# Radio-to-Submillimetre Spectral Energy Distributions of NGC 1365

Guangwen Chen1,2,3, George J. Bendo4, Gary A. Fuller4,5, Hong-Xin Zhang1,2,
and Xu Kong1,2,6
1Department of Astronomy, University of Science and Technology of China, Hefei
230026, China
2School of Astronomy and Space Sciences, University of Science and Technology
of China, Hefei 230026, China
3Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy,
The University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom
4UK ALMA Regional Centre Node, Jodrell Bank Centre for Astrophysics,
Department of Physics and Astronomy, The University of Manchester,
Oxford Road, Manchester M13 9PL, United Kingdom
5I. Physikalisches Institut, University of Cologne, Zülpicher Str. 77, 50937
Köln, Germany
6Frontiers Science Center for Planetary Exploration and Emerging Technologies,
University of Science and Technology of China, Hefei, Anhui, 230026, China
E-mail<EMAIL_ADDRESS>
(Accepted 2024 March 21. Received 2024 March 18; in original form 2024 January
24)

###### Abstract

We analyse the radio-to-submillimetre spectral energy distribution (SED) for
the central pseudobulge of NGC 1365 using archival data from the Atacama Large
Millimeter/submillimeter Array (ALMA) and the Very Large Array (VLA). This
analysis shows that free-free emission dominates the continuum emission at
50–120 GHz and produces about 75 per cent of the 103 GHz continuum emission.
However, the fraction of 103 GHz continuum emission originating from free-free
emission varies significantly among different subregions in the pseudobulge,
particularly for an outflow from the AGN on the eastern pseudobulge where the
synchrotron emission produces half of the 103 GHz continuum emission. Free-
free emission also dominates at 103 GHz within the central 400 pc diameter
region, but this emission is associated with the AGN rather than star
formation. The star formation rate (SFR) within the pseudobulge derived from
the ALMA free-free emission is $8.9\pm 1.1$ M⊙ yr-1. This is comparable to the
SFR from the mid-infrared emission but higher than the SFR from the
extinction-corrected H$\alpha$ line emission, mainly because the pseudobulge
is heavily dust obscured. The 1.5 GHz emission yields a comparable SFR for the
pseudobulge but may have lower SFRs within subregions of the pseudobulge
because of the diffusion outside of these regions of the electrons producing
the synchrotron radiation. We propose that applying a correction factor of 75
per cent to the 80–110 GHz continuum emission could provide valuable estimates
of the free-free emission without performing any SED decomposition, which
could derive extinction-free SFRs within 20 per cent accuracy.

###### keywords:

galaxies: individual: NGC 1365 – galaxies: starburst – galaxies: star
formation – radio continuum: galaxies.

††pubyear: 2024††pagerange: Radio-to-Submillimetre Spectral Energy
Distributions of NGC 1365–B

##  1 Introduction

The spectral energy distributions (SEDs) of galaxies from the radio to the
submillimetre include contributions from synchrotron emission, free-free
emission and thermal dust emission (as well as potentially more exotic forms
of emission such as anomalous microwave emission). Synchrotron emission mainly
originates from active galactic nuclei (AGNs) and/or supernova remnants, free-
free emission arises from the gas surrounding photoionizing stars, and thermal
dust emission emanates from interstellar dust (Condon, 1992). All of these
forms of emission within this frequency range are generally unaffected by dust
obscuration. Therefore, the decomposition of the SEDs can provide a direct
approach to understanding the AGNs and star formation within galaxies.

However, the analysis of SEDs from the radio to the submillimetre continuum
emission have only been published for a small number of galaxies using
relatively heterogeneous data, including M31 (Planck Collaboration et al.,
2015; Battistelli et al., 2019; Harper et al., 2023; Fernández-Torreiro et
al., 2024), M82 (Condon, 1992; Peel et al., 2011), NGC 253 (Peel et al., 2011;
Bendo et al., 2015), NGC 1808 (Salak et al., 2017; Chen et al., 2023), NGC
3256 (Michiyama et al., 2020) and NGC 4945 (Peel et al., 2011; Bendo et al.,
2016). Only Michiyama et al. (2020) and Chen et al. (2023) discussed
variations of the fractions of the total millimetre continuum emission from
synchrotron emission, free-free emission and thermal dust emission within
different regions in the individual galaxies that they studied. Linden et al.
(2019) and Linden et al. (2020) presented analyses of spectral indices for a
larger sample of local luminous infrared galaxies (LIRGs; $L_{\rm
IR}>10^{11}L_{\odot}$) based on 3, 15 and 33 GHz continuum emission and
revealed the connection between synchrotron radiation and free-free emission
within this frequency range for their galactic nuclei and for extranuclear
star-forming regions. However, their data could not fully constrain the shape
of the SEDs and the proportions of thermal dust emission, free-free emission
and synchrotron emission in the submillimetre and millimetre bands. Clearly,
the scientific literature still lacks detailed information on the radio-to-
submillimetre SEDs of galaxies in spite of the importance of these bands in
terms of measuring star formation rates and studying AGN activity.

In this paper, we report an analysis of the 2–246 GHz SED for the central
pseudobulge of NGC 1365 based on data from the Atacama Large
Millimeter/submillimeter Array (ALMA) and the Very Large Array (VLA). This
object is a barred spiral galaxy located at a distance of $19.6\pm 0.8$ Mpc
(Anand et al., 2021) within the Fornax Cluster (Lindblad, 1999). It hosts a
Seyfert 1.5 - 2 nucleus (e.g., Turner et al., 1993; Maiolino & Rieke, 1995;
Hjelm & Lindblad, 1996; Sakamoto et al., 2007) and, the AGN is well detected
in X-ray observations (e.g., Risaliti et al., 2005; Wang et al., 2009; Swain
et al., 2023). This galaxy has intense star formation and a biconical outflow
in the central region, which is heavily dust obscured (e.g., Venturi et al.,
2017; Fazeli et al., 2019; Gao et al., 2021). The SEDs from the far-infrared
to the near-infrared in the centre at a resolution of 20 arcsec has been
presented by Tabatabaei et al. (2013), and the SEDs from the infrared to the
X-ray for the nucleus has been presented by Swain et al. (2023), but the
radio-to-submillimetre SEDs are still unknown. The galaxy has been identified
as a LIRG (e.g., Armus et al., 2009), with most of the infrared emission
originating from the central pseudobulge. The high surface brightness of this
compact region across multiple bands makes it an ideal object for
submillimetre and millimetre observations. In addition, ancillary data from
the from the Spitzer Space Telescope (Werner et al., 2004) and the Herschel
Space Observatory (Pilbratt et al., 2010) and recent James Webb Space
Telescope (JWST) observations (Whitmore et al., 2023; Schinnerer et al., 2023;
Liu et al., 2023) provide supplemental infrared data for this object.

We present the description of data in Section 2. Continuum images for the
centre of NGC 1365 spanning radio to submillimetre wavelengths are presented
in Section 3. The analysis of the SEDs is described in Section 4, while
Section 5 presents the measurements of star formation rates (SFRs) and
comparisons of SFRs from the free-free emission with the SFRs from the radio
continuum emission and infrared dust emission. A brief summary of our main
results is given in Section 6.

##  2 Data

To accurately characterise the SEDs spanning from 2 to 246 GHz, we use the
National Radio Astronomy Observatory (NRAO) VLA L-, S-, C-, and X-band data
(covering 2–10 GHz) and the ALMA Band 3 and 6 data (covering 100–246 GHz).
These data had $uv$ ranges that overlapped significantly (in terms of
$k\lambda$) that could therefore be used to create SEDs with comparable beam
sizes and maximum recoverable scales. The ALMA and VLA archives as well as the
archive for the Australia Telescope Compact Array (ATCA) contained additional
data covering other frequencies, but these data differed from the data listed
above in terms of $uv$ coverage and thus yield images with either beam sizes
that are too large or maximum recoverable scales that are too small for the
analysis in the main part of this paper. We instead used these data in
supplemental SED analyses in Appendices A and B. 24 $\mu$m data from the
Spitzer Space Telescope are used to calculate alternate SFRs for comparisons.

###  2.1 ALMA data

Archival ALMA data111https://almascience.eso.org/aq/ from multiple projects
are used in our analysis; see Table 1 for details. We did not use the Band 7
and 8 data in constructing the SEDs that are analyzed in Section 4 because
their maximum recoverable scales are not large enough to cover the whole
circumnuclear region within the galaxy. As seen in that section, the thermal
dust emission is the dominant source of continuum emission in ALMA Band 6.
However, in Appendix A, we examined how including Band 7 and 8 data affected
the SED fitting. ALMA project 2019.2.00134.S also acquired Band 3 data, but we
did not use these data because the beam size of the data is much larger in
comparison to the other archival data in our analysis. In addition, we did not
include any ALMA total power observations in our analysis because those data
cannot be used for continuum imaging.

Table 1: General information of ALMA observationsa.

Band | Project | Array | Scheduling | 5%$-$80% of | Angular | Maximum | Frequency | casa
---|---|---|---|---|---|---|---|---
| code |  | Block | baselinesb | resolution | recoverable scale | range | version for
|  |  |  | (m) | (arcsec) | (arcsec) | (GHz) | calibrationc
3 | 2015.1.01135.S | 12 m | NGC1365_a_03_TE | 43$-$225 | 1.4 | 13.4 | 98.97-102.94, 111.52-114.88 | 4.7.0
| 2017.1.00129.S | 7 m | ESO358-G_a_03_7M | 9$-$32 | 10.0 | 77.8 | 99.90-103.76, 111.84-115.70 | 5.4.0
6 | 2013.1.01161.S | 12 m | NGC1365_a_06_TC | 25$-$144 | 1.0 | 11.5 | 228.46-231.90, 242.67-246.22 | 4.3.1
|  |  | NGC1365_a_06_TE | 44$-$241 | 0.6 | 6.0 | 228.46-231.90, 242.67-246.22 | 4.3.1
|  |  | NGC1365_b_06_TE | 134$-$785 | 0.2 | 1.9 | 228.46-231.90, 242.67-246.22 | 4.3.1
|  | 7 m | NGC1365_a_06_7M | 7$-$27 | 5.3 | 36.6 | 228.40-231.96, 242.61-246.28 | 4.3.1
| 2021.1.01150.S | 12 m | NGC_1365_a_06_TM1 | 82$-$739 | 0.2 | 3.1 | 225.79-230.23, 241.68-245.67 | 6.2.1
|  |  | NGC_1365_a_06_TM2 | 24$-$152 | 1.0 | 10.8 | 225.79-230.23, 241.68-245.67 | 6.2.1
| 2021.1.01295.S | 7 m | NGC_1365_a_06_7M | 9$-$29 | 4.9 | 30.7 | 226.27-230.30, 240.70-244.67 | 6.2.1

aThis information is retrieved from ALMA Science Archive
(https://almascience.eso.org/aq/).

bThese values are the 5th and 80th percentile of all projected baselines
calculated from the $uv$ distributions of the observations, but not all data
between the longest and shortest baseline are used in continuum imaging (see
also Table 2).

cThis is the specific version of casa used to calibrate the data, which
adheres to the general recommendations for pipeline-calibrated data. casa
version 6.4.1 was used for imaging.

