which is a very good tracer of low-mass, old stars.
We know that the central region contains a dense and compact star cluster at
its very center Lauer et al. (1993). Olsen et al. (2006) found that the M31
bulge and inner disk contain a great number of old($>5Gyr$), nearly solar-
metallicity stars.

The correlation parameters between radio-thermal or nonthermal and 3.6 $\mu m$
emission were found to be of 0.93 and 0.89, respectively, confirming the
expected strong correlations. For the central region, the correlation between
the free-free and the 24 $\mu$m emission gives a slope that is significantly
higher than 1. The deviation from linear in this region is in line with what
we find in Sect. 7: an excess of thermal emission in the center that is likely
not due to star formation activity. The absence of young ionizing stars does
not guarantee that the correlation between thermal and MIR emissions is
linear. The high density of stars in the bulge, although of small mass, can
provide an explanation to the correlation observed between the free-free and
24 $\mu$m emissions even if the explanation of the power-law exponent of close
to 2 remains an open question.
It is known that young stars and binary systems can generate synchrotron
emission (Blomme, 2011). Olsen et al. (2006) found that the M31 bulge only
hosts old stars, and so only binary systems can provide an explanation for
synchrotron emission. However, there is no evidence for the presence of a
large number of binary systems in the M31 bulge. Most probably, as mentioned
in Sect. 7, the presence of nonthermal emission within the central region of
M31 is due to the existence of shocks which accelerate cosmic ray electrons.
Another possibility is that the given nonthermal emission is generated by
energetic events at the center of the galaxy, similar to the ones that are
responsible for the Haze emission within our Galaxy (Finkbeiner, 2004; Planck
Collaboration et al., 2013). The physical mechanism at the origin of the Haze
nonthermal emission is not unanimously agreed upon in the literature, and a
variety of scenarios have been suggested. One of the most intriguing proposals
is that this signal is generated by cosmic rays injected in the galactic halo
by annihilation or decay of dark matter (Hooper et al., 2007; Egorov et al.,
2016). A radio emission from M31 Haze has never been detected, but its
$\gamma$-ray counterpart, the Fermi-Bubbles, have been observed (Pshirkov et
al., 2016). In addition to this, what was found for the Haze synchrotron
spectral index at the center of the Milky Way is $\alpha_{sync}\simeq 0.8$
(Guidi et al. in prep.), which is very similar to what we find for the central
region of M31 $\alpha_{sync}=0.83$ (see Sect. 6.1). This is also compatible
with previous single-dish telescope observations that measure, for the
Galactic center, a synchrotron emission characterized by a broken power law,
transitioning from $\alpha=0.25$ to $\alpha=1.1$ at a break frequency of
approximately $\sim$ 3 GHz (Heywood et al., 2019). Furthermore, this kind of
radio nonthermal emission due to energetic events at the center of the galaxy
has also been observed outside the Milky Way (Li et al., 2019), so it is not
only a Galactic mechanism. In this latter study, the best-fit spectral slope
in C-band was found to be $1.11\pm 0.34$, which is compatible with our value.
Nonthermal emission in the M31 central region seems to be supported by some
polarization in its emission measured with Effelsberg (Berkhuijsen et al.,
2003). This emission will be investigated further in a follow-up paper, which
will exploit polarized data (Murgia et al. in prep.).

##  9 Logarithmic infrared-to-radio ratio (q)

Another possible way to study the radio–IR correlation uses the logarithmic
IR–radio ratio, defined as:

$q_{\lambda}\equiv
log\left(\frac{F_{\lambda}(Jy/beam)}{F_{21cm}(Jy/beam)}\right).$ (12)

Below we present our study of this parameter for M31, correlating our and
EFF20 data with MIR data within different annular regions and the whole
galaxy. Our results are compared with those of Murphy et al. (2006), where
four galaxies were observed both at radio and MIR wavelengths. Then, we
compute the IR–radio correlation parameter $q_{FIR}$, following the convention
of Helou et al. (1985). Finally, the result is compared with the value found
in the literature over several galaxies.

