11institutetext: Astronomical Institute of the Czech Academy of Sciences,
Boční II 1401/1, 140 00 Prague
11email<EMAIL_ADDRESS>22institutetext: Department of Physics and
Astronomy, University of California, Los Angeles, CA 90095-1547, USA
33institutetext: Physics Institute, University of Szeged, Dóm tér 9, Szeged,
6723, Hungary

# A 3D model for the stellar populations in the nuclei of NGC 1433, NGC 1566,
and NGC 1808††thanks: The spectral library used in this paper is only
available in electronic form at the CDS via anonymous ftp to
cdsarc.cds.unistra.fr (130.79.128.5) or via https://cdsarc.cds.unistra.fr/cgi-
bin/qcat?J/A+A/

NIR photometry, CO absorption lines, and line-of-sight velocity and its
dispersion

P. Vermot 11 J. Palouš 11 B. Barna 33 S. Ehlerová 11 M. R. Morris 22 R. Wünsch
11

(Received September 15, 1996; accepted March 16, 1997)

###### Abstract

Aims. We aim to characterize the properties of the stellar populations in the
central few hundred parsecs of nearby galactic nuclei; specifically their age,
mass, and 3D geometry.

Methods. We use spatially resolved spectroscopic observations of NGC 1433, NGC
1566, and NGC 1808 obtained with SINFONI to constrain a 3D model composed of a
spherically symmetric nuclear star cluster (NSC) and an extended thick stellar
disk. We computed UV to mid-infrared single stellar population (UMISSP)
spectra to determine the age of the stellar populations and construct
synthetic observations for our model. To overcome degeneracies between key
parameters, we simultaneously fit the spatially resolved line-of-sight
velocity, line-of-sight-velocity-dispersion, low-spectral-resolution NIR
continuum, and high-spectral-resolution CO absorption features for each pixel.

Results. For the three objects, we derive the age and mass of the young and
old stellar populations in the NSC and surrounding disk, as well as their 3D
geometry: radius for the NSC; thickness, inclination, and position angle for
the disk. These results are consistent with published independent measurements
when available.

Conclusions. The proposed method allows us to derive a consistent 3D model of
the stellar populations in nearby galactic centers solely based on a near-
infrared IFU observation.

###### Key Words.:

Galaxies: nuclei – Galaxies: stellar content – Galaxies: kinematics and
dynamics

##  1 Introduction

In a series of papers (Palouš et al., 2020; Barna et al., 2022; Ehlerová et
al., 2022), we are using hydrodynamics simulations to investigate the
potential of supernova remnants to feed supermassive black holes (SMBH) in the
vicinity of the center of the Milky Way. We have discovered that nearby
supernovae can have a positive impact on the growth of Sgr A* by pushing
material from the interstellar medium (ISM) deeper into the central potential
well.

We wish to expand this investigation to other galactic nuclei, whose
properties are expected to vary (gravitational potential, ISM distribution,
supernova rate), potentially leading to different results. For this purpose,
we need a proper description of the stellar populations and ISM in these
regions.

In this paper, we aim to determine the gravitational potential for three
nearby galactic nuclei: NGC 1808, NGC 1433, and NGC 1566. These targets were
chosen because they have been observed with both SINFONI and ALMA. The former
is used in this paper to determine the gravitational potential, and the latter
will be used in an upcoming publication to determine the ISM distribution.
Recent publications on the same SINFONI data are associated with each of these
objects, but none of them have provided a sufficiently detailed description of
the gravitational potential: The analysis of NGC 1808 is presented in Busch et
al. (2017), where the authors fit a 2D circular Plummer model with an
inclination given by the Sérsic profile of the galaxy to reproduce the stellar
line-of-sight velocity (LOSV) map. However, the numerical values of the best
parameters are not provided, as the fit is only used to subtract the circular
motion from the map in order to detect noncircular motions of the gas and
stellar content. A qualitative analysis of the LOSV dispersion ($\sigma_{v}$)
is presented. NGC 1433 is described in Smajić et al. (2014). The stellar LOSV
is not fitted, but the authors present an analysis of the gaseous content,
where a combination of disk-like rotation and a one-sided outflow is used to
explain the measured LOSV. The observation of NGC 1566 is presented in Smajić
et al. (2015), where the authors also fit a 2D circular Plummer model with a
fixed inclination to reproduce the stellar LOSV map and qualitatively discuss
the $\sigma_{v}$ map. The numerical values of the best parameters for the
Plummer model are not provided, but it is indicated that no deviation from
circular motion is detected. Analyzing the same data, we aim to improve the
characterization of the stellar populations in these objects.

Two main issues can explain the limited description of the stellar mass
distribution in previous publications: Firstly, there is degeneracy between
mass, inclination, and age: when using photometry to determine the mass
distribution, a strong uncertainty is associated with the unknown age of the
stellar population, which can lead to differences in the mass-to-luminosity
ratio (M/L) of orders of magnitude; when using the LOSV distribution, a
perfect degeneracy appears between the mass and the inclination of the disk.
The second issue is the difficulty in determining the 3D mass distribution due
to projection effects: it is particularly challenging to determine the
geometrical thickness of a stellar disk with photometry and LOSV if not viewed
edge-on.

In order to overcome these difficulties, we use a high-resolution single
stellar population (SSP) library to determine the age of the stellar
populations and perform a simultaneous fit of four observables determined for
each pixel: a low-resolution near-infrared (NIR) spectral energy distribution
(SED), a high-resolution absorption spectrum of the CO bandheads, LOSV, and
$\sigma_{v}$. This allows us to break degeneracies, bring 3D information, and
obtain reliable estimates of the main parameters of the model.

The targets and observations are presented in Section 2, the construction of
the SSP spectra in Section 3, the mass distribution model in Section 4, the
results of the fitting procedure in Section 5, and a discussion and conclusion
in Sections 6 and 7. Additional figures supporting the analysis are presented
in the appendices of the paper.

##  2 Observations

###  2.1 Targets and observations

Three objects are analyzed in this paper: NGC 1808, NGC 1433, and NGC 1566.
The method was developed on the first of these, for which it produces the best
results, and was tentatively applied to the two other objects with good
results.

The modeling presented in this paper is done solely on archival SINFONI data
obtained in the high spectral resolution and medium spatial resolution modes.
For each observation, the official ESO pipeline was used to perform the data
reduction, and the flux calibration was performed on the entire field of view
with the 2MASS extended source survey as a reference.

NGC 1808 was observed as part of the 075.B-0648(A) ESO program on 24 March
2024, 2 April 2005 and 12 December 2005; each time in the three NIR J ($R\sim
2000$), H ($R\sim 3000$), and K ($R\sim 4000$) spectral bands. A $6"\times 6"$
field of view is extracted with a $125\times 250\ mas$ pixel scale and an
effective resolution of 0.7”. We assume a distance of $12.8\ \mathrm{Mpc}$,
corresponding to $z\sim 0.003$ and $62\ \mathrm{pc/arcsec}$. A first analysis
of this observation and a more detailed description of the host galaxy are
presented in Busch et al. (2017).

NGC 1433 was observed as part of the 90.B-0657(A) ESO program on 21 October
2012 in the H and K bands. A $4"\times 4"$ field of view is extracted, with a
$125\times 250\ \mathrm{mas}$ pixel scale and an effective resolution of 0.6”.
We assume a distance of $15.5\ \mathrm{Mpc}$, corresponding to $75\
\mathrm{pc/arcsec}$. A first analysis of this observation and a more detailed
description of the host galaxy are presented in Smajić et al. (2014).