These ALMA visibility data were reprocessed using the common astronomy
software applications (casa; CASA Team et al. 2022). The general
recommendation for recreating the pipeline-calibrated data is to use the same
version of casa that was originally used to calibrate the data; the specific
versions that we used are listed in Table 1. According to the ALMA Technical
Handbook222https://almascience.eso.org/documents-and-tools/cycle10/alma-
technical-handbook (Cortes et al., 2023), the typical flux calibration
uncertainties are 5 per cent for Band 3 and 10 per cent for Band 6 and 7 and
20 per cent for Band 8 data.

These processed visibility data were converted into continuum images using
tclean in casa version 6.4.1. We first created image cubes without any
continuum subtraction to identify the line-free regions in the data. After
this, continuum images were created using visibility data from adjacent
spectral windows with comparable frequencies. When creating these images, we
selected the Högbom cleaning algorithm because it is effective for imaging
data with relatively low signal-to-noise ratios. For the Band 3 and 6 data, we
used natural weighting to recover extended emission and to maximize the
signal-to-noise ratio. In addition, the $uv$ coverages were adjusted so that
all images have the same full width at half-maximum (FWHM) of the restoring
beam (2.1 arcsec) and the same maximum recoverable scale (36 arcsec). Data
with relatively long $uv$ baselines (Band 3 data with baselines $>360$ m and
Band 6 data with baselines $>145$ m) were not used in the continuum imaging.
Notably, the FWHM and maximum recoverable scale are not directly determined by
the upper and lower limits of the $uv$ coverage itself but rather by the
overall $uv$ distribution. For instance, the Band 3 data have a $uv$ coverage
with a large upper limit but fewer antennae on longer baselines, while the
Band 6 data have a $uv$ coverage with a lower upper limit but more antennae on
longer baselines; both dataset yield images that have restoring beams with
similar FWHMs. Consequently, the upper and lower limits of $uv$ coverages in
units of kilolambda (k$\lambda$) may vary sightly across different frequency
bands when creating images with comparable FWHMs and maximum recoverable
scales. Primary beam corrections were applied to all images. The pixel scales
are set as 0.2 arcsec to subsample the beam effectively, and the image
dimensions are set as $640\times 640$ spaxels ($128\times 128$
$\rm{arcsec^{2}}$ (corresponding to $\sim$12 kpc at a distance of 19.6 Mpc) to
cover the primary beam. Details related to the continuum imaging are listed in
Table 2.

Table 2: Information of ALMA continuum imaging. Band | Frequency |  $uv$ range | Beam FWHM
---|---|---|---
| (GHz) | (k$\lambda$) | ($\rm{arcsec^{2}}$)
3 | 102.5 | $5.4-123$ | 2.3 $\times$ 1.8
| 113.5 | $5.4-131$ | 2.3 $\times$ 1.8
6 | 230.5 | $5.4-111$ | 2.2 $\times$ 1.9
| 244.9 | $5.6-97$ | 2.2 $\times$ 1.9

###  2.2 VLA data

The VLA visibility data from projects 17B-331, 18A-418, SH0126, and 21A-282
used for our SED analysis were downloaded from the NRAO
archive333https://data.nrao.edu. We only used data with $uv$ ranges that
allowed us to create images with beam FWHMs and maximum recoverable scales
that are comparable to the ALMA continuum images. For projects 17B-331 and
18A-418, we used the pipeline-calibrated data provided by the NRAO archive,
and for projects SH0126 and 21A-282, we used casa version 5.3.0 with the VLA
scripted calibration pipeline
1.4.2444https://science.nrao.edu/facilities/vla/data-
processing/pipeline/scripted-pipeline to calibrate their visibility data. We
also used the 1.5 GHz visibility data from the NRAO VLA Archive Survey
(NVAS555http://www.vla.nrao.edu/astro/nvas/; Condon et al., 1998; Crossley et
al., 2007, 2008) to create a 1.5 GHz image that could be used to measure SFRs,
but we did not include 1.5 GHz data in our SED analysis in Section 4 for two
reasons. First, the beam is somewhat larger than the beam that we can produce
using the data from the other bands, and this mismatch in beam size will
affect the photometry. Second, data below 2 GHz may cause the synchrotron
emission to appear flatter because the cosmic rays producing the higher
frequency emission are more affected by aging effect (Davies et al., 2006).
The NVAS processed the raw visibility data from 1976 to 2006 with the
Astronomical Image Processing System VLA pipeline to make the archive data
easily accessible to the astronomical community. General information for these
observations is listed in Table 3.

Table 3: General information of VLA observation. Band | Project ID | Date | Frequency range
---|---|---|---
|  |  | (GHz)
L | AS133 | 1983 Nov 06 | 1.46$-$1.51
| AC348 | 1993 Feb 01 | 1.46$-$1.66
S | 21A-282 | 2021 Aug 07 | 2.0$-$3.9
|  | 2021 Dec 31 | 2.0$-$3.9
|  | 2022 Apr 16 | 2.0$-$3.9
C | 17B-331 | 2017 Nov 23 | 4.0$-$7.9
| 18A-418 | 2018 Apr 16 | 4.0$-$7.9
X | SH0126 | 2021 Sep 30 | 8.0$-$9.9

We used clean in casa version 5.6.1 to image the 1.5 GHz data because they
were not in a format that could be read by tclean. We used tclean in casa
version 6.4.1 to image the other VLA data. When creating these images, we
again used the Högbom cleaning algorithm. We only used data with baselines
with lengths (in terms of k$\lambda$) that were similar to what was used for
the ALMA data so as to obtain images with similar beam sizes and maximum
recoverable scales. Given the declination of this source relative to the
location of the VLA, the image exhibits an elongated beam, appearing extended
along the major axis and narrow along the minor axis. To adjust the shape of
beam to be comparable to ALMA Band 3 and 6 images, we first set the weighting
parameter to briggs and the robust parameter to 0.5 to improve the spatial
resolution. Subsequently, we used imsmooth in casa to smooth the beams so that
they were comparable in shape to the beams of ALMA images except for the 1.5
GHz data, which has a relatively large beam if we smooth it. All the final VLA
images have pixel sizes of 0.2 arcsec and dimensions of $640\times 640$
spaxels ($128\times 128$ arcsec2) to match the ALMA images. Based on the
information from the VLA Observational Status Summary
2023A666https://science.nrao.edu/facilities/vla/docs/manuals/oss/performance/fdscale
the flux scaling uncertainties are less than 10 per cent for the L-, C-, S-,
and X-band data. Detailed information related to the continuum imaging is
listed in Table 4.

Table 4: Information of VLA continuum imaging. Band | Frequency |  $uv$ range | Beam FWHM | Beam FWHM
---|---|---|---|---
| (GHz) | (k$\lambda$) | before smoothing | after smoothing
|  |  | ($\rm{arcsec^{2}}$) | ($\rm{arcsec^{2}}$)
L | 1.5 | 3.4$-$183 | 3.4 $\times$ 1.4 | -
S | 2.6 | 3.6$-$307 | 2.6 $\times$ 0.8 | 2.7 $\times$ 1.7
| 3.6 | 3.5$-$425 | 2.0 $\times$ 0.7 | 2.5 $\times$ 1.8
C | 5.0 | 3.4$-$250 | 2.1 $\times$ 0.7 | 2.2 $\times$ 1.9
| 7.0 | 3.4$-$187 | 1.8 $\times$ 0.8 | 2.2 $\times$ 1.9
X | 9.0 | 3.4$-$1054 | 2.3 $\times$ 0.6 | 2.5 $\times$ 1.8

###  2.3 Infrared data

We used mid-infrared data as an alternate star formation tracer for comparison
to those derived from our radio and millimetre data. JWST 21 $\mu$m data are
available for this galaxy (Lee et al., 2023), but those data are heavily
saturated, so we used 24 $\mu$m data from the Spitzer Space Telescope instead.
The specific 24 $\mu$m image that we used was originally created and published
by Bendo et al. (2020) using the MIPS data Analysis Tools version 3.10 (Gordon
et al., 2005) and additional data processing steps described by Bendo et al.
(2012). The Spitzer image has the beam FWHM of 6 arcsec and pixel sizes of 1.5
arcsec, and the flux calibration uncertainty is 4 per cent (Engelbracht et
al., 2007).

##  3 Images

Figure 1: The image of 102.5 GHz continuum emission in the central 48 arcsec
of NGC 1365. The solid green circles identify the locations where we measured
the continuum emission, including the total (T), north (N), east (E), centre
(C) and south (S) with diameters of 28 arcsec (2.7 kpc), 13$\times$8 arcsec
(1240$\times$760 pc), 12$\times$5 arcsec (1140$\times$480 pc), 4 arcsec (380
pc) and 8 arcsec (760 pc), respectively. Smaller apertures within the N, E and
S region, including the N1, N2, N3 and E1 regions with diameters of 4 arcsec
(380 pc) and the E2 region with 7$\times$4 arcsec (670$\times$380 pc), are
also marked as solid green circles. The cyan “X” symbol marks the location of
the AGN, indicated by the peak of X-ray emission (Wang et al., 2009). The
restoring beam is shown as the filled white ellipse in the lower-left corner.