###  9.1 $q_{24\mu m}$

Firstly, we focused on MIR, correlating radio and 24 $\mu$m data. To extract
$q_{24\mu m}$, we used the EFF20 map and the Spitzer 24 $\mu$m map both
convolved to $3.0$′. This ratio map is shown in Fig. 29 and was extracted
using a 3$\sigma$ cut-off in the EFF20 map convolved to $3^{\prime}$. We note
that within the ring, the logarithmic IR-to-radio ratio index assumes greater
values in the star forming regions. If MIR and radio emission are not linearly
correlated, $q_{24\mu m}$ depends on the radio intensity. Indeed, if
$F_{\lambda}\propto aF_{21cm}^{1+b}$ and b is nonzero, then q assumes the form
$q_{24\mu m}=log(a)+b\cdot log(F_{21cm})$. In Fig. 29 it is evident,
especially in the 10 kpc ring, that $q_{24\mu m}$ assumes higher values where
radio emission is greater, confirming that MIR and radio emissions are not
exactly linearly correlated.

Figure 29: Logarithmic IR–radio ratio map obtained with EFF20 and Spitzer
$24\mu m$ maps, both convolved to $3^{\prime}$.

Using the aperture photometry masks reported in Table 5, we extracted the mean
($q_{24\mu m}$) and dispersion ($\sigma_{24\mu m}$) of the logarithmic
IR–radio ratio for M31 at several galactocentic distances. The dispersion is
the standard deviation within the region normalized with the square root of
the pixel number in the single beam. The results are presented in Table 9 and
shown in Fig. 30. We also report in the last column of Table 9 the $q_{24\mu
m}$ radial values extracted using the SRT $6.6\,GHz$ map convolved to 3′ and
extrapolated to 21 cm.

Ring no. | $q_{24\mu m\,EFF20\,\,3.0^{\prime}}$ | $q_{24\mu m\,SRT_{extrap.}}$
---|---|---
1 | $1.62\pm 0.03$ | $1.62\pm 0.03$
2 | $1.65\pm 0.04$ | $1.63\pm 0.04$
3 | $1.42\pm 0.07$ | $1.45\pm 0.07$
4 | $1.34\pm 0.06$ | $1.41\pm 0.07$
5 | $1.34\pm 0.06$ | $1.38\pm 0.06$
6 | $1.22\pm 0.07$ | $1.26\pm 0.07$
7 | $1.10\pm 0.07$ | $1.10\pm 0.07$
Table 9: Log IR–RC estimate for different galaxy regions using EFF20 map
convolved to $3^{\prime}$ and SRT $6.6\,GHz$ map extrapolated to 1.4 GHz. A
decreasing $q_{24\mu m}$ trend can be seen as well as an increasing
$\sigma_{24\mu m}$ trend both for real and extrapolated data. Figure 30: The
$q_{24\mu m}$ parameter vs. galactocentric distance.

In both cases we find a clear radial trend, the logarithmic IR-to–radio ratio
decreases with increasing galactocentric distance. A similar trend was also
found by Murphy et al. (2006) who, using Spitzer MIPS and WSRT data, observed
a decrease in 24 $\mu m$ to 21 cm ratios with increasing radius. As explained
by these latter authors, this decreasing trend can be justified by smaller
scale lengths for the IR disks than the radio disks. The radio image of a
galaxy can be thought of as the smoothed version of the MIR one. Murphy et al.
(2006) also observed an increase in the logarithmic IR-to-radio ratio
dispersion with increasing galactocentric radius. This result is also observed
in M31. This seems to confirm that it is not an artefact due to lower S/N at
larger radii. This increasing trend is most likely due to the decrease in
surface brightness and Hii region density moving out radially into the disks.
Remarkably, in the outer regions, the presence of a small brighter region or
an Hii can cause a larger dispersion.

We estimated the global $q_{24\mu m}$ by repeating the aperture photometry
analysis on the whole galaxy. Using the SRT extrapolated map we find $q_{24\mu
m}=1.30\pm 0.09$, while using the EFF20 map at its proper angular resolution,
we find $q_{24\mu m}=1.26\pm 0.09$. These values are in agreement with the
ones derived by Murphy et al. (2006) within four different galaxies.

###  9.2 $q_{FIR}$

In this section we extract another correlation parameter between FIR and radio
continuum data. Following the convention of Helou et al. (1985):

$q_{FIR}\equiv log_{10}\left(\frac{FIR}{3.75\cdot
10^{12}\,Wm^{-2}}\right)-log_{10}\left(\frac{S_{1.4\,GHz}}{Wm^{-2}Hz}\right),$
(13)

where FIR, in units of W m-2, is the total FIR flux density calculated as:

$FIR\equiv 1.26\cdot 10^{-14}\left(\frac{2.58\,F_{60\mu m}+F_{100\mu
m}}{Jy}\right),$ (14)

and where $F_{60\mu m}$ and $F_{100\mu m}$ are flux densities. Helou et al.
(1985) found that the mean $q_{FIR}$ over a large sample of different galaxies
is $\sim 2.3$ with a scatter of $\sigma_{q_{FIR}}\leq 0.2$.

