# WISDOM project – XI. Star Formation Efficiency in the Bulge of the AGN-host
Galaxy NGC 3169 with SITELLE and ALMA

Anan Lu,1 Hope Boyce,1 Daryl Haggard,1 Martin Bureau,2, 3 Fu-Heng Liang,2
Lijie Liu,4 Woorak Choi,3 Michele Cappellari,2 Laurent Chemin,5 Mélanie
Chevance,6 Timothy A. Davis,7 Laurent Drissen,8 Jacob S. Elford,7 Jindra
Gensior,9 J. M. Diederik Kruijssen,6 Thomas Martin,8 Etienne Massé,8 Carmelle
Robert,8 Ilaria Ruffa,7 Laurie Rousseau-Nepton,10 Marc Sarzi,11 Gabriel
Savard,8 Thomas G. Williams12
1McGill Space Institute and Department of Physics, McGill University, 3600 rue
University, Montreal, QC H3A 2T8, Canada
2Sub-department of Astrophysics, Department of Physics, University of Oxford,
Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK
3Yonsei Frontier Lab and Department of Astronomy, Yonsei University, 50
Yonsei-ro, Seodaemun-gu, Seoul 03722, Republic of Korea
4DTU-Space, Technical University of Denmark, Elektrovej 327, DK-2800 Kgs.
Lyngby, Denmark
5Centro de Astronomía - CITEVA, Universidad de Antofagasta, Avenida Angamos
601, Antofagasta, 1270300, Chile
6Astronomisches Rechen-Institut, Zentrum für Astronomie der Universität
Heidelberg, Mönchhofstraße 12-14, 69120 Heidelberg, Germany
7Cardiff Hub for Astrophysics Research & Technology, School of Physics &
Astronomy, Cardiff University, Queens Buildings, The Parade, Cardiff, CF24
3AA,UK
8Département de Physique, de Génie Physique et d’Optique, Université Laval,
Québec, QC G1V 0A6, Canada
9Institute for Computational Science, University of Zürich,
Winterthurerstrasse 190, 8057 Zürich, Switzerland
10Canada–France–Hawaii Telescope, Kamuela, HI 96743, USA
11Armagh Observatory and Planetarium, College Hill, Armagh BT61 9DG, UK
12Max Planck Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The star formation efficiency (SFE) has been shown to vary across different
environments, particularly within galactic starbursts and deep within the
bulges of galaxies. Various quenching mechanisms may be responsible, ranging
from galactic dynamics to feedback from active galactic nuclei (AGN). Here, we
use spatially-resolved observations of warm ionised gas emission lines
(H$\beta$, [O iii] $\lambda\lambda$4959,5007, [N ii]
$\lambda\lambda$6548,6583, H$\alpha$ and [S ii] $\lambda\lambda$6716,6731)
from the imaging Fourier transform spectrograph SITELLE at the Canada-France-
Hawaii Telescope (CFHT) and cold molecular gas (12CO(2-1)) from the Atacama
Large Millimeter/sub-millimeter Array (ALMA) to study the SFE in the bulge of
the AGN-host galaxy NGC 3169. After distinguishing star-forming regions from
AGN-ionised regions using emission-line ratio diagnostics, we measure
spatially-resolved molecular gas depletion times ($\tau_{\rm dep}\equiv$1/SFE)
with a spatial resolution of $\approx 100$ pc within a galactocentric radius
of $1.8$ kpc. We identify a star-forming ring located at radii $1.25\pm 0.6$
kpc with an average $\tau_{\rm dep}$ of 0.3 Gyr. At radii $<0.9$ kpc, however,
the molecular gas surface densities and depletion times increase with
decreasing radius, the latter reaching approximately $2.3$ Gyr at a radius
$\approx 500$ pc. Based on analyses of the gas kinematics and comparisons with
simulations, we identify AGN feedback, bulge morphology and dynamics as the
possible causes of the radial profile of SFE observed in the central region of
NGC 3169.

###### keywords:

galaxies: bulge – galaxies: individual: NGC3169 – galaxies: nuclei – galaxies:
ISM – ISM: clouds – ISM: H II regions

††pubyear: 2021

##  1 Introduction

The tight correlation between the star-formation rates (SFRs) and stellar
masses ($M_{\star}$) of galaxies, referred to as the star-formation main
sequence (SFMS; e.g. Brinchmann et al. 2004; Noeske et al. 2007; Rodighiero et
al. 2011; Speagle et al. 2014), suggests a self-regulated evolution of galaxy
mass and size (Bouché et al., 2010; Genzel et al., 2010; Davé et al., 2012;
Feldmann & Mayer, 2015). However, many galaxies deviate from the SFMS,
particularly the "quenched" galaxy population, which has much lower specific
star-formation rates (sSFRs; i.e. ${\rm SFR}/M_{\star}$). Our understanding of
this star formation truncation remains poor, with galactic dynamics (e.g.
Martig et al., 2009, 2013; Gensior et al., 2020), mergers (e.g. Moustakas et
al., 2013) and/or active galactic nuclei (AGN) feedback (e.g. Pontzen et al.,
2017) all offering possible contributing mechanisms.

Fundamentally, stars should form from the cold and dense (typically molecular)
gas in the interstellar medium (ISM). Star-formation quenching is therefore
either the result of a low molecular gas fraction ($f_{\rm H_{2}}$; $f_{\rm
H_{2}}\equiv\frac{M_{\rm H_{2}}}{M_{\star}}$) and/or a low star formation
efficiency (SFE; ${\rm SFE}\equiv{\rm SFR}/M_{\rm H_{2}}$), as the reduction
of either of them would lead to a lower sSFR. In turn, the depletion time
($\tau_{\rm dep}$; $\tau_{\rm dep}\equiv{\rm SFE}^{-1}=\frac{M_{\rm
H_{2}}}{\rm SFR}$), i.e., the timescale required to exhaust the molecular gas
present at the current SFR, is a useful quantity to probe star formation. The
power-law relation between $M_{\rm H_{2}}$ and SFR on galactic scales
(Kennicutt Jr, 1998; Kennicutt Jr & De Los Reyes, 2021) shows that $\tau_{\rm
dep}$ of starburst galaxies is $10-10^{2}$ Myr, while those of spiral galaxies
tend to be much longer, reaching over $10^{3}$ Myr. On sub-galactic scales,
where $\tau_{\rm dep}$ is typically calculated as the ratio of the molecular
gas mass surface density ($\Sigma_{\rm H_{2}}$) to the SFR surface density
($\Sigma_{\rm SFR}$), Bigiel et al. (2008) revealed a depletion time of
$\approx 2$ Gyr.

Within individual galaxies, evidence also suggests that the SFE is not uniform
across the variety of local environments (e.g. disc, spheroid, spiral arms,
inter-arm regions; Querejeta et al. 2019). Contrary to the high SFEs of
starburst regions, the SFEs are reduced in galaxy bulges (e.g. Davis et al.,
2014; Kruijssen et al., 2014). The SFRs of early-type galaxies (ETGs) and
bulges are also suppressed by a factor of $\approx 2.5$ compared to those of
spiral galaxy discs with similar molecular gas mass surface densities (e.g.
Saintonge et al., 2011). It is possible that this star formation suppression
is analogous to that observed in the central regions of spirals and growing
spheroids at higher redshifts (Lang et al., 2014), and it may therefore be a
symptom of a more general process. Indeed, the central molecular zone (CMZ),
i.e, the inner $\approx 500$ pc in radius of our own Milky Way, also has a SFR
one order of magnitude smaller than the Galactic average, despite very high
molecular gas mass surface densities (e.g. Longmore et al., 2013). A multitude
of physical processes have been implicated, but the high turbulent pressure
within the CMZ is likely the most important (Kruijssen et al., 2014).

It is therefore crucial to compare different environments to validate
commonly-postulated quenching mechanisms, such as “morphological quenching”,
where a nearly spherical potential holds gas stable against star-formation
(e.g. Martig et al., 2009, 2013; Gensior et al., 2020), dynamical processes
such as shear and tides, that stabilise the gas clouds against gravitational
collapse (e.g. Davis et al., 2014; Kruijssen et al., 2019a; Liu et al., 2021),
and AGN feedback, whereby radiation and jets from AGN physically remove and/or
heat up the gas (e.g. Pontzen et al., 2017).

To distinguish these different mechanisms in the inner regions of galaxies, we
have initiated a project to measure the SFEs of galaxy bulges in a spatially-
resolved manner, combining high-spatial resolution 12CO(2-1) measurements (by
a cold gas indicator) from the Atacama Large Millimeter/sub-millimeter Array
(ALMA) with ionised hydrogen emission-line measurements (by a SFR indicator)
from the imaging Fourier transform spectrograph SITELLE at the Canada-France-
Hawaii Telescope (CFHT). Here we present a pilot study based on a single
galaxy, the nearby early type spiral galaxy NGC 3169 with a mean SFR of
$\approx 2$ M⊙ yr-1. This galaxy has a dense bulge (with an effective radius
$\approx 1.4$ kpc and a Sérsic index of $\approx 4.17$; from Dong & De
Robertis 2006) and a low-luminosity AGN (LLAGN; e.g. Ghosh et al. 2008; Mathur
et al. 2008), making it ideal for a deep investigation of various quenching
mechanisms. The target and observations are described in Section 2, followed
by surface-density and line-ratio maps in Section 3. In Section 4, we
calculate depletion times and examine the radial trend of SFE and its
correlation with galaxy dynamics (i.e. the virial parameters of the H ii
regions and galaxy velocity field). In Section 5, we discuss the mechanisms
contributing to the radial profile of SFE in the central region, including AGN
feedback, bulge morphology and dynamics. We conclude and discuss future work
in Section 6.

Table 1: Key parameters of NGC 3169.

Ref. | Quantity | Value
---|---|---
(1) | Type | SA(s)a pec
(2) | RA | $10^{\rm h}14^{\rm m}15\aas@@fstack{s}05$
(3) | DEC | $3^{\circ}27\arcmin 57\aas@@fstack{\prime\prime}90$
(4) | Distance (Mpc) | $18.7$
(5) | $\log({\rm M_{H_{2}}~{}/~{}M_{\odot}})$ | $9.32$
(6) | $\log({\rm\Sigma_{H_{2},~{}2~{}kpc}~{}/~{}M_{\odot}~{}pc^{-2}})$ | $2.31$
(7) | $\log({\rm M_{\star}~{}/~{}M_{\odot}})$ | $10.84$
(8) |  $\sigma_{\star}$ (km s-1) | $163$
(9) | $\log({\rm\mu_{\star}~{}/~{}M_{\odot}~{}kpc^{-2}})$ | $8.26$
(10) | $\log({\rm sSFR~{}/~{}yr^{-1}})$ | $-10.55$
(11) |  $R_{\rm e,KS}$ (″) | $85.7$
(12) | $\log({\rm SFR~{}/~{}M_{\odot}~{}yr^{-1}})$ | $0.29$
(13) | $\log({\rm\Sigma_{SFR,~{}2~{}kpc}~{}/~{}M_{\odot}~{}yr^{-1}~{}kpc^{-2}})$ | $-0.67$
(14) |  $L_{\rm Edd}$ (erg s-1) | $\approx 10^{46}$

Notes: (1) Morphology of the galaxy taken from de Vaucouleurs et al. (1991).
(2) and (3) are the RA-DEC coordinate of the galaxy center used for SITELLE
observations and referenced for all the maps in this work. (4) Assumed
distance taken from the NASA extragalactic database redshift independent
distance catalogue (Steer et al., 2016). (5) Molecular gas mass measured
within our ALMA field of view as described in Section 3.2 (6) Mean molecular
gas surface density measured within the inner $2$ kpc of the galaxy, as
described in Section 3.2. (7) Stellar mass of the galaxy taken from Leroy et
al. (2019). (8) Stellar velocity dispersion taken from Héraudeau et al.
(1999). (9) Stellar mass surface density ($\mu_{\star}={\rm M_{\star}}/(2\pi
R_{e}^{2})$) estimated within the effective radius ($R_{\rm e,KS}$) of the
galaxy. (10) Specific star formation rate (sSFR=${\rm M_{\star}/SFR}$)
calculated based on (7) and (12). (11) $K_{s}$-band effective radius estimated
from the $2~{}\mu$m all-sky survey (2MASS; Jarrett et al. 2003). (12) Total
star formation rate taken from Leroy et al. (2019). (13) Mean star formation
rate surface density measured within the inner $2$ kpc of the galaxy, as
described in Section 3.1. (14) Estimated Eddington luminosity of the
supermassive black hole, taken from Ghosh et al. (2008).