Figure 1 shows the 102.5 GHz continuum emission in the central region of NGC
1365 and the locations where we measured continuum emission from radio to
submillimetre bands for constructing SEDs. We also refer to these regions when
discussing the multiwavelength images qualitatively. The AGN is located within
the centre (C) region. The north (N) region represents the area where the gas
flows in from the northeast along the bar, the south (S) region is the
downstream location of the gas from the N region, and the east (E) region
shows the downstream location where the gas flowing in from the southern arm.
The N, S and E regions are comparable to the “Northern Arm”, “Mid-Southwest
Region” and “Mid-East Region” presented by Liu et al. (2023). In addition, the
N region includes three bright sources, which we labelled as “N1”, “N2” and
“N3”, and the E region includes two relatively faint sources, which we
labelled as “E1” and “E2”. The total (T) aperture with a diameter of 28 arcsec
(2.7 kpc) was used to measure the total detectable emission from the central
region.

Figure 2: Images of the VLA 2.6, 5.0, 9.0 GHz and ALMA 102.5, 113.5 and 244.9
GHz continuum in the central 48 arcsec of NGC 1365. In each panel, the
restoring beam is illustrated as the filled white ellipse at the lower-left
side.

The 2.6 to 244.9 GHz images of the central 48 arcsec of NGC 1365 are displayed
in Figure 2. The 2.6 GHz, 5 GHz and 9 GHz continuum are mainly dominated by
synchrotron radiation (see more in Section 4.2) that originates from the AGN
and supernova remnants. These three images show that the presence of
synchrotron emission from supernovae in the circumnuclear ring, particularly
in the N2, N3 and E2 regions, although fainter emission is seen from the N1
and S regions. Low frequency continuum emission is also detected from the AGN
in the C region, but it is fainter than the sources in the circumnuclear ring.
The C and E2 regions exhibit an outflow or jet-like feature, which is
consistent with previous radio and X-ray observations (e.g., Sandqvist et al.,
1995; Wang et al., 2009; Venturi et al., 2017). The 102.5 GHz and 113.5 GHz
continuum emission is mainly produced by photoionized gas associated with star
formation (Section 4.2). These two images show that the star formation in the
circumnuclear ring is concentrated in the N2 and N3 regions, which also harbor
several massive young star clusters (e.g., Whitmore et al., 2023). We also see
fainter free-free emission from the N1, E1 and S regions. The 244.9 GHz
continuum emission is dominated by thermal dust emission (Section 4.2). At
this frequency, the brightnesses of the various regions within the
circumnuclear ring are similar to what is seen in the 102.5 GHz and 113.5 GHz
images. This is probably because the dust is very closely associated with the
star-forming regions. However, the E1 region is notably bright relative to
other sources at 244.9 GHz, indicating that it is more dusty.

##  4 Spectral Energy Distribution

###  4.1 SED fitting

Table 5: Continuum measurements for SED analysis. Rest frame | Flux density
---|---
frequency | T region | N region | N1 region | N2 region | N3 region | E region | E1 region | E2 region | C region | S region
(GHz) | (mJy) | (mJy) | (mJy) | (mJy) | (mJy) | (mJy) | (mJy) | (mJy) | (mJy) | (mJy)
2.6 | 223 $\pm$ 22 | 64 $\pm$ 6 | 11.9 $\pm$ 1.2 | 13.1 $\pm$ 1.3 | 15.7 $\pm$ 1.6 | 39 $\pm$ 4 | 9.6 $\pm$ 1.0 | 22 $\pm$ 2 | 7.9 $\pm$ 0.8 | 32 $\pm$ 3
3.6 | 173 $\pm$ 17 | 52 $\pm$ 5 | 9.5 $\pm$ 0.9 | 11.2 $\pm$ 1.1 | 13.1 $\pm$ 1.3 | 30 $\pm$ 3 | 7.3 $\pm$ 0.7 | 17.0 $\pm$ 1.7 | 6.3 $\pm$ 0.6 | 25 $\pm$ 2
5.0 | 131 $\pm$ 13 | 42 $\pm$ 4 | 8.0 $\pm$ 0.8 | 9.6 $\pm$ 1.0 | 10.6 $\pm$ 1.1 | 25 $\pm$ 3 | 6.1 $\pm$ 0.6 | 13.6 $\pm$ 1.4 | 5.2 $\pm$ 0.5 | 20 $\pm$ 2
7.0 | 99 $\pm$ 10 | 32 $\pm$ 3 | 6.3 $\pm$ 0.6 | 7.8 $\pm$ 0.8 | 8.4 $\pm$ 0.8 | 19 $\pm$ 2 | 4.7 $\pm$ 0.5 | 10.6 $\pm$ 1.1 | 4.0 $\pm$ 0.4 | 15.4 $\pm$ 1.5
9.0 | 82 $\pm$ 8 | 24 $\pm$ 2 | 4.4 $\pm$ 0.4 | 6.1 $\pm$ 0.6 | 6.6 $\pm$ 0.7 | 14.4 $\pm$ 1.4 | 3.3 $\pm$ 0.3 | 8.6 $\pm$ 0.9 | 2.9 $\pm$ 0.3 | 14.2 $\pm$ 1.4
103.1 | 28.0 $\pm$ 1.4 | 11.9 $\pm$ 0.6 | 2.1 $\pm$ 0.1 | 3.5 $\pm$ 0.2 | 3.7 $\pm$ 0.2 | 5.1 $\pm$ 0.3 | 1.7 $\pm$ 0.1 | 2.4 $\pm$ 0.1 | 1.2 $\pm$ 0.1 | 5.0 $\pm$ 0.3
114.1 | 31.2 $\pm$ 1.6 | 13.1 $\pm$ 0.7 | 1.6 $\pm$ 0.1 | 4.0 $\pm$ 0.2 | 3.6 $\pm$ 0.2 | 4.9 $\pm$ 0.3 | 2.0 $\pm$ 0.1 | 1.9 $\pm$ 0.1 | 1.0 $\pm$ 0.1 | 4.9 $\pm$ 0.3
231.8 | 107 $\pm$ 11 | 44 $\pm$ 4 | 8.0 $\pm$ 0.8 | 10.3 $\pm$ 1.0 | 13.5 $\pm$ 1.4 | 20 $\pm$ 2 | 8.7 $\pm$ 0.9 | 7.8 $\pm$ 0.8 | 3.5 $\pm$ 0.4 | 16.7 $\pm$ 1.7
246.3 | 129 $\pm$ 13 | 52 $\pm$ 5 | 9.7 $\pm$ 1.0 | 12.2 $\pm$ 1.2 | 16.1 $\pm$ 1.6 | 24 $\pm$ 2 | 10.3 $\pm$ 1.0 | 9.7 $\pm$ 1.0 | 4.0 $\pm$ 0.4 | 20 $\pm$ 2

For our analysis, we use all the ALMA and VLA continuum data between 2 and 246
GHz in the regions shown in Figure 1 to construct SEDs. The continuum
measurements used in our SED analysis are listed in Table 5. We fit the SEDs
with functions representing synchrotron emission, free-free emission, and
thermal dust emission. The equation describing the overall emission is given
as

$\begin{split}f_{\nu}={}&f_{\nu}(\mathrm{syn})+f_{\nu}(\mathrm{ff})+f_{\nu}(\mathrm{dust})\\\
={}&A_{\mathrm{syn}}\left(\frac{\nu}{1\
\mathrm{GHz}}\right)^{\alpha_{\mathrm{syn}}}+A_{\mathrm{ff}}g_{\mathrm{ff}}+A_{\mathrm{dust}}\left(\frac{\nu}{200\
\mathrm{GHz}}\right)^{\alpha_{\mathrm{dust}}}\end{split}$ (1)

where the synchrotron emission and thermal dust emission are treated as power
laws and where the scales ($A_{\mathrm{syn}}$ and $A_{\mathrm{dust}}$) of
these components as well as the scale of the free-free emission
($A_{\mathrm{ff}}$) are also free parameters (only set to be non-negative,
adhering to physical priors). The spectral index of the synchrotron emission
($\alpha_{\mathrm{syn}}$) is additionally treated as a free parameter.
Following a separate analysis to derived the spectra slope of the dust
emission using ALMA Band 6, 7 and 8 data (which is described in Appendix A),
we fixed the spectral index of the thermal dust emission
($\alpha_{\mathrm{dust}}$) to 4.0, which is approximately equivalent to
$\beta=2.0$. We use the Gaunt factor ($g_{\mathrm{ff}}$) derived for $hv\ll
kT_{e}$ given by Draine (2011) to set the shape of the free–free emission.
This function is given as

$g_{\mathrm{ff}}=0.5535\ln\left|\left[\frac{T_{e}}{\mathrm{K}}\right]^{1.5}\left[\frac{\nu}{\mathrm{GHz}}\right]^{-1}Z^{-1}\right|-1.682,$
(2)

where $\nu$ is the frequency and $Z$ is the charge of the ions, which is set
to 1. For fitting the SED, we set $T_{e}$ to 7500 K, which is derived by the
average value from the H ii regions using auroral lines given by Bresolin et
al. (2005). This value is slightly larger than those in other nearby starburst
galaxies (e.g., Bendo et al., 2015; Chen et al., 2023); our measurements of
the flux density from free-free emission varies by $\leq 10$ per cent when
$T_{e}$ varies from 3000 to 10000 K.

Table 6: SED analysis results. | Synchrotron emission | Free-free emission | Dust emission
---|---|---|---
Region | Spectra | Fraction at | Fraction at | Fraction at
| index | 103 GHz | 103 GHz | 103 GHz
|  | (per cent) | (per cent) | (per cent)
T | $-1.06\pm 0.17$ | $13\pm 8$ | $75\pm 8$ | $12\pm 1$
N | $-1.19\pm 0.21$ | $5\pm 4$ | $84\pm 4$ | $11\pm 1$
N1 | $-0.92\pm 0.18$ | $19\pm 13$ | $67\pm 13$ | $14\pm 1$
N2 | $-1.21\pm 0.29$ | $3\pm 3$ | $89\pm 4$ | $8\pm 1$
N3 | $-1.15\pm 0.23$ | $5\pm 4$ | $83\pm 5$ | $12\pm 1$
E | $-0.94\pm 0.17$ | $22\pm 13$ | $65\pm 13$ | $13\pm 1$
E1 | $-1.23\pm 0.19$ | $5\pm 3$ | $79\pm 4$ | $16\pm 2$
E2 | $-0.80\pm 0.12$ | $50\pm 21$ | $38\pm 21$ | $12\pm 1$
C | $-0.98\pm 0.18$ | $17\pm 12$ | $74\pm 12$ | $9\pm 1$
S | $-0.87\pm 0.18$ | $22\pm 15$ | $67\pm 15$ | $11\pm 1$

With the lmfit package (Newville et al. 2022), we first fit the SED in
logarithmic space using the Levenberg-Marquardt algorithm, and then we sampled
the posterior distributions of the fitting parameters by using the maximum
likelihood via Markov chain Monte Carlo method in 1000 iterations. From these
parameters, we calculated the fractions of synchrotron emission, free-free
emission, and thermal dust emission relative to the total emission. Our
measurements, along with their associated uncertainties, were derived from the
medians and half the difference between the inner 68 per cent of the posterior
distributions.