We used the $IRAS$ data at 60 $\mu m$ with an angular resolution of $4.0$′,
the Herschel data at 100 $\mu$m with an angular resolution of $0.21$′, and the
SRT map at 1.385 GHz made by Melis et al. (2018) with an angular resolution of
$13.9$′. In Battistelli et al. (2019), SRT contours at 1.385 GHz were
overlayed on the 6.6 GHz map (their Fig. 1) and this map convolved to
$51.3\,\arcmin$ is shown in Fig. 2. After the resolution matching between the
previous two IR maps and the SRT L-band map, we extracted the galaxy flux
densities by integrating over an ellipse of major semi-axis $65^{\prime}$,
eccentricity 0.26, and position angle $-52$ deg in the $RA$-$DEC$ coordinate
system. We find $q_{FIR}=2.41\pm 0.04$, a value that is compatible with the
one reported in Helou et al. (1985) ($q_{FIR,1985}=2.37$).

##  10 Conclusions

We used the 64 m SRT to observe M31 at 6.6 GHz down to a sensitivity of 0.43
mJy/beam averaging over the whole bandwidth of 1.2 GHz. We mapped the galaxy
with an angular resolution of $2.9^{\prime}$ over a field of $RA\times
DEC=$$\times$$$ along the minor and major axis, respectively, corresponding to
a physical size of $32$ kpc $\times\,\,42$ kpc (Fig. 1). We subtracted a total
of 93 compact sources detected within the given field with a S/N greater than
3 (Fig. 4). The large number of detected point sources gave us the opportunity
to create a catalog and to study the differential source counts, comparing the
results with a theoretical model (Fig. 7). With more than 100 SNR or SNR
candidates known in the M31 field, we compared our source sample with the SNR
positions. We find 5 SNR candidates in our catalog within a $1.45^{\prime}$
search radius.
Our map at 6.6 GHz and the ancillary Effelsberg maps at 4.85 and 1.46 GHz were
used to extract the spectral index map at 3’ angular resolution (Fig. 10).
Including also the 2.7 GHz Effelsberg map we studied the spectral index trend
within the central region and over the whole galaxy (Figs. 11 and 12) and we
disentangled the thermal and nonthermal integrated emission in the northern
and southern ring regions, separately (Figs. 16 and 17). We find that the
nonthermal emission is relatively homogeneous and smooth over the whole ring,
while the thermal emission at 6.6 GHz is characterized by well-defined bright
regions following the distribution of Hii regions. Finally, we extracted a
thermal and nonthermal map with an angular resolution of $5^{\prime}$ (Fig.
18). Using the thermal map at 6.6 GHz, we extracted the SFR from the whole
galaxy (Fig. 20). We studied the SFR trend versus galactocentric radius and
the integrated SFR over the whole galaxy (Fig. 23). We compared our SFR map
with the one by Ford et al. (2013), finding a linear correlation within the
ring between the two maps (Fig. 22). We estimated the integrated M31 SFR to be
$0.19\pm 0.01M_{\odot}/yr$, a compatible value with Ford et al. (2013).
Finally, we correlated microwave data with IR observations of M31. Correlation
in the ring of M31 between thermal and MIR emission is characterized by a
power law of $0.96\pm 0.02$, indicating that the relationship is linear.
Interestingly, in the central region we find a deviation from linear in the
thermal–MIR correlation. This can be explained by the thermal emission
mechanism in the central region not being totally related to star formation
activity. We extracted the logarithmic IR-to-radio ratio $q_{24\mu m}$. We
find a decreasing trend with increasing galactocentric radius and an
increasing trend for the dispersion ratio (Fig. 30). Finally, we confirmed the
Helou et al. (1985) correlation with FIR data, using the maps at 60 $\mu m$
and 100 $\mu m$ and radio-continuum data at 1.4 GHz, finding $q_{FIR}=2.41\pm
0.04$ while the expected value is $\sim 2.3$.

We summarize the main results obtained in this work as follows.

  1. 1.