NGC 1566 was observed as part of the 90.B-0657(A) ESO program on 21 October
2012 in the H and K bands. A $4.8"\times 4.8"$ field of view is extracted with
a $125\times 250\ \mathrm{mas}$ pixel scale and an effective resolution of
0.6”. We assume a distance of $21.56\ \mathrm{Mpc}$, corresponding to $105\
\mathrm{pc/arcsec}$. A first analysis of this observation and a more detailed
description of the host galaxy are presented in Smajić et al. (2015).

###  2.2 The observables

For each of the objects mentioned above, we extract the following information:

  1. 1.

A low-spectral-resolution data cube, where the angular resolution, field of
view, and spectral coverage of the observations are preserved, but the
spectral resolution is lowered by a factor 50 by rebinning the original data
cube in the spectral dimension. As the continuum emission is dominated by
young stellar sources, this provides a spatially resolved SED for each pixel
that will constrain the age, mass, and spatial distribution of the young
stellar populations around the nuclei.

  2. 2.

A high-spectral-resolution data cube, where the angular resolution, field of
view, and spectral resolution of the observations are preserved, but the
spectral domain is restrained to $2.25-2.40\mathrm{\mu m}$ (after correction
of the redshift) and the flux normalized by the continuum. This provides the
absorption profile of the CO bandheads for each pixel, which can be used to
constrain the age and relative importance of the old and young stellar
populations.

  3. 3.

An LOSV map, obtained by fitting the above-mentioned absorption features with
the high-spectral-resolution SSP spectra described below. For each pixel, we
estimate the velocity shift as the median of 30 Doppler shifts measured with
SSPs of randomly chosen ages. We use the LOSV map to constrain the total mass
distribution of our model.

  4. 4.

A $\sigma_{v}$ map obtained with the same procedure as in point (3). We use
this measurement to constrain the thickness of the stellar disk and solve the
degeneracy between mass and inclination.

The simultaneous use of these four sources of information to constrain our
models will bring much stronger constraints than if fitted individually. In
addition to these observables, the flux from the [Fe II] emission at 1.64
$\mathrm{\mu}$m is measured in each pixel as the integrated emission line
after a linear continuum subtraction.

##  3 Stellar templates

To determine the age of the stellar populations and the LOSV and $\sigma_{v}$
maps, we need a high-spectral-resolution library of unresolved stellar
populations at various ages. Several spectral libraries computed with
evolutionary SSP models are publicly available. Such models are built with an
initial mass function, stellar evolution tracks, and spectral templates for
individual stars, offering a high degree of freedom.

Despite the relatively large number of possible libraries to choose from, we
found no public stellar library matching our requirements, which are:

  1. 1.

Solar metallicity. We lack sufficient information to simultaneously fit the
ages and metallicities of the stellar populations, and therefore we assume a
solar metallicity for the three objects as our best estimate for an elliptical
star-forming galactic center.

  2. 2.

Coverage of the NIR J, H, and K bands to match the spectral range of SINFONI.

  3. 3.

Spectral resolution of $R\gtrsim 3000$ ($\Delta v\lesssim 100\
\mathrm{km.s^{-1}}$) in the NIR to fit the shape of the absorption features
and measure LOSV and $\sigma_{v}$.

  4. 4.

Spectra for very young post-starburst populations down to 10 Myr to
investigate regions actively producing supernovæ.

Several public SSP libraries fulfill some of these criteria; MaStar (Maraston
et al., 2020), A-LIST (Ashok et al., 2021), MILES (Vazdekis et al., 2016), XSL
(Verro et al., 2022), and GALEV (Kotulla et al., 2009) being the most notable
ones. MaStar, which is based on the MaNGA Stellar Library, is constructed with
the empirical spectra of approximately $9000$ SDSS stars with a very wide
range of parameters. However, it only covers the shortest NIR wavelengths (up
to 1.03 $\mathrm{\mu m}$), does not provide sufficient spectral resolution to
fit the $\sigma_{v}$, and does not cover ages younger than 200 Myr. The APOGEE
Library of Infrared SSP Templates (A-LIST) provides high-spectral-resolution
templates, but only in a narrow band (1.51–1.70 $\mathrm{\mu m}$) and for
stellar populations older than 2 Gyr.

The MILES extended SSP model is based on a composite library of empirical
spectra covering a wide spectral range (NGSL for the UV, MILES for the
optical, IRTF for the infrared), with moderate to high spectral resolution.
Unfortunately, its spectral resolution in the NIR is R$\sim 2000-2500$,
corresponding to $\Delta\lambda\sim 1\mathrm{nm}$, or $\Delta v\sim 150\
\mathrm{km.s^{-1}}$, which is insufficient to fit stellar $\sigma_{v}$.
Although it provides spectra for stellar populations as young as 30 Myr, which
is close to our goal.

The X-shooter Spectral Library (XSL) from (Verro et al., 2022), which is based
on empirical spectra from the eponym instrument, provides spectra at high
resolution (R $\sim$ 10 000) over a wide spectral range (350–2480 nm) and
provides spectra for populations as young as 50 Myr. Lastly, the GALaxy
EVolutionary (GALEV) synthesis models are fast, publicly available models for
the computation of SSP spectra. These can be used to predict spectra for very
young ages, down to 4 Myr, and use the theoretical Basel Stellar Library to
cover a very large spectral domain from the extreme UV to the far-infrared
(FIR). However, their spectral resolution is rather low ($\Delta\lambda\sim
5-10\ \mathrm{nm}$ in the NIR) and therefore they are not suitable for the
analysis of absorption features.

Unable to find a spectral library matching our requirements, we constructed
our own using the Stellar Population Interface for Stellar Evolution and
Atmospheres (SPISEA, Hosek et al., 2020). Our library matches and surpasses
the above criteria, notably thanks to the use of synthetic spectra obtained
with the stellar atmosphere code PHOENIX (Allard et al., 2003, 2007, 2012). As
the construction of our library required a significant amount of resources and
can now be rapidly used for the analysis of other observations, we have chosen
to make it publicly available.

###  3.1 Initial mass function

We use a standard Kroupa IMF (Kroupa, 2001), which is a double-break power
law, defined as

$\xi(m)\propto m^{{-\alpha}}\cases{\alpha}=1.3\text{ for }0.1<m<0.5\\\
\alpha=2.3\text{ for }0.5<m<300\\\ {}\,,$ (1)

to generate an initial population of stars. No stellar multiplicity is
introduced.

Due to the stochastic construction of the initial population, there is
variability in the final spectra. For this reason, we compute ten $10^{6}\
\mathrm{M_{\odot}}$ models for each age, providing good precision on the
average spectrum as well as an estimate of the variability at a given age. The
standard deviation of each spectrum is provided in the library.

###  3.2 Stellar evolution

For each age, an isochrone is computed to be used to attribute physical
properties ($T_{eff}$, $log\ g$, and $L$) to each star of the initial
population. We chose to use the MESA isochrones and stellar tracks (MIST;
Dotter, 2016; Choi et al., 2016; Paxton et al., 2011, 2013, 2015), whose
evolutionary tracks are computed with the MESA 1D stellar evolution code and
cover a wide range of parameters: stellar mass (including high-mass stars,
[0.1; 300]$\ \mathrm{M_{\odot}}$), age (including very young stars, [$10^{6}$;
$10^{10}]\ $yr), and metallicity ([Fe/H], [-4; 0.5]). We used the version
defined as v1.2 in SPISEA, which was downloaded in August 2018 (solar
metallicity) and in April 2019 (other metallicities).

Each star is then attributed a synthetic spectrum from the PHOENIX Models for
Synphot111Described and available for download at
https://www.stsci.edu/hst/instrumentation/reference-data-for-calibration-and-
tools/astronomical-catalogs/phoenix-models-available-in-synphot (Allard et
al., 2003, 2007, 2012), which cover the 50-50000 nm spectral range. For
[Fe/H]=0, the library is at high spectral resolution ($R\sim 14000$) and was
constructed with no missing template except for the Wolf-Rayet stars.