##  2 Observations and data reduction

###  2.1 NGC 3169

NGC 3169 is an S(a) spiral galaxy at a distance $D=18.7$ Mpc, according to the
NASA Extragalactic Database redshift-independent distance catalogue (Steer et
al., 2016). It has an inclination $i=54^{\circ}$, a position angle
$PA=45^{\circ}$ and a systemic heliocentric velocity $V_{\rm sys}=1232$ km s-1
(from the H i $21$-cm observations of Lee-Waddell et al. 2012, consistent with
the best-fitting systemic velocity derived from our own CO data discussed in
Section 4.5). NGC 3169 is representative of early-type spiral galaxies, while
interacting with its close companion galaxy NGC 3166 (Sil’chenko & Afanasiev,
2006). It hosts a complex central region, consisting of a dense and clear
bulge, a small-scale bar, an AGN and a young nucleus, making it possible to
probe multiple components that can affect local SFE. Figure 1 shows an deep
image produced from our SITELLE observations of NGC 3169, overlaid with
contours of H$\alpha$ flux (measured with SITELLE) and 12CO(2-1) flux
(measured with ALMA).

Figure 1:  Deep image of NGC 3169 extracted from the CFHT SITELLE observations
discussed in Section 2.2, with the H$\alpha$ integrated intensity overlaid as
orange contours and the CO integrated intensity (extracted from the ALMA
observations discussed in Section 2.3) overlaid as blue contours. The white
box indicates the central $\approx 3.6\times 3.6$ kpc2 region, encompassing
the galaxy bulge, which is the main focus of this paper. The inset shows a
zoom-in view of the central region.

The H i gas mass of NGC 3169 is $(4.2\pm 0.5)\times 10^{9}$ M⊙ (Lee-Waddell et
al., 2012). The ultraviolet and infrared images from the Wide-field Infrared
Survey Explorer (WISE) and Galaxy Evolution Explorer (GALEX) missions (Leroy
et al., 2019) indicate a stellar mass of $\log(M_{\star}/{\rm
M_{\odot}})\approx 10.84$ and a star formation rate $\log(\rm SFR/{\rm
M_{\odot}~{}yr^{-1}})\approx 0.29$ within a central aperture of diameter
$\approx 19\arcmin$. Key parameters of NGC 3169 are summarised in Table 1.

Probing the centre of NGC 3169, Sil’chenko & Afanasiev (2006) obtained
spatially-resolved measurements in the wavelength range $4200$–$5600$ Å from
the Multi Pupil Fiber Spectrograph (MPFS) at the prime focus of the 6-m
Special Astrophysical Observatory (SAO) telescope. They reported a young
stellar population with an estimated age of $1$–$2$ Gyr in the central region,
including a star-like nucleus and a ring of young stars $6\arcsec$ ($\approx
600$ pc) from the centre. Within the nuclear region (radius $\lesssim 200$
pc), Dong & De Robertis (2006) reported an H$\alpha$ luminosity $\approx
10^{39}$ ergs s-1, based on a spectroscopic survey carried out with the 5-m
telescope at Mount Palomar by Ho et al. (1997). Integrated emission-line
ratios in this region were reported as H$\alpha$/H$\beta=5.03$, [O
iii]$\lambda$5007/H$\beta=2.88$ and [N ii]$\lambda$6583/H$\alpha=2.07$,
indicating the presence of an AGN.

Furthermore, Dong & De Robertis (2006) used the GALFIT software (Peng et al.,
2002) on Two Micron All Sky Survey (2MASS) $K_{\rm s}$-band images to
decompose the galaxy into a spheroidal bulge and an exponential disc. They
reported bulge and disc $K_{\rm s}$-band integrated apparent magnitudes
$m_{\rm bul}=7.52$ and $m_{\rm disc}=8.42$. The effective radius of the bulge
component is $15\aas@@fstack{\prime\prime}7$ or $1.4$ kpc at our assumed
distance. Based on this bulge-disc decomposition, we estimate the enclosed
bulge-to-disk luminosity ratio (B/D) as a function of galactocentric radius;
B/D approaches unity at a radius of $\approx 660$ pc. This suggests that the
bulge is dominating over the disc within $\approx 660$ pc radius, while the
impact from the bulge can reach over $1.4$ kpc.

The presence of a low-luminosity AGN (LLAGN) at the heart of NGC 3169 has been
confirmed at multiple wavelengths. Based on Chandra X-ray data, Ghosh et al.
(2008) reported a hard X-ray detection and fitted the spectrum with an
absorbed power law, deriving a power-law index $\Gamma\approx 2$ and a
hydrogen column density $N_{\rm H}\approx 10^{23}$ cm-2. They inferred an
unabsorbed broad-band X-ray flux $f_{\rm 0.3-8~{}keV}=1\times 10^{-11}$ erg
cm-2 s-1, equivalent to a luminosity $L_{\rm 0.3-8~{}keV}=1.1\times 10^{41}$
erg s-1 at our assumed distance. A milliarcsecond-scale radio source was also
detected, with a $5$-GHz flux $f_{\rm 5~{}GHz}=6.6\times 10^{-26}$ erg cm-2
s-1 Hz-1. The measured brightness temperature exceeds $10^{7.7}$ K, ruling out
a starburst and/or supernova remnant origin, thus confirming the presence of a
LLAGN (Nagar et al., 2005). Following the bulge-disc decomposition summarised
above, Dong & De Robertis (2006) estimated a central supermassive black hole
mass $\log(M_{\bullet}/\rm M_{\odot})=8.2$ using their derived bulge mass -
black hole mass correlation based on 117 galaxies. Separately, Héraudeau et
al. (1999) reported a central stellar velocity dispersion $\sigma_{\ast}=163$
km s-1, corresponding to $\log(M_{\bullet}/\rm M_{\odot})=7.8$ using the
stellar velocity dispersion - black hole mass correlation of Kormendy & Ho
(2013). Taking the average of these two black hole mass estimates, the
Eddington luminosity is estimated to be $L_{\rm Edd}\approx 10^{46}$ erg s-1
(Ghosh et al., 2008). With this well-characterised LLAGN at the centre, the
potential influence of AGN feedback on star formation in the bulge of NGC 3169
can be uniquely probed.

###  2.2 Ionised gas data

NGC 3169 was observed with SITELLE (Drissen et al., 2019) at CFHT during the
2020B semester (programme number 20Bc25). The observations were centred at
R.A. (J2000)$=10^{\rm h}14^{\rm m}15\aas@@fstack{s}05$ and Dec.
(J2000)$=3^{\circ}27\arcmin 57\aas@@fstack{\prime\prime}90$. SITELLE is an
optical imaging Fourier transform spectrograph (IFTS) equipped with two E2V
detectors each with $2048\times 2064$ pixels. The field of view (FOV) is
$11\arcmin\times 11\arcmin$, resulting in a mean spaxel size on the sky of
$0\aas@@fstack{\prime\prime}31\times 0\aas@@fstack{\prime\prime}31$ (or
$28\times 28$ pc2). Two datacubes were obtained: one centred on the emission
lines of [N ii]$\lambda 6548$, H$\alpha$, [N ii]$\lambda 6583$, [S ii]$\lambda
6716$ and [S ii]$\lambda 6731$ with the SN3 filter ($6480$–$6860$ Å) at a mean
spectral resolution of $R\approx 2000$; the other centred on the emission
lines of H$\beta$, [O iii]$\lambda 4959$ and [O iii]$\lambda 5007$ with the
SN2 filter ($4840$–$5120$ Å) at a mean spectral resolution $R\approx 900$.

The data reduction was performed with the ORBS software developed for SITELLE
(Martin et al., 2015, 2021). The seeing was measured to be
$1\aas@@fstack{\prime\prime}1$ (or $99$ pc) from the full-width at half-maxima
(FWHM) of Gaussian fits to foreground stars from the Gaia catalogue (Lindegren
et al., 2018). The data were further calibrated in wavelength based on
velocity measurements of the OH sky line. As a final pre-processing step, sky
background subtraction was performed using a median sky spectrum extracted
from a $200\times 200$ spaxels region located far away from the galaxy disc.
The emission lines in each spaxel were then fitted using the extraction
software ORCS (Martin et al., 2015). For each emission line, maps of flux,
mean velocity and velocity dispersion were generated from ORCS output
parameters including integrated flux within each spaxel, maximum intensity,
velocity, FWHM, continuum level and signal-to-noise ratio ($S/N$) of the
spectrum as well as their uncertainties. Modelling the sinc-Gauss profile of
the emission lines allows to measure velocity dispersions much smaller than
the nominal spectral resolution, provided that the $S/N$ is sufficient (Martin
et al., 2016). A detection threshold was then applied based on the $3\sigma$
noise level of the H$\alpha$+[N ii] flux. For illustration purposes, the full
set of H$\alpha$ maps (before extinction correction) and the [N ii]$\lambda
6583$ velocity dispersion map used for this work are presented in Figure 2.
The extinction correction and conversion of the H$\alpha$ flux to $\Sigma_{\rm
SFR}$ are discussed in Section 3.1. We use the ratios of these emission lines
to correctly identify the H ii regions and examine the extent of the AGN
impact in Sections 3.3 and 5.

Figure 2: Top row: H$\alpha$ maps of the central region of NGC 3169 ($\approx
3.6\times 3.6$ kpc2), encompassing the galaxy bulge, extracted from our
SITELLE SN3 datacube using the sinc-gauss line-fitting function in ORCS as
described in Section 2.2. (a) Integrated intensity map. (b) Mean velocity map,
with respect to the heliocentric velocity of the galaxy ($1232$ km s-1, see
Section 2.1). (c) Velocity dispersion map. Bottom row: (d) [N ii]$\lambda
6583$ velocity dispersion map of the same region. The orange and black
ellipses in panels a, b, c and d separate the inner disc and outer star-
forming ring discussed in Sections 4–5. (e) Sample spectrum (blue line)
containing the emission lines [N ii]$\lambda 6548$, H$\alpha$, [N ii]$\lambda
6583$, [S ii]$\lambda 6716$ and [S ii]$\lambda 6731$. The ORCS-fitted spectrum
is shown as a red line, while the residuals are shown as a grey line and
shifted lower for illustration purposes.

###  2.3 Molecular gas data

The NGC 3169 molecular gas data used were obtained in the 12CO(2-1) line (230
GHz) using ALMA in January 2015 (configuration: C36-4; programme:
2015.1.00598.S; PI: Bureau) and January 2016 (configuration: 7M; programme:
2016.2.00053.S; PI: Liu), as part of the mm-Wave Interferometric Survey of
Dark Object Masses (WISDOM) project (Onishi et al., 2017). The first
observation (2015.1.00598.S) obtained $605$ s on the source using thirty-eight
$12$-m antennae, resulting in a primary beam (FWHM) of
$27\aas@@fstack{\prime\prime}4$ and a maximum recoverable scale of
$5\aas@@fstack{\prime\prime}8$. The intrinsic channel width was $1.27$ km s-1
($977$ kHz). The line was covered by a correlator window of $1.875$ GHz,
equivalent to $2450$ km s-1. The second observation obtained $1120$ s on
source using eight $7$-m antennae, that extended the primary beam (FWHM) to
$45\aas@@fstack{\prime\prime}7$ and the maximum recoverable scale to
$36\aas@@fstack{\prime\prime}4$. The line was covered by a correlator window
of $2$ GHz.

