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# Metal-THINGS: a panchromatic analysis of the local scaling relationships of
the dwarf irregular galaxy NGC 1569

Garduño, L. E.1, Zaragoza-Cardiel, J.1,2, Lara-López, M. A.3, Zinchenko, I.
A.4,5, Zerbo, M.C.6,7, De Rossi, M. E.6,7, Fritz, Jacopo8, Dib, S.9, Pilyugin,
L. S.10,5, Sánchez-Cruces, M.11, Heesen, V.12, O’Sullivan, S. P.3, López-Cruz,
O.1, Valerdi, M.1,2, Rosado, M.11
1Instituto Nacional de Astrofísica, Óptica y Electrónica (INAOE), Luis Enrique
Erro No.1, Tonantzintla, Pue., C.P. 72840, México
2Consejo Nacional de Humanidades, Ciencias y Tecnologías, Av. Insurgentes Sur
1582, 03940, Ciudad de México, Mexico
3Departamento de Física de la Tierra y Astrofísica, Instituto de Física de
Partículas y del Cosmos, IPARCOS.
Universidad Complutense de Madrid (UCM), E-28040, Madrid, Spain
4Faculty of Physics, Ludwig-Maximilians-Universität, Scheinerstr. 1, 81679
Munich, Germany
5Main Astronomical Observatory, National Academy of Sciences of Ukraine, 27
Akademika Zabolotnoho St, 03680, Kiev, Ukraine
6Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales y Ciclo
Básico Común. Buenos Aires, Argentina
7CONICET-Universidad de Buenos Aires, Instituto de Astronomía y Física del
Espacio (IAFE). Buenos Aires, Argentina
8Instituto de Radioastronomia y Astrofisica, UNAM, Campus Morelia, A.P. 3-72,
C.P. 58089, Morelia, Mexico
9Max Planck Institute for Astronomy, Königstuhl 17, 69117, Heidelberg, Germany
10Institute of Theoretical Physics and Astronomy, Vilnius University,
Sauletekio av. 3, 10257, Vilnius, Lithuania
11Instituto de Astronomía, Universidad Nacional Autonoma de México, Apartado
Postal 70-264, CP 04510 México, CDMX, México
12Hamburg University, Hamburger Sternwarte, Gojenbergsweg 112, 21029 Hamburg,
Germany
E-mail<EMAIL_ADDRESS>
(Accepted 2023 August 31. Received 2023 August 28; in original form 2023 May
9)

###### Abstract

We investigate several panchromatic scaling relations (SRs) for the dwarf
irregular galaxy NGC 1569 using IFU data from the Metal-THINGS Survey. Among
the spatially resolved properties analyzed, we explore SRs between the stellar
mass, SFR, molecular gas, total gas, baryonic mass, gas metallicity, gas
fraction, SFE and effective oxygen yields. Such multiwavelength SRs are
analyzed at a spatial resolution of 180 pc, by combining our IFU observations
with data from the surveys THINGS, CARMA, and archival data from DustPedia.
Although we recover several known relations, our slopes are different to
previously reported ones. Our star formation main sequence, Kennicutt-Schmidt
(KS) and molecular KS relations show higher SFRs, lower scatter, and higher
correlations, with steeper (1.21), and flatter slopes (0.96, 0.58)
respectively. The shape of the SRs including metallicity, stellar mass, and
gas fraction are flat, with an average value of 12+log(O/H) $\sim$ 8.12 dex.
The baryonic mass vs effective oxygen yields, and the stellar, gas and
baryonic mass vs SFE show higher dispersions and lower correlations. Since we
use the dust mass as a tracer of gas mass, we derive the Dust-to-Gas Ratio and
the CO luminosity-to-molecular gas mass conversion factors, showing
differences of 0.16 and 0.95 dex for the total and molecular gas surface
density, respectively, in comparison to previously reported values. We use a
self regulated feedback model to conclude that stellar feedback plays an
important role generating outflows in NGC 1569.

###### keywords:

galaxies: abundances, galaxies: dwarf, galaxies: irregular, galaxies:
starburst, galaxies: star formation, galaxies: statistics

††pubyear: 2023††pagerange: Metal-THINGS: a panchromatic analysis of the local
scaling relationships of the dwarf irregular galaxy NGC 1569–G

##  1 Introduction

The evolutionary path of galaxies is driven by distinct processes, at
different time scales. The history of star formation in galaxies, gas
accretion, mergers or gas inflows/outflows -among others- can be studied
through the current galaxy properties, such as star formation rate (SFR), gas
metallicity (Z), stellar mass (M⋆), gas mass (Mgas), baryonic mass (Mbar),
star formation efficiency (SFE) and effective yields (Yeff). Since all of
these properties contain information about the past and current evolution of
galaxies, scaling relations (SRs) between these properties are an important
tool to understand the most important mechanisms that drive galaxy evolution.

During the last few decades, several global SRs were explored using fiber
spectroscopic surveys, showing critical physical properties of galaxies in the
local universe. On one hand, using the Sloan Digital Sky Survey (SDSS, York et
al. 2000) and the Galaxy and Mass Assembly Survey (GAMA, Driver et al. 2011)
data, the relation between stellar mass vs. SFR (M⋆-SFR, Brinchmann et al.
2004; Lara-López et al. 2013b), and stellar mass vs. metallicity (M⋆-Z,
Tremonti et al. 2004; Lara-Lopez et al. 2013a) were established for thousands
of galaxies. On the other hand, additional SRs were established such as the
baryonic mass-effective yield (Mbar-Yeff, Tremonti et al. 2004; Lara-López et
al. 2019), the gas fraction-metallicity ($\upmu$-Z, Pilyugin et al. 2004;
Lara-López et al. 2019), the gas fraction-effective yield ($\upmu$-Yeff,
Dalcanton 2007) or the relation between baryonic mass, stellar mass and gas
mass with the star formation efficiency (Mbar-SFE, M⋆-SFE, Mgas-SFE, Haynes et
al. 2011; Lara-López et al. 2019). All the SRs mentioned above help to
understand a part of the galaxy evolution process by analyzing the gas inside
galaxies and how different feedback processes affect their evolutionary path.

Since stars form out of collapsing gas clouds, a correlation is expected
between the surface densities of star formation and gas. Indeed, this is
described by the Kennicutt-Schmidt (KS) relation, an empirical scaling
relation between the gas surface density and the SFR surface density given by
$\Sigma_{\rm SFR}$ = $\rm a~{}{\Sigma_{\rm gas}}^{\rm n}$ (Kennicutt 1989)
which was first proposed in the pioneering work of Schmidt (1959). A classic
example of the global KS relation is presented in Kennicutt (1998), who also
demonstrate the importance of radio and infrared observations in order to get
robust estimations of gas mass and SFRs, respectively.

With the advent of new instrumentation such as the Integral Field Units (IFU),
new possibilities of study have opened up; what was first done on a global
scale, now can be done on a spatially resolved scale. Recently, IFU surveys
have established local SRs, such as the stellar mass surface density vs.
metallicity relation ($\Sigma_{\rm*}-$ $\rm Z$, Rosales-Ortega et al. 2012;
Barrera-Ballesteros et al. 2016; Baker et al. 2023), the stellar mass surface
density vs. SFR surface density relation ($\Sigma_{\rm*}-$ $\Sigma_{\rm SFR}$,
Sánchez et al. 2021; Pessa et al. 2022; Baker et al. 2023) and the spatially
resolved gas fraction vs. metallicity relation ($\upmu$ \- Z, Barrera-
Ballesteros et al. 2018). Although the spatially resolved oxygen yields have
not been studied by many authors, Vílchez et al. (2019) explored resolved
effective yield profiles for two galaxies (NGC 5457 and NGC 628). In all of
the above cases, it is important to note that local SRs mimic the global ones,
which suggests that processes understood at local scales could be the key to
understand global galaxy evolution processes.

The KS relation has also been analyzed locally and it preserves a similar
shape to the global relation (Kennicutt et al. 2007; Blanc et al. 2009;
Casasola et al. 2022; Pessa et al. 2022). Other studies have extended the
relation by considering different star formation regions in a vast range of
galactic environments, from the outer disks of dwarf galaxies to spiral disks,
merging galaxies and individual molecular clouds (Dib 2011; Dib et al. 2017;
Shi et al. 2018; Pessa et al. 2021).

By analyzing the spatially resolved KS relation, some authors conclude that
the gas mass (usually molecular gas) could play the most important role in the
SFR regulation process, suggesting that this could be a more fundamental
relation instead of the $\Sigma_{\rm*}-$ $\Sigma_{\rm SFR}$ relation (Morselli
et al. 2020; Ellison et al. 2021; Pessa et al. 2021, 2022). However it is
still a matter of debate. Indeed, other authors (e.g., Dib et al. 2017)
explain the importance of the $\Sigma_{\rm*}-$ $\Sigma_{\rm SFR}$ relation
since the gravity of stars, over scales of a few hundred parsec in galactic
disks, is as important (and very often dominates) as that of gas. As Dib et
al. (2017) (and references there in) mentioned, this implies that existing
stars can play a fundamental role in generating large scale gravitational
instabilities which in turn lead to star formation.

The correlation between the M⋆, Z and SFR led to the simultaneous discovery of
a 3-dimensional structure (whose shape is still debatable), refereed to the
Fundamental Plane (Lara-López et al. 2010), and the Fundamental Metallicity
Relation (FMR, Mannucci et al. 2010), highlighting the importance of the
already known galactic gas inflows and outflows. Once the observational
capabilities of radio wavelength reached the point at which large surveys
could be pursued, the consequent analysis was to explore the relation between
the gas mass (atomic, molecular or both) and the metallicity (e.g., Lara-Lopez
et al. 2013a). Some studies confirmed the correlation between such properties
and even showed the possibility of being more fundamental by driving the FMR
(Bothwell et al. 2013). Deeper studies set that the dependence with the gas
mass is probably stronger than with SFR, and thus the underlying FMR is
between stellar mass, metallicity and gas mass (Bothwell et al. 2016a). This
latter is supported by Bothwell et al. (2016b), who affirm that the
metallicity dependence on SFR is a derivation of the dependence on the
molecular gas content given by the KS relation. However, this is also a matter
of debate. As other author mentioned, there is a dual dependence of the
SFR/SFE on metallicity. One is due to the metal content in the star forming
gas which governs its ability to cool, and the second is via metallicity
dependent feedback (stellar winds, Dib 2011; Dib et al. 2011). Both of the
effects play against each other. A lower metallicity implies less cooling and
thus less molecular gas. A lower metallicity also means weaker stellar winds
and so collapsing clouds encounter less gas expulsion, which leads to more
SFR. Which of these effects wins or what is their relative importance is yet
to be quantified in detail.

The total gas mass is obtained by the mass in atomic gas, MHI, which is
derived from HI observations, in combination with the molecular gas, M${}_{\rm
H_{2}}$, whose determination relies on measurement of the CO molecule
emission, and on the CO luminosity-to-molecular gas mass conversion factor,
$\rm\upalpha_{CO}$. For simplicity, $\rm\upalpha_{CO}$ factor was considered
constant for all type of galaxies (e.g., Kennicutt 1998). However, there is
evidence proving that $\rm\upalpha_{CO}$ can vary with gas density and
metallicity (Rubio et al. 1993; Lequeux et al. 1994; Bolatto et al. 2013). In
particular, low metallicity galaxies have very faint CO emission, which makes
its detection difficult and hinders a reliable calibration for the conversion
factors. As pointed out also in Bolatto et al. (2013), gas rich galaxies with
active star formation and low metallicity usually have a very faint CO
emission. A very particular case is seen for low mass dwarf irregular
galaxies, that show almost null CO emission, and have regularly low
metallicities (Draine et al. 2007).