###  4.2 SED analysis

Figure 3: The SED for the T region of NGC 1365 illustrated in Figure 1. The
black lines show the best fitting SED model for the data. The red line
represents synchrotron emission. The green line represents free–free emission
based on the Gaunt factor with $T_{e}$ fixed at 7500 K. The blue line
corresponds to thermal dust emission. Figure 4: The corner plot showcasing the
posterior distributions for parameters characterizing the SED in Figure 3. In
each 2-dimensional panel, solid contours enclose central 1, 2, and 3$\sigma$
levels of data points, with the median value denoted by a blue square
intersected by two solid lines. In each 1-dimensional histogram, vertical
dashed lines represent the 16th and 84th percentile values. Figure 5: The SEDs
for the subregions of the T region illustrated in Figure 1. The lines with
different colors are the same as those used in Figure 3.

The best fitting parameters from the SED fits are listed in Table 6, and the
SEDs, along with the best-fitting functions, are shown in Figure 3 and 5. In
the central 2.6 kpc diameter region (T region), the best fitting SED model
reveals that free-free emission dominates between 30 and 150 GHz and produces
75 per cent of the total continuum emission at 103 GHz, while the remaining 25
per cent of 103 GHz continuum emission is produced by the synchrotron and dust
emission. The posterior distributions and correlation diagrams for parameters
characterizing the SED are shown in Figure 4. The scale of the free-free
emission ($A_{\mathrm{ff}}$) is correlated with the spectral index of
synchrotron emission ($\alpha_{\mathrm{syn}}$), but the variation of
$A_{\mathrm{ff}}$ with $\alpha_{\mathrm{syn}}$ is not significant;
$A_{\mathrm{ff}}$ ranges from 0.005 to 0.008 with $\alpha_{\mathrm{syn}}$
varying from $-1.4$ to $-0.8$. The free-free emission also dominates the
millimetre continuum emission in the N and S regions, while the fractions of
the 103 GHz continuum emission originating from free-free emission vary
between $\sim$ 60 and $\sim$ 90 per cent between these regions and among their
subregions.

The SED model for the central 380 pc diameter region (C region) shows that the
free-free emission is dominated at the 103 GHz continuum emission as well,
which seems unexpected given that the AGN within this region would be expected
to produce significantly stronger synchrotron emission rather than free-free
emission. Notably, significant broad Pa$\alpha$ line emission, which traces
the same photoionized gas as free-free emission, is detected within the
nucleus (Fazeli et al., 2019), though the emission at near-infrared
wavelengths is still affected by dust obscuration. Nonetheless, we can use the
measured Pa$\alpha$ line emission to calculate the expected free-free emission
in millimetre bands, which we can then compare to our SED. To calculate the
Gaunt factor, we used

$g_{\mathrm{ff}}=\ln\left|{0.1907\left\lgroup\left[\frac{T_{e}}{\mathrm{K}}\right]^{1.5}\left[\frac{\nu}{\mathrm{GHz}}\right]^{-1}\mathrm{Z}^{-1}\right\rgroup^{0.5513}}+2.718\right|,$
(3)

given by Draine (2011) for situations where the frequency of Pa$\alpha$ line
is comparable to $kT_{e}/h$. We then derived the free–free emission using

$\begin{split}\left[\frac{f_{\nu}(\mathrm{ff})}{\mathrm{Jy}}\right]=\left[\frac{\int{f_{\nu}(\mathrm{Pa}\alpha)dv}}{\mathrm{Jykms}^{-1}}\right]\exp\left\lgroup-0.048\left[\frac{\nu}{\mathrm{GHz}}\right]\left[\frac{T_{e}}{\mathrm{K}}\right]^{-1}\right\rgroup\\\
\times\left\lgroup 4.38\times
10^{33}g_{\mathrm{ff}}^{-1}\left[\frac{\epsilon_{\nu}}{\mathrm{ergs}^{-1}\mathrm{cm}^{-3}}\right]\left[\frac{\nu}{\mathrm{~{}GHz}}\right]^{-1}\left[\frac{T_{e}}{\mathrm{~{}K}}\right]^{0.5}\right\rgroup^{-1},\end{split}$
(4)

based on equations from Draine (2011) and Scoville & Murchikova (2013) where
$\epsilon_{\nu}$ is the emissivity from Storey & Hummer (1995), the hydrogen
recombination line emission integrated over velocity is represented by
$\int{f_{\nu}(\mathrm{Pa}\alpha)dv}$ and the free–free emission is represented
by $f_{\nu}$(ff). Using the extinction-corrected Pa$\alpha$ flux of
$2.45\times 10^{-13}$ erg cm-2 s-1 as calculated from the measurements from
Fazeli et al. (2019) and assuming an electron temperature of 7500 K, we
derived the free-free flux density of 1.4 mJy at 103 GHz. Although this
measurement is 1.5 times larger than our measurement of 0.9 mJy, it is crucial
to note that the free-free flux density derived from the Pa$\alpha$ line is
dependent on $T_{e}$. If we use an electron temperature of 6000 K, the free-
free flux density at 103 GHz derived from the Pa$\alpha$ line would be is 0.9
mJy, which would match our measurement. This comparison also indicates that
exotic forms of emission, such as “submillmetre excess emission”, do not
significantly affect the overall flux densities measured in ALMA Band 3 (This
is discussed further in Appendix A). To summarize, this calculation
demonstrates that the level of free-free emission that we are measuring from
the nuclear region is indeed consistent with the previously observed
Pa$\alpha$ line emission.

As an additional note about the C region, the thermal dust emission within
this region, indicated by the 240 GHz flux density, is relatively low compared
to other regions within the central starburst ring. Combes et al. (2019)
demonstrated that the 350 GHz continuum emission within the C region is
concentrated within a torus with a diameter of $\sim 50$ pc. The dust is
relatively hot ($\sim$1300 K; Fazeli et al. 2019), but apparently the dust
mass is very low, thus leading to the relatively low thermal dust emission in
the submillimetre and millimetre bands.

In the E region, the free-free emission dominates between 50 and 120 GHz, but
only produces 65 per cent of the 103 GHz continuum emission, while the
synchrotron and dust emission produce the remaining 35 per cent continuum
emission. This region can be divided into two distinct subregions with very
different SEDs. Within the E1 region, $\sim 80$ per cent of the 103 GHz
emission originates from free-free emission, which likely arises mainly from
the two newly identified massive young star clusters in this region (Whitmore
et al., 2023). In the neighboring E2 region, however, the synchrotron emission
produces approximately half of the 103 GHz continuum emission. This enhanced
synchrotron emission may be attributed to outflows driven by the AGN or
starburst as also observed in [O iii] emission (e.g., Venturi et al., 2017),
X-ray emission (e.g., Wang et al., 2009) and radio emission (see, for example,
Sandqvist et al. 1995 as well as the top panels of Figure 2), although
morphology of these outflows varies in different wavebands. Despite the
presence of young star clusters within the E2 region as noted by Whitmore et
al. (2023), the free-free emission from these clusters is still relatively
weak in comparison to the synchrotron emission.

Above all, the radio-to-submillimetre SEDs of the pseudobulge of NGC1365 can
be well characterised by the combination of synchrotron emission, free-free
emission, and thermal dust emission. However, these SED fits have not included
either submillimetre excess emission, which has been detected in the 230 to
350 GHz frequency range in some nearby galaxies (e.g., Galliano et al., 2003,
2005; Bendo et al., 2006; Galametz et al., 2009; Galliano et al., 2011; Planck
Collaboration et al., 2011; Hermelo et al., 2016; Chang et al., 2020), or
anomalous microwave emission (AME), which has been detected in the 20 to 60
GHz range in other nearby galaxies (e.g., Murphy et al., 2010; Hensley et al.,
2015; Planck Collaboration et al., 2015; Murphy et al., 2018; Battistelli et
al., 2019; Harper et al., 2023; Fernández-Torreiro et al., 2024). To
investigate the possible contributions of these forms of emission to the SED,
we performed additional analyses on the SED of NGC 1365 in Appendices A and B.
We ultimately determined that these sources of emission are generally
negligible compared to the synchrotron, free-free, and thermal dust components
used in our SED fits.

###  4.3 Comparisons of SEDs to previous studies

The spectral indices of the radio continuum emission in the central
pseudobulge of NGC 1365 were previously measured by Sandqvist et al. (1995)
using VLA 20 cm (1.5 GHz), 6 cm (4.9 GHz) and 2 cm (14.9 GHz) data with a
resolution of 2.36 $\times$ 0.95 arcsec2 and by Stevens et al. (1999) using
the Australia Telescope Compact Array 3 cm and 6 cm data with a 2 arcsec
diameter aperture. Despite variations in our data and methodology, our
measured spectral index of $-0.98\pm 0.18$ in the C region from our SED
analysis falls within the range of their measurements, which were $-0.87$ and
$-1.1$ in the nucleus. Similarly, the spectral index of $-0.80\pm 0.12$ in the
E2 region from our SED analysis aligns with their measurements of $-0.94$ and
$-1.0$ in the comparable regions. However, their spectral indices for the N2
and N3 regions, which range from $-0.6$ to $0.07$, are notably larger than the
synchrotron spectral indices of $-1.21$ and $-1.15$ from our SED analysis.
This disparity arises simply because the spectral indices measured by
Sandqvist et al. (1995) and Stevens et al. (1999) are empirical measurements
of the spectral slope that are not based on any spectral decomposition and
therefore reflect the blending of synchrotron and free-free emission in the
SED at these frequencies. Without any spectral decomposition, the spectral
indices based on our data from 2.6 to 9 GHz for the N2 and N3 regions are
$-0.59\pm 0.05$ and $-0.68\pm-0.04$, respectively. These values fall at the
lower limit of the range of Sandqvist et al. (1995) and Stevens et al. (1999)
measurements.