The SRT map at 6.6 GHz over $\sim 7.4$ deg2 contains 93 compact sources with
$S/N>3$. We compared the normalized differential counts with the theoretical
model by Bonato et al. (2017), and found excellent agreement between
experimental data and theoretical predictions.

  2. 2.

In the central region, we find $\alpha_{sync}\sim 0.8$, and therefore a
younger accelerated electrons population with respect to the ring. The
population of cosmic ray electrons was confirmed to be old in the M31 ring,
$\alpha_{sync}\sim 1.1$.

  3. 3.

We find that the galaxy presents an active injection of relativistic electrons
in the northeast region of the ring and also in the central region, while the
southwest region of the ring mainly hosts old electrons.

  4. 4.

We find smooth nonthermal emission in the ring which confirms that nonthermal
emission profiles are uniform along the ring and that magnetic fields in the
ring must be almost constant in strength. Thermal emission was found to be
much more irregular, with variations even of two orders of magnitude in the
ring.

  5. 5.

The total SFR integrated within a radius of 15 kpc is $0.19$ M⊙ yr-1, which is
in agreement with the findings of Ford et al. (2013). The different star
formation morphology in the inner regions between 2 kpc and 5 kpc is due to
the nondetection of the M31 spiral arms with radio observations of the M31
spiral arms. On the other hand, the disagreement in the central region at
galactocentric distances smaller than 2 kpc is due to an excess of radio
thermal emission in this region of the galaxy. This probably originates from
old low-mass stars rather than star forming regions.

  6. 6.

Thermal emission in the M31 central region was found to be strongly correlated
with IR 24 $\mu m$ data and 3.6 $\mu m$. These correlations seem to suggest
that thermal emission within the central region is due to a high number of old
stars. In any case, old stars cannot be responsible for nonthermal emission,
so this is most likely due to shocks.

  7. 7.

The correlation between MIR and free-free emission within the ring and the
central region shows two different slopes, which could mean that these
emissions have different origins. Radio-thermal emission in the ring is
confirmed as being due to star forming processes, while in the central region,
free-free emission can be correlated to a high density of Sun-like old stars.

  8. 8.

We found a high correlation between nonthermal and MIR emission in the ring
region, with a correlation parameter of $r=0.78$. This correlation is most
likely linked to the formation of massive stars.

  9. 9.

We found a nonthermal–MIR correlation in the central region that is even
stronger than the same correlation found in the ring, with $r=0.93$. This was
unexpected, as the bulge is a low-SFR region. Polarized data and further radio
observations will be necessary to investigate the origin of this nonthermal
emission in the central region of M31.

  10. 10.

The correlation between FIR and radio-continuum was computed following the
convention by Helou et al. (1985). We find $q_{FIR}=2.41\pm 0.04$, in
agreement with 2.3 (i.e., the average value over a sample of galaxies).

This work provides a brand new insight into M31, unveiling the galaxy
morphology and physical properties at a frequency that is unprecedented. The
study will be continued using new data, already in our hands, for the same
region of sky.

## Acknowledgements

We acknowledge the support by 2016 Sapienza Ateneo project.
The Sardinia Radio Telescope (SRT) is funded by the Ministry of University and
Research (MIUR), Italian Space Agency (ASI), and the Autonomous Region of
Sardinia (RAS), the European Union (EU) and is operated for the National
Institute for Astrophysics (INAF) by Cagliari Observatory (OAC).
This research made use of Astropy,111111http://www.astropy.org a community-
developed core Python package for Astronomy (Astropy Collaboration et al.,
2013, 2018).
This research made use of Montage. It is funded by the National Science
Foundation under Grant Number ACI-1440620, and was previously funded by the
National Aeronautics and Space Administration’s Earth Science Technology
Office, Computation Technologies Project, under Cooperative Agreement Number
NCC5-626 between NASA and the California Institute of Technology.
J.F. acknowledges financial support from the UNAM- DGAPA-PAPIIT IN111620
grant, México.
We thank the referee for the useful comments that helped us improving the
paper.
We thank Mark Halpern for useful suggestions which improved the paper.