Both the metallicity and the dust grains have been highly studied since they
provide important information for understanding the chemical evolution of
galaxies. Indeed, it is thought that metals in the ISM are encapsulated inside
of dust grains. Under the proper physical processes, e.g., supernovae shocks,
the dust grains can be destroyed, carrying metals back to the ISM. As
mentioned in Rémy-Ruyer et al. (2014), the amount of metals that are locked up
in the dust grains can be quantify by the Dust-to-Gas Ratio (DGR). Thus, it is
expected that the DGR depends strongly on metallicity and changes from one
galaxy to another (e.g., because the ISM density also changes, Clark et al.
2023), especially in the low metallicity regime (Rémy-Ruyer et al. 2014). Some
studies, such as Leroy et al. (2011) and Sandstrom et al. (2013), developed
methodologies to compute not only the DGR but the $\upalpha_{\rm CO}$, taking
into consideration the dependence of each other with the metallicity,
obtaining reliable results.

In the local Universe, dwarf galaxies look like scaled down versions of high
redshift massive galaxies due to their similarity in star formation,
metallicity, physical size, high mass fractions or morphology (e.g., thickness
and clumpiness) (Elmegreen et al. 2012, 2013; Motiño Flores et al. 2021). The
variety of dwarf galaxies is extensive and there are different types
(elliptical, spheroidal, irregular, blue compact, ultra faint, ultra compact).
It is thought that the dwarf irregular galaxies are the most common in the
universe, regularly found in isolation, with the particularity that they are
still forming stars (Ellis 1997; Spaans & Norman 1997; Tolstoy 2000; Gallart
et al. 2015; Simon 2019). Nowadays, there is an effort trying to find a
connection among dwarf galaxies, establishing that irregular dwarfs evolve to
Blue Compact Dwarf (BCD) through several starbursts, that enrich the ISM until
their gas is exhausted to finally fade to gas free dwarf spheroidals (Kong et
al. 2019). Dwarf irregular galaxies are important since they could reach the
extreme low regime of mass and metallicity. Since possibly they were the first
structures that were formed (Belokurov et al. 2006; Mateo 1998), they provide
unique clues to explore the early galaxy formation and evolution.

We present an analysis of the irregular dwarf galaxy NGC 1569 using local
scaling relationships. This galaxy is a particularly interesting object since
it is a starburst with a low stellar mass and low metallicity. A summary of
the relevant physical properties can be seen in Table 1. We use IFU
spectroscopy observations of the Metal-THINGS Survey (Lara-López et al. 2021,
2023) in combination with other multiwavelength data. Such combination of data
allows us to study, in a spatially resolved way, SRs involving both the
stellar and the gas component: $\Sigma_{\rm*}-$ $\Sigma_{\rm SFR}-$ Z,
$\Sigma_{\rm gas}-$ $\Sigma_{\rm SFR}-$ Z. Given that NGC 1569 has a very
limited CO emission that results in a poor estimation of M${}_{\rm H_{2}}$ and
Mgas, we compute the dust mass (Mdust) by fitting of the SED using
multiwavelength data, and then we estimate the DGR according to methods
already reported in the literature to obtain a reliable Mgas.

NGC 1569
---
Distance | 3.25 Mpc | Tully et al. (2013)
Stellar mass | 8.61 log(M⊙) | Leroy et al. (2019)
8.6 log(M⊙) | This work
Dust mass | 4.88 log(M⊙) | Young et al. (1989)
5.3 log(M⊙) | This work
Gas mass | 8.7 log(M⊙) | Young et al. (1989)
8.33 log(M⊙) | This work
Metallicity 12+log(O/H) | 8.16-8.19 | Kobulnicky & Skillman (1997)
8.12 | This work
SFR | 0.95 M⊙ yr-1 | Leroy et al. (2019)
0.91 M⊙ yr-1 | DustPedia
0.48 M⊙ yr-1 | This work
0.27 M⊙ yr-1 | Outputs CIGALE
Scale | 0.015 kpc/arcsec | This work
Table 1: Global important properties of NGC 1569. The adopted distance is
using TRGBs. We compare some values computed by our own data with values in
the literature. A SFR estimation is taken from the reported data of DustPedia.
The other SFR estimation is taken from the outputs modules of CIGALE in our
own runs. The scale is computed according to the distance.

Since the galaxies from the Metal-THINGS survey are nearby (D < 15 Mpc), the
physical spatial resolution of the survey is up to two orders of magnitude
better than other surveys such as CALIFA (Sánchez et al. 2012) and MANGA
(Bundy et al. 2015) at the same wavelength coverage, which implies that galaxy
properties are analyzed at scales of parsecs. The Field of View (FOV) of the
George Mitchel Spectrograph allows us to cover almost the entire galaxy, in
contrast with previous studies in which only the central structure was
analyzed (Westmoquette et al. 2007a, b; Mayya et al. 2020). Another important
key to note is that neither MANGA nor CALIFA have studied the metallicity and
SFR in detail for the regime of low mass galaxies because they are biased
towards higher mass ones (log(M/M⊙) > 9 ). By analyzing galaxies such as NGC
1569, we are filling the gap to the low mass regime.

The structure of this work is as follows. In §2, we describe our data
acquisition and in §3, the estimation of the main galaxy properties. In §4.1,
we derive the spatially resolved SRs for the galaxy properties:
$\Sigma_{\rm*}$, $\Sigma_{\rm{gas}}$, $\Sigma_{\rm SFR}$ and Z. We mainly
explore the local relations for oxygen yields (Yeff) and SFE in §4.2. Finally
in §5 and 6, we present a general discussion of this work and a summary of our
conclusions, respectively.

##  2 Observations and data reduction

###  2.1 Metal-THINGS

In this paper, we use data from the Metal-THINGS survey (Lara-López et al.
2021, 2023), which is obtaining IFU spectroscopy of 34 nearby galaxies mapped
in HI with the Very Large Array (VLA) (The THINGS survey, Walter et al. 2008).

Metal-THINGS is an ongoing survey currently observing at the McDonald
Observatory with the 2.7m Harlan Schmidt telescope using the George Mitchel
Spectrograph (GMS), formerly known as VIRUS-P (Hill et al. 2008). GMS is an
IFU with 246 fibers, each one with a diameter of 4.2 arcsecs, disposed on a
square array of 100 x 102 arcsecs. The planning and development of the
observing program as well as the technical details and general procedures are
described in Lara-López et al. 2023 (in preparation). For the case of NGC
1569, our observations and methodology are the same as to those described in
Lara-López et al. (2021).

We observed two pointings of NGC 1569 (Fig. 1), the first one (left) was
observed in January 2018 and the second one (right) in October 2020, both
using the red setup, with a spectral coverage from 4400 to 6800 Å and spectral
resolution of 5.3 Å (low resolution grating VP1). Since GMS has a 1/3 filling
factor, each pointing is observed in three dither positions to ensure a high
surface coverage of the galaxy. The motivation for taking the second pointing
is to investigate the faint tail in this galaxy (in the upper left corner of
the pointing 2 can be seen the tail structure that goes towards the center).

Figure 1: NGC 1569 image showing the field of view of the GMS in red boxes, 2
pointings were observed. The archival image in B band was taken from the
Palomar Observatory Sky Survey, NGS-POSS.

During the observations, we integrated 15 minutes per dither, followed by 5
minutes of sky exposure. This process is repeated three times until a total
integration time of 45 minutes is reached per dither (2.25 hours of total time
for the three dithers). The observing nights for NGC 1569 were clear and had
an average seeing of 2 arcsecs.

As usual, every night exposures of bias, calibration lamps (Neon + Argon), and
sky flats were taken, as well as a calibration star using a six dither
position in order to get a full flux coverage for flux calibration.

For the data reduction we use P3D 111http://p3d.sourceforge.io, for bias
subtraction, flat frame correction, and wavelength calibration. Next, our own
routines in Python are used to make the sky subtraction and dither
combination. Finally, we use the common tasks of IRAF
222http://iraf.nao.ac.jp/ (Tody 1986) to make the flux calibration. First, we
use the _Standard_ task to extract the spectrum of the standard star observed
the same night of the observations with the same observational conditions.
Second, we use the _Sensfunc_ task to compute both, the sensitivity and
extinction functions, and finally, for the flux calibration, we use the
_Calibrate_ task to apply the sensitivity curve and extinction to the spectra.

We fit the stellar continuum of all flux-calibrated spectra using STARLIGHT
(Cid Fernandes et al. 2005; Mateus et al. 2006; Asari et al. 2007), by setting
45 simple stellar populations (SSP) models from the evolutionary synthesis
models of Bruzual & Charlot (2003) with ages from 1 Myr up to 13 Gyr and
metallicities Z = 0.005, 0.02 and 0.05. Thus, the stellar mass is extracted as
one of the STARLIGHT outputs. After subtracting the fitted stellar continuum
from the spectra, the emission lines are measured using Gaussian line-profile
fittings, according to the methodology of Zinchenko et al. (2016, 2019). Such
fitting allow us to analyze the spectral regions that contain important
emission lines such as H$\upbeta$; [O${\rm{III}}$] $\uplambda$$\uplambda$4959,
5007; [O${\rm{I}}$] $\uplambda$$\uplambda$6300, 6364; H$\upalpha$,
[N${\rm{II}}$] $\uplambda$$\uplambda$6548, 6583; [S${\rm{II}}$]
$\uplambda$$\uplambda$6716, 6731.

In Fig. 2, we show the BPT diagnostics (Baldwin et al. 1981) of our fibers
using the classification of Kauffmann et al. (2003) and Kewley et al. (2001),
imposing a cut of S/N $>$ 3 for the H$\upalpha$, H$\upbeta$, [O${\rm{III}}$]
$\uplambda$5007 and [N${\rm{II}}$] $\uplambda$6583 emission lines. As can be
seen, we obtain a total of 750 (98.5%) SF, 5 (0.6%) Composite and 6 (0.8%)
Active Galactic Nuclei (AGN) - like fibers.

Figure 2: BPT diagram for fibers with a S/N $>$ 3 in the four emission lines
used. The solid line corresponds to the Kauffmann et al. (2003) limit, the
dashed line to the Kewley et al. (2001) limit. Red, green and blue colors
correspond to SF, compound and AGN fibers, respectively.

As a part of our analysis -described later- we use data at other wavelengths
provided by the THINGS Survey (Walter et al. 2008), the CARMA survey (Rahman
et al. 2012) and DustPedia (Davies et al. 2017). DustPedia is a collaboration
that provides access to multiwavelength imagery (and photometry) for nearby
galaxies and model physical parameters for such galaxies. Thus, we get images
of NGC 1569 in different bands, from UV (152.8, 227.1 nm, GALEX) to the mid-
far infrared (3.6, 4.5, 5.8, 8, 24 $\upmu$m Spitzer; 70, 100, 160 $\upmu$m
Hershel-PACS; 250 $\upmu$m Hershel-SPIRE).

##  3 Estimation of physical properties

We create a tridimensional datacube by associating each fiber to its relative
position on the sky and convolving them by a gaussian of FWHM = 4.2" (the
diameter of the fiber) at each wavelength. Thus, we get spaxel maps of our
different emission lines. Then, we convolve and reproject such maps to a
common spatial resolution (see §3.2 below). According to the pixel size, we
get a pixel area which allows to compute surface densities of some properties.
Since we have an associated stellar mass at each fiber, this procedure is also
applied to all the individual fibers, getting as a result a stellar mass
surface density map (left panel of Fig. 3). We identify several stars in the
field of view and use the star positions from the Two Micron All-Sky Survey
(2MASS, Skrutskie et al. 2006) to obtain the astrometry.

The total number of spaxels for NGC 1569 before of corrections, binning and
cleaning tasks is $\sim$400 (pixel size is $\sim$6", see §3.2). Then, after
applying a 2x2 binning and a similar S/N cut for the previously mentioned
emission lines, we get a total of $\sim$100 spaxels with a pixel size of
$\sim$12" that corresponds to $\sim$180 pc.

The measured emission line spaxels are corrected for interstellar reddening
using the Balmer Decrement (H$\upalpha$/H$\upbeta$) with the theoretical value
of 2.86, using the extinction curve of Cardelli et al. (1989).

###  3.1 Stellar mass, SFR and metallicity estimation

The SFR was computed following Kennicutt et al. (2009) using the L${}_{\rm
H\upalpha}$.