This model was used to compute a library that we call UMISSP (UV to mid-
infrared single stellar population), which consists of synthetic spectra of
$10^{6}\ \mathrm{M_{\odot}}$ solar-metallicity stellar populations with ages
ranging from $10^{6}$ to $10^{10}$ yr with $10^{0.05}$ steps222The library is
publicly available at http://galaxy.asu.cas.cz/page/umisspr. Spectra at lower
resolution ($R\sim 280$) are also computed for other metallicities ([Fe/H] =
0.5, -0.5, -1, -2, -3, -4), although not used in this paper. The mean spectra
and standard deviation spectra for all ages and metallicities are provided.
They are flux calibrated and expressed in $\mathrm{W.\mu m^{-1}}$,
corresponding to the total cluster emission in a 4$\pi$ sr solid angle. In
addition, for each stellar population, we provide: (a) the initial mass
histogram, (b) the MIST isochrone, (c) the parameters of the isochrone and the
ones used for the spectrum attribution of each star.. In Appendix B, we
perform several basic tests to verify that our spectra are in general
agreement with other SSP models.

##  4 Model

In order to model the observations, we choose a two-component model composed
of a spherical nuclear cluster and an extended thick disk. For each set of
parameters, we construct 3D maps of the various observables, which we stack
along the line of sight. These are then used to compute synthetic
observations, which are directly compared to the above-mentioned set of
observables (Subsection 2.2). The construction of these synthetic observations
is described in more detail in the following paragraphs.

###  4.1 3D maps of mass, velocity, and velocity dispersion

We first construct 3D maps of the mass distribution, circular velocity, and
velocity dispersion for each component using two self-consistent potential
density function pairs: the Plummer spherical cluster (Plummer, 1911) and the
Miyamoto-Nagai thick disk (Miyamoto & Nagai, 1975).

The mass distribution is computed as

$\rho_{NSC}(R,z)=\frac{3M_{NSC}}{4\pi
b_{NSC}^{3}}\left(1+\frac{R^{2}+z^{2}}{b_{NSC}^{2}}\right)^{-\frac{5}{2}},$
(2)

for the nuclear star cluster (NSC) Plummer model, and

$\displaystyle\rho_{disk}(R,z)=$ (3)
$\displaystyle\frac{b^{2}M}{4\pi}\frac{a_{MN}R^{2}+[a_{disk}+3(z^{2}+b_{disk}^{2})^{1/2}][a_{disk}+(z^{2}+b_{disk}^{2})^{1/2}]^{2}}{(R^{2}+[a_{disk}+(z^{2}+b_{disk}^{2})^{1/2}]^{2})^{5/2}(z^{2}+b_{disk}^{2})^{3/2}}$

for the extended Miyamoto-Nagai disk.

The corresponding potentials are

$\Phi_{NSC}(R,z)=-\frac{GM_{NSC}}{\sqrt{R^{2}+z^{2}+b_{NSC}^{2}}},$ (4)

$\Phi_{disk}(R,z)=-\frac{GM_{disk}}{\sqrt{R^{2}+[a_{disk}+(z^{2}+b_{disk}^{2})^{1/2}]^{2}}},$
(5)

where (R, $\theta$, z) are the cylindrical coordinates in the disk.

The resulting total gravitational potential is

$\Phi(R,z)=\Phi_{NSC}(R,z)+\Phi_{disk}(R,z).$ (6)

We assume that the central cluster is spherical and has a rest-frame LOSV. For
the extended disk, we compute the circular velocity as the derivative of the
total gravitational potential (NSC + disk):

$\overrightarrow{v}_{disk}(R,z)=\sqrt{r\frac{d\Phi}{dr}}\left(\frac{R}{r}\right)\overrightarrow{e}_{\theta},$
(7)

where $r=\sqrt{R^{2}+z^{2}}$ and $\overrightarrow{e}_{\theta}$ is the unit
vector in the direction of rotation.

For the NSC Plummer cluster (Dejonghe, 1987), the local isotropic velocity
dispersion is given by

$\sigma_{NSC}^{2}(r)=\frac{GM_{NSC}}{6\sqrt{r^{2}+b_{NSC}^{2}}},$ (8)

while for the extended MN disk, we assume that the local velocity dispersion
is isotropic and equal to the product of the orbital velocity by the aspect
ratio $(h/r)^{*}$ of the disk:

$\sigma_{disk}(R,z)=(h/r)^{*}\sqrt{r\frac{d\Phi_{NSC+disk}}{dr}}\left(\frac{R}{r}\right),$
(9)

where $(h/r)^{*}$ is the scale height of the disk, that is, the ratio between
the vertical and radial spatial scales. We note that this ratio is not equal
to $\frac{b_{disk}}{a_{disk}}$, but was measured by fitting an elongated
Gaussian on $\rho_{disk}$ distributions for a wide range of
$(a_{disk},b_{disk})$ parameters, and is interpolated at the desired values
(see Fig. D).

In the end, the LOSV of each component is given by

$\cases{v}_{disk,los}(R,z)=\overrightarrow{v}_{disk}(R,z)\cdot\overrightarrow{e}_{los}=v_{disk}(R,z)\times
sin(i)sin(\theta)\\\ v_{NSC,los}(R,z)=0,\\\ {}$ (10)

where $\overrightarrow{e}_{los}$ is the unit vector along the direction of the
LOS, $i$ is the inclination of the disk (null for a face-on disk), and
$\theta$ the azimuth angle in the disk (null in the direction of the
observer).

###  4.2 Spectral templates

With the model described above, we compute 3D grids for each component, with a
mass, LOSV, and LOSV dispersion attributed to each cell. We use this
information to attribute a spectrum to each cell of the grid, as follows:

  1. 1.

Based on the high-resolution SSP spectra presented above, we attribute two
spectra to each component: a low-resolution component with the full spectral
domain, and a high-resolution component limited to the $2.25-2.40\ \mathrm{\mu
m}$ wavelength interval. Both of them are scaled according to mass enclosed in
each cell.

  2. 2.

We perform a Doppler shift of the high-resolution spectrum based on the
projected velocity along the LOS.

  3. 3.

We convolve the high-resolution spectrum with a Gaussian whose width is given
by the $\sigma_{v}$.

  4. 4.

We convolve the high-resolution spectrum with the SINFONI line spread function
(measured on strong OH lines from the SKY observation, which are assumed to be
constant within the Field of View FoV).

###  4.3 Synthetic observations

At last, we compute the synthetic observation corresponding to our four
observables:

  1. 1.

A low-resolution datacube obtained by summing the low-resolution spectra along
the LOS.

  2. 2.

A high-resolution datacube obtained by summing the shifted and convolved high-
resolution spectra along the LOS, and normalizing it to the measured
continuum.

  3. 3.

A LOSV map obtained by calculating the average of the velocity projected along
the LOS weighted by the luminosity of each cell at 2.3 $\mathrm{\mu m.}$

  4. 4.

A LOSV dispersion map obtained by computing the square root of the quadratic
sum of (i) the average of the local velocity dispersion projected along the
LOS, and (ii) the standard deviation of the velocity projected along the LOS,
with both weighted by the luminosity in each cell at 2.3 $\mathrm{\mu m}$.

All the synthetic observations are convolved with a Gaussian kernel matching
the angular resolution of the observation.