The raw ALMA data were reduced using the Common Astronomy Software
Applications (CASA) pipeline (version 5.3), which automatically processes the
data by performing basic flagging and calibration (McMullin et al., 2007). The
two ALMA observations were then combined using the CASA task concat. Continuum
emission was detected, measured over the full line-free bandwidth, and then
subtracted from the data in the $uv$ plane using the CASA task uvcontsub. The
data were then imaged using the CASA task tclean (version 5.6.1), adopting a
channel width of $2$ km s-1, a Briggs’ weighting robust parameter of $2.0$ and
a $uv$-taper of $0\aas@@fstack{\prime\prime}7$, thus achieving a synthesised
beam size (FWHM) of $0\aas@@fstack{\prime\prime}99\times
0\aas@@fstack{\prime\prime}84$ ($89$ pc$\times 76$ pc) at a position angle of
$-58^{\circ}$, roughly matching the seeing of the H$\alpha$ observations.
Finally, we convolved this datacube spatially by a narrow and slightly
elongated two-dimensional Gaussian, to achieve a perfectly circular
synthesized beam of FWHM $1\aas@@fstack{\prime\prime}1$, thus perfectly
matching the seeing of the H$\alpha$ observations. This produced our final
fully calibrated and continuum-subtracted 12CO(2-1) datacube of NGC 3169, with
a mean RMS noise level of $3.06$ mJy beam-1 per $2$ km s-1 channel, although
we also performed primary beam correction for the imaging of moment maps. Sky
coordinates and spaxel size of the datacube were also matched to those of the
SITELLE datacubes. Finally, another datacube was produced following the same
procedure but with a channel width of $15$ km s-1, to enhance the S/N for the
purpose of the gas kinematics analyses carried out in Section 4.5.

The integrated 12CO(2-1) spectrum and moment-0 (integrated flux), moment-1
(mean velocity) and moment-2 (velocity dispersion) maps are shown in Figure 3.
The $230$ GHz continuum map is also shown in Figure 3 and reveals a bright
region at the galaxy centre, slightly extended along the disc major axis and
with an integrated flux of $8.5$ mJy, that could be explained by dust or a jet
from the LLAGN. The one-sided structure of this continuum source favors the
explanation of a jet, although some VLA radio maps (in e.g. Nagar et al.,
2005) show extended structure in the perpendicular direction. Higher
resolution observations are needed to resolve the existence and morphology of
the jet. Nevertheless, the strong and broadened N ii emission lines at this
location indicate the existence of AGN-driven outflows or a supernova very
close to the LLAGN. A recent spectral energy distribution (SED) analysis of
the source also reveals a negative spectral index consistent with synchrotron
emission (Elford et al. ,in prep).

Figure 3:  Top row: 12CO(2-1) maps of the central region of NGC 3169 ($\approx
3.6\times 3.6$ kpc2), encompassing the galaxy bulge, extracted from our ALMA
observations as described in Section 2.3. (a) Moment-0 (integrated intensity)
map. (b) Moment-1 (mean velocity) map, with respect to the heliocentric
velocity of the galaxy of ($1232$ km s-1; see Section 2.1). (c) Moment-2
(velocity dispersion) map. Bottom row: (d) $230$-GHz continuum emission map.
The orange and black ellipses in panels a, b, c and d separate the inner disc
and outer star-forming ring discussed in Sections 4–5. (e) 12CO(2-1) spectrum
integrated over the whole field of view of the $15$ km s-1 per channel
datacube.

The physical scale of the synthesised beam ($99$ pc) allows to estimate the
molecular gas mass surface density and thus (combined with H$\alpha$) the SFE
at the typical spatial scale of giant molecular clouds (GMCs; $\approx 50$ pc;
Rosolowsky et al. 2003; 2007) and H ii regions of similar sizes. Stand-alone
GMC analyses should however be performed at the highest angular resolution
possible for the CO data, here $0\aas@@fstack{\prime\prime}58\times
0\aas@@fstack{\prime\prime}58$ ($52$ pc$\times 52$ pc), when adopting a more
standard Briggs’ robust parameter of 0.5 and no taper.

##  3 Surface densities and line ratio

###  3.1 SFR surface densities

We correct the observed H$\alpha$ fluxes ($F_{\rm H\alpha,obs}$) for
extinction using the observed H$\beta$ fluxes ($F_{\rm H\beta,obs}$) and the
assumed Balmer decrement described below. Due to differences in the spectral
resolutions and observing conditions of the two SITELLE datacubes, the
H$\beta$ spaxels that satisfy a $3\sigma$ detection threshold only constitute
$52\%$ of the H$\alpha$ spaxels satisfying that same condition. For each
spaxel with both the H$\alpha$ and H$\beta$ lines satisfying the $3\sigma$
detection threshold, the "color excess" of H$\alpha$ over H$\beta$ is defined
as

$E({\rm H}\beta-{\rm H}\alpha)\equiv 2.5\log\left(\frac{(F_{\rm
H\alpha,obs}/F_{\rm H\beta,obs})}{({\rm H\alpha}/{\rm H\beta})_{\rm
init}}\right)~{},$ (1)

where we assume $({\rm H\alpha}/{\rm H\beta})_{\rm init}=2.86$, as expected
for the case B recombination at a temperature of $10^{4}$ K (Osterbrock &
Ferland, 2006).

The H$\alpha$ extinction is then calculated as

$A_{\rm H\alpha}=\left(\frac{E({\rm H\beta}-{\rm H\alpha})}{k(\lambda_{\rm
H\beta})-k(\lambda_{\rm H\alpha})}\right)\,k(\lambda_{\rm H\alpha})$ (2)

following Nelson et al. (2016), where $k(\lambda)$ is the reddening curve of
Fitzpatrick (1986) and $k(\lambda_{\rm H\alpha})$ and $k(\lambda_{\rm
H\beta})$ are evaluated at the wavelengths of H$\alpha$ and H$\beta$,
respectively. For each spaxel with a $3\sigma$ H$\alpha$ detection but no
$3\sigma$ H$\beta$ detection, we adopt the H$\alpha$ extinction of the nearest
reliable spaxel. The extinction-corrected H$\alpha$ flux of each spaxel is
then obtained as

$F_{\rm H\alpha}=F_{\rm H\alpha,obs}\,e^{A_{\rm H\alpha}/1.086}~{}.$ (3)

This extinction-corrected H$\alpha$ flux is converted to a SFR using the
relation of Kennicutt Jr & Evans (2012):

$\log({\rm SFR}\,/\,{\rm M}_{\odot}\,{\rm yr}^{-1})=\log(L_{\rm
H\alpha}\,/\,{\rm ergs}\,{\rm s}^{-1})-41.27~{},$ (4)

where $L_{\rm H\alpha}=F_{\rm H\alpha}(4\pi D^{2})$ is the extinction-
corrected H$\alpha$ luminosity. We note that when probing spatial scales
smaller than $500$ pc, this conversion relation can break down, as seen in
examples of Kennicutt Jr & Evans (2012). Local SFR depend on environments and
age of the stellar population that we can only estimate here. Nevertheless,
this conversion holds true for the radial profile of SFE (see Section 4.3),
which is calculated within apertures large enough for confident H$\alpha-$SFR
conversions.

The $\Sigma_{\rm SFR}$ is then calculated as the SFR within each spaxel
divided by the spaxel area. The mean $\Sigma_{\rm SFR}$ in the central region
of NGC 3169,
$\log({\rm\Sigma_{SFR,~{}2~{}kpc}~{}/~{}M_{\odot}~{}yr^{-1}~{}kpc^{-2}})=-0.67$,
is calculated from the total SFR within a radius of $\approx 1.8$ kpc, as
listed in Table 1.

Figure 4:  Ionisation-mechanism classification over the whole disc of NGC
3169. Top-left: O3N2 diagram, showing the [O iii]/H$\beta$ versus [N
ii]/H$\alpha$ line ratios. AGN/LINER-ionised (blue), H ii (purple) and
composite (green) regions are separated by the classification criteria of
Kewley et al. (2006). The corresponding regions are colour-coded on the
H$\alpha$ flux map in the top-right, indicating the extent of the AGN impact.
Bottom-left: O3S2 diagram, showing the [O iii]/H$\beta$ versus [S
ii]/H$\alpha$ line ratios. AGN/LINER-ionised (blue) and H ii (orange) regions
are again separated by the classification criteria of Kewley et al. (2006),
and the H$\alpha$ flux map in the bottom-right is colour-coded accordingly.
The classification lines of Law et al. (2021) are overlaid as red lines on the
left panels, showing slightly different definitions of the regions and an
additional separation of AGN-like and LINER-like ionisation. The black boxes
indicates the central $\approx 3.6\times 3.6$ kpc2 region, encompassing the
galaxy bulge. The black ellipses separate the inner disc and outer star-
forming ring discussed in Sections 4–5.

###  3.2 Molecular gas mass surface densities

A map of the molecular gas mass surface density ($\Sigma_{\rm H_{2}}$) can be
obtained from the 12CO(2-1) observations described in Section 2.3. First, the
CO flux ($F_{\rm CO(2-1)}$) within each spaxel is obtained from the moment-0
map fluxes (in Jy beam-1 km s-1) by dividing by the synthesised beam area in
spaxels. $F_{\rm CO(2-1)}$ is then converted to luminosity according to

$\frac{L_{\rm CO(2-1)}}{\rm K~{}km~{}s^{-1}~{}pc^{2}}=\left(\frac{3.25\times
10^{7}}{{(1+z)}^{3}}\right)\,\left(\frac{F_{\rm CO(2-1)}}{\rm
Jy~{}km~{}s^{-1}}\right)\,{\left(\frac{\nu_{\rm obs}}{\rm
GHz}\right)}^{-2}\,\left(\frac{D}{\rm Mpc}\right)^{2}$ (5)

(e.g. Decarli et al., 2016), where $z$ is the galaxy redshift and $\nu_{\rm
obs}$ is the observed frequency (i.e. the redshifted frequency of the CO(2-1)
line). The luminosity-based molecular gas mass within each spaxel is then
calculated using

$\frac{M_{\rm H_{2}}}{{\rm M}_{\odot}}=4.4\,\left(\frac{L_{\rm CO}}{\rm
K~{}km~{}s^{-1}~{}pc^{2}}\right)\,\left(\frac{X_{\rm CO}}{\rm 2\times
10^{20}\,cm^{-2}~{}(K~{}km~{}s^{-1})^{-1}}\right)~{},$ (6)

where $L_{\rm CO}$ is the CO(1-0) luminosity and $X_{\rm CO}$ is the CO-to-H2
conversion factor. We adopt a CO(2-1)/CO(1-0) ratio of $0.8$ (in brightness
temperature units), typical of spiral galaxies (e.g. Lamperti et al., 2020),
and $X_{\rm CO}=2.3\times 10^{20}$ cm-2 (K km s-1)-1. The adopted $X_{\rm CO}$
is commonly used in extragalactic studies (e.g. Hughes et al., 2013; Utomo et
al., 2015; Sun et al., 2018), although it can depend on the environment of
molecular clouds, such as metallicity and radiation field (Bolatto et al.,
2013). The mass contribution from helium is considered and included in the
coefficient (Strong et al., 1988; Bolatto et al., 2013). The complete
conversion thus becomes

$\frac{M_{\rm H_{2}}}{{\rm M}_{\odot}}=6.325\,\left(\frac{L_{\rm CO(2-1)}}{\rm
K~{}km~{}s^{-1}~{}pc^{2}}\right).$ (7)

The molecular gas mass surface density is then calculated as $M_{\rm H_{2}}$
divided by the spaxel area. The mean $\Sigma_{\rm H_{2}}$ in the central
region of NGC 3169, $\log({\rm\Sigma_{\rm
H_{2},~{}2~{}kpc}~{}/~{}M_{\odot}~{}pc^{-2}})=2.31$, is calculated from the
total molecular gas mass within a radius of $\approx 1.8$ kpc, as listed in
Table 1.