$\rm{SFR}\left[M_{\odot}\;yr^{-1}\right]={5.5\times
10^{-42}}\times{L_{\rm{H\upalpha}}}$ (1)

The above SFR formula considers a Kroupa IMF (Kroupa 2001), which is very
similar to a Chabrier IMF (Chabrier 2003), with which our stellar masses are
computed. Our star formation rate surface density map is shown in the middle
panel of Fig. 3.

The gas metallicities are estimated using the S-calibration described in
Pilyugin & Grebel (2016) which also use the emission line spectra. It is
noteworthy that the computed oxygen abundances are gas-phase abundances. They
are based on the definition of the strong line ratios as $\rm N_{2}$ =
$\rm{[N\,II]\ \uplambda 6548+\uplambda 6584}/{\rm H}\upbeta$, $\rm S_{2}$ =
$\rm[S\,II]\ \uplambda 6717+\uplambda 6731/{\rm H}\upbeta$ and $\rm R_{3}$ =
${\rm[O\,III]\ \uplambda 4959+\uplambda 5007}/{\rm H}\upbeta$. Then, a
relation is given according to the value of log $\rm N_{2}$. The so-called
upper branch ($\log\rm N_{2}\geq-0.6$) is defined by the relation:

$\displaystyle\begin{array}[]{lll}{\rm
12+log(O/H)}_{S,U}&=&8.424+0.030\,\log(R_{3}/S_{2})+0.751\,\log N_{2}\\\
&+&(-0.349+0.182\,\log(R_{3}/S_{2})+0.508\log N_{2})\\\ &\times&\log S_{2}\\\
\end{array}$ (5)

While the so-called lower branch ($\log N_{2}<-0.6$) by the relation:

$\displaystyle\begin{array}[]{lll}{\rm
12+log(O/H)}_{S,L}&=&8.072+0.789\,\log(R_{3}/S_{2})+0.726\,\log N_{2}\\\
&+&(1.069-0.170\,\log(R_{3}/S_{2})+0.022\log N_{2})\\\ &\times&\log S_{2}\\\
\end{array}$ (9)

Throughout all the paper we use the term metallicity when referring to the
gas-phase oxygen abundances. Our metallicity map can be seen in the right
panel of Fig. 3.

Figure 3: The stellar mass surface density, $\Sigma_{\rm*}$ (left), the SFR
surface density, $\Sigma_{\rm SFR}$ (middle) and the gas metallicity (right)
maps. The black contours correspond to the galaxy structure in the J band from
2MASS. We show the 2 observed pointings in this map; after the S/N and BPT
selection criteria, just a few spaxels were recovered from the second
pointing. Each spaxel in the three maps corresponds to a scale of $\sim$12"
($\sim$180 pc).

###  3.2 Dust mass estimates

For the dust mass estimates, we follow Calzetti et al. (2018), in which the
global dust properties of a galaxy are used to compute the resolved dust
masses, by fitting the entire Spectral Energy Distribution (SED) (to the
available bands) and using the dust models from Draine & Li (2007).

To fit the complete SED, we use the Code Investigating GALaxy Emission
(CIGALE, Burgarella et al. 2005; Nersesian et al. 2019). CIGALE is a fitting
code that reproduces the observed panchromatic photometry of galaxies, by
means of stellar and dust emission models, imposing an energy balance to
simultaneously account for UV/optical extinction, and IR dust emission. Among
the most important input parameters are the Simple Stellar Population (SSP)
set, the Star Formation History (SFH) and dust properties affecting both
extinction and IR emission. The input parameters that we used were taken from
DustPedia (see Table 2), and as input observations, we take the data of NGC
1569 also from DustPedia, in the mentioned bands at the end of §2 (from GALEX
(UV) up to Hershel-SPIRE (FIR) at 250 $\mu\rm{m}$.) The optical photometry,
also included in the SED, are centered at 5170, 5426, 5725 and 6080 Å
(obtained from our spectroscopic data, as described in §2). CIGALE provides
several output estimates (in which the dust mass can be found) according to
the best computed SED fitting and a specific choice of the initial mass
function. In this analysis, we use the IMF of Chabrier (2003) and the
synthesis models of Bruzual & Charlot (2003).

We apply color corrections to all the Spitzer, Hershel-PACS and Hershel-SPIRE
bands that we used, as Lianou et al. (2014) recommend. Spitzer bands are also
corrected by aperture effects due to the condition of extended objects.

Since NGC 1569 is in a sky projection to the galactic plane, it has a high
extinction magnitude AV=1.58. Thus, we correct for galactic extinction all the
GALEX and optical bands using the extinction curve of Cardelli et al. (1989),
and for the Spitzer bands, we use the curve extinction of Indebetouw et al.
(2005).

Finally, since all bands have different spatial resolutions, a re-projection
and convolution to the worst resolution band (Hershel-SPIRE, 250 $\upmu$m) is
applied in order to have a correct and fair comparison between all image data.
The pixel size of such image data results in $\sim$6". Then, we also apply a
2x2 binning to the images to have a pixel size of $\sim$12" ($\sim$180 pc).
The re-projection is performed using the kernels from Aniano et al. (2011).
Before the re-projection and convolution process, we mask the stars that
appear in the field of view of the galaxy to avoid flux contamination.

We run CIGALE on each pixel to get a dust mass surface density map (left upper
panel of Fig. 6). We have estimated dust masses on each pixel twice: first, we
use data up to Hershel-PACS 160 $\upmu\rm{m}$ (without any binning) in order
to have several pixels and apply the method described in §3.3 to estimate the
DGR. Second, we also use data up to Hershel-SPIRE 250 $\upmu\rm{m}$ to
estimate the dust masses used in the following sections with a 2x2 binning. As
mentioned, the specific input parameters that we use are shown in Table 2.
Dust mass estimates using data up to Hershel-SPIRE 250 $\rm{\upmu m}$ would be
more sensitive to the bulk of the cold dust than the Hershel-SPIRE 160
$\rm{\upmu m}$. For this galaxy the dust masses are very similar when using IR
data up to the Hershel-PACS 160 $\rm{\upmu m}$ wavelength, compared to using
up to the Hershel-PACS 500 $\rm{\mu m}$ wavelength (see Appendix A).

CIGALE Parameters
---
Module | Parameter | Value
Delayed SFH |  | $\tau_{\rm{main}}$
---
Agemain
$\tau_{\rm{burst}}$
Ageburst
$f_{\thinspace\rm{burst}}$
SFRA
| 7000, 8000, 10000, 12000, 13000
---
9500, 13000
5.0,10.0,20.0, 50.0, 80.0, 110.0
10,30,50,70,90,150,200,250,300,400
0.1, 0.2, 0.3, 0.4, 0.5, 0.75
1.0
SSP |  |
Bruzual & Charlot (2003) |  | IMF
---
Stellar metallicity
| Chabrier (2003)
---
0.004, 0.008
Dust attenuation |  |
Calzetti et al. (2000) and Leitherer et al. (2002) |  | $E(B-V)_{\rm{young}}$
---
$E(B-V)_{\rm{old\thinspace factor}}$
Central wavelength of the UV bump
Width (FWHM) of the UV bump
Amplitude of the UV bump
Slope delta of the power law
Filters
| 0.0075, 0.011, 0.017, 0.026, 0.038, 0.058, 0.087, 0.13, 0.20, 0.29, 0.44
---
0.50
217.5
35.00.0
0.0
0.0
B$\rm B90$ & V$\rm B90$ & FUV
Dust emission |  |
Draine et al. (2014) |  | $q_{\rm PAH}$
---
$U_{\rm min}$
$\alpha$
$\gamma$
| 0.47
---
35
2.0
0.015, 0.025, 0.035
Table 2: Modules and physical parameter values used as input to CIGALE.
$\tau_{\rm{main}}$, Agemain, $\tau_{\rm{burst}}$ and Ageburst are expressed in
106yr. The SSP, dust attenuation models and dust emission model are described
in the cited references.

###  3.3 Gas mass estimates

In order to properly take into account the low gas metallicity of this galaxy,
we followed the method presented in Leroy et al. (2011) and Sandstrom et al.
(2013) to compute the DGR and the CO to molecular gas mass factor. This method
combines the masses of the atomic (MHI) and molecular (M${}_{\rm{H_{2}}}$)
gas, together with the dust masses, to compute the DGR and
$\upalpha_{\rm{CO}}$, where $\upalpha_{\rm{CO}}$ is the factor to convert
observed CO luminosity (LCO) to molecular gas mass,
M${}_{\rm{H_{2}}}=\upalpha_{\rm{CO}}\ {\rm{L_{CO}}}$. Briefly, the method is
based on the relation between these three masses:

$\rm
M_{\rm{gas}}=\frac{M_{\rm{dust}}}{\rm{DGR}}=\upmu_{\rm{gal}}(M_{\rm{HI}}+\upalpha_{\rm{CO}}\
L_{\rm{CO}}),$ (10)

where $\upmu_{\rm{gal}}$ is the mean atomic weight, so the gas mass term
includes the contribution of hydrogen and other elements, as defined in Rémy-
Ruyer et al. (2014). We use the Galactic mass fraction of Helium,
Y${}_{\odot}=0.270$ (Asplund et al. 2009), and the metallicity derived from
the observations presented in this work, $12+\log(\rm{O/H})=8.12$, in order to
estimate $\upmu_{\rm{gal}}$ and take into account the mass of the different
elements. Assuming a unique solution for similar metallicity regions to Eq.
10, one can find the best values of DGR and $\upalpha_{\rm{CO}}$ varying the
value of $\upalpha_{\rm{CO}}$ while deriving the solution for DGR, using Eq.
10 for all the pixels inside an specific region, and estimating the standard
deviation of DGR of all those pixels inside that region. The best value of
$\upalpha_{\rm{CO}}$, and then the solution, is that which minimizes the
scatter of DGR.

We obtain the HI intensity from the THINGS survey (Walter et al. 2008) that we
convert to HI gas mass using the equation:

$\frac{\rm M_{\rm{HI}}}{\rm{M_{\odot}}}=2.36\times 10^{5}\left(\frac{\rm
D}{\rm{Mpc}}\right)^{2}\times\frac{\rm F_{\rm{HI}}}{\rm{\rm{Jy\thinspace
km/s}}}$ (11)

where FHI is the flux measured from the moment-0 map, and D is the adopted
distance according to Table 1.

The CO intensity is obtained using the data from the CARMA survey (Rahman et
al. 2012). We derived the CO(1-0) moment-0 map from the CARMA datacube
estimating the noise in the emission free channels, and looking for the peak
which fulfills a signal to noise ratio larger than 3. We estimate the CO
luminosity using the following equation (Solomon et al. 1992):

$\begin{split}\frac{\rm
L_{\rm{CO}}}{\mathrm{K}\thinspace\mathrm{km}\thinspace\mathrm{s}^{-1}\thinspace\mathrm{pc}^{2}}&=3.25\thinspace\times
10^{7}\left(\frac{\upnu_{\rm{rest}}}{\mathrm{GHz}}\right)^{-2}(1+\rm
z)^{-1}\\\ &\times\left(\frac{\rm D}{\mathrm{Mpc}}\right)^{2}\left(\frac{\rm
F_{\rm{CO}}}{\mathrm{Jy}\thinspace\mathrm{km}\thinspace\mathrm{s}^{-1}}\right),\end{split}$
(12)

where $\upnu_{\mathrm{rest}}$ is the rest frequency of the line
($115.27\thinspace\mathrm{GHz}$ in the case of CO(1-0)), D is the distance
(seen in Table. 1), z is the redshift of the galaxy (in this case it is
negligible due to the proximity of the galaxy), and FCO is the velocity-
integrated flux measured from the moment-0 map.

Both, the CO and HI flux maps were reprojected and convolved to the worst
resolution when deriving the dust mass, which we choose to be that of PACS 160
$\rm{\mu m}$ (FWHM = 11.2") in order to have enough pixels in the CO flux map
and being able to apply the method.