##  5 Results

###  5.1 Fitting procedure

The model consists of an NSC with two stellar populations —one young and one
old— characterized by their masses $M_{Y,NSC}$ and $M_{O,NSC}$ and ages
$age_{Y,NSC}$ and $age_{O,NSC}$, respectively; and similarly for the disk with
$M_{Y,disk}$ and $M_{O,disk}$ and ages $age_{Y,disk}$ and $age_{O,disk}$. The
two stellar populations of each component are assumed to be extinguished by
the same amount of foreground material, which are defined by $A_{K,NSC}$ and
$A_{K,disk}$. Additionally, the geometry of the cluster is given by only one
parameter, its radius $b_{NSC}$; while the geometry of the disk is described
by four parameters, namely its radius $a_{disk}$, its scale height
$(h/r)^{*}$, its inclination $i,$ and its position angle $PA$. Overall, the
model has 15 free parameters, and the computation time for each set of
parameters is between 100 ms and 1000 ms. This prevents us from fully
exploring the parameter space, and requires the use of a specific two-step
fitting procedure. We first determine the eight parameters corresponding to
the ages of the stellar populations, namely $age_{Y,NSC}$, $age_{O,NSC}$,
$M_{Y,NSC}/M_{O,NSC}$, $A_{K,NSC}$, $age_{Y,disk}$, $age_{O,disk}$,
$M_{Y,disk}/M_{O,disk}$, and $A_{K,disk}$, as follows:

  1. 1.

We extract integrated spectra of the observed NSC and extended disk with
aperture spectrometry. The average disk spectrum is subtracted from the NSC
one (masks are determined based on the continuum images, where the NSC is
clearly distinguishable).

  2. 2.

For each component (NSC and disk), we measure one low-resolution spectrum
covering the JHK bands, and a high-resolution spectrum covering the CO
bandheads at 2.4 $\mathrm{\mu m}$.

  3. 3.

For each component, we fit an extinction $A_{K}$ and the relative mass
contribution ($M_{Y}/M_{O}$) between two stellar populations with different
ages. We try every combination of young ($10^{6}\ \mathrm{yr}\leq age_{Y}\leq
10^{9}\ \mathrm{yr}$ in steps of $10^{0.025}$) and old ($10^{9}\
\mathrm{yr}\leq age_{O}\leq 10^{10}\ \mathrm{yr}$ in steps of $10^{0.1}$)
stellar populations, measure the sum of the squared residuals both on the
observed low-resolution continuum and on the normalized high-resolution
absorption spectrum (weighted by the inverse of the average value of each of
these), and keep the best combination for the second step of the fit
(secondary solutions have also been tried).

Once we have fixed the ages of the two stellar populations, we fit the masses
and geometrical parameters, $M_{NSC}$, $b_{NSC}$, $M_{disk}$, $a_{disk}$,
$b_{disk}$, $i$ and $PA$:

  1. 1.

At first, we find an initial solution by manually adjusting the parameters and
visually inspecting the synthetic observables, as follows:

     * •

$M_{NSC,0}$ and $r_{NSC,0}$ are set to reproduce the continuum images and the
$\sigma_{v}$ map.

     * •

$M_{disk,0}$ and $a_{disk,0}$ are chosen to approximately match the continuum
images.

     * •

$PA_{0}$ is easily obtained by the line of nodes of the LOSV map.

     * •

Keeping the previous estimate for $M_{disk,0}$ and $a_{disk,0}$, we fit
$i_{0}$ and $b_{disk,0}$ to best reproduce the observed LOSV and $\sigma_{v}$.

  2. 2.

We use the previous solution ($M_{NSC,0}$, $r_{NSC,0}$, $M_{disk,0}$,
$a_{disk,0}$, $b_{disk,0}$, $i_{0}$, $PA_{0}$) as an initial guess for the
automated Trust Region Reflective least-square algorithm implemented in
scipy.optimize, which provides the optimal solution around the initial guess.
For each object, the weights associated to the four observables (continuum, CO
absorption features, LOSV, and $\sigma_{v}$) are determined iteratively to
give them similar importance in the fitting procedure.

  3. 3.

Finally, we estimate the uncertainties on the parameters by exploring each of
them individually, except for $M_{disk}$ and $i,$ which can be strongly
correlated and for which a 2D map of the summed squared residuals is computed
to determine the uncertainties (see Figs. 9, 12, and 15).

The best parameters for the three objects are presented in the following
tables, and the maps associated to the best models are presented in the
appendices.

###  5.2 Supernova rate

With MIST, we can estimate the spatial distribution of the average supernova
rate in each object. For a SSP at a given age, the specific supernova rate can
be estimated by counting the number of stars with $M>8\mathrm{M_{\odot}}$
(smoothed over three temporal bins) and taking its derivative (see Fig. 1).
This simplified model assumes that all stars with $M>8\mathrm{M_{\odot}}$ will
eventually explode as supernovæ, and does not take into account type I. We use
this information and the mass distribution of each stellar population to
construct a 3D map of the supernova rate.

Figure 1: Evolution of the UMISSP supernova rate

Another way to estimate the average supernova rate is to measure the [Fe II]
emission line flux, because the flux of these lines is observed to be strongly
correlated in integrated galaxy spectra and on a pixel-to-pixel basis
(Rosenberg et al., 2012):

$\log\frac{v_{\mathrm{SNrate}}}{\mathrm{yr}^{-1}\mathrm{pc}^{-2}}=(1.01\pm
0.2)*\log\frac{[\mathrm{FeII}]_{1.26}}{\mathrm{erg}.\mathrm{s}^{-1}\mathrm{pc}^{-2}}-41.17\pm
0.9.$

We also use this scaling relation to compute 2D supernova rate maps. A similar
relation is presented by Alonso-Herrero et al. (2003), which is based on the
number of supernova remnant radio counts in M82 and NGC 253:

$v_{\mathrm{SNrate}}=0.8\frac{L_{\mathrm{[FeII]}}}{W}\times
10^{-34}\mathrm{yr}^{-1}.$

###  5.3 NGC 1808

Table 1: NGC 1808: Results of the age-retrieval procedure log($age_{Y,NSC}$) | 7.2 | log(Myr)
---|---|---
log($age_{O,NSC}$) | 9.0 | log(Myr)
$M_{Y,NSC}/M_{O,NSC}$ | 0.10 |
$A_{K,NSC}$ | 0.30 |
log($age_{Y,disk}$) | 7.2 | log(Myr)
log($age_{O,disk}$) | 9.5 | log(Myr)
$M_{Y,disk}/M_{O,disk}$ | 0.04 |
$A_{K,disk}$ | 0.20 |
Table 2: NGC 1808: Mass and geometry parameters $M_{NSC}$ | 8.4 $\pm$ 0.2 | $log(\mathrm{M_{\odot}})$
---|---|---
$M_{disk}$ | 8.5 $\pm$ 0.3 | $log(\mathrm{M_{\odot}})$
$i$ | 46.3 $\pm$ 13.5 | $\degree$
$PA$ | 233.6 $\pm$ 40.9 | $\degree$
$b_{NSC}$ | 20.0 $\pm$ 5.0 | pc
$a_{disk}$ | 23.5 $\pm$ 5.9 | pc
$(h/r)_{disk}$ | 0.67 $\pm$ 0.07 |
Figure 2: NGC 1808: Observation (right) and best model (left). From top to
bottom: Averaged CO absorption lines over the entire FoV, H band image, LOSV,
and LOSV dispersion.