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##  Appendix A Compact sources

We present in this section the fit of the spectra of 93 point sources detected
in the SRT C-band map; see Fig. 31. Each panel is composed of ancillary radio
flux densities (red points) and SRT C-band flux densities extracted in this
work (green points), and the best fit is represented by the blue line. The
experimental data have been fitted with the modified power law reported in Eq.
2. Moreover, at the top of each panel we report the best-fit parameters for
each spectra, at the bottom the point source coordinates, and on the right the
“local” spectral indexes at 0.15, 1.4, and 6.6 GHz. Above each panel is
reported the point source identification in the NVSS catalog that we
constructed during the point sources extraction; see Sect. 4 for further
details.

Table LABEL:Tab:SRT_Sources_catalogue shows all the details of our point-
source sample. The SRT catalog includes the source identification in our NVSS
catalog in the M31 field, the name of the nearest object (i.e., the most
probable match), the source position on the sky, the SRT C-band flux density,
the modified power-law fit parameters, and the spectral index measured between
1.4 and 6.6 GHz $\alpha_{1.4GHz}^{6.6\,GHz}$. The data at 1.4 GHz refer to the
NVSS catalog. The given position is the position of the peak of the 2D
Gaussian that best fits the SRT source emission. The value of the flux density
corresponds to the value of the Gaussian peak.

Finally, the $Nearest-object$ is the name of the closest object that appears
in other catalogs, together with the object type ( [*] for star or point
source, [G] for galaxy, [UvS] for ultraviolet source, [IrS] for infrared
source, [SNR] for supernova remnant, and [*Cl] for star cluster). The search
was made using the NED/IPAC121212https://ned.ipac.caltech.edu/ extragalactic
database and using a search radius of 1 beam ($2.9^{\arcmin}$). We decided to
report the B3, 4C, or NVSS source name if it is available, otherwise we used
the WISEA, SSTLS2, or 2MASS catalog.

Figure 31:  Spectral fit of the 93 compact sources detected in our C-band map.
Green points are the new flux densities extracted in this work. Red points are
the ancillary data. Each figure shows the compact source RA-DEC coordinates
(bottom left), the fit parameters (upper side), and the model spectral indices
at 150, 1400, and 6600 MHz (right side).