###  3.3 Emission-line ratios

Considering the existence of the LLAGN at the centre of NGC 3169, it is
unclear what fraction of the H$\alpha$ emission originates from star
formation, especially within a $1$ kpc radius of the galaxy centre. To
accurately determine the SFEs in this region, we must distinguish the
different ionisation mechanisms and confidently identify the H ii regions.
Here, we utilise Baldwin, Phillips $\&$ Terlevich (BPT) diagrams (Baldwin et
al., 1981) to assess the extent of the AGN impact. We consider the ionised-gas
emission-line ratios of [O iii]/H$\beta$, [N ii]/H$\alpha$ and [S
ii]/H$\alpha$. These ratios are most relevant when distinguishing H ii regions
from AGN-ionised regions (as described in Baldwin et al., 1981) and they are
readily available from our SITELLE datacubes. Figure 4 shows two different BPT
diagrams: the "O3N2" diagram, which compares the [N ii]/H$\alpha$ ratios with
[O iii]/H$\beta$, and the "O3S2" diagram, which compares the [S ii]/H$\alpha$
ratios with [O iii]/H$\beta$. We applied the classification lines of Kewley et
al. (2006) to separate and thus uniquely identify H ii, AGN-ionised and
composite regions using the O3N2 diagram, and to separate H ii and
Seyfert/LINER-ionised regions using the O3S2 diagram. Spaxels are colour-coded
accordingly to these ionisation mechanisms in the flux maps shown in Figure 4.
The classification lines of Law et al. (2021) are overlaid on the left panels
of Figure 4 for reference, as they additionally show the separation of AGN-
like and Seyfert-like regions.

From the O3S2 diagram, the majority of spaxels can be identified as belonging
to H ii regions, except within the central $650$ pc radius where only a small
fraction of the spaxels contain gas definitely ionised by star formation. On
the other hand, the O3N2 diagram shows more ambiguous results. While the
spaxels within the central $650$ pc radius also appear to be affected by AGN
ionisation, the composite regions are much more extended, encompassing
$\approx 50\%$ of the spaxels in the central region and reaching beyond a
radius of $2$ kpc from the galaxy centre. The exact fraction of star-formation
ionisation in these composite and AGN-ionised regions is thus unclear.
Nevertheless, it is suggested from the maps that AGN-ionisation occupies the
diffuse emissions surrounding H$\alpha$ emission peaks.

The SFE analyses in Sections 4 and 5 thus systematically consider the three
cases based on the O3N2 diagram separately. Spaxels belonging to H ii regions
(colour-coded purple) are referred to as star-forming regions and their SFRs
can be confidently derived from their H$\alpha$ emissions. For spaxels that
occupy composite regions (colour-coded green) and AGN-ionised regions (colour-
coded blue), upper limits to the SFRs are estimated. The distinction between
AGN-like and LINER-like ionisation mechanisms is unclear, due to insufficient
$S/N$ of the H$\beta$ and O iii lines. Nevertheless, the majority of these
regions can be classified as AGN-like based on the O3S2 diagram, supporting
the hypothesis that AGN photoionisation is the dominant ionisation mechanism
within the inner $1$ kpc radius of NGC 3169. However, there are also a few
LINER-like regions presumably due to old stellar populations.

We also note that recent studies (e.g. Law et al., 2021) proposed using 3D
diagnostic diagrams in the parameter space of [O iii]/H$\beta$, [N
ii]/H$\alpha$ and [S ii]/H$\alpha$, to address the difference between the O3S2
and O3N2 diagrams. We include this alternative analysis in Appendix A, which
confirms the nature of H ii regions while showing that the composite regions
are likely AGN-ionised.

##  4 Analysis: depletion time and gas dynamics

###  4.1 Spatially-resolved depletion times

Figure 5 shows the spatially-resolved depletion time map, constructed from the
ratio of the SFR and molecular gas mass surface densities ($\tau_{\rm
dep}\equiv\Sigma_{\rm H_{2}}/\Sigma_{\rm SFR}$). For spaxels where both
$\Sigma_{\rm SFR}$ and $\Sigma_{\rm H_{2}}$ are reliably measured (i.e. where
both H$\alpha$ and CO are detected with $S/N>3$), $\tau_{\rm dep}$ is
calculated by simply dividing the two surface densities. This ratio is shown
in Figure 5 with an orange-purple colour scale. However, many spaxels have
only one tracer brighter than our adopted detection threshold. When only
$\Sigma_{\rm SFR}$ is reliably measured, we estimate a $\tau_{\rm dep}$ upper
limit by dividing the $3\sigma$ upper limit on the molecular gas mass surface
density by the reliably measured SFR surface density, shown in Figure 5 with a
light-orange corlour scale. An analogous approach is applied to spaxels where
only $\Sigma_{\rm H_{2}}$ is reliably measured, to estimate a $\tau_{\rm dep}$
lower limit, shown in Figure 5 with a grey-purple colour scale.

Figure 5: Depletion time map of the central region of NGC 3169 ($\approx
3.6\times 3.6$ kpc2), encompassing the galaxy bulge, calculated from the
$\Sigma_{\rm H_{2}}$ and $\Sigma_{\rm SFR}$ maps (see Section 4.1). The
orange-purple colour scale shows actual measurements, for spaxels where both
CO and H$\alpha$ are above our detection threshold. The light-orange color
scale shows upper limits, for spaxels where only H$\alpha$ is above our
detection threshold. The grey-purple color scale shows lower limits, for
spaxels where only CO is above our detection threshold. Blank spaxels are
either outside the ALMA field of view or have both H$\alpha$ and CO below our
detection threshold. The black ellipse separates the inner disc and outer
star-forming ring discussed in Sections 4–5.

We note that Figure 5 purposefully shows all spaxels where a tracer is
detected, without distinguishing them based on the dominant ionisation
mechanism. For spaxels identified as composite or AGN-ionised based on their
emission-line ratios (see Section 3.3), the SFR is therefore likely an upper
limit. In turn, the derived depletion times measurements are likely lower
limits. We return to this issue in Sections 4.2 and 4.3.

The region of NGC 3169 within the ALMA FOV has a radius of $\approx 1.8$ kpc,
and will be referred to here as the "central region". It is nearly completely
filled with molecular gas, while the ionised-gas emission is more patchy and
concentrated within a ring-like structure encompassing radii of $\approx 0.65$
kpc to $\approx 1.8$ kpc. The average depletion time within the ring is
$\approx 0.3$ Gyr, calculated from the ratio of total molecular gas mass to
total SFR. It is consistent with typical extragalactic starburst regions (e.g.
Barrera-Ballesteros et al., 2021). The ring may constitute a resonance ring
(e.g. Buta & Combes, 1996; Buta & Purcell, 1998; Comerón et al., 2014) or may
simply be a pseudo-ring formed by winding spiral arms. Inside the ring (radii
$\lesssim 650$ pc), the molecular gas is more concentrated and star formation
appears to be suppressed. This inner disk also has stronger impact from the
bulge, with B/D$>1$. The radial variation of the depletion time is discussed
in Section 4.3.

Figure 6:  Plots of $\Sigma_{\rm H_{2}}$-$\Sigma_{\rm SFR}$, overlaid with the
power-law relations of Kennicutt Jr (1998) and Bigiel et al. (2008), shown as
light-purple and light-pink shaded regions, respectively. The power-law
relation of Davis et al. (2014) for early-type galaxies is shown as a yellow
line. Using the BPT analysis of Section 3.3, from left to right the panels
include only spaxels classified as star-forming, composite and AGN-ionised,
respectively. All data points are from spaxels with both H$\alpha$ and CO
above our detection threshold. The mean $\Sigma_{\rm H_{2}}$-$\Sigma_{\rm
SFR}$ of within the ALMA FOV is shown as an orange star in each panel.
Representative depletion timescales of $0.1$, $1$ and $10$ Gyr are shown as
black dashed lines.

###  4.2 SFR to molecular gas mass relation

Figure 6 shows the $\Sigma_{\rm SFR}$-$\Sigma_{\rm H_{2}}$ relation of all the
spaxels with reliably-detected H$\alpha$ and CO (orange-purple colour scale in
Figure 5). However, contrary to Figure 5, we separate and compare the three
cases discussed in Section 3.3: star-forming (orange scale), composite (green
scale) and AGN-ionised (blue scale) spaxels, based on the O3N2 type BPT
diagnostic described in Section 3.3. It is important to recall that for the
composite and AGN-ionised cases, $\Sigma_{\rm SFR}$ has large uncertainties
and should really be treated as an upper limit. The mean $\Sigma_{\rm SFR}$
and $\Sigma_{\rm H_{2}}$ within the ALMA field of view is shown with an orange
star in Figure 5. The distributions are compared to characteristic timescales
($0.1$, $1$ and $10$ Gyr) and the power-law relations of Kennicutt Jr (1998)
and Bigiel et al. (2008), shown as light-purple and light-pink shaded areas,
respectively. The power-law relation of Davis et al. (2014) for early-type
galaxies is also shown as yellow lines for reference.

Because ionised-gas and molecular gas peaks are generally not exactly co-
spatial (e.g. Schruba et al., 2010; Schinnerer et al., 2019; Kruijssen et al.,
2019b; Pan et al., 2022), it is common to have broad $\Sigma_{\rm
SFR}$-$\Sigma_{\rm H_{2}}$ distributions at sub-kpc scales. In fact, this
scatter can be interpreted as a sign of a rapid cycling between gas and stars
on the sub-kpc scales, which allows us to constrain molecular gas lifetimes,
as shown in Section 4.6. Figure 6 reveals that this is indeed the case in NGC
3169, but the $\Sigma_{\rm SFR}$-$\Sigma_{\rm H_{2}}$ measurements roughly
follow the Kennicutt Jr (1998) and Bigiel et al. (2008) relations in each
case, encompassing depletion timescales of $0.1$ to $10$ Gyr. Nevertheless,
the three distributions peak at different $\Sigma_{\rm SFR}$ and $\Sigma_{\rm
H_{2}}$. Although some star-forming spaxels have short depletion timescales
(see left panel of Fig. 6), many star-forming spaxels occupy a part of the
$\Sigma_{\rm SFR}$-$\Sigma_{\rm H_{2}}$ space centred at $\Sigma_{\rm
SFR}\approx 0.04$ M⊙ yr-1 kpc-2 and $\Sigma_{\rm H_{2}}\approx 100$ M⊙ pc-2
($\tau_{\rm dep}\approx 2.5$ Gyr), slightly larger than the average $\tau_{\rm
dep}\approx 2$ Gyr reported by Bigiel et al. (2008). In the composite regions,
shown in the middle panel of Figure 6, the distribution is centred at
$\Sigma_{\rm SFR}\approx 0.3$ M⊙ yr-1 kpc-2 and $\Sigma_{\rm H_{2}}\approx
160$ M⊙ pc-2 ($\tau_{\rm dep}\approx 0.5$ Gyr). However, this results from
ionised-gas emission that is only partly due to star formation (i.e. the
derived $\Sigma_{\rm SFR}$ of composite regions are really upper limits), so
this peak would be lowered in the figure (higher $\tau_{\rm dep}$) if we knew
the exact fraction of the star-forming radiation. As shown in the right panel
of Figure 6, the $\Sigma_{\rm SFR}$-$\Sigma_{\rm H_{2}}$ distribution of AGN-
ionised regions is similar to that of composite regions, but more scattered
and extended.

###  4.3 Radial trend of depletion time

Figure 7 shows $\tau_{\rm dep}$ as a function of deprojected galactocentric
radius, where individual spaxels are shown as purple datapoints, and the
running mean for each case discussed in Section 3.3 is shown in the associated
colour, using annuli of width $200$ pc and a step size of $100$ pc. Only
spaxels reliably-detected in both H$\alpha$ and CO are included in this
figure. For reference, the global depletion time ($0.9$ Gyr) is indicated by a
grey horizontal line, calculated from the ratio of the total SFR to the total
molecular gas mass within a radius of $\approx 1.8$ kpc that is the ALMA FOV.

Figure 7:  Depletion time radial profile in the central region of NGC 3169,
encompassing the galaxy bulge. Purple datapoints are calculated from all the
spaxels with both H$\alpha$ and CO above our detection threshold in Figure 5.
The black line is the running mean of the depletion times of all datapoints as
a function of galatocentric radius. The red, green and blue lines are the
running means of the star-forming, composite and AGN-ionised regions only,
respectively. The vertical black dashed line separates the inner disc and
outer star-forming ring at a radius of $650$ pc discussed in Sections 4–5. The
grey horizontal line shows the mean depletion time within the ALMA FOV,
corresponding to the orange star in Figure 6.