We show in Fig. 4 the HI flux (left), CO(1-0) flux (middle), and the dust mass
surface density (right), $\Sigma_{\rm{dust}}$. The CO map of NGC 1569 does not
present much CO emission, probably due to its low metallicity. However, the CO
emission is enough to compute the best DGR as a function of
$\upalpha_{\rm{CO}}$ using 84 pixel values, where we assume a unique solution
of Eq. 10, since the metallicity of this galaxy is pretty constant (as we see
in this work).

Figure 4: HI intensity (left), CO (1-0) intensity (middle), and dust mass
surface density, $\Sigma_{\rm{dust}}$ (right), of NGC 1569. The three maps are
reprojected and convolved to the worst resolution observation
(FWHM$=11.2\rm{"}$, Hershel-PACS 160$\rm{\mu m}$) used in the dust mass
estimation (see §3.3).

We show the result of the method in Fig. 5, where we plot the standard
deviation of the DGRs, versus $\upalpha_{\rm{CO}}$ for the whole set of
pixels. We obtain the best solution, i.e., where the minimum value of the
standard deviation of log(DGR) occurs. This solution corresponds to
$\log(\rm{DGR})=-3.08\pm 0.18$ and $\log(\upalpha_{\rm{CO}})=1.6\pm
0.4\thinspace\rm{M_{\odot}pc^{2}(K\thinspace km/s)^{-1}}$. The change of the
DGR standard deviations is very small in comparison with the large variation
of $\log(\upalpha_{\rm{CO}})$ in Fig. 5. Therefore, the estimated uncertainty
of $\upalpha_{\rm{CO}}$ shows a significant margin of error of 40%. However,
this does not affect our results since we will use the DGR value to compute
gas masses presented in this work.

The uncertainties were obtained following Leroy et al. (2011); Sandstrom et
al. (2013). We first estimated two different noises: i) adding random noise to
the observed properties according to their errors, and ii) via bootstrapping.
Each estimation is repeated 100 times independently. The errors for the two
different noises were inferred with the standard deviation of the derived
values, and finally we added in quadrature to obtain the final error.

The DGR and $\log(\upalpha_{\rm{CO}})$ reported values are in agreement with
those expected for the metallicity of NGC 1569 (Z = 0.0044) (e.g., Sandstrom
et al. 2013; Bolatto et al. 2013, for $\upalpha_{\rm{CO}}$) and (e.g., Rémy-
Ruyer et al. 2014; De Vis et al. 2019, for DGR).

At the end, using the Eq. 10 we compute the gas mass surface density map,
displayed in the upper middle panel of Fig. 6.

Figure 5: Standard deviation of the Dust-to-Gas Ratios (DGRs), as a function
of the CO Luminosity-to-H2 factor, $\upalpha_{\rm{CO}}$. The DGRs for each
value of $\upalpha_{\rm{CO}}$ were estimated using Eq. 10 for all the pixels
where CO in NGC1569 is detected.

###  3.4 The baryonic mass, gas fraction, oxygen yields, SFE and depletion
time estimates

The physical properties that we obtained in previous sections (stellar mass,
total gas mass and SFR) allow us to compute other galaxy properties following
previous works (e.g., Lara-López et al. 2019). The baryonic mass is given by:

$\rm M_{\rm bar}=\rm M_{\rm\star}+\rm M_{\rm gas}$ (13)

The $\rm M_{\rm bar}$ surface density map is shown in the right upper panel of
Fig. 6

Thus, the gas fraction is:

$\upmu=\frac{\rm M_{\rm gas}}{\rm M_{\rm gas}+\rm M_{\rm\star}}$ (14)

For the effective oxygen yields (Yeff), we follow the classic formalism
described in Pagel & Patchett (1975); Searle & Sargent (1972). This is:

$\rm Y_{\rm eff}=\frac{\rm Z_{\rm gas}}{\ln(1/\upmu)}$ (15)

We adopt the relation given by Garnett (2002) for the value of Zgas, in which
the oxygen abundance O/H is expressed in units of the number of oxygen atoms
relative to hydrogen:

$\rm Z_{\rm gas}=12\times\rm{(O/H)}$ (16)

Finally, we define the SFE as:

$\rm SFE=\frac{\rm SFR}{\rm M_{\rm gas}}$ (17)

And the depletion time as:

$\rm t_{\rm dep}=\frac{\rm M_{\rm gas}}{\rm SFR}$ (18)

Since our information is spatially resolved, we have maps for all of these
properties. Thus, the effective oxygen yield map, the gas mass fraction map
and the SFE map are shown in the left, middle and right bottom panel of Fig.
6, respectively.

Figure 6: Upper panels: dust mass surface density $\Sigma_{\rm{dust}}$ (left),
gas mass surface density $\Sigma_{\rm{gas}}$ (middle) and baryonic mass
surface density $\Sigma_{\rm{bar}}$ (right) maps. Bottom panels: spatially
resolved Yeff (left), spatially resolved gas fraction (middle) and spatially
resolved SFE (right) maps. In these maps we show the two observed pointings;
after the S/N and BPT selection criteria, just a few spaxel were recovered
from the second pointing. The black contours correspond to the galaxy
structure in the 2MASS J band. Each spaxel corresponds to a scale of $\sim$12"
($\sim$180 pc).

##  4 Results

###  4.1 The local relationships between the stellar mass, gas mass, star
formation rate and metallicity

In this section, we analyze the spatially resolved maps that we get for the
different galaxy properties previously estimated. As mentioned in §3.2, we
convolved and reprojected our multiwavelength maps to the lowest resolution
band (Herhsel-SPIRE 250 $\upmu\rm{m}$). The convolution and reprojection is
also done to the emission lines maps, which are used to obtain the physical
properties described in §3. Thus, we establish a common and equivalent scale
to compare all the properties with each other. A binning of 2x2 was also
applied to avoid pixel correlations and to have a larger spatial scale. All
the maps in Figs. 3 and 6 have a pixel size of 12" corresponding to a scale of
180 pc. We also compute the global galaxy properties by adding the values of
the individual spaxels in the flux emission lines, and then apply the
equations in §2.

The spatially resolved map of stellar mass $\Sigma_{\rm*}$ (left panel of Fig.
3) shows that the highest values of mass are located in the center of the
galaxy, where the super stellar clusters (SSC) A and B are located. Around
these regions the values of mass decreases in a gradual way up to the outer
parts. We compute a global stellar mass of log(M⋆) = 8.6 M⊙. The SFR surface
density map (middle panel of Fig. 3) shows similar features than the stellar
mass map. Our estimation for the global SFR is log(SFR) = -0.32 M⊙ yr-1. The
metallicity map does not show any pattern but it highlights some spaxels in
the center of the galaxy with a lower metallicity than others in the outer
part. For example, the central spaxel with a 12+log(O/H) $\sim$ 8.05 has a
metallicity 0.1 dex lower than other spaxels in the outer zone with
12+log(O/H) $\sim$ 8.15. This slight change could be an evidence of outflows
in the outer parts and a possible inflow just in the center of the galaxy, as
we mention in the discussion of this work. In general, we have 65.3% of
spaxels with higher Z and 34.7% lower Z, than the global value (12+log(O/H) =
8.12), respectively.

The $\Sigma_{\rm*}-$ $\Sigma_{\rm SFR}$ relation is shown in the panel A of
Fig. 7, where it can be observed that we recover the well known correlation
between the two properties. The solid magenta line in the figure is the power
law fit reported for galaxies with SBc-Irr morphology in the MANGA survey
(Cano-Díaz et al. 2019). Our reported $\Sigma_{\rm*}-$ $\Sigma_{\rm SFR}$
relation is color-coded by the gas metallicity, without a clear pattern or
correlation. In the same figure, the dashed red line represents the limit
below which the SFR value is lower than 10-3 M⊙ yr-1. This is a typical value
for which the SFR can become very uncertain due to poor sampling of the IMF.
The use of synthesis stellar population (SSP) modeling to derive stellar
population properties (among them the SFR), is intrinsically limited by the
fact that the models are constructed using fully sampled IMFs. This means that
by applying such models to an observed spectra, it is implicitly assumed that
the mass distribution of stars that are producing the spectra are sufficiently
complete (Haydon et al. 2020a). It has been shown that this happens when the
total stellar mass relative to this spectrum is of at least 104 M⊙ (El-Badry
et al. 2017). As the SFR prescriptions are calculated by assuming a constant
SFR over $\sim$107 yr, this means that values below 104 M⊙/107 yr-1$\sim$ 10-3
M⊙ yr-1 can possibly suffer from incomplete IMF sampling, resulting in an
uncertain SFR determination.

The $\Sigma_{\rm*}-$ $\rm Z$ relation can be seen in the panel B of Fig. 7. In
this case we do not recover the classic polynomial shape of the global M⋆-Z
relation. Instead, the metallicity spans across a range of $\sim$8.05-8.15 dex
with respect to the stellar mass. This implies that spaxels have a metallicity
variation of up to 0.1 dex (25%). In general, the metallicity is not constant
throughout the galaxy but the variation is small. The plot is color-coded with
the SFR and, as expected, the SF main sequence is seen (i.e., $\Sigma_{\rm*}$
scales with $\Sigma_{\rm SFR}$).

Figure 7: The $\Sigma_{\rm*}-$ $\Sigma_{\rm SFR}$ (panel A), $\Sigma_{\rm*}-$
$\rm Z$ (panel B) and the $\Sigma_{\rm{gas}}-$ $\Sigma_{\rm SFR}$ (panel C)
scaling relationships. The fits to the data are shown in black solid lines.
The purple solid lines correspond to the fits of Cano-Díaz et al. (2019) and
Bigiel et al. (2008) for their respective relation. The purple solid line in
the $\Sigma_{\rm*}-$ $\rm Z$ relation is our own fitting for the MANGA
galaxies (see Appendix G for details). The horizontal dashed red line is the
limit below which there is a valid SFR. The circles with black contours
correspond to the spaxels in the tail of NGC 1569. The data is color coded by
the property shown in the colorbar of each panel.

The surface density of dust and gas mass maps (upper left and bottom panels of
Fig. 6, respectively) share similar features with respect to the previous map
of stellar mass, the highest values are in the center of the galaxy and the
regions follows the structure of the galaxy. The global value of dust mass
that we compute is log(Mdust) = 5.3M⊙. Since the dust masses are multiplied by
a DGR, it is expected to have the same spatial distribution in the dust and
gas surface density maps. The global estimation for the gas mass that we
computed is log(Mgas) = 8.33M⊙. We take the values of surface density of HI,
H2 and the CO/H2 conversion factor reported by Kennicutt (1998) to estimate
the luminosity of CO. This value is multiplied by the $\upalpha_{\rm{CO}}$
factor that we get in §3.3 to have our own value of H2 and thus we derive a
new global gas mass surface density (log($\Sigma_{\rm{gas}}$) = 1.49M⊙ pc-2).
In comparison with the surface density gas mass estimates of Kennicutt (1998),
log($\Sigma_{\rm{gas}}$)=1.33M⊙ pc-2, the difference between these 2
estimations is 0.16 dex. We also derive a new value for log($\Sigma_{\rm
H_{2}}$) = 1.05 M⊙ pc-2, in contrast with the old value log($\Sigma_{\rm
H_{2}}$) = 0.10 M⊙ pc-2 reported by Kennicutt (1998). The large difference
between these values is due to the used different $\upalpha_{\rm{CO}}$.
Kennicutt (1998) assumed a constant Milky Way value
$\log(\upalpha_{\rm{CO}})=0.65\pm 0.4\thinspace\rm{M_{\odot}pc^{2}(K\thinspace
km/s)^{-1}}$ for all galaxies in the sample, while we applied our own estimate
in this work of $\log(\upalpha_{\rm{CO}})=1.6\pm
0.4\thinspace\rm{M_{\odot}pc^{2}(K\thinspace km/s)^{-1}}$.

Figure 7 (panel C) displays the spatially resolved KS relation color-coded by
metallicity. We clearly recover the linear shape of the scaling relation,
although with a different slope than previous works. It is also important to
note the high dispersion towards the low mass range, specifically below the
horizontal red line, which is due to the sampling problem of the IMF. When we
compare our results with the previous work of Bigiel et al. (2008), the
difference between both slopes is clear. We discuss the possible origin of
such differences in §5.