The nucleus of NGC 1808 appears to have undergone a recent starburst event, as
evidenced by the estimated ages of young stellar populations in the disk and
the NSC of $\sim 30\ \mathrm{Myr}$. Analysis of residuals in the continuum and
CO absorption features (see Figs. 7 and 8) shows that the extinction is well
constrained to a low value and several ages provide a good fit, with the
deepest and widest minimum around the selected ages. We find similar ages for
the populations in the disk and in the NSC, which supports the idea of a
common star formation history. Two secondary solutions are observed at
$10^{8.3}$ and $10^{8.5}$ yr, corresponding to the late apparition of red
giants from $2-3.5\ \mathrm{M_{\odot}}$ stars, which briefly make the NIR
spectrum similar to that of a young stellar population.

A large NSC surrounded by a compact thick disk inclined by $i\sim 45\degree$
and oriented along $PA\sim 230\degree$ is the best model for NGC 1808.
Although a larger disk radius and mass can improve the LOSV map fit, this
leads to a poor fit for the $\sigma_{v}$ map and a slightly poorer fit for the
continuum images. It is possible that two disk-like structures coexist, but
for modeling the inner region, we focus on the compact disk solution.

The model offers a relatively good fit to all observations, with some
noticeable differences. The colors and morphology of the central bright
continuum source match the observations, but its size is hard to estimate due
to limited angular resolution. The CO absorption features are well
represented. The LOSV is slightly underestimated, and our model predictions
suggest its maximum is closer to the nucleus than actually observed,
indicating that it is missing mass at large radii. However, the model
correctly predicts the position of the maximum of $\sigma_{v}$, as well as the
central drop and its average value.

###  5.4 NGC 1433

Table 3: NGC 1433: Results of the age-retrieval procedure log($age_{Y,NSC}$) | 8.6 | log(Myr)
---|---|---
log($age_{O,NSC}$) | 9.8 | log(Myr)
$M_{Y,NSC}/M_{O,NSC}$ | 0.11 |
$A_{K,NSC}$ | 0.60 |
log($age_{Y,disk}$) | 8.2 | log(Myr)
log($age_{O,disk}$) | 9.8 | log(Myr)
$M_{Y,disk}/M_{O,disk}$ | 0.04 |
$A_{K,disk}$ | 0.25 |
Table 4: NGC 1433: Mass and geometry parameters $M_{NSC}$ | 9.3 $\pm$ 0.4 | $log(\mathrm{M_{\odot}})$
---|---|---
$M_{disk}$ | 10.4 $\pm$ 0.25 | $log(\mathrm{M_{\odot}})$
$i$ | 2.3 $\pm$ 2.2 | $\degree$
$PA$ | 335.1 $\pm$ 54.5 | $\degree$
$b_{NSC}$ | 35.8 $\pm$ 8.9 | pc
$a_{disk}$ | 200.0 $\pm$ 17.5 | pc
$(h/r)_{disk}$ | 0.23 $\pm$ 0.06 |
Figure 3: NGC 1433: Observation (right) and best model (left). From top to
bottom: Averaged CO absorption lines over the entire FoV, H band image, LOSV,
and LOSV dispersion.

NGC 1433 shows no evidence of recent star formation, with the youngest
populations estimated to be over 100 million years old in both the disk and
NSC. The spectra of both components are similar, and so are the residual maps
in Figures 10 and 11, despite the NSC being more extincted.

The best model features a massive NSC of $\gtrsim 10^{9}\ \mathrm{M_{\odot}}$
with a radius of $\sim 25$ pc surrounded by an extended, almost face-on disk
of $\sim 300\ \mathrm{pc}$ in diameter with a height-to-radius ratio of $\sim
0.25$ and an inclination of $\lesssim 10\degree$.

The model offers a fair representation of all observables, particularly the CO
absorption lines, continuum images, and LOSV map, which match the observations
well. The $\sigma_{v}$ map geometry is less convincing, but is still
reasonable: the observed $\sigma_{v}$ map is mostly featureless, and the model
accurately reproduces its average value, which is relatively high. The model
would predict a slight drop around the NSC, but this is not observed.

###  5.5 NGC 1566

Table 5: NGC 1566: Results of the age-retrieval procedure log($age_{Y,NSC}$) | 6.6 | log(Myr)
---|---|---
log($age_{O,NSC}$) | 9.8 | log(Myr)
$M_{Y,NSC}/M_{O,NSC}$ | 0.08 |
$A_{K,NSC}$ | 0.90 |
log($age_{Y,disk}$) | 6.1 | log(Myr)
log($age_{O,disk}$) | 9.2 | log(Myr)
$M_{Y,disk}/M_{O,disk}$ | 0.10 |
$A_{K,disk}$ | 0.40 |
Table 6: NGC 1566: Mass and geometry parameters $M_{NSC}$ | 8.5 $\pm$ 0.4 | $\mathrm{log(\mathrm{M_{\odot}})}$
---|---|---
$M_{disk}$ | 10.1 $\pm$ 0.3 | $\mathrm{log(\mathrm{M_{\odot}})}$
$i$ | 5.4 $\pm$ 5.0 | $\degree$
$PA$ | 322.1 $\pm$ 40.3 | $\degree$
$b_{NSC}$ | 7.9 $\pm$ 2.0 | pc
$a_{disk}$ | 110.0 $\pm$ 28.9 | pc
$(h/r)_{disk}$ | 0.22 $\pm$ 0.05 |
Figure 4: NGC 1566: Observation (right) and best model (left). From top to
bottom: Averaged CO absorption lines over the entire FoV, H band image, LOSV,
and LOSV dispersion.

The best model for NGC 1566 reveals recent star formation in both the NSC and
the circumnuclear disk. The residual maps in Figures 13 and 14 indicate that
star formation in the disk occurred very recently or is ongoing, with minimum
residuals for ages of less than $3\ \mathrm{Myr}$. The NSC also shows evidence
of recent star formation, with ages younger than $10\ \mathrm{Myr}$, and a
minimum of between $3$ and $10\ \mathrm{Myr}$. The extinction values are well
constrained and higher compared to the other two objects, particularly in the
NSC, which supports the proposed recent star formation in these structures.

The NSC in NGC 1566 is compact and relatively massive, and the disk has
similar geometry to NGC 1433 with a size of $\sim 150\ \mathrm{pc}$, an aspect
ratio of $(h/r)_{disk}\sim 0.25$, and a nearly face-on orientation ($i\lesssim
10\degree$). The central bright source of continuum emission is well
represented by the model, but the depth of the CO absorption features is
underestimated. This could be due to a large LOSV dispersion ($\sigma_{v}$),
imprecision in determining the ages of the stars, or imperfections in the
stellar templates. The observed LOSV map is well reproduced by the model, but
the $\sigma_{v}$ map is more difficult to compare, as the observed map is
mostly uniform. However, the model accurately represents the average value of
the $\sigma_{v}$.

In order to obtain a better fit of the CO absorption depth, we tried to
constrain the ages of the stellar population in the disk and/or in the NSC to
be older than 10 Myr. This slightly increased the CO depth. However, the
improvement was minor, and this constraint had a negative impact on the rest
of the fit: the continuum was not well reproduced by the template, making the
fitting procedure converge toward a solution with a very large NSC, replacing
part of the disk.

##  6 Discussion

###  6.1 Age and extinction

By fitting the low-resolution continuum and CO absorption features, we
determined ages and extinction for the stellar populations of the two
components solely based on the SINFONI observations.

For NGC 1808, we find evidence of past star formation with young stellar
populations of ages between 10 and 50 Myr. These estimates are in good
agreement with previous results, which all point toward the presence of recent
stellar populations in the central region: Kotilainen et al. (1996) also
estimate the age of the stellar populations in the central region to be
between $\sim 10$ and $\sim 40$ Myr from NIR line and continuum analysis,
Galliano et al. (2005); Galliano & Alloin (2008) observed the presence of a
very embedded young stellar cluster in the MIR, and Busch et al. (2017)
measured the age of the 1 kpc circumnuclear ring to range between 5 and 8 Myr
(the age of the central cluster is not given, but is indicated as having
experienced less recent star formation).