Between galactocentric radii of $0.65$ and $1.8$ kpc, i.e. within the
previously-discussed star-forming ring, the depletion times of individual
spaxels cover a wide temporal range ($0.03$ – $10$ Gyr). The mean $\tau_{\rm
dep}$ within this region is $\approx 0.3$ Gyr, lower than the global
$\tau_{\rm dep}$ of $0.9$ Gyr. Larger variations of the average depletion
times occur beyond a radius of $1.8$ kpc, marking the outer edge of the star-
formation ring. As shown in Figure 5, GMCs and ionised-gas clouds are
particularly not co-spatial beyond this radius.

Within a galactocentric radius of $\approx 0.9$ kpc, the average $\tau_{\rm
dep}$ rapidly increases with decreasing radius, with or without the inclusion
of composite and AGN-ionised regions. For exclusively star-forming regions
(orange curve in Figure 7), the mean $\tau_{\rm dep}$ reaches $\approx 2.3$
Gyr at a radius of $\approx 500$ pc. The further inclusion of composite and
AGN-ionised regions (blue curve) is considered merely for illustration
purposes, but clearly shows that there is H$\alpha$ emission down to a radius
of $\approx 100$ pc, partly as the result of AGN ionisation. The measurements
nevertheless clearly indicate depletion times of at least the age of the
Universe (most likely more) in the central few hundred parsecs. Therefore,
although the average $\Sigma_{\rm SFR}$ and $\Sigma_{\rm H_{2}}$ of this
central region follows the SF relations of star-forming regions, we observe a
clear transition from efficient SF in the star-forming ring to quenched SF
within the central $\approx 500$ pc in radius. These low SFEs result from
feeble star formation in that region, despite plentiful molecular gas,
potentially due to accretion onto the central regions.

The radial extent of AGN ionisation is also illustrated in Figure 7. While
H$\alpha$ emissions reaches the centre of the galaxy, the inner $400$ pc in
radius are dominated by AGN ionisation. Individual spaxels that are
confidently classified as exclusively star-forming are located at a
galactocentric radius of at least $650$ pc. The size of the AGN-ionised region
is consistent with that of other nearby galaxies. For example, it is within
the broad range of $0.3$–$10$ kpc reported by Chen et al. (2019) based on 152
AGN-host galaxies using the Mapping Nearby Galaxies at Apache Point
Observatory (MaNGA) survey.

###  4.4 Virial parameters

To take a more in-depth look at the star formation on cloud scales, we
identified H ii regions using the SITELLE H$\alpha$ data and algorithms
developed by Rousseau-Nepton et al. (2018), summarised below. The H$\alpha$
flux peaks are first identified, based on the following criteria: (1) the flux
of the peak spaxel is greater than the flux of at least five immediately
surrounding spaxels, (2) the total flux in a $3\times 3$ spaxels box centred
on the emission peak is above the $3\sigma$ detection threshold and (3) if two
emission peaks are separated by a distance smaller than the seeing
($1\aas@@fstack{\prime\prime}1$ or $99$ pc in this case), only the brightest
peak is preserved. Given an estimate of the characteristic radius of H ii
regions, spaxels are then classified into zones surrounding the peaks, and the
boundaries of the spatial zones are defined. Figure 13 in Appendix B shows
these peaks and region boundaries overlaid on the H$\alpha$ flux map of NGC
3169. For each H ii region, a two-dimensional (2D) Gaussian fit is then
carried out and the luminosity ($L_{\rm\text{H\,{ii}}}$ based on H$\alpha$
flux) and size (half-width at half-maximum $R_{\rm\text{H\,{ii}}}$) of the
best-fitting Gaussian are calculated (and taken as those of the H ii region).
Furthermore, the spectra from the SITELLE H$\alpha$ datacube are binned around
the best-fitting peak of each H ii region, according to the size of the
region, and the resulting spectrum is fit with ORCS to derive the velocity
dispersion ($\sigma_{\rm\text{H\,{ii}}}$) of the region. We note that not all
H ii regions identified are necessarily also star-forming regions, as some may
be partially or primarily ionised by the AGN. The full H ii region catalogue
is presented in Table 3 in Appendix B.

Based on these measurements, the virial parameter of each region is calculated
as

$\alpha_{\rm vir,\text{H\,{ii}}}\equiv
5\,\frac{\sigma_{\rm\text{H\,{ii}}}^{2}R_{\rm\text{H\,{ii}}}}{GM_{\rm\text{H\,{ii}}}}$
(8)

(Bertoldi & McKee, 1992), where $M_{\rm\text{H\,{ii}}}$ is the ionised-gas
mass of the region, estimated from

$\frac{M_{\rm\text{H\,{ii}}}}{\rm M_{\odot}}=1.57\times
10^{-17}\,\left(\frac{L_{\rm\text{H\,{ii}}}}{\rm
erg~{}s^{-1}}\right)^{\frac{1}{2}}\,\left(\frac{R_{\rm\text{H\,{ii}}}}{\rm
pc}\right)^{\frac{3}{2}}$ (9)

(Zaragoza-Cardiel et al., 2015). This ionised-gas mass assumes that the mean
electron number volume density can be derived from $L_{\rm\text{H\,{ii}}}$
assuming a spherical geometry, and that all ionising photons are from
hot/massive stars.

As the ionised-gas mass is usually only a fraction of the total gas mass
(Zaragoza-Cardiel et al., 2015), the virial parameter calculated above is
typically larger than that based on molecular gas. Therefore, it is not fair
to use a criterion of about unity to diagnose whether a cloud is
gravitationally bound (often stated as $\alpha_{\rm vir}\lesssim 2$). However,
it is meaningful to compare virial parameters measured for different H ii
regions. A larger virial parameter implies a higher kinetic energy, that is
harder to counter-balance with gravity and is more likely to require external
pressure to maintain equilibrium. As should then be expected, there is a weak
correlation between $\alpha_{\rm vir,\text{H\,{ii}}}$ and $\tau_{\rm dep}$,
shown in the left panel of Figure 8. The $\tau_{\rm dep}$ in each H ii region
is calculated based on the ratio of molecular gas mass and SFR within the
region. According to equations (8), variations of $\alpha_{\rm
vir,\text{H\,{ii}}}$ are in fact primarily due to the velocity dispersions.
This suggests that large $\tau_{\rm dep}$ in the inner $1$ kpc radius can be
associated with high turbulence.

Figure 8:  (a) H ii region depletion time versus virial parameter, with the
best-fitting power-law relation overlaid as a red line. (b) Depletion time
versus virial parameter calculated from the molecular gas at each spaxel. The
orange datapoints and associated red density contours are based on spaxels
classified as star-forming only, while the grey datapoints and associated
black density contours are based on all the spaxels with reliable measurements
of $\tau_{\rm dep}$ and $\alpha_{\rm vir,\,gas}$.

As a comparison, we also estimate the molecular gas virial parameter on the
cloud scale, which is equivalent to the CO spaxel size of $\approx 35$ pc. The
molecular gas virial parameter ($\alpha_{\rm vir,\,H_{2}}$) can be calculated
following the method from Rosolowsky & Leroy (2006) that is recently adopted
by Sun et al. (2020):

$\alpha_{\rm vir,\,H_{2}}=3.1\times\,\left(\frac{\Sigma_{\rm H_{2}}}{\rm
10^{2}\,M_{\odot}\,{\rm pc}^{2}}\right)^{-1}\left(\frac{\sigma_{\rm
H_{2}}}{\rm 10\,km\,s^{-1}}\right)^{2}\left(\frac{D_{\rm spaxel}}{\rm
150\,pc}\right)~{},$ (10)

where $\Sigma_{\rm H_{2}}$ is calculated in Section 3.2, $\sigma_{\rm H_{2}}$
is taken from the moment-2 map shown in Figure 3 and $D_{\rm spaxel}$ is the
CO spaxel size of $\approx 35$ pc, comparable to typical GMC sizes. As shown
in the right panel of Figure 8 with orange datapoints and associated red
distribution density contours, the majority of spaxels belonging to star-
forming regions appear to be roughly virialised, with virial parameters
varying between $0$ and $10$ and depletion times less than $1$ Gyr. A virial
parameter equals 2 is shown with a vertical dotted line on the right panel of
Figure 8, for illustration purposes. On the other hand, a wider range of
$\alpha_{\rm vir,\,gas}-\tau_{\rm dep}$ parameter space is covered by all
reliably measured spaxels (i.e. including those with composite and AGN
ionisation), shown with grey datapoints and associated black distribution
density contours in the right panel of Figure 8. This suggests that AGN-
ionisation increases both $\alpha_{\rm vir,\,gas}$ and $\tau_{\rm dep}$.
However, none of the mean molecular mass surface density, velocity dispersion
and/or size is accurately estimated from that within a single spaxel, so the
virial parameters evaluated using equation (10) should not be taken nor
interpreted literally. Instead, they are rough indicators of the significance
of the kinetic energy (turbulence) relative to the gravitational potential
(gravity), which varies among spaxels.

###  4.5 Non-circular motions

We now probe both molecular gas and ionised gas kinematics in the central
region of NGC 3169, to understand more about the SFEs in the inner disc and
the star-forming ring.

For the molecular gas, we use the 3DBarolo
software111https://editeodoro.github.io/Bbarolo/ of Teodoro & Fraternali
(2015), that fits tilted-ring models to three-dimensional (3D) emission-line
datacubes, taking advantage of the high spectral and spatial resolutions of
the ALMA CO observations. Following a first fit with all parameters free, we
fix the dynamical centre (spatially and spectrally)and inclination, while
letting the position angle (PA) vary to account for warping of the disc. The
rotation and radial velocities from the resulting best-fitting model are shown
in Figure 9 with blue lines. As a reference, the corresponding position-
velocity (PV) profile and best-fitting reults of all the relevant parameters
are in Appendix C.

Figure 9: (a) Rotation curve (left) and radial velocity profile (right) for
both CO (blue line) and H$\alpha$ (red line) gas. The green and orange
backgrounds are separated at the dotted black line, indicating a radius of
$650$ pc that is the intersection of the inner disc and the star-forming ring.
(b) CO velocity residual map, defined as the difference between the observed
velocity, shown in Figure 3b, and the best-fitting circular velocity model
from 3DBarolo.

We use another method to determine the rotation and radial velocity profiles
of the ionised gas, by performing Markov Chain Monte Carlo (MCMC) fits of the
tangential and radial velocity components. A nuisance parameter is also
applied, which corresponds to the line-of-sight velocity scatter in the axi-
symmetric modelling of the observed asymmetric kinematics. The Bayesian
inferences are done using the Python library emcee (Foreman-Mackey et al.,
2013), fixing the dynamical centre, inclination and PA of the H$\alpha$
velocity field as the best-fitting values obtained from the 3DBarolo for the
CO gas. This way, we use the same set of kinematic parameters for both gaseous
components to assist the comparisons, while taking into account the individual
errors on the line-of-sight H$\alpha$ velocities, which are unique to the
SITELLE datacubes. The best-fitting rotation and radial velocities are shown
in Figure 9 with red lines.

The rotation and radial velocity profiles of CO and H$\alpha$ gas show
different behaviors within the inner disc and in the star-forming ring, marked
in Figure 9 with green and orange shaded areas, respectively. The inner edge
of the star-forming ring at $650$ pc radius is marked with a black dotted
line. Beyond the radius of $\approx 650$ pc, the derived rotation curve of CO
gas suggests roughly constant rotation velocities of $240$–$260$ km s-1 in the
star-forming ring. The rotation curve of H$\alpha$ gas, however, increases
from $200$ km s-1 at $650$ pc to $280$ km s-1 at $1700$ pc.