In all the SRs shown in this section, the black contours dots represent the
tail of NGC 1569 as can be appreciated in the second pointing of Fig. 1. As
mentioned, from the second pointing we just recover a few spaxels due to the
selection criteria. Such tail is an important zone of the galaxy because it is
a possible sign of interaction. However, the spaxels related to this zone, for
the plots of this section, do not show a particular behaviour or strong
feature to be taken into consideration.

The coefficient values of the fittings in this section are reported in Table
3.

###  4.2 The local relationships between the gas fraction, baryonic mass,
oxygen yields and star formation efficieny.

The baryonic mass surface density, gas fraction, Yeff and SFE maps are shown
in the lower panel of Fig. 6. The baryonic surface density does not show big
differences with respect to the former maps. We estimate a global value for
the baryonic mass of log(Mbar/M⊙) = 8.8.

In contrast with the maps in which the spaxels have high values in the center
with a gradually decreasing distribution to the edges, the resolved Yeff map
(left lower panel of Fig. 6) does not show neither a pattern or a structure.
For this map, the highest values are in the center of the galaxy, and also
along the horizontal plane of the map. Although it is a common practice to use
the gas-phase oxygen abundance to specify the galaxy metallicity in the
construction of the different relations, in the estimation of the oxygen yield
the total oxygen abundance (gas + dust) must be used. Peimbert & Peimbert
(2010) estimated the oxygen dust depletions in galactic and extragalactic HII
regions finding that the fraction of oxygen atoms embedded in dust grains are
a function of the oxygen abundance, being around 0.10 dex for metal-poor HII
regions; thus, the total oxygen abundance for NGC 1569 is around 12 + log(O/H)
= 8.22. Using the gas mass fraction $\upmu$ = 0.34, the integrated total
oxygen yield is Yeff= 0.00185 or log(Yeff) = -2.73.

The resolved gas fraction map (middle bottom panel of Fig. 6) shows a relevant
fact at first impression, the values go up to $\upmu=0.5$, which implies that
all the spaxels are probably dominated by stellar mass. Our estimate of the
integrated gas fraction is $\upmu$ = 0.34, which reveals its low gas fraction.
Later, we discuss the origin of the low gas fraction values in §5.

The surface density SFE map (right lower panel of Fig. 6) does not show a
clear pattern, some regions clearly show low values of SFE (log(SFE) < -8.8
yr-1), while the high values do not correspond to a particular part of the
galaxy. The integrated property for this galaxy is log(SFE) = -8.65 yr-1 which
directly corresponds to a depletion time of log(tdep) = 8.65 yr ($\sim$450
Myr).

The relation $\upmu$-Z (panel A of Fig. 8) is flat, similar to the
$\Sigma_{\rm*}-$ $\rm Z$ relation. As mentioned earlier, all the data are
concentrated towards $\upmu$ < 0.5 which implies a stellar dominated regime
across the whole galaxy. The points with black contours (tail of NGC 1569) do
not show a special location in the plot with respect to the rest of the data.
A high percentage of the data correspond to values of log(tdep) > 8.5 yr.

For the relation $\Sigma_{\rm{bar}}$-Yeff (panel B of Fig. 8), we recover a
correlation between both properties. The points corresponding to the tail of
the galaxy do not show evidence of a particular location.

The plots of Fig. 8 show the relation between $\Sigma_{\rm*}$(panel C),
$\upmu$ (panel D) and $\Sigma_{\rm{bar}}$ (panel E) with the SFE. There is no
special dependence with the metallicity for the three plots, indeed, since the
metallicity range is of only 0.2 dex, there is little dependence with
metallicity. For the relations $\Sigma_{\rm*}$\- SFE and $\Sigma_{\rm{bar}}$\-
SFE, we appreciate an important feature. In general, if we do not consider the
points of the tail (points with black contours), we have a mostly flat
relation. However, when we consider only the points with black contours, our
data suggest a possible anticorrelation between both properties. These two
relations are the only ones in which the points corresponding to the tail have
a clear difference with respect to the others. It is clear how such points
have a limit up to log($\Sigma_{\rm*}$) < 2.0 and log($\Sigma_{\rm{bar}}$) <
2.25. Finally, we see a negative correlation between $\upmu$ and SFE. The
regions of the tail are distributed throughout the whole range of $\upmu$ with
no special concentration.

Figure 8: The $\upmu$-Z (panel A), the $\Sigma_{\rm{bar}}$\- Yeff (panel B),
the $\Sigma_{\rm*}$\- SFE (panel C), the $\upmu$ \- SFE (panel D) and the
$\Sigma_{\rm{bar}}$\- SFE (panel E) SRs. The fits to the observed data are
shown as black lines. The circles with black contours correspond to the
spaxels in the tail of NGC 1569. The data is color coded by the properties
shown in the colorbar.

The coefficient values of the fittings of all the SRs mentioned in this
section are shown in Table 3.

Fitting coefficients
---
|  $\Sigma_{\rm*}-$ $\Sigma_{\rm SFR}$ |  $\Sigma_{\rm{gas}}-$ $\Sigma_{\rm SFR}$ |  $\Sigma_{\rm{bar}}$\- Yeff |  $\Sigma_{\rm*}$\- SFE |  $\upmu$ \- SFE |  $\Sigma_{\rm{bar}}$\- SFE |  $\Sigma_{\rm{HI}}-$ $\Sigma_{\rm SFR}$ |  $\Sigma_{\rm H_{2}}-$ $\Sigma_{\rm SFR}$
m | 1.21 | 0.96 | 0.11 | 0.08 | -0.77 | 0.03 | 1 | 0.58
y0 | -10.64 | -2.68 | -3.04 | -8.88 | -8.46 | -8.8 | -2.55 | -1.62
$\upsigma_{\rm SD}$ | 0.64 | 0.64 | 0.17 | 0.22 | 0.22 | 0.22 | 0.64 | 0.42
xRMS | 1.44 | 1.44 | 2.82 | 8.74 | 8.74 | 8.74 | 1.44 | 0.92
xRMSE | 0.25 | 0.22 | 0.16 | 0.22 | 0.2 | 0.22 | 0.58 | 0.13
$\uprho$ | 0.92 | 0.94 | 0.36 | 0.08 | -0.46 | 0 | 0.39 | 0.95
p-value |  $<$ 2.2$\times 10^{-16}$ |  $<$ 2.2$\times 10^{-16}$ | 9.9$\times 10^{-4}$ | 0.48 | 1.7$\times 10^{-5}$ | 0.99 | 3.3$\times 10^{-4}$ |  $<$ 2.2$\times 10^{-16}$
Table 3: The slopes (m), zero points (y0), standard deviations ($\upsigma_{\rm
SD}$), root mean squared (xRMS), root mean squared error (xRMSE) and Pearson
correlation coefficient ($\uprho$) of all fittings reported in this work. The
last row corresponds to the p-value of the Pearson correlation.

Finally, it is worth to mention that the systematic offsets that can be
observed depend on the method used for metallicity estimates (e.g., Zurita et
al. 2021; Groves et al. 2023). The temperature inhomogeneities within the gas
can explain the differences between strong line methods (Méndez-Delgado et al.
2023) and thus the proper corrections can be taken into consideration (Peña-
Guerrero et al. 2012b, a). Our work relies in the S-calibrator that takes into
account the ionization parameter, the nitrogen-to-oxygen ratio and produces
abundances compatible with those computed by the direct Te method. Since our
metallicities are mostly constant across all the galaxy with a very low
dispersion ($\rm\sigma\sim$ 0.05), we do not expect changes in the slopes of
our SRs.

##  5 Discussion

###  5.1 Global properties of NGC 1569

NGC 1569 is an interesting case of study for its high star formation rate
(Kennicutt 1998), low metallicity (Kobulnicky & Skillman 1997) and complex
star formation history (Angeretti et al. 2005). Additionally, an extended
emission has been observed beyond the optical galaxy data as warm and hot
ionized and atomic hydrogen gas (Lianou et al. 2014). NGC 1569 is part of a
small group of galaxies, that experienced a possible recent interaction with
its close companion (Johnson 2013).

The position of NGC 1569 on some global SRs using SDSS galaxies (red asterisk
in the plots of Fig. 9) also reveals the importance of analyzing dwarf galaxy
systems. Because NGC 1569 has a relatively high SFR, but low metallicity and
low stellar mass, this work becomes relevant since it can help to fill in the
gap towards the low mass regime. Although NGC 1569 seems to be gas rich, we
propose this galaxy should have had even more gas. Later we discuss if this
extra quantity of gas was lost via outflows.

Figure 9: The global M⋆\- SFR (panel A), M⋆\- Z (panel B) and
$\Sigma_{\rm{gas}}$\- $\Sigma_{\rm SFR}$(panel C) SRs using SDSS galaxies. The
red asterisk shows our global estimation for NGC 1569. The red asterisk in the
panel C corresponds to our estimation of gas mass, preserving the HI mass and
CO data reported by Kennicutt (1998), but using our own
$\upalpha_{\rm{CO}}$(the error bars are plotted following Kennicutt (1998)).
In this plot the black dot and the line are the gas mass and the fitting
reported by Kennicutt (1998), respectively (the dashed lines are the one sigma
curves). The blue dots are normal spiral galaxies and the gray squares are
galaxies considered as starburst. The samples are described in Kennicutt
(1998).

By analyzing the classical SRs with the respective position of NGC 1569 (Fig.
9), we can test the idea that the presence of pristine inflows of gas enhanced
the SFR, while at the same time diluted the gas metallicity; also, the
position of NGC 1569 on the KS relation suggests a high SFR with respect to
the main sequence of spiral and starburst galaxies. However, when the relation
$\Sigma_{\rm*}-$ $\Sigma_{\rm SFR}$ is analyzed (see panel A of Fig. 7), the
role of inflows is not completely clear. Besides, even in the flat
$\Sigma_{\rm*}-$ $\rm Z$ relation (panel B of Fig. 7), a high SFR is not seen
for the spaxels with lower metallicity.

SRs such as $\upmu$-Z or Mbar-Yeff help to explore the presence of inflows or
outflows by analyzing the changes in effective yields and comparisons with
models of galactic chemical evolution. In principle, NGC 1569 does not appear
to be comparable with the spiral SDSS galaxies in the $\upmu$-Z or Mbar-Yeff
relations (Tremonti et al. 2004; Lara-López et al. 2019), indeed NGC 1569
would clearly be an outlier in comparison with the those papers. However,
Pilyugin et al. (2004) reported different distributions between spiral and
irregular galaxies in the $\upmu$-Z diagram. Precisely in the irregular
galaxies sample of Pilyugin et al. (2004), NGC 1569 follows the distribution
of such irregular galaxies. The scenario of gas outflows takes high relevance
since our global estimation of the total oxygen yield is Yeff= 0.00185 or
log(Yeff) = -2.73. Empirical estimations of the true oxygen yield ($\rm
Y_{o}$), using oxygen abundances defined with the electronic temperature in
the H II regions, result in $\rm Y_{o}$ = 0.0027 (Pilyugin et al. 2004), $\rm
Y_{o}$ = 0.0032 (Bresolin et al. 2004) or $\rm Y_{o}$ = 0.0030 / 0.035
(Pilyugin et al. 2007). The ratio Yeff/$\rm Y_{o}$ provides the estimation for
the fraction of the produced oxygen that is rejected from the galaxy through
galactic winds. Thus, NGC 1569 would lose around 38% of the produced oxygen if
the $\rm Y_{o}$ were 0.0030 or would lose around 47% of the produced oxygen if
the $\rm Y_{o}$ were 0.0035.