For NGC 1433, we find no evidence of star formation in the last 100 Myr. This
result is supported by previous publications on the object (Cid Fernandes et
al., 1998; Sánchez-Blázquez et al., 2011; Smajić et al., 2014), which do not
detect any significant star formation or extinction outside the 1 kpc
circumnuclear ring.

For NGC 1566, we find evidence of very recent or ongoing star formation, both
in the NSC and in the 150 pc ring. This result is not in strong agreement with
previous publications, which tend to attribute a significant portion of the
nuclear featureless continuum to AGN activity (accretion disk + hot dust)
instead of hot stars (Smajić et al., 2015; da Silva et al., 2017). This
contribution from the AGN is supported by the observation of associated
emission lines, and is not taken into account in our model, which probably
overestimates the contribution from young stars. However, these latter two
papers still point toward the presence of recent star formation: Smajić et al.
(2015) measure the presence of significant star formation and supernova rates
through $Br_{\gamma}$ and [Fe II] emission lines, and the stellar spectral
synthesis done with the STARLIGHT software in da Silva et al. (2017)
attributes a major part of the featureless continuum to a young stellar
population. The AGN interpretation alone cannot explain the observations; in
particular, the spatial extent of the continuum source, nondiluted nuclear CO
absorption features, and the continuum in the circumnuclear disk. The stellar
population interpretation can explain most of the SINFONI observation, but
does not take into account observations at other wavelengths, which reliably
identify an AGN component. The nucleus of NGC 1566 most probably hosts both
phenomena, and the contribution of a young stellar population predicted by our
model should be interpreted as an upper limit.

Overall, our method to determine the age of the young stellar populations in
galactic centers provides good results. For NGC 1808 and NGC 1433, the ages
obtained are fully consistent with observations and previous publications. For
NGC 1566, the depth of the CO absorption features is not well reproduced by
our model, and the contribution of young stellar populations might be
overestimated by our model because of the presence of the featureless AGN.

###  6.2 Supernova rates

Figure 5: NGC 1808: 2D maps of the supernova rate (number of supernova per
year and per pixel): (top) as estimated from our modeling and (bottom) as
estimated from the [Fe II] emission line with the Rosenberg et al. (2012)
conversion factor.

Figure 6: NGC 1566: 2D maps of the supernova rate (number of supernova per
year and per pixel): (top) as estimated from our modeling and (bottom) as
estimated from the [Fe II] emission line with the Rosenberg et al. (2012)
conversion factor.

Figures 5 and 6 show the supernova rate maps for NGC 1808 and NGC 1566,
respectively, obtained from (1) our modeling; (2) the [Fe II] scaling relation
from Rosenberg et al. (2012). For NGC 1433, our model does not predict any
supernova rate, and no [Fe II] emission line is detected.

For NGC 1808, the total supernova rate in the considered field of view amounts
to 0.038, 0.028, and 0.007 supernovæ per year with the three estimators. The
relative distribution between the NSC and the disk is also well reproduced
(see 2). The consistency between the estimate from our modeling and [Fe II]
emission lines confirms the age and mass deduced from our model for the young
stellar population. According to Fig. 1, the stellar population is reaching
the end of its supernova episode.

For NGC 1566, the results are very similar, but with a slightly larger
discrepancy between our modeling and the [Fe II] estimators, in particular for
the NSC where the difference between methods (1) and (3) is of an order of
magnitude. However, even with the lowest estimator, the supernova rate is very
significant in the central region, which supports the diagnostic of a young
stellar population discussed above. According to Fig. 1 and our very young age
estimates, this corresponds to the beginning of the supernova episode for this
stellar population, and the high supernova rate should be sustained for
several tens of millions of years.

The absence of [Fe II] emission in NGC 1433 is consistent with the prediction
of our model, which does not detect any stellar population younger than 100
Myr, and so does not predict any core-collapse supernova. Overall, the
estimates of the supernova rate are of the same magnitude with the three
methods. The consistency (same order of magnitude) between the estimate from
our modeling and the direct [Fe II] measurements confirms the age
determination we performed based on CO and continuum.

###  6.3 Mass and geometry

The simultaneous fit of the continuum of emission, LOSV, and $\sigma_{v}$
allows us to obtain an estimate of the 3D geometry of the object, in
particular the inclination and scale height of the disk. For NGC 1808, the
model converges toward a rather compact and very thick disk inclined by $\sim
50\degree$. The inclination found for the disk is fully consistent with values
derived from the geometry of larger scale structures, such as the 1 kpc
starburst ring or the de Vaucouleurs profile of the galaxy. With this
inclination, the high aspect ratio of the disk ($r/h\sim 0.65$) is imposed by
the LOSV to $\sigma_{v}$ maps ratio. The required mass to explain both LOSV
and $\sigma_{v}$ with this inclination is consistent with the observed
continuum emission flux given the assumed stellar populations. This consistent
description of the four observables gives us confidence that such a compact
thick disk is present around the NSC. Nevertheless, as already mentioned, a
more standard larger and flatter disk is also probably present.

The geometry of our model for the disk of NGC 1433 and NGC 1566 consists of a
larger and flatter disk seen almost face-on. The inclinations differ
significantly from the estimates found from a large-scale fit of the galaxies,
which indicates inclinations of $\sim 30\degree$ for both objects from HI
geometry and kinematics (Elagali et al., 2019; Ryder et al., 1996,
respectively). However, as noted in these latter papers, inclination is
difficult to accurately measure so close to a face-on orientation, especially
considering that strongly barred late-type galaxies could have an intrinsic
elongation on average. Both results (from large-scale HI and small-scale
stellar content) could be simultaneously correct in a tilted disk
configuration.

##  7 Conclusions

We used SINFONI spatially resolved spectroscopic observations of three nearby
galactic nuclei (NGC 1433, NGC 1566, and NGC 1808) to derive the properties of
the stellar populations in their central few hundred parsecs. Our main goal is
to determine the age, mass, and 3D geometry of the nuclear star cluster (NSC)
and the surrounding extended thick stellar disk.

To achieve this, we propose a method that uses single stellar population (SSP)
spectra to determine the age of the stellar populations and construct
synthetic observations for our model. We simultaneously fitted the spatially
resolved line-of-sight-velocity (LOSV) and its dispersion ($\sigma_{v}$), as
well as the low-spectral-resolution NIR continuum and high-spectral-resolution
CO absorption features for each pixel of the datacube in order to overcome
common degeneracies between key parameters.

We determined the ages of the various stellar populations by fitting the low-
resolution continuum of emission and CO absorption features with UMISSP, an
SSP library developed specifically for this purpose. We then determined the 3D
mass distribution of the stellar populations with a two-component model
composed of a Plummer NSC surrounded by a Miyamoto-Nagai disk, which produces
synthetic continuum and CO datacubes, as well as LOSV and $\sigma_{v}$ maps,
which we directly compared to observations.

For each object, we obtain a model that is consistent with all the SINFONI
observables, and is in general agreement with previously published studies:

  * •

NGC 1808 experienced relatively recent star formation —that is, around 10 Myr
ago in the NSC and 30 Myr ago in a surrounding compact thick disk. A strong
supernova rate is suggested by both the [Fe II] flux and our age modeling.

  * •

NGC 1433 has not experienced recent star formation. It is composed of the most
massive NSC with $M_{NSC}\sim 1.3\times 10^{9}\ \mathrm{M_{\odot}}$ and is
surrounded by a large and flat disk seen almost face-on. No significant
supernova rate is measured.