Higher CO rotation velocities are found in many galaxies (Davis et al., 2013;
Levy et al., 2018). If H$\alpha$ is tracing recent star formation, it would
typically co-rotate with CO gas in the mid-plane of the galaxy disc. The
discrepancy in NGC 3169 can be explained by a different scale height of
molecular and ionised gas, due to the fact that the H$\alpha$ gas is
associated with a mixture of ionisation mechanisms, including old stellar
population and shock-ionisation from the AGN. Based on our ionised gas
emission line ratio analysis in Section 3.3, this is likely the case. The
positive slope of H$\alpha$ rotation velocities from inner parts of the star-
forming ring to outer parts, where they merge with CO rotation velocities,
also indicate that the fraction of recent star-forming ionisation is dominant
as we move away from the centre of the galaxy.

The kinematics in the inner disc appear to be much more complex. Both CO and
H$\alpha$ rotation velocities rise to $\approx 300$ km s-1 and then rapidly
drop to below $200$ km s-1. The resulting rotation curve resembles those of
many barred galaxies, with a characteristic “double-hump” (i.e., a steep inner
rise of the rotation velocities, followed by a decrease and local minimum
before a second increase to the flat part of the rotation curve; e.g. Bureau &
Freeman 1999; Athanassoula & Bureau 1999). Although NGC 3169 does not harbour
a large-scale bar, Figs. 1 and 3 suggest a small-scale bar within the star-
forming ring (half-length $\lesssim 650$ pc). This structure may be
responsible for transporting material to the inner regions, causing high
turbulence.

As shown in the middle panel of Figure 9, the radial velocities of the CO and
H$\alpha$ gas oscillate from positive to negative, tracing expansion and
contraction at different locations. In particular, the oscillations of CO and
H$\alpha$ are out of phase, indicating complex non-circular motions. The right
panel of Figure 9 shows the residual velocity of CO gas, which is the
difference between observed velocity (as shown in Figure 3 ) and the best-fit
model from 3DBarolo. It shows the presence of significant spiral arm-like
residuals. Given the presence of a small-scale bar, such residuals may trace
inflow/outflow streaming motions induced by the inner bar (see e.g. Domínguez-
Fernández et al., 2020; Ruffa et al., 2022). Another source of non-circular
motion is outflows. In Figure 9 (c), there appear to be particularly large CO
residual velocities near the intersection of the major axis and the $650$ pc
line. At this location, we also find secondary emission peaks on the CO
spectra, as well as asymmetric broadening of ionized gas emission lines. As
these features are close to the galaxy centre, they suggest the presence of
gas blobs ejected from the galaxy plane, as a result of AGN-driven outflows.
However, the signal-to-noise at this location is not robust enough to draw
concrete conclusions.

Regardless of the sources of non-circular motions, high velocity dispersions
are observed within the inner disc as a result. As shown in Figure 14, the
molecular gas velocity dispersions are high throughout ($\sigma_{\rm
gas}\gtrsim 20$ km s-1) and rise rapidly with decreasing radius within the
inner disc, reaching $\approx 70$ km s-1 in the very centre. Since 3DBarolo
takes into consideration beam smearing effects, the best-fitting velocity
dispersion result can assist the moment-2 maps in Figure 3 to illustrate the
turbulent gas motions within the inner region of NGC 3169.

###  4.6 Molecular cloud timescales

A spatial de-correlation between CO and H$\alpha$ emission regions is commonly
observed in nearby galaxies (e.g. Kruijssen et al., 2019b; Chevance et al.,
2020; Zabel et al., 2020), due to the different life cycles of different
regions (e.g. Kruijssen & Longmore, 2014), i.e. at least partially independent
cloud assembly, collapse, star formation and disruption due to stellar
feedback. A statistical analysis of CO and H$\alpha$ snapshots of these
regions therefore allows us to estimate the characteristic lifetime of the
molecular gas ($t_{\rm gas}$), the duration of the young stellar phase
($t_{\rm star}$), duration of the overlapping/feedback phase ($t_{\rm fb}$),
as well as the characteristic separation length of individual regions
($\lambda$). To infer these quantities, we adopt here the methods developed by
Kruijssen et al. (2018) and use the associated HEISENBERG code. This approach
has been used previously to characterise the GMC life-cycle for a wide range
of observed galaxies (e.g. Kruijssen et al., 2019b; Chevance et al., 2020;
Ward et al., 2020; Zabel et al., 2020; Kim et al., 2021; Chevance et al.,
2022), as well as for numerical simulations (e.g. Fujimoto et al., 2019;
Jeffreson et al., 2021).

We briefly summarise the methodology below, while the details can be found in
Section 3 of Kruijssen et al. (2018). The first step is to measure CO-
to-H$\alpha$ flux ratios in apertures of a range of sizes, all centred on the
peaks of the CO and the H$\alpha$ emission, respectively. The identification
of the molecular gas and ionised-gas peaks and the total flux calculations use
the same maps as for the depletion time calculations (see Section 4). The
relative changes of the flux ratios compared to the galactic average, as a
function of the aperture size, are governed by the three aforementioned
timescales ($t_{\rm gas}$, $t_{\rm star}$ and $t_{\rm fb}$) as well as
$\lambda$, as shown in Kruijssen & Longmore (2014). We fit the observed CO-
to-H$\alpha$ flux ratio with the analytical model presented in Kruijssen et
al. (2018) to constrain $\lambda$ and the ratios $t_{\rm gas}/t_{\rm star}$
and $t_{\rm fb}/t_{\rm star}$. Assuming a reference timescale $t_{\rm
star,~{}ref}=4.3$ Myr for the isolated stellar phase (the appropriate
timescale when using continuum-subtracted H$\alpha$ emission as the SFR
tracer; see Haydon et al. 2020), we convert these timescale ratios into
absolute values.

Table 2: Best-fitting timescales and separation length.

| Case A | Case B | Case C
---|---|---|---
$t_{\rm gas}$ (Myr) | $7.1^{+5.3}_{-2.1}$ | $10.5^{+4.2}_{-2.6}$ | $6.7^{+5.0}_{-1.4}$
$t_{\rm fb}$ (Myr) | $\leq 2.9$ | $\leq 3.3$ | $\leq 1.5$
$t_{\rm star}$ (Myr) | $5.5^{+1.5}_{-0.9}$ | $5.9^{+1.7}_{-0.9}$ | $4.4^{+1.4}_{-0.16}$
$\lambda$ (pc) | $145^{+59}_{-38}$ | $148^{+53}_{-28}$ | $163^{+33}_{-21}$

Notes: all values are associated with their $1\sigma$ uncertainties

As the input maps for this analysis require contiguous emission across each
regions, a naive separation of star-forming and AGN-ionised regions based on
BPT diagram diagnostics would not work. Considering this constraint, we adopt
a different approach to nevertheless attempt to distinguish emission arising
from each of the three ionisation mechanisms discussed in Section 4 (i.e.
star-forming, composite and AGN-ionised regions). For Case A listed in Table
2, all emission within the field of view is considered, implying that all
three ionisation mechanisms are included. For Case B, a central region of
$650$ pc radius is masked out. As the majority of AGN-ionised regions is
within this central region, Case B includes mostly star-forming and composite
regions. For Case C, in addition to removing the central region, diffuse
emissions is also removed across the field, achieved by tapering large-scale
emission in Fourier space. More specifically, the emission in each line is
transformed into Fourier space, and a Gaussian high-pass filter is then
applied centred at zero Fourier frequency, using the best-fitting $\lambda$
(the separation lengthscale) in an iterative process (Hygate et al., 2019). As
can be inferred from Figure 4 and the discussions in Sections 3.3 and 5.2,
diffuse emission is dominated by ionisation from the AGN, generally leading to
a composite classification. Case C thus mostly contains star-forming regions,
as required by the adopted methodology, and therefore constitutes our
reference case. The Case C maps, relative flux ratios and best-fitting model
from the HEISENBERG code are shown in Figure 10, while those for Cases A and B
are shown in Figure 15 for reference.

Figure 10:  Left: relative change of the CO-to-H$\alpha$ flux ratio as a
function of aperture size for Case C (see Section 4.6, with a focus on the
molecular gas peaks (red datapoints) and the ionised-gas peaks (blue
datapoints). The best-fitting model based on the uncertainty principle for
star formation of Kruijssen et al. (2018) is shown as a green solid line (see
Section 4.6). The horizontal grey dotted line indicates the galactic average
of the flux ratio normalized to the mean $\tau_{\rm dep}=0.9$ Gyr within the
ALMA FOV. The associated best-fitting parameters with $1\sigma$ uncertainties
are listed at the bottom of the panel. Right: corresponding CO (top) and
H$\alpha$ (bottom) maps.

As listed in Table 2, the fitting results from the three cases agree with one
another within the $1\sigma$ uncertainties. The fact that the separation
lengths are $1.5$ times larger than the spatial resolution of our data (i.e.
$\lambda>100$ pc) gives us confidence that the spatial resolution of our maps
is sufficient for this analysis. The separation length of $\approx 160$ pc is
also typical of nearby spirals ($100$–$300$ pc; Chevance et al. 2020). The
molecular gas lifetime is rather short ($t_{\rm gas}=6.7^{+5.0}_{-1.4}$ Myr),
but with high upper limit. A gas lifetime of $\approx 10$ Myr, comparable to
the representative cloud free-fall time, typically indicates that the cloud is
regulated by internal dynamical processes. Comparing to the GMC lifetimes of
other nearby galaxies (e.g. Kruijssen et al., 2019b; Zabel et al., 2020;
Chevance et al., 2022), the cloud lifetime is reasonable, given the small
galactocentric radii where the dynamical timescales are typically small.
However, the number of flux peaks identified, 77 for CO flux and 80 for
H$\alpha$ flux, is not large enough to constrain timescale radial trends. The
blending of the CO and H$\alpha$ peaks also results in significant
uncertainties on the feedback timescale $t_{\rm fb}$. This is characterized
with a region filling factor $\zeta$ described in Kruijssen et al. (2018),
which is $\approx 0.7$ in our best-fitting result that prevent the accurate
measurement of $t_{\rm fb}$ within a few Myr (see e.g. Chevance et al., 2022).
Therefore, only the upper limits of $t_{\rm fb}$ for each case are listed in
Table 2. These factors unfortunately prohibit firm conclusions about the
nature of the dominant GMC-scale feedback and quenching mechanisms in the
central region of NGC 3169. Observations with higher spatial resolution and
larger FOV can help to resolve some of the issues.

##  5 Discussion

###  5.1 Quenching from bulge dynamics

In Section 4.3, we showed that the short depletion times in the star-forming
ring gradually increase as we probe close to the centre of the galaxy, and
identified it as a quench of star-formation. A common explanation for star-
formation quenching in galaxy bulges is the bulge dynamics itself, which can
enhance velocity dispersions in the central regions of the galaxy (e.g. Martig
et al., 2009, 2013). In particular, Gensior et al. (2020) and Gensior &
Kruijssen (2021) simulated the influence of a bulge on the radial variations
of the SFE. A power-law relation normally exists between $\Sigma_{\rm SFR}$
and $\Sigma_{\rm H_{2}}$ in bulgeless galaxies (i.e. the usual “spatially-
resolved” Kennicutt-Schmidt relation; Kennicutt Jr & Evans 2012; Bigiel et al.
2008). A similar power-law relation is reproduced by the simulations of
bulgeless galaxies in Gensior et al. (2020), but with a different slope.
However, adding bulges to the galaxy models results in $\Sigma_{\rm
SFR}/\Sigma_{\rm H_{2}}$ ratios that are systematically lower at small
galactocentric radii, and the decrease is more extreme in denser bulges.