The slopes of synchrotron radiation exhibit a range between $-1.59$ and
$-0.74$ among different nearby galaxies (Peel et al., 2011; Bendo et al.,
2015, 2016; Salak et al., 2017; Michiyama et al., 2020; Chen et al., 2023).
However, despite these variations, the free-free emission has been found to
still dominate the continuum emission within the frequency range of 50–150 GHz
in other galaxies (Peel et al., 2011; Bendo et al., 2015, 2016; Salak et al.,
2017; Michiyama et al., 2020; Chen et al., 2023). Our SED result of the
central pseudobulge (T region) in NGC 1365 is consistent with these findings,
as shown in Figure 3. Nonetheless, the fractions of millimetre continuum
emission originating from free-free emission varies significantly across
different regions, as illustrated in Figure 5. The fraction of the 103 GHz
emission originating from free-free emission varies from 38 to 89 per cent
among the subregions that we examined within the pseudobulge, and the dominant
source of millimetre continuum emission may shift from free-free emission to
synchrotron emission. Smaller variations in the fraction of millimetre
emission originating from free-free emission have also been observed between
the two neighboring nuclei of NGC 3256 (from 76 to 90 per cent; Michiyama et
al., 2020), and between the nucleus and the second brightest star-forming
region of NGC 1808 (from 62 to 86 per cent; Chen et al., 2023). In both cases,
however, the fraction of millimeter continuum emission originating from free-
free emission is still significantly higher than that from synchrotron
radiation, which differs from what we see in the E2 region. Synchrotron
emission that is notably strong relative to the free-free emission from star-
forming regions would only be expected in situations where an AGN jet is the
dominant energy source and where star formation is relatively weak or largely
absent.

Notably, Michiyama et al. (2023) published an analysis of the SED of the AGN
in NGC 1068 in which they found emission associated with an X-ray corona that
did not follow a standard synchrotron power law but exhibited significant
self-absorption at $~{}\hbox to0.0pt{$>$\hss}{\lower
4.30554pt\hbox{$\sim$}}20$ GHz. However, we do not observe any such phenomena
in any of the regions where we extracted SEDs, including the AGN, although we
may need additional data between 10 and 100 GHz to properly detect this type
of coronal emission.

##  5 Star Formation Rates

###  5.1 SFR measurements from free-free emission

The SFR can be calculated from the photoionizing photon production rate ($Q$)
using

$\frac{\mathrm{SFR}}{\mathrm{M}_{\odot}~{}\mathrm{yr}^{-1}}=7.29\times
10^{-54}\frac{Q}{\mathrm{s}^{-1}},$ (5)

where the conversion was derived from Murphy et al. (2011) using the
Starburst99 model (Leitherer et al., 1999) with solar metallicity ($Z_{\odot}$
= 0.020) and a Kroupa (2002) initial mass function for a mass range of 0.1–100
M⊙. While different conversions that account for the stellar rotation from
Leitherer et al. (2014) were used in previous studies (Bendo et al., 2015,
2016, 2017), these conversions are not widely used to compare with the SFRs
from other conversion equations, and thus we use a standard conversion that
includes no stellar rotation effects to simplify our comparison of SFRs.

The free–free emission can be related to the values to $Q$ by applying

$Q=\alpha_{B}n_{e}n_{p}V$ (6)

from Scoville & Murchikova (2013), where $n_{e}$ and $n_{p}$ represent the
electron and proton densities. $\alpha_{B}$ is the effective recombination
coefficient from Storey & Hummer (1995), which varies significantly with
$T_{e}$, but remains relatively stable for $n_{e}$ values spanning from
$10^{2}$ to $10^{5}$ cm-3. We therefore keep the electron density
$n_{e}$=$10^{3}$ cm-3 constant and use an $\alpha_{B}$ corresponding to the
$T_{e}$ of 7500 K set in Section 4. Following this, we can convert the free-
free flux density, $f_{\nu}$(ff), to SFRs using

$\begin{split}\frac{\mathrm{SFR}(\mathrm{ff})}{\mathrm{M}_{\odot}\mathrm{yr}^{-1}}=1.28\times
10^{11}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\\ \times
g_{\mathrm{ff}}^{-1}\left[\frac{\alpha_{B}}{\mathrm{cm}^{3}\mathrm{s}^{-1}}\right]\left[\frac{T_{e}}{\mathrm{~{}K}}\right]^{0.5}\left[\frac{D}{\mathrm{Mpc}}\right]^{2}\left[\frac{f_{\nu}(\mathrm{ff})}{\mathrm{Jy}}\right],\end{split}$
(7)

where $D$ is the distance of 19.6 Mpc. Notably, the SFRs are dependent upon
the values of $T_{e}$, which we have set to 7500 K; the SFRs may vary by $\sim
30$ per cent from our values for $T_{e}$ ranging from 5000 K to 15000 K.

Table 7: SFR measurements from 103 GHz free-free emission. Region | $f_{\nu}(\mbox{ff})$ | SFR | Area | $\Sigma_{\rm SFR}$
---|---|---|---|---
| (mJy) | (M⊙ yr-1) | (kpc2) | (M⊙ yr-1kpc-2)
T | 20.2 $\pm$ 1.1 | 8.9 $\pm$ 1.1 | 5.56 | 1.6 $\pm$ 0.2
N | 9.9 $\pm$ 0.5 | 4.4 $\pm$ 0.3 | 0.74 | 5.9 $\pm$ 0.4
N1 | 1.44 $\pm$ 0.07 | 0.63 $\pm$ 0.13 | 0.11 | 5.6 $\pm$ 1.2
N2 | 3.14 $\pm$ 0.16 | 1.39 $\pm$ 0.09 | 0.11 | 12.2 $\pm$ 0.8
N3 | 3.06 $\pm$ 0.15 | 1.35 $\pm$ 0.10 | 0.11 | 11.9 $\pm$ 0.9
E | 3.33 $\pm$ 0.17 | 1.5 $\pm$ 0.3 | 0.43 | 3.5 $\pm$ 0.7
E1 | 1.34 $\pm$ 0.07 | 0.59 $\pm$ 0.04 | 0.11 | 5.2 $\pm$ 0.4
E2 | 0.92 $\pm$ 0.05 | 0.4 $\pm$ 0.2 | 0.20 | 2.0 $\pm$ 1.2
C | 0.85 $\pm$ 0.04 | 0.38 $\pm$ 0.06 | 0.11 | 3.3 $\pm$ 0.6
S | 3.34 $\pm$ 0.17 | 1.5 $\pm$ 0.4 | 0.45 | 3.2 $\pm$ 0.8

The SFRs measured from the 103 GHz free-free emission (i.e., the 103 GHz
continuum emission multiplied by the free-free fractions at 103 GHz listed in
Table 6) are listed in Table 7. Since the free-free emission from the C region
is mainly related to the AGN (as discussed in Section 4.2), the SFRs for the T
region are calculated after subtracting the emission from the C region. These
values are used for comparisons to the SFRs obtained from other bands.

The resulting SFR from the central pseudobulge (excluding the AGN) is $8.9\pm
1.1$ M⊙ yr-1. Remarkably, nearly half of the star formation is concentrated
within the N region, which occupies only 13 per cent of the area of the T
region. The star formation in the N region is predominantly within two bright
sources (N2, N3) where the star formation surface densities reach 12 M⊙
yr-1kpc-2. This is seven times larger than the star formation surface density
averaged over the T region. The SFRs of the E and S regions are both about 15
per cent of the entire pseudobulge, and the corresponding star formation
surface densities are twice that of the T region.

###  5.2 Comparisons of millimetre and previously-published H$\alpha$ SFRs

Schinnerer et al. (2023) derived a SFR of 5 M⊙ yr-1 from the attenuation-
corrected H$\alpha$ line emission within the central 3.6 kpc diameter region.
Even though Schinnerer et al. (2023) used a larger aperture, their SFR is
still lower than our measurement within our T region, probably because of
inaccuracies in the dust attenuation correction applied to the H$\alpha$ data
within this dusty starburst. The dust extinction within the central regions of
NGC 1365 ranges from 4 to 9 (Fazeli et al., 2019), which indicates that the
observed H$\alpha$ luminosity will only be 0.025 per cent to 2.5 per cent of
the total H$\alpha$ emission from the stellar population. A similar situation
is also seen in other dusty starburst regions (e.g., Chen et al., 2023).

Many researchers have calculated SFRs using ultraviolet or H$\alpha$ emission
added to rescaled versions of the infrared emission from dust measured in
either a specific band or integrated over a large frequency range (e.g.,
Kennicutt & Evans, 2012). The infrared emission in these situations represent
an extinction correction to the ultraviolet or H$\alpha$ data. However, given
the large effect of dust extinction within dusty starbursts like NGC 1365, the
ultraviolet and H$\alpha$ emission has been sufficiently obscured that the
calculated SFRs end up being highly dependent on the infrared terms in these
equations. Hence, we will not attempt to calculate SFRs using combinations of
ultraviolet or H$\alpha$ data with infrared data.

###  5.3 Comparisons of millimetre and radio SFRs

Table 8: SFR measurements from 1.5 GHz emission.