Previous works such as Sánchez-Cruces et al. (2022) and references there in,
not only reported an extended emission of ionized hydrogen, but also the
presence of a high number of supernovae remnants. This latter supports the
idea that NGC 1569 is experimenting gas outflows via stellar feedback, in
particular high speed galactic winds that are powered by supernova explosions,
which is consistent with the conclusions of previous works (Tremonti et al.
2004; Pilyugin et al. 2004). As mentioned in Tremonti et al. (2004), a very
high infall of pristine gas in a galaxy would i) reduce its metallicity, ii)
enhance its gas fraction and iii) reduce slightly its effective yield. Sánchez
Almeida et al. (2015) also analyzed how the very extreme low metallicity in
galaxies is attributed to infalls of metal poor gas. The presence of metal
poor inflows have been also invoked to explain the strong anticorelation in
the local SFR-Zg relation seen in the metal poor MANGA galaxies (Sánchez-
Menguiano et al. 2019). With all these latter facts, the idea of the presence
of inflows is not consistent with the observations reported here. We think
that outflows are more consistent in this case, since for instance, the gas
fraction of NGC 1569 is low and dominated by stellar mass ($\upmu$ = 0.34), as
we computed previously. The results of Dalcanton (2007) also support the
reduction of the effective yields by outflows -via stellar feedback-.

We show the position of NGC 1569 in the KS relation (panel C of Fig. 9). The
plot is taken from Kennicutt (1998), and shows two distinguishable samples:
spiral galaxies (blue dots) and IR galaxies considered as starbursts
(squares). It is worth noting that the position of NGC 1569 shows a high
$\Sigma_{\rm SFR}$ and $\Sigma_{\rm{gas}}$ with respect to all the spirals. At
least in this diagram, NGC 1569 would be in the regime of starburst if it had
1 dex (maybe 1.5 dex) extra of gas mass surface density. With this, we want to
propose the idea that such possible extra gas may have been lost via outflows
(instead of directly assuming that such gas was converted efficiently into
stars). In the next paragraphs, we show how this possible extra gas mass could
have been lost by establishing the use of a simple self-regulator model of the
SF.

###  5.2 Spatially resolved properties of NGC 1569

Back to our spatially resolved SRs, it is clear in the $\Sigma_{\rm*}-$
$\Sigma_{\rm SFR}$ relation (panel A of Fig. 7), that the SFRs for the regions
of NGC 1569 are on average 1 dex higher compared to other reported surveys
(e.g., MANGA survey); as mentioned previously the fitting of MANGA (Cano-Díaz
et al. 2019) in Fig. 7 (panel A) is for irregular galaxies. As also mentioned,
the regime of low mass galaxies has not been analyzed due to observational
limitations. Surveys such as MANGA or CALIFA, do not reach low values of
stellar mass and metallicity, and hence this difference becomes noticeable
when compared with their results. Note that the local $\Sigma_{\rm*}-$ $\rm Z$
for MANGA galaxies spans from 12+log(O/H) 8.10-8.65 (or 7.95-8.70 once they
are converted to our metallicity scale) and NGC 1569 has a metallicity
12+log(O/H) = 8.12. Besides, our $\Sigma_{\rm*}-$ $\rm Z$ does not have the
classic polynomial shape of such relationship, which makes the study of low
metallicity galaxies relevant, because it could be evidence of a extremely
homogeneous star formation history across the galaxy.

The spatially resolved KS relation for this galaxy (panel C of Fig. 7), shows
a remarkable similarity to the $\Sigma_{\rm*}-$ $\Sigma_{\rm SFR}$ relation,
something not usually observed, since the KS is usually tighter. We compare
our data with the fitting reported in Bigiel et al. (2008) which is a mean of
all the slopes and zero points computed in that work (see panel C of Fig. 7).
In comparison with Bigiel et al. (2008), it becomes relevant the large
difference in the slopes between the fitting in that work and ours; this
behaviour could be attributed to the hypothetical extra gas mass that was lost
by stellar feedback.

From a spatially resolved point of view, some works in the literature have
attempted to study the SR $\upmu$-Z (see panel A of Fig. 8). This relation is
mainly important because it is a via to compare observations with models of
galactic chemical evolution. Barrera-Ballesteros et al. (2018) show a study
for MANGA galaxies which are contrasted with a couple of analytical models
(gas regulator and leaky-box); at the end, the best fitted model is the gas
regulator which is also supported by the presence of outflows driven by
stellar feedback. Barrera-Ballesteros et al. (2018) also highlight the concept
of escape velocity that is of particular relevance for low mass galaxies;
indeed, such galaxies have a weak gravitational potential that results in a
low escape velocities that at the end facilitates the metal loss via stellar
feedback. At this point, the case of NGC 1569 is one more time relevant
because Barrera-Ballesteros et al. (2018) do not have a direct estimation of
gas mass and the lower metallicity limit of 12+log(O/H) = 8.10 (7.95 in our
metallicity scale) of the study is not low enough.

The work of Vílchez et al. (2019) is relevant since they study spatially
resolved oxygen yields in two spiral galaxies, M101 and NGC 628. The novelty
of their work is that they establish a reference range of a typical empirical
estimation for the oxygen yields under a closed boxed model; then, they
compared the values of oxygen yields of such spiral galaxies as a function of
the galactocentric radius, finding that at large galactocentric radius the
oxygen yields values of M101 are deviated from the given reference range in
the beginning. At the end, the authors conclude that there could be gas flows
in the outer parts of this galaxy.

Another important point to take into consideration is that our analysis relies
on the methodology of Calzetti et al. (2018), who reconstructed the SED of NGC
4449 to estimate the global dust masses properties using different analytical
models and taking the pertinent assumptions. Such global properties are then
used to compute the spatially resolved dust masses and, more importantly, the
gas masses using the so called factor D/H dust-to-hydrogen ratio -which is
similar to the DGR-. This opens a new way of analysis when either the CO is
limited or there is a lack of observations in CO (M${}_{\rm H_{2}}$) or atomic
hydrogen (MHI), which is particularly common for dwarf galaxies. First, the
M${}_{\rm H_{2}}$ implies the use of a $\rm\upalpha_{CO}$ which is different
galaxy to galaxy; the case of study NGC 1569 helps to fill the values of
$\rm\alpha_{CO}$ or XCO particularly in the low mass and metallicity regime
for dwarf galaxies, as mentioned in Bolatto et al. (2013). Second, since the
DGR or D/H strongly depends on metallicity (especially also in the low
metallicity regime, Rémy-Ruyer et al. 2014), NGC 1569 also helps to reduce the
large scatter in the observed DGR precisely at such metallicity. Other studies
such as Relaño et al. (2018) or Vílchez et al. (2019) show the variation of
the DGR as a function of the galaxy radius and metallicity, revealing that
using a common DGR for all the galaxy must be done carefully.

###  5.3 The interplay between the atomic gas, molecular gas, and SFR surface
densities

Figure 10 shows a comparison between 3 SRs: in the panel A, the
$\Sigma_{\rm{gas}}-$ $\Sigma_{\rm SFR}$ (KS relation), in the panel B the
$\Sigma_{\rm{HI}}-$ $\Sigma_{\rm SFR}$ , and in the panel C the $\Sigma_{\rm
H_{2}}-$ $\Sigma_{\rm SFR}$(molecular KS relation). The $\Sigma_{\rm HI}$ is
computed directly from the THINGS data. We follow Calzetti et al. (2018) to
get an estimation of $\Sigma_{\rm H_{2}}$using eq. 10. Indeed, once we have a
$\Sigma_{\rm{gas}}$ estimation, and since $\upmu_{\rm{gal}}$ and $\Sigma_{\rm
HI}$ are known, a value of $\Sigma_{\rm H_{2}}$can be computed. There are
fewer spaxels since we remove the negative values.

Previous studies pointed out that some relations can be more fundamental than
others. The origin of this fact is still a matter of debate since there are
works that support different scenarios. For example, Kennicutt (1998) found a
stronger correlation between the $\Sigma_{\rm{HI}}-$ $\Sigma_{\rm SFR}$
relation, instead of the $\Sigma_{\rm H_{2}}-$ $\Sigma_{\rm SFR}$ relation,
while Wong & Blitz (2002) found a stronger correlation between the
$\Sigma_{\rm H_{2}}-$ $\Sigma_{\rm SFR}$ instead of the $\Sigma_{\rm{HI}}-$
$\Sigma_{\rm SFR}$ relation. However, studies such as Schuster et al. (2007)
and Crosthwaite & Turner (2007) found that the $\Sigma_{\rm{gas}}-$
$\Sigma_{\rm SFR}$ relation had a tighter correlation than the $\Sigma_{\rm
H_{2}}-$ $\Sigma_{\rm SFR}$ relation.

For the particular case of NGC 1569, our $\Sigma_{\rm H_{2}}-$ $\Sigma_{\rm
SFR}$ (panel C of Fig. 10) relation is only very marginally tighter ($\uprho$
= 0.94 and $\uprho$ = 0.95) than our $\Sigma_{\rm{gas}}-$ $\Sigma_{\rm SFR}$
relation (panel A of Fig. 10). For our $\Sigma_{\rm{HI}}-$ $\Sigma_{\rm SFR}$
relation (panel B of Fig. 10), we can not conclude that a correlation exists
due to the high dispersion of the data.

Leroy et al. (2008) analyzed the results of several galaxy surveys in
combination with several theoretical models that were called Star Forming
Laws, i.e., different scenarios in which star formation occurs. The work of
Leroy et al. (2008) indicates, for example, a clear difference between dwarf
and spiral galaxies. Dwarf galaxies form stars at their average rate, while
spiral galaxies form stars at about half of their average rate. This work
supports the idea that star formation could depend only on the presence of
molecular Hydrogen.

Bigiel et al. (2008) show that the range of the possible power law values for
the KS relation is N = 1.1-2.7, while the range of the possible power law
values for the molecular KS relation is N = 0.8-1.1. The molecular KS relation
reported by Bigiel et al. (2008) is also tighter than the KS relation.

It is noteworthy that our KS and molecular KS slopes (power law index) are
different in comparison with Bigiel et al. (2008) and the molecular KS
relation for 80 nearby galaxies from Sun et al. (2023), in which they obtained
the coefficient N taking different assumptions and finding a linear molecular
KS relation. However, Shetty et al. (2014) (and references there in) discuss
the origin of a sub-linear $\Sigma_{\rm H_{mol}}-$ $\Sigma_{\rm SFR}$ relation
with observational support. They explain that a possible origin of such
relation could be due to the presence of an important amount of diffuse
molecular gas, which is not forming stars. Casasola et al. (2015) also found
sub-linear $\Sigma_{\rm H_{mol}}-$ $\Sigma_{\rm SFR}$ relations, although for
nearby active galactic nuclei. Confronting with our results, our molecular KS
relation suggests a deficit of molecular gas in comparison with Bigiel et al.
(2008). Thus, the origin of the slope in our relation is not due to a diffuse
molecular gas but a possible gas expulsion by stellar feedback (see below).

Since the slopes in the KS relations are associated to the Star Formation
Efficiency (SFE), this concept becomes relevant at the moment of interpreting
the physical conditions of a galaxy. Leroy et al. (2008) showed that the SFE
is more useful than the $\Sigma_{\rm SFR}$ alone to identify where the
conditions propitiate the star formation. Ellison et al. (2020) suggested that
the $\Sigma_{\rm H_{2}}-$ $\Sigma_{\rm SFR}$ relation could be primarily
driven by changes in the SFE, and secondary with a weaker dependence on the
gas fraction.

In conclusion, the SRs for NGC 1569 displayed in Fig. 10 are driven by the
$\Sigma_{\rm H_{2}}$ supported by the idea that the stars are formed directly
from molecular clouds, and thus it seems logic that $\Sigma_{\rm H_{2}}$ and
$\Sigma_{\rm SFR}$ are immediately more related than $\Sigma_{\rm HI}$or the
total gas mass ($\Sigma_{\rm{gas}}$) with the $\Sigma_{\rm SFR}$. This idea is
also supported by Leroy et al. (2008) who mentioned that under certain SF
Laws, indirect evidence for abundant M${}_{\rm H_{2}}$ in the central parts of
dwarf galaxies can be estimated.