Here, we attempt to compare the radial trend of $\Sigma_{\rm SFR}/\Sigma_{\rm
H_{2}}$ of both our NGC 3169 data and simulations of galaxies with dense
bulges. The $\Sigma_{\rm SFR}-\Sigma_{\rm H_{2}}$ ratio of NGC 3169 is shown
in Figure 11 as a function of galactocentric radius (using running means;
another way to visualise Figs. 5–7). The estimated power-law relation of
bulgeless galaxies from the simulation of Gensior et al. (2020) is shown as a
red dashed line, and representative depletion timescales shown as black dashed
lines. The bulge simulations of Gensior & Kruijssen (2021), with an effective
radius of $\approx 1$ kpc and a bulge-to-total mass ratio of $\approx 0.9$,
are shown with triangles. As the simulations are idealised cases, with smooth
rotation curves and a limited range of $\Sigma_{\rm H_{2}}$, they do not
accurately represent the bulge environment of NGC 3169. Nevertheless, the
simulations do show that the presence of a bulge can produce a radial trend of
the $\Sigma_{\rm SFR}/\Sigma_{\rm H_{2}}$ ratio similar to that observed in
NGC 3169. We also notice that the simulated bulges are much denser than the
bulge of NGC 3169, with an effective radius $\approx 1.4$ kpc, a bulge-to-
total mass ratio $\approx 0.1$ and B/D $>1$ only within a radius of $660$ pc.
This suggests that other mechanisms may be required to quench SF within the
central kpc of this galaxy.

Figure 11:  Relation of $\Sigma_{\rm H_{2}}$-$\Sigma_{\rm SFR}$. The
galactocentric radius of each datapoint is indicated by its colour scales.
Filled circles are azimuthal averages from our NGC 3169 data, for spaxels
classified as star-forming only. Filled triangles are azimuthal averages from
the simulations of Gensior & Kruijssen (2021). The red dashed line shows the
power-law relation best-fitting the results from a bulgeless galaxy simulation
(Gensior et al., 2020). Representative depletion timescales of $0.1$, $1$ and
$10$ Gyr are shown as black dashed lines.

Using the relation found by Gensior & Kruijssen (2021), the sSFR predicted
from bulge dynamics can be inferred by relating it to the total molecular gas-
to-stellar mass fraction ($f_{\rm gas}$) and the stellar mass surface density
($\mu_{\star}$):

$\begin{split}\log({\rm
sSFR_{sim}\,/\,Gyr^{-1}})=-1.79+4.00\log\left(\frac{f_{\rm
gas}}{0.05}\right)\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-0.99\log\left(\frac{\mu_{\star}}{10^{8}\,{\rm
M_{\odot}\,kpc^{-2}}}\right)~{}.\end{split}$ (11)

Using the total molecular gas mass of $\approx 10^{9.3}~{}{\rm M_{\odot}}$,
the total stellar mass of $\approx 10^{10.8}~{}{\rm M_{\odot}}$ and the
stellar surface density of $\approx 10^{8.3}~{}{\rm M_{\odot}}$ kpc-2 (all
listed in Table 1), we obtained a prediction of $\log({\rm
sSFR_{sim}\,/\,Gyr^{-1}})=-0.68$. Comparing to the observed $\log({\rm
sSFR[Gyr^{-1}]})=-1.55$, using the total SFR of $\approx 10^{0.29}~{}{\rm
M_{\odot}}$ yr-1 (in Table 1), this suggests that bulge dynamics are not the
sole source of SF suppression. Other factors could be acting in concert with
the dynamics to further decrease the sSFR.

###  5.2 AGN feedback

In Section 3.3, with the aid of BPT diagrams, we analysed the optical emission
line ratios, across the whole disc of NGC 3169. In addition to using these to
separate star-formation, composite and AGN ionisation, which help to interpret
the (often large) inferred molecular gas depletion times, we established that
AGN ionisation is pervasive up to a galactocentric radius of $\approx 2$ kpc.
It is therefore natural to investigate whether other lines of evidence support
a causal relationship between the LLAGN and SFE.

One important observable is the coincidence of a large velocity dispersion (in
both ionised and molecular gas) and AGN ionisation in many spaxels. Indeed, as
shown in Figures 2 and 3, the velocity dispersions of both H$\alpha$ and CO
are significantly larger in the inner disc than in the star-forming ring (i.e.
at radii $\lesssim 650$ pc), while Figure 4 clearly shows that these same
regions are AGN ionised, potentially indicating shock heating and/or
ionisation from the AGN radiation. We can further probe this velocity
dispersion-ionisation mechanism correlation with star-formation quenching by
looking back at the H ii region dynamics discussed in Section 4.4. There, we
uncovered a weak correlation between the H ii regions’ virial parameters and
the molecular gas depletion times, higher virial parameters being associated
with longer depletion times (or equivalently lower SFEs). As the velocity
dispersion is the dominant parameter in virial parameter calculations, there
is in turn a correlation between these high velocity dispersions and
suppressed star formation. However, it is important to note that beam smearing
may have a large impact here due to the rapid changes of rotation curve slopes
shown in Section 4.5.

As discussed in Section 2.3 and 4.5, there are traces of AGN-driven outflows
that can contribute to the enhanced velocity dispersions discussed above,
quenching the star formation in the innermost regions of NGC 3169. Similar
negative feedback mechanisms are discussed in recent works of Venturi et al.
(2021) and Gao et al. (2021). On the other hand, we also identified a fraction
of AGN-ionisation in the star-forming ring, associated with diffuse emissions
surrounding the H$\alpha$ peaks (see Section 3.3). It suggests positive
feedback from the AGN that enhanced SF in the star-forming ring. Overall, AGN
feedback could thus be partly responsible for the radial profile of SFE from
the star-forming ring to the central $100$ pc in radius of NGC 3169.

###  5.3 Quenching from the bar

As shown in Section 4.5, there are multiple sources of non-circular motions in
the central region of NGC 3169. An outstanding one is significant spiral-like
structures that can be associated with inflow/outflow streaming motions as a
result of the small-scale bar. Barred galaxies often have characteristic star-
formation features, as studied in great details in the recent work of Sato et
al. (2021). In particular, there are often cold gas overdensities and enhanced
star formation both in a nuclear ring, caused by resonances with the bar, and
at the nodes where the spiral arms and bar connect. With material tranported
into the bulge by inflows within the bar, turbulence can be enhanced in the
central region and star formation may be suppressed. However, simulations of
clouds and star formation in galactic bars (in e.g. Renaud et al., 2015) have
also shown that enhanced turbulence can, in some cases, induce strong local
compression and lead to star bursts. These simulations also illustrated that
strong shear along the elongated orbits in galactic bars may instead
suppresses turbulence and the associated local compression, while also
stabilizing against gravitational instabilities.

The presence of the small-scale bar may explain the radial trend of $\tau_{\rm
dep}$ observed in NGC 3169, although there is no corresponding overdensity of
GMCs at the nodes. We should also note that the velocity dispersions of both
the ionised and the molecular gas in NGC 3169 are not particularly high. As
illustrated in many works (e.g. Sun et al., 2020; Deconto-Machado et al.,
2022), in the central kpc in radius, AGN-host galaxies can have velocity
dispersions over $100$ km s-1 higher than those of other galaxies with similar
morphologies.

A more detailed analysis of the spatially-resolved properties of GMCs in NGC
3169 could also provide more information about the influence of the bulge
dynamics and shear. For example, as part of the WISDOM project, Liu et al.
(2021) analysed individual GMCs in NGC 4429 and revisited the conventional
Virial theorem, including the effects of the galatic gravitational potential
in addition to the usual cloud self-gravity. Applying such a model to the
bulge region of NGC 3169 may help to better understand the dynamics of
individual GMCs and H ii regions, and to examine the impact of energy
injection from the AGN. However, the nominal spatial resolution of the current
CO observations ($\approx 65$ pc) is not sufficient for this purpose. It may
still be worthwhile to probe giant molecular associations (GMAs; i.e.
collections of GMCs), but wider spatial coverage would be necessary to draw
firm statistical conclusions.

##  6 Conclusions

In this work, we study the star formation efficiency of the central region of
NGC 3169 ($\approx 3.6\times 3.6$ kpc2). The molecular gas is traced by
12CO(2-1) observations from ALMA, while the star formation rate is traced by
extinction-corrected H$\alpha$ observations from SITELLE at CFHT. The surface
densities of molecular and ionised gas are measured at a spatial resolution of
$<100$ pc. The depletion times and $\Sigma_{\rm SFR}$-$\Sigma_{\rm H_{2}}$
relation are measured accordingly and probed for variations with
galactocentric distance and bulge dynamics. To explain the enhanced star-
formation suppression closer to the galactic centre, we examined the potential
impact of AGN feedback and other mechanisms. Our key findings are summarised
as follows:

  1. 1.

We identified star-forming, AGN-ionised and composite regions based on
emission-line ratios, and showed that the AGN impact is strongest within the
inner disc, while composite regions reach over $2$ kpc from the galaxy centre.
For the AGN-ionised and composite regions, we estimated upper limits of
$\Sigma_{\rm SFR}$ and hence lower limits of depletion times.

  2. 2.

The overall star-formation relation follows closely the power-law relations of
Kennicutt Jr (1998) and Bigiel et al. (2008), with depletion times ranging
from $0.1$ to $10$ Gyr. For star-forming regions, the peak of $\Sigma_{\rm
SFR}-\Sigma_{\rm H_{2}}$ distribution is slightly above 2 Gyr, indicating
star-formation quenching. For composite and AGN-ionised regions, the
distributions have different peaks with much larger $\Sigma_{\rm SFR}$ and
slightly larger $\Sigma_{\rm H_{2}}$, which could however result from
$\Sigma_{\rm SFR}$ overestimates due to the unkown fraction of star-forming
ionisation in these spaxels.

  3. 3.

Ionised-gas emission appears to be concentrated within a star-forming ring,
while molecular gas fills up both a smaller central disc and the ring. The
depletion times in the star-forming ring are $\approx 0.3$ Gyr, comparable to
typical starbursts. As the radius decreases within a galactocentric distance
of $\approx 0.9$ kpc, star formation becomes scarce while molecular gas gets
denser, resulting in a longer depletion time. In star-forming regions,
$\tau_{\rm dep}$ increases gradually from $\approx 0.3$ Gyr in the starburst
ring to $\approx 2.3$ Gyr at a radius of $500$ pc.

  4. 4.

There is a weak correlation between the molecular gas depletion times and the
virial parameters of the associated H ii regions. As large virial parameters
are generally associated with high velocity dispersions, this suggests that
the SFEs in the bulge of NGC 3169 are affected by turbulence. Potential
sources of turbulence in this region are outflows from the AGN and inflows
from the spiral arms to the inner bar. The molecular gas lifetime and feedback
durations are constrained by the uncertainty principle for star formation of
Kruijssen et al. (2018) are reasonable comparing to other nearby galaxies.
Bulge dynamics may also contribute to the star-formation quenching in the
central region.

Spatially resolved analyses of GMCs and H ii regions are underway to better
understand the kinematics and local environments of the bulge of NGC 3169 and
similar galaxies. We are also exploiting other SFR indicators such as free-
free emission from VLA, to minimise any uncertainty caused by dust extinction.
Most importantly, to piece together the puzzle of star formation quenching at
the heart of galaxies, we require more observational data probing galaxy
bulges and ETGs.

## Acknowledgements

We thank the referee, Frederic Bournaud, for useful comments that improved the
analyses and manuscript. We thank Federico Lelli for the help with 3DBarolo
and valuable discussions. This research is based on observations obtained with
the SITELLE instrument on the Canada-France-Hawaii Telescope (CFHT) which is
operated from the summit of Maunakea, and the Atacama Large
Millimeter/submillimeter Array (ALMA) in the Atacama desert. We are grateful
to the CFHT and ALMA scheduling, data processing and archive teams. We also
wish to acknowledge that the summit of Maunakea is a significant cultural and
historic site for the indigenous Hawaiian community, while the the high-
altitude plateau Chajnantor on which the ALMA telescope sits is sacred to
indigenous Likanantai people. We are most grateful to have the opportunity of
observing there. This paper also makes use of observations made with the
NASA/ESA Hubble Space Telescope and the NASA/IPAC Extragalactic Database
(NED).