  * •

NGC 1566 has recently experienced or is currently experiencing a starburst,
both in the disk and the central NSC. Its geometry is similar to that of NGC
1433, with a massive NSC surrounded by a large and flat disk seen almost face-
on. The nucleus is at the beginning of a long episode of supernova.

For the three objects, we find good agreement between the supernova rate
estimated with our model and that estimated from [Fe II] scaling relations,
confirming the age and mass deduced from our model for the young stellar
population. We conclude that, based on an IFU observation alone, our method
can be used to characterize the properties of the stellar populations in the
central few hundred parsecs of nearby galactic nuclei.

###### Acknowledgements.

We thank the anonymous referee for his useful comments. This work was made
possible by the support of the international collaboration in astronomy (ASU
mobility) with the number $CZ.02.2.69/0.0/0.0/18_{0}53/0016972$ and the
institutional project RVO:67985815. ASU mobility is co-financed by the
European Union.

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##  Appendix A  Age determination of stellar populations

In this Appendix, we present figures associated to the procedure carried out
to recover the ages of the three objects: NGC 1808 (Figs. 7, 8 and 9), NGC
1433 (Figs. 10, 11 and 12), and NGC 1566 (Figs. 13, 14 and 15).

Figure 7: NGC 1808 (NSC): (top) 2D map of the summed residuals as a function
of age and extinction between the model and the extracted NSC spectrum,
(middle) minimum residuals as a function of age, and (bottom) minimum
residuals as a function of extinction.

Figure 8: NGC 1808 (disk): (top) 2D map of the summed residuals as a function
of age and extinction between the model and the extracted disk spectrum,
(middle) minimum residuals as a function of age, and (bottom) minimum
residuals as a function of extinction Figure 9: NGC 1808 (disk): 2D map of the
summed residuals as a function of mass and inclination for the final model.

Figure 10: NGC 1433 (NSC): (top) 2D map of the summed residuals as a function
of age and extinction between the model and the extracted NSC spectrum,
(middle) minimum residuals as a function of age, (bottom) minimum residuals as
a function of extinction.

Figure 11: NGC 1433 (disk): (top) 2D map of the summed residuals as a function
of age and extinction between the model and the extracted disk spectrum,
(middle) minimum residuals as a function of age, and (bottom) minimum
residuals as a function of extinction. Figure 12: NGC 1433 (disk): 2D map of
the summed residuals as a function of mass and inclination for the final
model.

Figure 13: NGC 1566 (NSC): (top) 2D map of the summed residuals as a function
of age and extinction between the model and the extracted NSC spectrum,
(middle) minimum residuals as a function of age, and (bottom) minimum
residuals as a function of extinction.

Figure 14: NGC 1566 (disk): (top) 2D map of the summed residuals as a function
of age and extinction between the model and the extracted disk spectrum,
(middle) minimum residuals as a function of age, and (bottom) minimum
residuals as a function of extinction. Figure 15: NGC 1566 (disk): 2D map of
the summed residuals as a function of mass and inclination for the final
model.

##  Appendix B Comparisons

The construction of SSP is a complex process where a great number of implicit
astrophysical assumptions are made through the choice of the IMF, isochrones,
and spectral library, leading to important variations between the output from
one model and that of another. In order to check that no dramatic error has
been made in the process, and to compare the typical age predictions of our
models to others, we made a series of comparisons with three similar SSP
models that also cover relatively young ages over a large spectral domain. The
first one is GALEV (Kotulla et al. 2009), a user-friendly evolutionary model
that produces low-resolution spectra over a wide range of ages similar to
ours; the second is EMILES (Vazdekis et al. 2016), an SSP library with
relatively high spectral resolution; the third is UNISSP, computed with the
same isochrones as UMISSP, but using the PHOENIX v16 stellar spectra presented
by Husser et al. (2013). Please note that all the comparisons are done at
solar metallicity, and thus do not include age–metallicity degeneracies.

###  B.1 SED fitting

On the overlapping part of the parameter space, i.e., SSP from 100 Myr to 10
Gyr with solar metallicity, we extracted low-resolution SEDs from UMISSP at
various ages and tried to retrieve the original age by fitting the spectra
with the three other SSPs: UNISSP, EMILES, and GALEV. The results are
presented in Fig. 16. While a visible difference is observed between the
original and the recovered ages, we consider the result of this test to be
conclusive, as the retrieved ages are systematically in the correct order of
magnitude with a moderate spread. The largest discrepancies are observed for
the youngest ages of UMISSPR (between 100 and 300 Myr), in particular when
retrieved with the EMILES library. However, all models recognize a young
stellar population, and these ages are outside the explicitly safe range of
the EMILES SSP (which can be found on the EMILES website, and result from
various limitations of the BaSTI and Padova00 models).

###  B.2 NIR absorption line fitting

Similarly, we checked the consistency of our two SSPs by retrieving the age of
UMISSP solely by fitting its NIR absorption features with UNISSP. For this
purpose, we extracted and normalized the spectra in two spectral domains (K
band between 2.1 and 2.5 $\mathrm{\mu m}$, and JHK bands between 1. and 2.5
$\mathrm{\mu m}$), convolved them with a 1 nm Gaussian, and added a 5% normal
noise to simulate a $R\sim 2000$ spectroscopic observation. The results are
presented in Fig. 17. While the fit of the K band absorption features can
produce good results in some cases, it can also lead to very large errors in
the estimate of the age of a stellar population. Indeed, the CO bandheads,
which arise from cold evolved stars and dominate the K band absorption
spectrum, are remarkably similar over a wide range of ages. On the other hand,
when simultaneously fitting the absorption features from the three NIR
spectral bands (J, H and K), we are able to reliably retrieve the age of the
original population. However, this procedure remains less precise than the SED
fitting, and we suggest to couple the two strategies when fitting the age of
an unresolved stellar population.

###  B.3 Visible magnitude fitting

We then checked the ability of our libraries to recover the ages of star
clusters based on visible photometry, and compared the results with GALEV and
EMILES. For this purpose, we used the M33 star cluster catalog presented in
Fan & de Grijs (2014), which provides UBVRI photometry and age estimates for
188 clusters.

For each cluster in the catalog, we compared the [(U-B), (U-V), (U-R), (U-I)]
colors to the ones predicted by the four libraries, and looked for the age
providing the best fit. For each library, we present the retrieved age
distribution in Figs. 18, 19, 20, and 21, and Table 7 shows the residuals as
well as the associated Pearson’s correlation coefficient between the published
ages and retrieved ones. The results are similar for the four libraries: with
only a few exceptions (less than 10 outliers) we are able to distinguish the
very young, young, intermediate, and old stellar populations, but with a large
spread in the age estimate ($0.6\ dex\lesssim\sigma_{\tau}\lesssim 0.9\ dex$).

Acknowledging this large dispersion, our two libraries perform similarly to
GALEV and EMILES, with a large spread, but are still able to determine the age
group. We consider this sanity check to be fulfilled.

Table 7: Residuals and Pearson’s correlation coefficient for the age retrieval based on UBVRI photometry with the four spectral libraries. Library | $residuals_{\tau}$ | PCC
---|---|---
UMISSP | 0.62 | 0.75
UNISSP | 0.58 | 0.39
GALEV | 0.61 | 0.73
EMILES | 0.62 | 0.38
Figure 16: Retrieval of the age of UMISSP with the UNISSP, GALEV, and EMILES
spectral libraries based on a SED fitting Figure 17: Retrieval of the age of
UMISSP with the UNISSP library, solely based on the normalized absorption
features between 1 and 2.5 $\mathrm{\mu m}$. Figure 18: Retrieval of the M33
cluster age (Fan & de Grijs 2014) with the UMISSP library based on UBVRI
photometry. Figure 19: Retrieval of the M33 cluster age (Fan & de Grijs 2014)
with the UNISSP library based on UBVRI photometry. Figure 20: Retrieval of the
M33 cluster age (Fan & de Grijs 2014) with the GALEV library based on UBVRI
photometry. Figure 21: Retrieval of the M33 clusters age (Fan & de Grijs 2014)
with the EMILES library based on UBVRI photometry.