Table 10: The SRT C-band Compact Source catalog. N. | Nearest Obj. | RA(∘) | DEC(∘) | Flux density(Jy) |  $A$(Jy) | $\alpha$ | $k$ | $\alpha_{1.4\,GHz}^{6.6\,GHz}$
---|---|---|---|---|---|---|---|---
3 | 4C +39.03 [RadioS] | 10.57 | 40.16 | $0.153\pm 0.015$ | 2.397 | 1.26 | -0.40 | 1.01
4 | B3 0035+413 [QSO] | 9.60 | 41.62 | $0.388\pm 0.039$ | 0.559 | 0.08 | 0.00 | 0.37
5 | 4C +42.03 [RadioS] | 11.16 | 42.80 | $0.094\pm 0.009$ | 1.067 | 1.16 | -0.25 | 1.02
6 | 4C +40.03 [G] | 12.05 | 40.36 | $0.117\pm 0.012$ | 1.223 | 1.15 | -0.20 | 1.06
7 | B3 0039+412 [G] | 10.58 | 41.49 | $0.096\pm 0.01$ | 0.682 | 0.96 | -0.15 | 0.86
23 | B3 0041+405 [RadioS] | 10.98 | 40.78 | $0.062\pm 0.006$ | 0.226 | 0.75 | -0.01 | 0.64
27 | B3 0045+404 RadioS] | 12.18 | 40.75 | $0.056\pm 0.006$ | 0.343 | 0.98 | -0.11 | 0.89
33 | B3 0044+418 [G] | 11.70 | 42.15 | $0.032\pm 0.003$ | 0.249 | 1.01 | -0.12 | 0.95
39 | B2 0046+39 [G] | 12.19 | 39.89 | $0.039\pm 0.004$ | 0.218 | 0.91 | -0.12 | 0.81
41 | B3 0036+409 [RadioS] | 9.70 | 41.27 | $0.03\pm 0.003$ | 1.171 | 1.44 | -0.96 | 0.86
48 | B3 0041+416 [G] | 11.00 | 41.95 | $0.019\pm 0.002$ | 0.227 | 1.20 | -0.16 | 1.14
49 | GALEXMSC J004755.61+394903.6 [UvS] | 11.98 | 39.82 | $0.036\pm 0.004$ | 0.060 | 0.25 | 0.20 | 0.62
52 | B3 0043+405 [RadioS] | 11.51 | 40.80 | $0.01\pm 0.002$ | 0.403 | 1.58 | -0.56 | 1.34
56 | B3 0039+416 [*] | 10.65 | 41.96 | $0.018\pm 0.002$ | 0.095 | 0.83 | -0.02 | 0.94
58 | B3 0036+415 [RadioS] | 9.87 | 41.77 | $0.016\pm 0.002$ | 0.121 | 1.01 | -0.06 | 1.05
63 | B3 0037+409 [RadioS] | 9.98 | 41.19 | $0.019\pm 0.002$ | 0.128 | 0.92 | -0.11 | 0.93
74 | B3 0040+406 [RadioS] | 10.92 | 40.91 | $0.016\pm 0.002$ | 0.099 | 0.98 | -0.04 | 0.94
79 | B3 0035+401 [RadioS] | 9.44 | 40.42 | $0.011\pm 0.002$ | 0.183 | 1.39 | -0.27 | 1.14
81 | B3 0042+406 [RadioS] | 11.27 | 40.90 | $0.011\pm 0.002$ | 0.110 | 1.15 | -0.11 | 1.11
85 | B3 0037+396 [RadioS] | 10.07 | 39.92 | $0.011\pm 0.002$ | 0.189 | 1.34 | -0.40 | 1.07
87 | NVSS J004653+403219 [RadioS] | 11.72 | 40.54 | $0.011\pm 0.002$ | 0.080 | 1.00 | -0.03 | 1.09
88 | B3 0034+413 [RadioS] | 9.27 | 41.60 | $0.018\pm 0.002$ | 0.062 | 0.74 | 0.15 | 0.82
93 | B3 0043+422 [RadioS] | 11.65 | 42.53 | $0.008\pm 0.002$ | 0.138 | 1.25 | -0.34 | 1.17
97 | B3 0036+398 [G] | 9.89 | 40.14 | $0.014\pm 0.002$ | 0.065 | 0.82 | -0.01 | 0.82
101 | NVSS J004141+410333 [RadioS] | 10.42 | 41.06 | $0.026\pm 0.003$ | 0.031 | 0.27 | 0.31 | 0.40
105 | NVSS J004147+411848 [RadioS] | 10.45 | 41.31 | $0.008\pm 0.002$ | 0.128 | 1.32 | -0.23 | 1.15
106 | B3 0043+410 [RadioS] | 11.61 | 41.33 | $0.013\pm 0.002$ | 0.058 | 0.82 | 0.09 | 0.90
110 | WISEA J004848.46+424112.5 [RadioS] | 12.20 | 42.69 | $0.017\pm 0.002$ | 0.102 | 0.84 | -0.17 | 0.78
111 | B3 0035+416 [RadioS] | 9.63 | 41.88 | $0.01\pm 0.002$ | 0.095 | 1.04 | -0.18 | 1.05
112 | B3 0039+420 [RadioS] | 10.55 | 42.28 | $0.013\pm 0.002$ | 0.135 | 1.08 | -0.34 | 0.83
113 | B3 0044+404 [RadioS] | 11.85 | 40.74 | $0.034\pm 0.003$ | 0.025 | 0.06 | 0.39 | 0.20
116 | B3 0038+399 [G] | 10.35 | 40.21 | $0.021\pm 0.002$ | 0.245 | 1.06 | -0.37 | 0.89