AL, HB, DH and LD acknowledge funding from the NSERC Discovery Grant and the
Canada Research Chairs (CRC) programme. HB is grateful for support from the
Natural Sciences and Engineering Research Council of Canada (NSERC) Alexander
Graham Bell Canada Graduate Scholarship. LC acknowledges support by the
ANID/FONDECYT Regular Project 1210992. MC and JMDK gratefully acknowledge
funding from the German Research Foundation (DFG) through an Emmy Noether
Grant (grant number KR4801/1-1), as well as from the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation
programme via the ERC Starting Grant MUSTANG (grant agreement number 714907).
TAD and IR acknowledge support from the UK Science and Technology Facilities
Council through grants ST/S00033X/1 and ST/W000830/1. JG gratefully
acknowledges financial support from the Swiss National Science Foundation
(grant no CRSII5_193826). TGW acknowledges funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No. 694343).

## Data Availability

The raw data underlying this article are publicly available on NRAO and CFHT
archive. All analysed data are available in the article and upon request.

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##  Appendix A Ionisation-mechanism classification in a three-dimensional line
ratio space

Figure 12:  Ionisation-mechanism classification over the whole disc of NGC
3169 in a three-dimensional line ratio space. Left: three-dimensional plot of
the line ratios [O iii]/H$\beta$, [N ii]/H$\alpha$ and [S ii]/H$\alpha$.
AGN/LINER-ionised (blue) and H ii (pink) regions are separated using a plane
defined by equation (11) and Table 2 in Law et al. (2021). The color gradient
shows the broadening of the H$\alpha$ line, indicative of gas temperature and
associated ionisation mechanism. The top and bottom plots illustrate different
directions of viewing the same parameter space. Right: the H$\alpha$ flux map
colour-coded according to the region separation shown in the left panels, as a
comparison to the right panels of Figure 4.

##  Appendix B HII catalogue

Table 3: Catalogue of all H ii region parameters discussed in Section 4.4.

ID | R.A. | Dec. | $R_{\rm\text{H\,{ii}}}$ | $R_{\rm G}$ |  $\log(L_{\rm\text{H\,{ii}}}/$ergs s-1) | $M_{\rm\text{H\,{ii}}}$ | $V_{\rm avg,\text{H\,{ii}}}$ | $\sigma_{\rm\text{H\,{ii}}}$ | $\log(\alpha_{\rm vir,\text{H\,{ii}}})$ | $\tau_{\rm dep,\text{H\,{ii}}}$
---|---|---|---|---|---|---|---|---|---|---
– | (deg) | (deg) | (pc) | (kpc) | – | ($\times 10^{3}$ M⊙) | (km s-1) | (km s-1) | – | (Gyr)
0 | 153.565 | 3.463 | <100 | 2.61 | 36.8 | 46 | 1360 | N/A | N/A | N/A
1 | 153.569 | 3.463 | <100 | 2.71 | 36.8 | 46 | 1548 | 68 | 4.11 | N/A
2 | 153.568 | 3.463 | <100 | 2.42 | 36.9 | 50 | 1492 | 30 | 3.36 | N/A
3 | 153.562 | 3.464 | <100 | 3.26 | 37.0 | 55 | 1303 | N/A | N/A | 0.39
4 | 153.569 | 3.464 | 145 | 2.53 | 37.7 | 197 | 1497 | 32 | 2.94 | N/A
5 | 153.568 | 3.464 | 237 | 2.07 | 37.6 | 349 | 1495 | 39 | 3.08 | 0.69
6 | 153.567 | 3.464 | 193 | 1.79 | 38.6 | 845 | 1470 | 28 | 2.33 | 0.19
7 | 153.568 | 3.465 | 146 | 2.08 | 38.3 | 373 | 1498 | 21 | 2.29 | 0.45
8 | 153.564 | 3.465 | 187 | 1.73 | 38.2 | 489 | 1330 | 21 | 2.31 | 0.16
9 | 153.562 | 3.465 | 138 | 2.42 | 38.1 | 272 | 1278 | 21 | 2.42 | 0.14
10 | 153.565 | 3.465 | 249 | 1.34 | 38.3 | 861 | 1397 | 33 | 2.57 | 0.12
11 | 153.567 | 3.465 | 154 | 1.43 | 37.7 | 206 | 1493 | 33 | 2.96 | 1.85
12 | 153.569 | 3.465 | 138 | 2.07 | 38.0 | 270 | 1491 | 37 | 2.91 | 0.20
13 | 153.563 | 3.466 | 204 | 1.59 | 38.0 | 452 | 1303 | 19 | 2.30 | 0.14
14 | 153.565 | 3.466 | 205 | 1.15 | 38.3 | 658 | 1393 | 28 | 2.46 | 0.15
15 | 153.566 | 3.466 | 227 | 1.13 | 37.6 | 339 | 1474 | 52 | 3.32 | 1.67
16 | 153.570 | 3.466 | 147 | 2.56 | 38.2 | 351 | 1444 | 27 | 2.55 | N/A
17 | 153.564 | 3.466 | 210 | 1.30 | 37.9 | 421 | 1350 | 19 | 2.31 | 0.26
18 | 153.568 | 3.466 | 104 | 1.78 | 38.8 | 406 | 1466 | 33 | 2.52 | 0.28
19 | 153.561 | 3.466 | 167 | 2.26 | 37.9 | 305 | 1259 | 24 | 2.57 | 0.20
20 | 153.562 | 3.467 | 309 | 1.77 | 37.9 | 722 | 1251 | 24 | 2.45 | 0.15
21 | 153.560 | 3.466 | <100 | 2.62 | 36.8 | 46 | 1230 | N/A | N/A | 0.56
22 | 153.567 | 3.467 | 226 | 1.42 | 37.5 | 298 | 1423 | 29 | 2.88 | 0.40
23 | 153.567 | 3.467 | 179 | 1.00 | 37.7 | 273 | 1410 | 38 | 3.05 | 1.12
24 | 153.563 | 3.467 | 272 | 1.25 | 37.5 | 389 | 1284 | 22 | 2.61 | 0.70
25 | 153.568 | 3.467 | 186 | 1.63 | 37.7 | 289 | 1392 | 24 | 2.65 | 0.53
26 | 153.560 | 3.467 | 178 | 2.81 | 38.1 | 404 | 1226 | 16 | 2.10 | 0.21
27 | 153.561 | 3.467 | <100 | 1.83 | 37.2 | 74 | 1208 | 23 | 2.96 | 0.32
28 | 153.567 | 3.467 | <100 | 1.45 | 37.2 | 70 | 1379 | 37 | 3.39 | 1.29
29 | 153.562 | 3.467 | 335 | 1.53 | 37.4 | 491 | 1199 | 22 | 2.57 | 0.30
30 | 153.567 | 3.467 | <100 | 1.08 | 36.9 | 51 | 1389 | 17 | 2.88 | 1.06
31 | 153.560 | 3.468 | <100 | 2.18 | 36.8 | 46 | 1175 | N/A | N/A | 1.58
32 | 153.563 | 3.468 | <100 | 0.67 | 36.8 | 47 | 1190 | 90 | 4.34 | 0.79
33 | 153.564 | 3.468 | <100 | 0.11 | 36.9 | 50 | 1356 | 108 | 4.47 | 11.77
34 | 153.567 | 3.468 | <100 | 1.52 | 37.1 | 62 | 1348 | 39 | 3.49 | 0.94
35 | 153.566 | 3.468 | 181 | 0.81 | 38.7 | 808 | 1322 | 45 | 2.73 | 0.48
36 | 153.564 | 3.468 | <100 | 0.25 | 36.8 | 46 | 1158 | N/A | N/A | 5.40
37 | 153.567 | 3.469 | 200 | 1.56 | 38.4 | 740 | 1308 | 32 | 2.49 | 0.15
38 | 153.561 | 3.469 | <100 | 1.53 | 37.3 | 81 | 1130 | 26 | 3.02 | 1.23
39 | 153.561 | 3.469 | 227 | 1.86 | 38.2 | 657 | <1108 | 27 | 2.46 | 0.15
40 | 153.566 | 3.469 | 226 | 1.52 | 38.4 | 852 | 1278 | 29 | 2.40 | 0.14
41 | 153.565 | 3.470 | 213 | 1.24 | 37.7 | 360 | 1238 | 33 | 2.87 | 0.15
42 | 153.560 | 3.470 | 337 | 2.28 | 37.5 | 557 | 1123 | 39 | 3.02 | 0.09
43 | 153.565 | 3.470 | 281 | 1.37 | 37.5 | 440 | 1206 | 38 | 3.03 | 0.26
44 | 153.560 | 3.470 | 321 | 2.33 | 37.6 | 588 | 1079 | 29 | 2.73 | 0.01
45 | 153.569 | 3.470 | <100 | 3.55 | 36.8 | 46 | 1298 | N/A | N/A | N/A
46 | 153.567 | 3.470 | 182 | 2.47 | 37.6 | 244 | 1259 | 28 | 2.85 | 0.24
47 | 153.563 | 3.471 | 342 | 1.42 | 37.4 | 470 | 1149 | 24 | 2.69 | 0.68
48 | 153.564 | 3.471 | <100 | 1.79 | 36.9 | 49 | 1195 | 33 | 3.46 | 0.56
49 | 153.560 | 3.471 | 198 | 2.29 | 38.6 | 829 | 1050 | 30 | 2.40 | 0.15
50 | 153.566 | 3.471 | <100 | 2.49 | 37.3 | 84 | 1251 | N/A | N/A | 0.02
51 | 153.559 | 3.472 | <100 | 2.66 | 37.8 | 146 | 1042 | 27 | 2.81 | N/A
52 | 153.570 | 3.472 | <100 | 4.36 | 37.0 | 58 | 1303 | N/A | N/A | N/A
53 | 153.563 | 3.472 | <100 | 2.06 | 36.8 | 47 | 1145 | 32 | 3.45 | 0.52
54 | 153.560 | 3.472 | <100 | 2.38 | 37.4 | 89 | 1046 | 14 | 2.43 | N/A
55 | 153.570 | 3.472 | <100 | 4.80 | 36.8 | 46 | 1290 | N/A | N/A | N/A
56 | 153.561 | 3.473 | <100 | 2.77 | 37.0 | 56 | 1088 | 23 | 3.09 | N/A
57 | 153.562 | 3.473 | <100 | 2.74 | 36.8 | 46 | 1112 | 12 | 2.58 | N/A

Notes: Column 2 and 3: R.A. and Dec. sky coordinates. Column 4: galactocentric
radius ($R_{\rm G}$). Column 5: H$\alpha$ luminosity of the H ii regions
($\log(L_{\rm\text{H\,{ii}}})$). Column 6: ionised gas mass
($M_{\rm\text{H\,{ii}}}$). Column 7: Mean line of sight velocity ($V_{\rm
avg,\text{H\,{ii}}}$). Column 8: velocity dispersion
($\sigma_{\rm\text{H\,{ii}}}$). Column 9: virial parameter ($\log(\alpha_{\rm
vir,\text{H\,{ii}}})$). Column 10: depletion time $\tau_{\rm
dep,\text{H\,{ii}}}$. More parameters and/or parameters for the whole field of
view of SITELLE can be provided upon request.

Figure 13: Boundaries (white lines) and centres (blue crosses) of the H ii
regions identified in Section 4.4, overlaid on the H$\alpha$ flux map. The red
ellipse separates the inner disc and outer star-forming ring discussed in
Sections 4–5.

##  Appendix C The best-fitting molecular gas kinematics model

Figure 14: Left: position–velocity diagrams of the CO emission along the major
axis (top; at PA=$224^{\circ}$) and minor axis (bottom; at $314^{\circ}$).
Shaded blue contours show the observed velocities, while red contours show the
best-fitting models from 3DBarolo. Overlaid yellow circles indicate the best-
fitting rotation velocities. Right: radial profiles of the velocity dispersion
($\sigma_{\rm gas}$ (top), inclination ($i$; middle-right) and position angle
($PA$; bottom-right), from the best-fitting 3DBarolo model assuming a fixed
systemic (heliocentric) velocity of $1230$ km s-1. Grey datapoints are from a
fit with all these parameters free, while red datapoints are from a fit with
$i$ and $PA$ kept fixed.

##  Appendix D Lifecycle of molecular clouds

Figure 15: Same as Figure 10 but for Cases A (top) and B (bottom).