Figure 10: The $\Sigma_{\rm{gas}}-$ $\Sigma_{\rm SFR}$ (panel A), the
$\Sigma_{\rm{HI}}-$ $\Sigma_{\rm SFR}$ (panel B) and the $\Sigma_{\rm H_{2}}-$
$\Sigma_{\rm SFR}$ (panel C) SRs. The fits to the data are shown in black
solid lines. The purple dashed lines correspond to the fits of Bigiel et al.
(2008) for their respective relation. The horizontal dashed red line is the
limit in which there is a valid SFR. The circles with black contours
correspond to the spaxels in the tail of NGC 1569. The data is color coded by
the property shown in the colorbar of each panel.

###  5.4 Possible origin of inflows

In this section, we compare our data with the general trend found by Bigiel et
al. (2008) and propose the possible presence of inflows in NGC 1569. Since
this result is not statistically significant, we do not mention it as a part
of our main results. However, a methodology will be implemented in future
analysis.

Our KS relation (panel A of Fig. 10 and also panel C of Fig. 7) has a very
different slope with respect to Bigiel et al. (2008). Following the magenta
line (fitting of Bigiel et al. 2008), we divide this plot into three zones.
The first one with log($\Sigma_{\rm{gas}}$) < 1.5 is called the zone of
outflows (see next subsection), the second one with 1.5 <
log($\Sigma_{\rm{gas}}$) < 2 is called the normal zone, and the third one with
log($\Sigma_{\rm{gas}}$) > 2 is called the inflow zone.

Now, we focus on the inflow zone and particularly on the points that lie to
the right of the magenta fit. We define the offset $\upDelta\Sigma_{\rm gas}$
as the difference between $\Sigma_{\rm gas,obs}$ (the observed
$\Sigma_{\rm{gas}}$ value) and $\Sigma_{\rm gas,KS~{}B08}$ (the Bigiel et al.
(2008) fit value at $\Sigma_{\rm{gas}}$). In other words, $\upDelta\Sigma_{\rm
gas}$ is the distance between each data point and the Bigiel et al. (2008) fit
in the $\Sigma_{\rm{gas}}$ axis.

The offset $\upDelta\Sigma_{\rm gas}$ is plotted versus metallicity in Fig.
11, where we show that the larger the gas excess, the lower is the
metallicity. This is probably because the points that lie to the right of
Bigiel et al. (2008) fit could have a slight gas excess due to the presence of
inflows, supported by the idea that pristine gas dilutes the metallicity.
These points correspond only to the center part of the galaxy.

The $\Sigma_{\rm{HI}}-$ $\Sigma_{\rm SFR}$ relation show evidence of an excess
of neutral gas, since the HI gas surface density values are larger than in
other galaxies, where $\Sigma_{\rm HI}$ is not larger than
$9\rm{M_{\odot}~{}pc^{-2}}$ (Bigiel et al. 2008). In Fig. 10 (panel B), we
show that the majority of $\Sigma_{\rm HI}$ values are larger than
$10\rm{M_{\odot}~{}pc^{-2}}$. These large $\Sigma_{\rm HI}$ values are still
present using higher spatial resolution elements of 700 pc (see Appendix D).

Therefore, the data suggests that the possible excess of gas seen in the KS
relation for log($\Sigma_{\rm{gas}}$) > 2 (in the central part of the galaxy)
could be due to an inflow of neutral gas.

We consider that this type of analysis is important in order to examine the
interplay between inflows and outflows in the same galaxy. Although our
metallicity variation (0.1 dex) due to the presence of inflows is not
statistically significant and it is within the errors, the method allows to
test changes in metallicity related to inflows. We also tested this
methodology using different metallicity calibrators (O3N2, N2, Pettini & Pagel
2004; Marino et al. 2013) and found that the shape of the $\upDelta\Sigma_{\rm
gas}$ versus metallicity relationship (see Fig. 11) is preserved for those
calibrations. We will implement this idea in future analysis with a larger
sample and appropriate fits for comparison.

Figure 11: The offset $\upDelta\Sigma_{\rm gas}$ vs. metallicity. Such offset
is defined as the difference between $\Sigma_{\rm gas,obs}$ (the observed
$\Sigma_{\rm{gas}}$value) and $\Sigma_{\rm gas,KS~{}B08}$.

###  5.5 Possible origin of outflows: a self regulated feedback model

As mentioned previously, another interesting fact of NGC 1569 is its low gas
fraction given its metallicity. The global gas fraction that we report is
$\upmu$ = 0.34, which in general means that the total mass of the galaxy is
dominated by the stellar mass. Common values of gas fractions ($\upmu$) can
also be found in Pilyugin et al. (2004); Lara-López et al. (2019) for galaxies
of different types. For spiral galaxies, the mean global gas fraction values
are in the range $\upmu$ = 0.50 - 0.85 with a mean 12+log(O/H) in the range
8.4 - 8.7. A particular case is seen for the sample of irregular galaxies in
Table 7 of Pilyugin et al. (2004), for which they report $\upmu$ = 0.20 - 0.83
with a 12+log(O/H) between 7.22 - 8.35 dex. When we compare galaxies with
almost the same MB than NGC 1569 (MB $\sim$ -15.7) we find galaxies with
$\upmu$ from 0.22 to 0.66. The hypothetical 1 - 1.5 dex of extra gas mass that
we previously mentioned, could drastically enhance the value of gas fraction.

We propose two scenarios to explain the low gas fraction in NGC 1569. The
first one is related to the possible interaction of NGC 1569 with other
galaxy. As mentioned in Johnson (2013), NGC 1569 is in a system with other 3
galaxies with a possible recent interaction with the nearest companion UGCA
92. Geha et al. (2006) mention that the galaxy internal processes reduce the
gas fraction of galaxies, but the presence of a massive or luminous galaxy
within 0.5 Mpc could totally remove the gas from a dwarf galaxy. Precisely,
one of the companion galaxies of NGC 1569 is a relatively luminous galaxy (IC
342, Johnson 2013), which could imply that the interaction with IC 342 is more
relevant than with UGCA 92.

The removal of gas by an external massive companion should affect also to the
neutral gas. However, we report an excess of neutral gas, so the scenario
where removal of gas by massive companion occurs is not supported by our
analysis. We propose an alternative scenario. A simple model of stellar
feedback can explain the deficit of gas. While stellar feedback is due to
different phenomena (e.g., stellar radiation, stellar winds, supernova
explosions), occurring from stellar to galactic scales, the spatially resolved
star formation self regulator model can be used to parametrize such a complex
process. This simple model has been used in previous studies (e.g., Zhu et al.
2017; Barrera-Ballesteros et al. 2018; Zaragoza-Cardiel et al. 2020) and can
be described by:

$\dot{\Sigma}_{\rm{out}}=\upeta\cdot\Sigma_{\rm{SFR}},$ (19)

where $\rm\upeta$ is the so-called mass loading factor,
$\dot{\Sigma}_{\rm{out}}$ the gas outflow rate surface density due to stellar
feedback and $\Sigma_{\rm SFR}$ is the SFR surface density. Therefore, the
mass loading factor, $\upeta$, is the mass surface density outflow rate per
unit of SFR surface density provoked by stellar feedback.

Such expression can be written as a function of the gas outflow surface
density, $\Delta\Sigma_{\rm{out}}$:

$\Delta\Sigma_{\rm{out}}=\upeta\cdot\Delta t\cdot\Sigma_{\rm{SFR}}$ (20)

Cosmological and hydrodynamical simulations predict a relation between
$\upeta$ and total stellar mass (Muratov et al. 2015; Hayward & Hopkins 2017)
or local gas surface density (Li et al. 2017). Observationally, the value of
$\upeta$ is uncertain, but recent works have made progress to quantify it as a
function of galactocentric radius (Kruijssen et al. 2019), stellar mass
surface density and total stellar mass (Roberts-Borsani et al. 2020; Zaragoza-
Cardiel et al. 2020), or the local escape velocity (Barrera-Ballesteros et al.
2018). Since we have estimations of $\Sigma_{*}$, we can estimate $\upeta$
using the relation between $\Sigma_{*}$ and $\upeta$ from Zaragoza-Cardiel et
al. (2020):

$\log(\upeta+1)=(-0.32\pm 0.03)\log(\Sigma_{*})+(3.2\pm 0.3),$ (21)

which is in agreement with models and theory (Muratov et al. 2015; Li et al.
2017; Hayward & Hopkins 2017).

The median value of the stellar mass surface density is $\log\Sigma_{*}=7.9\
\rm{M_{\odot}kpc^{-2}}$, then $\upeta$ = 3.7 according to Eq. 21. We assume a
value of $\Delta\rm t=4.3\ \rm{Myr}$ in concordance with the characteristic
H$\upalpha$ time scale (Haydon et al. 2020b) and $\Sigma_{\rm
SFR}$$=0.15$M⊙yr-1 kpc-2 (Kennicutt 1998). We estimate a gas outflow of
$\Delta\Sigma_{\rm{out}}$ = 2.4 M⊙pc-2.

We define $\Delta\Sigma_{\rm{gas\thinspace}}$ as the difference between the
global gas surface densities, that of the global KS relation, and that
observed in NGC1569 (our corrected estimation), which are shown as a black
dashed line, and as a red symbol in Fig. 9 (panel C), respectively. We found
$\Delta\Sigma_{\rm{gas\thinspace}}=3\ \rm{M_{\odot}pc^{-2}}$. Therefore,
stellar feedback due to the recent star formation can explain the gas deficit
that we propose.

We plot the spaxels in the zone of outflows, where log($\Sigma_{\rm{gas}}$) <
1.5, in Fig.12. We show that these spaxels are in the outer parts of the
galaxy, where filaments associated with outflows are usually seen (Johnson et
al. 2012).

Figure 12: The gas mass surface density, $\Sigma_{\rm{gas}}$ maps for values
log($\Sigma_{\rm{gas}}$) < 1.5 $\rm{M_{\odot}pc^{-2}}$, this is the spaxels in
the zone of outflows. The black contours is the H$\upalpha$ emission. The blue
crosses correspond to supernovae in NGC 1569 (coordinates are taken from
(Sánchez-Cruces et al. 2022)).

Outflows can be also analyzed via X-ray emission, stellar and gas kinematics.
For the particular case of NGC 1569, an important quantity of X-ray emission
is associated to a diffuse halo and a metal enriched outflows (Heckman et al.
1995; Martin et al. 2002). In some particular scenarios, the X-ray emission
can be linked to the kinematics of a galaxy. Indeed, for NGC 1569, components
of velocity relative to the systemic velocity (vsys) reveal the presence of
superbubbles or expanding shells of ionized gas (Heckman et al. 1995; Martin
1998). Particularly, gas kinematics using HI has allowed the detection of an
unusual high mean HI velocity dispersion possibly due to outflows (Stil &
Israel 2002). Other studies (e.g., Johnson et al. 2012) have found evidence of
an outflow with a potential expanding shell near the supermassive star cluster
A of NGC 1569, supported by both, gas kinematics and the high velocity
dispersion of star kinematics. Finally, recent studies (e.g., Sánchez-Cruces
et al. 2022) performed analysis of physical phenomena in NGC 1569 (e.g.,
shocks and supernova remnants) using high spectral resolution.

##  6 Conclusions

We compute a set of scaling relations (SRs) and analyze $\sim$100 spaxels of
the dwarf galaxy NGC 1569 using the Metal-THINGS, THINGS and CARMA Surveys in
combination with DustPedia archival data. Emission line fluxes were derived
using STARLIGHT and Gaussian line-profile fittings. We estimated spatially
resolved physical properties such as star formation rate surface density
($\Sigma_{\rm{SFR}}$), oxygen gas metallicity (Z) and total gas mass surface
density ($\Sigma_{\rm{gas}}$), this latter computed by using the braodband
spectral energy distribution modelling code -CIGALE- to get the dust mass
surface densities that were converted to total gas surface densities adopting
a Dust-to-Gas Ratio (DGR). Such DGR was estimated simultaneously with the CO
luminosity-to-molecular gas mass conversion factor ($\upalpha_{\rm{CO}}$)
using the method presented in Leroy et al. (2011) and Sandstrom et al. (2013).
With the $\Sigma_{\rm{gas}}$, it was possible to derive other properties such
as the baryonic mass surface density ($\Sigma_{\rm{bar}}$), the local gas
fraction ($\upmu$), the local effective oxygen yields (Yeff), the local star
formation efficiency (SFE) and the local depletion time (tdep).