Region | $f_{\nu}(\mbox{1.5 GHz})$ | SFR | SFR$-$SFR(ff) |  $\Delta$SFRa/SFR(ff)
---|---|---|---|---
| (mJy) | (M⊙ yr-1) | (M⊙ yr-1) | (per cent)
T | $285\pm 30$ | $8.3\pm 0.9$ | $-0.6\pm 1.4$ | $-7\pm 1$
N | $85\pm 9$ | $2.5\pm 0.2$ | $-1.9\pm 0.4$ | $-44\pm 5$
N1 | $15.1\pm 1.5$ | $0.44\pm 0.04$ | $-0.20\pm 0.14$ | $-31\pm 7$
N2 | $17.0\pm 1.7$ | $0.49\pm 0.05$ | $-0.89\pm 0.10$ | $-64\pm 8$
N3 | $19.5\pm 1.9$ | $0.56\pm 0.06$ | $-0.78\pm 0.12$ | $-58\pm 7$
E | $55\pm 6$ | $1.61\pm 0.16$ | $0.14\pm 0.35$ | $9\pm 2$
E1 | $14.2\pm 1.4$ | $0.41\pm 0.04$ | $-0.18\pm 0.06$ | $-30\pm 4$
E2 | $30\pm 3$ | $0.86\pm 0.09$ | $0.5\pm 0.2$ | $112\pm 65$
S | $46\pm 5$ | $1.33\pm 0.13$ | $-0.15\pm 0.37$ | $-10\pm 3$

aThe difference between the SFRs measured from 1.5 GHz radio emission and 103
GHz free-free emission.

For calculating SFRs based on the radio emission, we first measure the 1.5 GHz
flux densities in the same apertures used in Section 5.1 from the VLA 1.5 GHz
data described in Section 2.2 (which also meant subtracting the emission from
the C region from the emission within the T region). Following this, we
convert the 1.5 GHz flux densities to SFRs using

$\frac{\mathrm{SFR}(1.5\mathrm{GHz})}{\mathrm{M}_{\odot}\mathrm{yr}^{-1}}=0.0760\left[\frac{D}{\mathrm{Mpc}}\right]^{2}\left[\frac{f_{\nu}(1.5\mathrm{GHz})}{\mathrm{Jy}}\right]$
(8)

given by Murphy et al. (2011). The SFRs measured from the 1.5 GHz emission are
listed in Table 8.

The difference between the SFRs measured from the 1.5 GHz radio emission and
103 GHz free-free emission is illustrated as filled circles in Figure 6. The
SFR averaged over the T region estimated from the 1.5 GHz emission is
comparable to the SFR from the free-free emission; the relative discrepancy
between them is 10 per cent. However, the 1.5 GHz emission in the N region
yields a SFR much lower than the SFR measured from the free-free emission,
particularly in its two starburst dominated subregions, N2 and N3. This
difference is likely attributed by the diffusion through the interstellar
medium of the cosmic rays producing the synchrotron radiation (e.g., Murphy et
al., 2006), which would make the radio emission appear more extended than
other star formation tracers. Since the T region is much larger than the N
region, it covers both the star-forming regions producing the free-free
emission and the more diffuse regions emitting the synchrotron emission
associated with those regions, so the SFRs in the T region are more consistent
than in the smaller N region.

The E1 region is very similar to the subregions of the N region in that the
SFR from the 1.5 GHz emission is low relative to the 103 GHz free-free
emission, which again is likely to be related to the diffusion of the
electrons producing the synchrotron emission from the star-forming region.
Conversely, in the E2 region, the SFR measured from the 1.5 GHz emission is
much higher than the SFRs derived from the free-free emission; the relative
discrepancy between them is more than 100 per cent. This is likely because the
1.5 GHz emission in the E2 region is not entirely related to star formation
but includes significant emission from the outflow within that region (e.g.,
the top panels of Figure 2). The consistency in SFRs measured between the 1.5
GHz emission and free-free emission within the E region arises from mixing the
emission from E1 and E2.

The difference in SFRs measured from the 1.5 GHz emission and free-free
emission is not significant despite the fact that the S region is smaller than
the N region and we would expect the electrons producing the synchrotron
emission to diffuse outwards from the S region in the same way that they have
seemed to diffuse outwards from the N region. Given that the S region does not
have an outflow like the E2 region, the consistency in SFRs derived from the
1.5 GHz emission and free-free emission within the S region may imply the
presence of possibly more supernova remnants than in the N region. However, we
cannot draw this conclusion because of the lack of direct evidence of
supernovae remnants within the S region. Further high angular resolution radio
observations of the pseudobulge would be instrumental in providing a clearer
understanding of what is happening.

Figure 6: The differences between the SFRs from free-free emission and from
1.5 GHz radio emission (red circles) and the difference between the SFRs from
free-free emission and from 24 $\mu$m continuum emission (blue squares).

###  5.4 Comparisons of millimetre and mid-infrared SFRs

Figure 7: The image of the Spitzer 24 $\mu$m emission in the central 144
arcsec of NGC 1365. The color is set as linear scale to show the infrared
emission in the central pseudobulge region. The solid green circles show the
T, N, C and S regions, and the dashed green circles show the annuli used for
the local background subtraction. The restoring beam is illustrated as the
filled white ellipse at the lower-left side. Table 9: SFR measurements from
mid-infrared flux.

Region | Aperture | $f_{\nu}(\mbox{MIR})$ | SFR | SFR$-$SFR(ff) |  $\Delta$SFRa$/$SFR(ff)
---|---|---|---|---|---
| correction | (Jy) | (M⊙ yr-1) | (M⊙ yr-1) | (per cent)
T | 1.16 | $8.1\pm 0.3$ | $9.6\pm 0.4$ | $0.7\pm 1.2$ | $8\pm 1$
N | 1.61 | $3.27\pm 0.13$ | $3.86\pm 0.15$ | $-0.5\pm 0.4$ | $-12\pm 1$
S | 1.61 | $1.45\pm 0.06$ | $1.71\pm 0.07$ | $0.2\pm 0.4$ | $16\pm 4$

aThe difference between the SFRs measured from 24 $\mu$m emission and 103 GHz
free-free emission.

For this comparison, we used the Spitzer 24 $\mu$m image to derive the mid-
infrared (MIR) SFRs for the central pseudobulge of NGC 1365. Given that the
beam FWHM of Spitzer 24 $\mu$m image is larger than those of the ALMA images
(as described in Section 2), our comparison focuses on the SFRs from the T, N
and S regions, which are larger than the beam FWHM of Spitzer 24 $\mu$m image.

We first measured the flux densities in the T, N and S apertures, and then
subtracted the flux density within the C region (which contains the AGN) from
the T region. We then subtracted the median background within an annulus with
radii of 40–50 arcsec, and we applied aperture corrections given by
Engelbracht et al. (2007). The image of the Spitzer 24 $\mu$m emission
associated with the apertures is shown in Figure 7, and the aperture
corrections factors and the corrected flux densities are listed in Table 9. To
convert Spitzer 24 $\mu$m flux densities to SFRs, we use

$\frac{\mathrm{SFR}(24\mu\mathrm{m})}{\mathrm{M}_{\odot}\mathrm{yr}^{-1}}=2.44\times
10^{-16}\left[\frac{D}{\mathrm{Mpc}}\right]^{2}\left[\frac{\nu
f_{\nu}(24\mu\mathrm{m})}{\mathrm{JyHz}}\right]$ (9)

given by Rieke et al. (2009). The SFRs measured in the T, N and S regions
listed in Table 9. Notably, the corresponding MIR luminosities range from
$2.2\times 10^{9}$ to $1.2\times 10^{10}\mathrm{L}_{\odot}$, falling within
the applicable range of $6\times 10^{8}$ to $1.3\times
10^{10}\mathrm{L}_{\odot}$ for this relation.

The difference between the SFRs measured from 24 $\mu$m emission and 103 GHz
free-free emission are shown as the filled squares in Figure 6. The relative
discrepancy between the SFRs from the T, N and S regions estimated from the
MIR emission is less than 1 $\sigma$. The conversions from MIR emission to
SFRs are based on assumptions that the shape of infrared SED remains unchanged
and that the monochromatic infrared emission scales linearly with the total
infrared emission. The strong correspondence between the 24 $\mu$m and 103 GHz
SFRs indicate that these assumptions are applicable to the pseudobulge
(excluding the AGN) within NGC 1365. Note that the 24 $\mu$m flux density does
not always yield the same SFR as measured from millimetre free-free emission.
In many low-metallicity dwarf galaxies, for example, the dust is unusually hot
(Hunt et al., 2005; Engelbracht et al., 2008; Rosenberg et al., 2008;
Hirashita & Ichikawa, 2009; Rémy-Ruyer et al., 2013), which affects the
conversion of MIR flux densities to SFR and the comparison of SFRs from MIR
flux densities to SFRs from other tracers (e.g., Calzetti et al., 2010; Bendo
et al., 2017). Conversely, Bendo et al. (2016) found that the dust densities
in the AGN/starburst composite galaxy NGC 4945 were so high that the MIR
emission was optically thick and therefore suppressed relative to millimetre
star formation tracers. Hence, it is still important to identify situations
where SFRs from MIR emission correspond to SFRs from millimetre continuum
emission, which is why it is important to show the agreement between these two
SFRs in NGC 1365.

###  5.5 Variations of free-free emission fractions with frequencies

Free-free emission is a valuable SFR metrics because it can directly trace
young stellar populations while remaining generally unaffected by dust
attenuation (e.g., Bendo et al., 2015, 2016; Chen et al., 2023). Nevertheless,
free-free emission is not widely used to measure SFRs. This is not only
because the millimetre continuum emission is so weak that only powerful
millimetre instruments like ALMA can detect this form of emission from
extragalactic sources but also because of the lack of comprehensive radio-to-
submillimetre SED templates for identifying the fractions of continuum
emission originating from the free-free emission. Conducting SED analyses, as
applied to this and other galaxies (e.g., Condon, 1992; Peel et al., 2011;
Bendo et al., 2015, 2016; Salak et al., 2017; Michiyama et al., 2020; Chen et
al., 2023), enable us to quantify and model free-free, synchrotron and dust
emission among galaxies in general and provide an direct approach to measure
SFRs accurately using millimetre emission from the ALMA data.

Figure 8: The fraction of the continuum emission originating from free-free
emission calculated within the different regions in the pseudobulge as a
function of frequency. The squares show the frequencies where the fraction of
emission from free-free emission reaches maximum values.

We can use our results from NGC 1365 to at least show the best frequencies at
which to detect free-free emission. The variations in the fractions of
continuum emission originating from free-free emission with frequencies for
different regions of NGC 1365 are shown in Figure 8. Notably, the maximum
fractions of the continuum emission from free-free emission fall within the
range of 60–100 GHz, which lies within the frequency range of ALMA Band 2 and
3 data. In addition, these maximum fractions are only marginally higher than
the fractions measured at 103 GHz. For instance, in the T region, the maximum
fraction is 78 per cent at 78 GHz while the fraction at 103 GHz is 75 per
cent; the difference between these fractions is less than 5 per cent. Hence,
using the continuum emission in the 80–110 GHz range from ALMA Band 3 data for
measuring the SFR of millimetre bright star-forming regions appears to be a
viable option.