119 | B3 0042+411 [RadioS] | 11.26 | 41.41 | $0.01\pm 0.002$ | 0.073 | 0.94 | -0.11 | 0.98
120 | NVSS J004013+410845 [RadioS] | 10.06 | 41.15 | $0.014\pm 0.002$ | 0.068 | 0.80 | -0.03 | 0.87
126 | B3 0037+405 [QSO] | 10.06 | 40.83 | $0.018\pm 0.002$ | 0.047 | 0.57 | 0.10 | 0.64
136 | NVSS J003909+422736 [RadioS] | 9.79 | 42.46 | $0.008\pm 0.002$ | 0.064 | 0.96 | -0.15 | 0.99
139 | B3 0034+417 [RadioS] | 9.34 | 42.02 | $0.006\pm 0.002$ | 0.048 | 0.96 | -0.01 | 1.12
145 | WISEA J004139.28+413034.9 [G] | 10.41 | 41.51 | $0.012\pm 0.002$ | 0.059 | 0.77 | -0.13 | 0.66
146 | B3 0042+409A [RadioS] | 11.30 | 41.19 | $0.008\pm 0.002$ | 0.046 | 0.86 | 0.01 | 0.95
154 | B3 0042+409B [RadioS] | 11.42 | 41.20 | $0.007\pm 0.002$ | 0.062 | 1.08 | -0.10 | 1.08
160 | WISEA J004638.18+405425.1 [RadioS] | 11.66 | 40.91 | $0.008\pm 0.002$ | 0.039 | 0.78 | 0.01 | 0.88
177 | B3 0038+406 [RadioS] | 10.23 | 40.96 | $0.007\pm 0.002$ | 0.042 | 1.04 | 0.01 | 0.90
184 | WISEA J004536.73+403003.6 [G] | 11.40 | 40.50 | $0.009\pm 0.002$ | 0.012 | 0.23 | 0.32 | 0.68
185 | SSTSL2 J004150.78+411439.1 [IrS] | 10.46 | 41.24 | $0.005\pm 0.002$ | 0.024 | 0.84 | 0.19 | 1.03
196 | NVSS J004624+402215 [RadioS] | 11.60 | 40.37 | $0.006\pm 0.002$ | 0.041 | 0.86 | -0.13 | 0.88
204 | NVSS J004815+402341 [RadioS] | 12.07 | 40.39 | $0.01\pm 0.002$ | 0.029 | 0.55 | 0.00 | 0.54
208 | WISEA J004344.69+412843.7 [*Cl] | 10.94 | 41.48 | $0.004\pm 0.002$ | 0.034 | 0.98 | -0.02 | 1.20
216 | NVSS J004737+405522 [RadioS] | 11.91 | 40.92 | $0.007\pm 0.002$ | 0.020 | 0.66 | 0.20 | 0.81
224 | WISEA J004410.47+422549.3 [RadioS] | 11.04 | 42.43 | $0.009\pm 0.002$ | 0.022 | 0.36 | 0.00 | 0.56
232 | B3 0045+423 [RadioS] | 12.04 | 42.62 | $0.009\pm 0.002$ | 0.055 | 0.90 | -0.09 | 0.90
235 | 2MASS J00425141+4126344 [RadioS] | 10.71 | 41.44 | $0.013\pm 0.002$ | 0.023 | 0.25 | -0.05 | 0.42
241 | WISEA J003959.01+401343.1 [RadioS] | 10.00 | 40.23 | $0.007\pm 0.002$ | 0.035 | 0.91 | -0.09 | 0.71
245 | NVSS J003807+422413 [RadioS] | 9.53 | 42.40 | $0.01\pm 0.002$ | 0.055 | 0.81 | -0.08 | 0.95
248 | B3 0043+398 [G] | 11.48 | 40.11 | $0.003\pm 0.002$ | 0.128 | 1.26 | -0.51 | 1.48
253 | 2MASX J00473217+4004212 [G] | 11.88 | 40.07 | $0.008\pm 0.002$ | 0.036 | 0.66 | -0.23 | 0.61
254 | NVSS J004648+411408 [RadioS] | 11.70 | 41.24 | $0.006\pm 0.002$ | 0.178 | 1.49 | -0.73 | 0.96
255 | NVSS J003643+404009 [RadioS] | 9.18 | 40.67 | $0.006\pm 0.002$ | 0.064 | 1.00 | -0.44 | 0.76
258 | NVSS J003645+402700 [RadioS] | 9.19 | 40.45 | $0.006\pm 0.002$ | 0.051 | 0.94 | -0.13 | 1.09
266 | NVSS J004119+411043 [RadioS] | 10.33 | 41.18 | $0.008\pm 0.002$ | 0.013 | 0.41 | 0.32 | 0.60
272 | NVSS J003838+412838 [RadioS] | 9.66 | 41.48 | $0.007\pm 0.002$ | 0.016 | 0.51 | 0.14 | 0.64
273 | SSTSL2 J004035.81+413510.7 [RadioS] | 10.15 | 41.59 | $0.012\pm 0.002$ | 0.020 | 0.28 | 0.00 | 0.27
274 | WISEA J004835.31+401435.8 [RadioS] | 12.15 | 40.24 | $0.01\pm 0.002$ | 0.017 | 0.34 | 0.10 | 0.37
284 | WISEA J004054.80+395317.2 [RadioS] | 10.23 | 39.89 | $0.008\pm 0.002$ | 0.012 | 0.34 | 0.23 | 0.55