For all the mentioned properties, we get spaxel maps with 12" of resolution
($\sim$180 pc). By comparing such properties with each other, we estimate
different local SRs.

Our study can be divided into two main sections. In the first one, we analyze
the classic $\Sigma_{\rm*}-$ $\Sigma_{\rm SFR}$, $\Sigma_{\rm*}-$ $\rm Z$ and
$\Sigma_{\rm{gas}}-$ $\Sigma_{\rm SFR}$ relations. We recover the known
correlations ($\uprho$ > 0.92) and scatterings (up to 0.25 dex) except for the
$\Sigma_{\rm*}-$ $\rm Z$.

In the second one, we derive the relations $\upmu$ \- Z, $\Sigma_{\rm{bar}}$\-
Yeff and $\Sigma_{\rm*}$, $\upmu$, $\Sigma_{\rm{bar}}$ \- SFE. For all SRs
(except for the $\upmu$ \- Z), we have low correlations ($\uprho$ < 0.36) and
a relative low dispersion (up to 0.22 dex).

We discuss the global and local properties of NGC 1569, since the global ones
reveal the presence of inflows, but the local ones the presence of outflows.
Thus, we propose two methodologies to explore both scenarios. Our local
estimation for the oxygen yields and local gas fractions reveal a deficiency
of gas mass possibly ejected by outflows. For this, one of our methodologies
is based on a self regulated feedback model, which show that the stellar
feedback plays a stronger role to propitiate outflows.

Finally, given our multiwavelength data, we compute the atomic and molecular
Kennicutt-Schmidt (KS) relation in order to discuss the slopes and the SFRs of
each one, in comparison with our $\Sigma_{\rm{gas}}-$ $\Sigma_{\rm SFR}$.

Our main results are summarised as follows:

  1. 1.

We recover the known classic shapes of the $\Sigma_{\rm*}-$ $\Sigma_{\rm SFR}$
, $\Sigma_{\rm{gas}}-$ $\Sigma_{\rm SFR}$ and $\Sigma_{\rm H_{2}}-$
$\Sigma_{\rm SFR}$ , all with a low scatter up to 0.25 dex, and high
correlation of $\uprho$ > 0.92.

  2. 2.

The SFRs in NGC 1569 are $\sim$1.26 dex higher than the SFRs in MANGA
galaxies. The slope of our $\Sigma_{\rm*}-$ $\Sigma_{\rm SFR}$ is $\sim$1.6
times steeper than the slope in MANGA galaxies.

  3. 3.

Our fittings of the $\Sigma_{\rm{gas}}-$ $\Sigma_{\rm SFR}$ (m = 0.96) and the
$\Sigma_{\rm H_{2}}-$ $\Sigma_{\rm SFR}$ (m = 0.58) are flatter than the
reported in previous works (e.g., Bigiel et al. 2008).

  4. 4.

The shape of our $\Sigma_{\rm*}-$ $\rm Z$ is not similar to the global one,
rather we obtain a flat relation mainly due to the constant metallicity
12+log(O/H) $\sim$ 8.12. This, in combination with the flat gradient, differ
from what it is observed in spiral galaxies. In fact, flat metallicity
gradients are rather common in young low mass galaxies that are still
assembling their mass.

  5. 5.

We report a log(DGR) = -3.08 and log($\upalpha_{\rm{CO}}$) = 1.6 M⊙pc-2 for
NGC 1569, establishing new estimations for the gas mass
log($\Sigma_{\rm{gas}}$ ) = 1.49 M⊙pc-2 and molecular gas log($\Sigma_{\rm
H_{2}}$ ) = 1.05 M⊙pc-2. This is 0.16 and 0.95 dex higher than the reported in
Kennicutt (1998), respectively.

  6. 6.

We show in the $\Sigma_{\rm*}$\- SFE and $\Sigma_{\rm{bar}}$\- SFE relations
that the tail of NGC 1569 could play an important role in establishing the
slope of the fittings.

  7. 7.

Although the global SRs show a possible evidence of inflows, our local
relations do not confirm such scenario. On the contrary, they support the idea
of the presence of outflows because NGC 1569 has a lower oxygen yield
(log(Yeff) = -2.83) and a lower gas fraction ($\upmu$ = 0.34) compared with
galaxies of similar masses. As pointed out in Dalcanton (2007), outflows can
be the mechanisms that really reduce the oxygen yield. Also, as mentioned in
Tremonti et al. (2004), an inflow could produce an enhancement in the gas
fraction but it is not enough to explain low values in the oxygen yields.

  8. 8.

The position of NGC 1569 in the global KS relation, the shape of the local
relation, and the values of $\upmu$ and Yeff could reveal a deficiency of gas
mass -possibly ejected by outflows-. We show how the simple spatially resolved
star formation self-regulator model can explain the absence of gas mass that
is lost by stellar feedback. We conclude that the stellar feedback plays a
stronger role to propitiate outflows and thus to explain the low gas fraction
of NGC 1569.

The origin of SRs go down to the star formation process itself, and its wide
range of star formation efficiencies for galaxies. However, the analysis have
never been very precise because star formation spans a wide range of scales,
from cluster-forming cores to molecular clouds to the whole interstellar
medium. Moreover, the combination with other physical parameters, for
instance, dust distribution as a tracer of gas mass, is important in
understanding the evolution and formation of stars from galaxy to galaxy, or
the conversion of gas into stars, that control not only the formation history
of stars within galaxies, but also their chemical enrichment.

We highlight the importance of dwarf galaxies in the cosmological context.
Indeed, it is thought that dwarf galaxies have similar properties to galaxies
at high redshift; hence, the analysis of the local ones could uncover insights
of evolution and physical processes of the first galaxies. With the advent of
the James Webb Space Telescope (JWST), a new era of IFU observations of dwarf
galaxies is possible, not only for local galaxies but also for high redshift
ones.

In a future study we will contrast our results with simulated low mass
galaxies to explore the role of inflows, outflows and the possible changes in
global SRs.

## Acknowledgments

This research was supported by the International Space Science Institute
(ISSI) in Bern, through ISSI International Team project No. 505 (The Metal-
THINGS Survey of Nearby Galaxies). LEG and MV gratefully acknowledge to the
Consejo Nacional de Ciencia y Tecnología del Estado de Puebla (CONCYTEP) for
their generous financial support for telescope observations during this
project. MALL acknowledges support from the Spanish grant
PID2021-123417OB-I00, and the Ramón y Cajal program funded by the Spanish
Government (RYC2020-029354-I). MR acknowledge financial support from the
CONACYT project CF-86-367. LSP acknowledges support from the Research Council
of Lithuania (LMTLT) grant P-LU-PAR-23-28. MEDR acknowledge support from
PICT-2021-GRF-TI-00290 of ANPCyT (Argentina). SPO acknowledges support from
the Comunidad de Madrid Atracción de Talento program via grant
2022-T1/TIC-23797.

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author. All the data cubes, images and spectra will be made
available on the Metal-THINGS website (in preparation).

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##  Appendix A Dust masses through Hershel-SPIRE bands

This section shows the correlation between the different values of dust masses
using CIGALE, computed using different bands of HERSCHEL-PACS/SPIRE. As shown
in Fig. 13, the differences between them are not critical, hence we can use
the dust masses up to different bands. For example, to compute the DGR we use
the values of dust masses up to 160 $\upmu$m, because we have more values to
have statistics. All our work is based on the reprojection and convolution of
all our spaxels maps up to 250 $\upmu$m, because we have enough scale and
number of values to plot our SRs.

Figure 13: Comparison between different dust mass estimates using different
Hershel-SPIRE bands.

##  Appendix B CIGALE: SEDs examples

In order to evaluate the quality of our SEDs, which we use to compute the dust
masses, we show some SEDs fits. The first example correspond to the spaxel
closest to the center of the galaxy (upper right panel of Fig. 14). We also
show a couple of extra SEDs corresponding to the right edge of the galaxy
(bottom panels of Fig. 14). The global SED of NGC 1569 is shown in the upper
left panel of the same figure.

Figure 14: Different spectra computed by CIGALE. The integrated spectrum for
all the galaxy is shown in the upper left panel. The rest of the spectra were
selected across a horizontal axis from the center of the galaxy to the edge.

##  Appendix C Gas excess up to 500 microns HERSHEL-SPIRE band

To evaluate if the inverse relation between excess of gas and metallicity
shown in Fig. 11 is preserved for larger spatial scales, we show the same
relation at the maximum scale available (that of 500 $\upmu$m). We show in
Fig. 15 that the same offset of 0.05 dex is obtained.

Figure 15: The offset $\upDelta~{}\Sigma_{\rm gas}$ vs. metallicity. Such
offset is defined as the difference between $\Sigma_{\rm gas,obs}$ (the
observed $\Sigma_{\rm{gas}}$value) and $\Sigma_{\rm gas,KS~{}B08}$. These
values correspond to the gas/dust mass estimated up to 500 $\upmu$m.

##  Appendix D The HI KS relation at higher spatial scales (700 pc)

Fig. 16 shows the $\Sigma_{\rm{HI}}-$ $\Sigma_{\rm SFR}$ relation at a similar
scale (750pc) as in Bigiel et al. (2008). This probe that even at larger
scales, our result of HI excess is preserved.

Figure 16: The $\Sigma_{\rm{HI}}-$ $\Sigma_{\rm SFR}$ scaling relation up to
$\sim$700 pc. The horizontal dashed red line is the limit in which there is a
valid SFR. The circles with black contours correspond to the spaxels in the
tail of NGC 1569. The data is color coded by the oxygen abundance. The
vertical dashed blue line shows the values larger than
$9\rm{M_{\odot}/pc^{2}}$.

##  Appendix E Coefficients for global scaling relations

In Table 4, we report the coefficient values of the fits for the global SRs
shown in Fig. 9. We use a sample of SDSS galaxies for the M⋆\- SFR and M⋆\- Z,
and the sample of Kennicutt (1998) to plot the $\Sigma_{\rm M_{gas}}-$
$\Sigma_{\rm SFR}$ relation.

Global fitting coefficients
---
| M⋆\- SFR | M⋆\- Z |  $\Sigma_{\rm M_{gas}}-$ $\Sigma_{\rm SFR}$
m | 0.73 | -0.03, 0.83, -7.10 | 1.4
y0 | -6.97 | 27.97 | 0.00025
Table 4: The slopes (m) and zero points (y0) for the global scaling relations
reported in this work. For the M⋆\- Z relation, a third order polynomial of
was fitted, the values shown in the table correspond to the x3, x2 and x
coefficients respectively. The values for the global $\Sigma_{\rm M_{gas}}-$
$\Sigma_{\rm SFR}$ relation are taken from Kennicutt (1998).

##  Appendix F Gas metallicity recalibration

Since we compare our metallicities with MANGA galaxies, we compute the
recalibration of metallicities between Pilyugin & Grebel (2016) and Marino et
al. (2013), also using SDSS galaxies. In Fig. 17 is shown such recalibration
as well as the value of NGC 1569 in both metallicity scales.

Figure 17: The metallicity recalibration between Pilyugin & Grebel (2016) and
Marino et al. (2013). The red asterisk shows the metallicity of NGC 1569 in
both scales.

##  Appendix G Gas metallicity estimates for the MANGA galaxies

In order to have a fair comparison of our $\Sigma_{\rm*}-$ $\rm Z$ with other
works, we use the MANGA DR15 data from Zinchenko et al. (2021) who also derive
the oxygen abundance (O/H) using the S-calibrator (Pilyugin & Grebel 2016). We
take the SF spaxels enclosed in the 60% area of the density contour plot (Fig.
18). Then, we fit a logarithmic curve with the following parameters:
y=0.28ln(x) +8.29. This fitting is used in panel B of Fig. 7.

Figure 18: The local $\Sigma_{\rm*}-$ $\rm Z$ for the MANGA galaxies using the
S-calibrator Pilyugin & Grebel (2016).