The fractions of the continuum emission originating from millimetre free-free
emission typically range between 65 and 85 per cent, as observed in the N and
S regions of NGC 1365, as well as the central starburst regions of NGC 253
(Bendo et al., 2015), NGC 4945 (Bendo et al., 2016), NGC 3256 (Michiyama et
al., 2020) and NGC 1808 (Salak et al., 2017; Chen et al., 2023). Consequently,
it is reasonable to assume the fraction of 80–100 GHz continuum emission from
free-free sources is $75\pm 15$ per cent for typical star-forming regions
($\sim 0.5$–2 kpc). This fraction may increase by about 15 per cent in an
intensely starburst region, such as the N2 region in NGC 1365 and the southern
starburst region in NGC 1808 (Chen et al., 2023). Conversely, in regions where
star formation is less significant, like the N1 and S regions, this fraction
may decrease by about 10 per cent in the region, and it could decrease further
if there exist significant outflows or radio jets, as observed in the E2
region, where this calibration method unsuitable for accurate SFR
measurements.

As noted throughout our analysis, variations in the electron temperature could
also affect the SFRs derived from millimetre free-free emission. Nonetheless,
our work and the body of other work in the literature suggest that, assuming
electron temperatures of $6000-10000$ K and free-free emission fractions of 75
per cent, it should be possible to measure extinction-free SFRs that are
accurate to within 20 per cent.

##  6 Conclusions

We have presented an analysis of the radio-to-submillimetre SEDs for the
central pseudobulge of NGC 1365 based on archival ALMA and VLA data. Through
the decomposition of the SED from 2 to 246 GHz, we demonstrate that free-free
emission dominates of 50–120 GHz continuum emission from the pseudobulge, with
synchrotron emission primarily observed at lower frequencies and thermal dust
emission as the dominant emission source at higher frequencies. In addition,
no significant anomalous microwave emission or submillimetre excess emission
is detected. The free-free emission from the photoionized gas primarily
originates from the northern star-forming region as well as the fainter
eastern and southwest star-forming regions. The thermal dust emission
corresponds closely with the millimetre continuum emission at $\sim 100$ GHz
with the exception of a dusty star-forming region in the northeastern part of
the pseudobulge. The synchrotron radiation is primarily associated with
supernovae in the circumnuclear ring, although synchrotron emission is also
detected from the AGN. An outflow to the east of the AGN produces notably high
synchrotron emission that has also been observed in [O iii] emission (e.g.,
Venturi et al., 2017) and X-ray emission (e.g., Wang et al., 2009). Notably,
the fraction of the millimetre continuum emission originating from free-free
emission varies significantly among the different subregions within the
central pseudobulge, particularly for the eastern region that contains the
outflow producing strong synchrotron emission.

Free-free emission also dominates at the 103 GHz continuum emission in the
central 380 pc region, but the photoionized gas associated with this emission
is associated with the AGN rather than star formation. We calculated that the
observed free-free flux density is consistent with the broad Pa$\alpha$ line
emission presented by Fazeli et al. (2019). However, the millimetre dust
emission from the area around the AGN is relatively low because the dust is
relatively hot and concentrated within a $\sim 50$ pc region (Combes et al.,
2019).

The SFR within the central pseudobulge derived from the 103 GHz free-free
emission is $8.9\pm 1.1$ M⊙ yr-1, and the star formation surface density
averaged over this area is $1.6\pm 0.2$ M⊙ yr-1kpc-2. The most intense star
formation is observed in two star-forming regions in the northern part of the
pseudobulge, where the star formation surface densities reach $12$ M⊙
yr-1kpc-2.

We treated the SFRs measured from the 103 GHz free-free emission as the best
SFR measurements possible and compared them to other star formation metrics to
test the effectiveness of the other metrics and understand how other physical
processes could affect the accuracy of these metrics. The SFRs measured from
the ALMA data are consistent with the SFRs from the mid-infrared data but
significantly higher than the SFR derived from the extinction-corrected
H$\alpha$ line emission. This is likely because the central pseudobulge is so
heavily dust obscured, so applying extinction corrections to the line emission
is problematic. The SFR averaged over the central pseudobulge estimated from
the 1.5 GHz emission matches the SFR from the free-free emission, but the 1.5
GHz emission in the much smaller northern star-forming region yields a SFR
much lower than the SFR measured from the free-free emission. This difference
is likely because synchrotron radiation produced by cosmic rays tend to be
more extended than the star-forming regions (e.g., Murphy et al., 2006), while
the central pseudobulge region is large enough to cover both the star-forming
and more diffuse regions emitting the synchrotron emission. Conversely, the
1.5 GHz emission in another smaller eastern region yields a SFR much higher
than the SFR derived from the free-free emission, but this is because a
significant fraction of the 1.5 GHz synchrotron emission from this region
comes from an outflow from the AGN.

Free-free emission stands out as a valuable SFR metrics because of its ability
to directly trace young stellar populations while remaining generally
unaffected by dust attenuation. Radio-to-submillimetre SED analyses like what
we have presented here for NGC 1365 could enable us to quantitatively model
free-free emission among other galaxies in general and provide a direct
approach to measuring SFRs accurately using ALMA millimetre data. Based on
both the results from our analysis on NGC 1365 and other published analyses of
the SEDs of other nearby starbursts, we propose that continuum emission from
the 80–110 GHz range (which falls within ALMA Bands 2 and 3) multiplied by a
75 per cent correction factor could yield useful measurements of the free-free
emission without performing any SED decomposition. The SFRs based on these
measurements would be accurate to within 20 per cent. Several CMB survey
telescopes, (e.g., the South Pole Telescope, the Atacama Cosmology Telescope,
the Simons Observatory), the Green Bank Telescope, the Sardinia Radio
Telescope, and the upcoming Square Kilometre Array (SKA) and Next Generation
Very Large Array will provide abundant data for constructing more radio-to-
submillimetre SEDs. In addition, ALMA Band 1 and future Band 2 observations,
which will extend the frequency coverage of ALMA to 35 GHz, would place
additional constraints on the SEDs measured for these galaxies and thus
provide more accurate measurements of the free-free emission from
extragalactic objects.

## Acknowledgements

We thank the reviewer and Dr. Patrick J. Leahy for their very useful comments
on this paper. GC acknowledges support from the SKA exchange programme of STFC
and CSC (No. 201906340241). GJB is supported by STFC grant ST/T001488/1. GAF
acknowledges support from the Collaborative Research Centre 956, funded by the
Deutsche Forschungsgemeinschaft (DFG) project ID 184018867. This work is also
supported by the National Science Foundation of China (NSFC, Grant No.
12233008, 11973038 and 11973039), the National Key R&D Program of China
(2023YFA1608100), the Strategic Priority Research Program of Chinese Academy
of Sciences (Grant No. XDB 41000000), the China Manned Space Project (No. CMS-
CSST-2021-A07, CMS-CSST-2021-B02), the Cyrus Chun Ying Tang Foundations, the
111 Project for "Observational and Theoretical Research on Dark Matter and
Dark Energy" (B23042), and the CAS Pioneer Hundred Talents Program.

This paper makes use of the following ALMA data: ADS/JAO.ALMA#2013.1.01161.S,
2015.1.01135.S, 2017.1.00129.S, 2019.1.01635.S, 2021.1.01150.S,
2021.1.01295.S, and 2021.2.00079.S. ALMA is a partnership of ESO (representing
its member states), NSF (USA) and NINS (Japan), together with NRC (Canada),
MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the
Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and
NAOJ. The Australia Telescope Compact Array is part of the Australia Telescope
National Facility (grid.421683.a) which is funded by the Australian Government
for operation as a National Facility managed by CSIRO.

## Data Availability

The reduced, calibrated and science-ready ALMA data is available from the ALMA
Science Archive at https://almascience.eso.org/alma-data/archive, and the
science-ready VLA data is available from the National Radio Astronomy
Observatory Data Archive at https://data.nrao.edu .

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##  Appendix A Spectra index of thermal dust emission

Figure 9: Continuum emission measurements across the ALMA Band 6, 7 and 8 for the T, N and S regions (from left to right). The best fits for the Band 6, 7 and 8 data are depicted with black solid lines. The red dashed lines represent the best fit based on the Band 6 and 7 data, while the blue dashed line represent the best fit from the Band 7 and 8 data. The best-fitting spectra indices, along with their $1\sigma$ uncertainties, are indicated as text that matches the colour of the corresponding best fit. Table 10: ALMA observations and continuum measurements used in fitting the spectra index of dust emission. Band | Rest-frame | Beam FWHM | Beam FWHM |  | Flux density |  |
---|---|---|---|---|---|---|---
| frequency | before smoothing | after smoothing | T region | N region | S region |
| (GHz) | ($\rm{arcsec^{2}}$) | ($\rm{arcsec^{2}}$) | (mJy) | (mJy) | (mJy) |
6 | 231.8 | 2.2 $\times$ 1.9 | 4.5 $\times$ 2.5 | 68 $\pm$ 7 | 35 $\pm$ 4 | 9 $\pm$ 1 |
7 | 345.3 | 4.7 $\times$ 2.4 | - | 331 $\pm$ 33 | 165 $\pm$ 16 | 40 $\pm$ 4 |
8 | 463.6 | 3.2 $\times$ 2.3 | 4.5 $\times$ 2.6 | 1130 $\pm$ 226 | 553 $\pm$ 111 | 141 $\pm$ 28 |

To constrain the spectral slope of the Rayleigh-Jeans side of the dust
emission within NGC 1365 and to assess whether the slope may flatten, which
would be indicative of “submillimetre excess” emission (e.g., Galliano et al.,
2003, 2005; Bendo et al., 2006; Galametz et al., 2009; Galliano et al., 2011;
Planck Collaboration et al., 2011; Hermelo et al., 2016; Chang et al., 2020),
we performed additional analyses of the SED of NGC 1365 using the ALMA Band 6
data listed in Table 1 as well as ALMA Band 7 from project 2021.2.00079.S and
Band 8 data from project 2019.1.01635.S. Compared to the Band 6 data, the Band
7 and 8 data have larger beam FWHMs and smaller maximum recoverable scales,
which is why we did not incorporate those data into the analysis in Section 4.
To create images of these data for this analysis, we first adjusted the $uv$
coverage of the data in each band to match the maximum recoverable scale from
the Band 8 data before imaging the data. Next, when imaging the Band 7 and 8
data using tclean, we set the weighting parameter to briggs and the robust
parameter to 0.5. After this, we used imsmooth in casa version 6.4.1 to smooth
the beams from the Band 6 and 8 data to match the resolution in the Band 7
data. The continuum measurements for the T, N and S regions used in this
analysis are listed in Table 10.

The continuum measurements were fitted as a single power law model in
logarithmic space. Figure 9 show the best-fitting results for the T, N and S
regions. The best fits for the Band 6, 7 and 8 data are indicated by black
lines, while the best fit for the combined Band 6 and 7 data is shown in red,
and the Band 7 and 8 data in blue. These comparable spectral indices suggest
that the continuum emission within this frequency range is well characterized
by a single power law with a spectral index of $\approx$4.0. This would be
consistent with $\beta\geq 2.0$, which is what is expected for standard dust
emission. If submillimetre excess emission was present, the spectral slope of
these data would be expected to be shallower. Additionally, we see no
inflection in the slope of the data between ALMA Bands 6 and 8, which
indicates that no significant submillimetre excess emission appears between
these bands.

Based on the results from this analysis, we adopted a spectral index of 4.0
for the thermal dust emission in our SED fits in Section 4. The uncertainty in
the spectral slope is 0.3, but this will have a minimal effect on our main
results. When accounting for this uncertainty in the SED fits, the
contribution of free-free emission to the 103 GHz continuum emission
fluctuated by less than 5 per cent.

##  Appendix B Non detection of AME on the pseudobulge of NGC 1365

Table 11: General information of observations and continuum measurements for the T region. Instrument | Band | Rest-frame | Beam FWHM | Beam FWHM | Flux density
---|---|---|---|---|---
|  | frequency | before smoothing | after smoothing |
|  | (GHz) | ($\rm{arcsec^{2}}$) | ($\rm{arcsec^{2}}$) | (mJy)
VLA | S | 2.6 | 2.6 $\times$ 0.8 | 10.3 $\times$ 10.0 | 202 $\pm$ 20
|  | 3.6 | 2.2 $\times$ 0.8 | 10.2 $\times$ 10.0 | 156 $\pm$ 16
|  | 4.9 | 5.5 $\times$ 1.5 | 10.6 $\times$ 9.4 | 127 $\pm$ 13
| C | 5.0 | 1.7 $\times$ 0.5 | 10.2 $\times$ 10.0 | 118 $\pm$ 12
|  | 7.0 | 1.3 $\times$ 0.4 | 10.2 $\times$ 10.0 | 90 $\pm$ 9
| X | 9.0 | 2.3 $\times$ 0.6 | 10.3 $\times$ 10.0 | 73 $\pm$ 7
| Ku | 15.0 | 5.5 $\times$ 1.5 | 10.6 $\times$ 9.4 | 54 $\pm$ 5
| K | 22.6 | 11.1 $\times$ 1.2 | 13.7 $\times$ 8.3 | 33 $\pm$ 5
ATCA | C | 5.6 | 3.5 $\times$ 1.2 | 10.6 $\times$ 10.1 | 125 $\pm$ 13
| K | 23.4 | 6.1 $\times$ 1.2 | 10.9 $\times$ 9.1 | 36 $\pm$ 5
ALMA | 3 | 103.1 | 2.3 $\times$ 1.8 | 10.3 $\times$ 10.2 | 25.1 $\pm$ 1.3
|  | 114.1 | 2.3 $\times$ 1.8 | 10.3 $\times$ 10.2 | 27.7 $\pm$ 1.4
| 6 | 231.8 | 2.2 $\times$ 1.9 | 10.2 $\times$ 10.1 | 93 $\pm$ 9
|  | 246.3 | 2.2 $\times$ 1.9 | 10.2 $\times$ 10.2 | 114 $\pm$ 11

Anomalous microwave emission (AME) at 20 to 60 GHz has been detected from
several nearby galaxies, including M31 (Planck Collaboration et al., 2015;
Battistelli et al., 2019; Harper et al., 2023; Fernández-Torreiro et al.,
2024), NGC 2903 (Poojon et al., 2024), NGC 4725 (Murphy et al., 2018) and NGC
6946 (Murphy et al., 2010; Hensley et al., 2015). To investigate the potential
contribution of AME within the pseudobulge of NGC 1365 (the T region), we
performed additional analyses of the SED using the ALMA Band 3 and 6 data; the
VLA S-, C-, X-, Ku- and K-band data; and the ATCA C- and K-band data listed in
Table 11. The VLA Ku- and K-band data were processed using the the same steps
described in Sections 2.2. We followed the the ATCA Users
Guide777https://www.narrabri.atnf.csiro.au/observing/users_guide/html/, to
calibrate the ATCA data using the miriad
package888https://www.atnf.csiro.au/computing/software/miriad/. The calibrated
data were then converted into continuum images using tclean in casa version
6.4.1, with the weighting parameter set to briggs and the robust parameter set
to 0.5 to more closely match the beam FWHM to the FWHM of the other data. When
creating all of the continuum images for this analysis, the $uv$ coverages
were adjusted so that all images have the same maximum recoverable scale (36
arcsec). After this, we used imsmooth in casa version 6.4.1 to smooth the
beams from all the data to match the resolution with that of the VLA K-band
data.

The spinning dust model effectively describe the majority of AME (Draine &
Lazarian, 1998; Ali-Haïmoud et al., 2009), yet the specific mechanisms for
this emission remains ambiguous. AME has been widely characterized through an
empirical lognormal approximation, which was first introduced by Stevenson
(2014) and later developed by Cepeda-Arroita et al. (2021). This approach
simplifies the SPDUST model (Ali-Haïmoud et al., 2009; Silsbee et al., 2011)
by introducing only three free parameters, thus avoiding degeneracies
associated with the models. To include AME within our SED models (as presented
in Equation 1), the equation describing the overall SED is given as

$\begin{split}f_{\nu}={}&f_{\nu}(\mathrm{syn})+f_{\nu}(\mathrm{ff})+f_{\nu}(\mathrm{dust})+f_{\nu}(\mathrm{AME})\\\
={}&A_{\mathrm{syn}}\left(\frac{\nu}{1\
\mathrm{GHz}}\right)^{\alpha_{\mathrm{syn}}}+A_{\mathrm{ff}}g_{\mathrm{ff}}+A_{\mathrm{dust}}\left(\frac{\nu}{200\
\mathrm{GHz}}\right)^{\alpha_{\mathrm{dust}}}\\\
{}&+A_{\mathrm{AME}}\mathrm{exp}\left[-\frac{1}{2W_{\mathrm{AME}}^{2}}\ln^{2}\left(\frac{\nu}{\nu_{\mathrm{AME}}}\right)\right].\end{split}$
(10)

For the AME peak, the flux density ($A_{\rm AME}$) is set to be a non-negative
parameter, the frequency ($\nu_{\rm AME}$) is constrained to a range of 10 to
60 GHz, and the width ($W_{\rm AME}$) is constrained to between 0.2 and 1.
These constraints, obtained from the comparisons with SPDUST models, have been
extensively applied in detecting AME (Cepeda-Arroita et al., 2021; Poidevin et
al., 2023; Fernández-Torreiro et al., 2023, 2024; Poojon et al., 2024).
Following the methodology described in Section 4, we used the Levenberg-
Marquardt algorithm with the LMFIT package to fit the SED in logarithmic
space, and then we sampled the posterior distributions of the fitting
parameters.

The SED, along with the best fitting functions and the corner plot for the
marginalized posterior distributions and correlation diagrams are illustrated
in Figure 10. The SED is predominantly represented through a combination of
synchrotron, free-free and thermal dust emission, and free-free emission still
dominates between 30 and 150 GHz; $78\pm 5$ per cent of the 103 GHz continuum
emission is produced by the free-free emission. However, no significant AME is
detected; $A_{\rm AME}$ is $\approx 0$, and $\nu_{\rm AME}$ and $W_{\rm AME}$
are poorly constrained, as illustrated in the posterior distributions. The
fraction of the millimetre continuum emission originating from AME is less
than 1 per cent. Furthermore, the Bayesian information criterion (BIC) value
for this SED model is 9.7, in contrast to a significantly lower BIC value of
1.7 when the same SED is fitted without including the AME component in the
model.

Observations of AME usually correlate with the presence of abundant diffuse
dust within nearby galaxies (e.g., Dickinson et al., 2018). Additionally,
these galaxies or the regions within these galaxies exhibit weak star
formation. In contrast, the pseudobulge of NGC 1365 contains a starburst that
is notably associated with significant free-free emission. This difference
might explain why AME could be detected in galaxies like M31 but not in
starburst galaxies such as M82 and NGC 253 (Peel et al., 2011) as well as
LIRGs such as Arp 220, Mrk 231, NGC 3690, NGC 6240 (Dickinson et al., 2018)
and NGC 1365 in this work. Given the lack of data within the 30-80 GHz
frequency range, the presence of AME within the pseudobulge of NGC 1365 cannot
be entirely ruled out. However, our SED analysis suggests that if AME is
present, the AME would be different from those detected in other nearby
galaxies; either the peak of AME would be at frequencies $\gtrsim$ 60 GHz or
the width of the AME ($W_{\mathrm{AME}}$) is outside the range of 0.2 to 1.0
used in SED fits applied to other galaxies. Consequently, given the negligible
contribution of AME, we did not include the contribution of AME in the SED
model presented in Section 4.

Figure 10:  The SED (top-right) and the corresponding corner plot (left) for
the T region of NGC 1365. The SED is similar to the one shown in Figure 3, but
it also includes VLA Ku- and K-band data and the ATCA C- and K-band data to
assist in constraining the potential contribution of AME. The corner plot
shows the posterior distributions for parameters in Equation 10 characterizing
the SED.