##  Appendix C Impact of missing stellar templates

In order to estimate the maximum error caused by the missing stellar
templates, we measured the bolometric luminosity of the stars whose parameters
were outside the model grid, and which were interpolated to the nearest
available stellar template. In Fig. 22, we plot the ratio of this flux over
the total flux, for UNISSP and UMISSP ([Fe/H] = 0).

Figure 22: Percentage of the UMISSP and UNISSP flux corresponding to stars
with parameters outside the grid ([Fe/H] = 0).

With UMISSP, UNISSP, GALEV, and EMILES, we fitted the age of 188 clusters
observed in M33 based on their UBVRI photometry. The results are presented in
Figs. 18, 19, 20, and 21. In each case, the x-axis corresponds to age of the
cluster as published in Fan & de Grijs (2014) and the y-axis corresponds to
the fitted age.

##  Appendix D Miyamoto Nagai disk

In our model, we define the velocity dispersion for each position in the
Miyamoto-Nagai disk as purely vertical, and equal in amplitude to the product
of the local circular velocity to the scale height of the disk. However, there
is no analytical expression for the scale height of a Miyamoto-Nagai disk
because the parameter $b_{disk}$ is not directly providing the size of the
disk in the vertical direction. In order to get an estimate of the scale
height $(h/r)^{*}$ of the disk as a function of $a_{disk}$ and $b_{disk}$, we
computed a large number of Miyamoto-Nagai mass distributions with variable
$b_{disk}/a_{disk}$, and measured the scale height of the disk by fitting a 2D
Gaussian on the mass distribution in a vertical plane passing through the
center. The resulting table is presented in Fig. 23. Then, during our
modeling, the $(h/r)^{*}$ value for any $b_{disk}/a_{disk}$ ratio is
interpolated from this table.

Figure 23: Scale height of the Miyamoto-Nagai disk as a function of
$b_{disk}/a_{disk}$ determined by fitting an elongated Gaussian on
$\rho_{disk}(a_{disk},b_{disk})$ density distributions.


11institutetext: University of Perugia, Dept. of Engineering, Perugia,
Italy22institutetext: University of Florence, Dept. of Mathematics and
Computer Science, Florence, Italy33institutetext: University of Perugia, Dept.
of Mathematics and Computer Science, Perugia, Italy

# Skin Cancer Classification using Inception Network and Transfer Learning

Priscilla Benedetti ORCID:0000-0002-3029-0681
Damiano Perri ORCID:0000-0001-6815-6659
Marco Simonetti ORCID:0000-0003-2923-5519
Osvaldo Gervasi ORCID:0000-0003-4327-520X
Gianluca Reali ORCID:0000-0002-8567-5917
Mauro Femminella ORCID:0000-0002-6695-5956 112222331111

###### Abstract

Medical data classification is typically a challenging task due to imbalance
between classes. In this paper, we propose an approach to classify
dermatoscopic images from HAM10000 (Human Against Machine with 10000 training
images) dataset, consisting of seven imbalanced types of skin lesions, with
good precision and low resources requirements. Classification is done by using
a pretrained convolutional neural network. We evaluate the accuracy and
performance of the proposal and illustrate possible extensions.

###### Keywords:

Machine Learning, Convolutional Neural Network, Keras, TensorFlow

##  1 Introduction

Training of neural networks for automated diagnosis of pigmented skin lesions
can be a difficult process due to the small size and lack of diversity of
available datasets of dermatoscopic images. The HAM10000 (“Human Against
Machine with 10000 training images”) dataset is a collection of dermatoscopic
images from different populations acquired and stored by different modalities.
We used the benchmark dataset, with a small number of images and a strong
imbalance among the 7 different types of lesions, to prove the validity of our
approach, which is characterized by good results and light usage of resources.

Exploiting a highly engineered convolutional neural network with transfer
learning, customized data augmentation and a non-adaptive optimization
algorithm, we show the possibility of obtaining a final model able to
precisely recognize multiple categories, although scarcely represented in the
dataset. The whole training process has a limited impact on computational
resources, requiring no more than 20 GB of RAM space. The rest of paper is
structured as follows: Section 2 describes the related work in the field of
medical images processing. Section 3 illustrates the dataset of interest.
Section 4 gives an overview of the model architecture. Section 5 includes the
training process and shows experimental results. Finally, some final comments
and future research directions are reported in Section 6.

##  2 Related work

Processing of biomedical images has always been a field strongly beaten by CNN
pioneers. The first related papers date back to 1991 [1], with a strong
impulse in the following years in the search for methods for automating the
classification of pathologies and related diagnosis [2, 3].
Nowadays, almost thirty years later, reliability of networks reached a rather
high level, as well as intrinsic complexity. This reliability allowed a wide
diffusion of the approach of subjecting diagnostic images to automatic
classification systems, from evolutionary algorithms [4, 5, 6, 7] to deep
networks [8, 9, 10, 11], being them either convolutive or not. Even in the
medical sector of dermatology, automatic image recognition and classification
was used for decades to detect tumor skin lesions [12, 13].

Recent and promising research has highlighted the possibility that properly
trained machines can exceed the human recognition and classification
capability to recognize skin cancers. The scores obtained are very encouraging
[14] and we are confident that in the near future the recognition capacity of
these forms of pathologies will become almost total.

Today CNNs are used for image feature extraction. Features are used for image
classification[15, 16].

##  3 The Dataset

Dermatoscopy is often used to get better diagnoses of pigmented skin lesions,
either benign or malignant. With dermatoscopic images is also possible to
train artificial neural networks to recognize pigmented skin lesions
automatically. Nevertheless, training requires the usage of a large number of
samples, although the number of high quality images with reliable labels is
either limited or restricted to only a few classes of diseases, often
unbalanced.

Due to these limitations, some previous research activities focused on
melanocytic lesions (in order to differentiate between a benign and malignant
sample) and disregarded non-melanocytic pigmented lesions, even if very
common. In order to boost research on automated diagnosis of dermatoscopic
images, HAM10000 has been providing the participant of the ISIC 2018
classification challenge, hosted by the annual MICCAI conference in Granada,
Spain [17], specific images.

The set of 10015 8-bit RGB color images were collected in 20 years from
populations from two different sites, specifically the Department of
Dermatology of the Medical University of Vienna, and the skin cancer practice
of Cliff Rosendahl in Queensland. Relevant cases include a representative
collection of all important diagnostic categories of pigmented lesions[17]:

  * •

akiec: Actinic Keratoses and Intraepithelial Carcinoma, common noninvasive
variants of squamous cell carcinoma that can be treated locally without
surgery. [327 images]

  * •

bcc: Basal cell carcinoma, a cancer that rarely metastasizes but grows
destructively if untreated. [514 images]

  * •

bkl: Generic label that includes seborrheic keratoses, solar lentigo and
lichen-planus like keratoses (LPLK), which corresponds to a seborrheic
keratosis or a solar lentigo with inflammation and regression, often mistaken
for melanoma. [1099 images]

  * •

df: Dermatofibroma, a benign skin lesion. [115 images]

  * •

nv: Melanocytic nevi are benign neoplasms of melanocytes. [6705 images]

  * •

mel: Melanoma, if diagnosed in an early stage, it can be cured by simple
surgical excision. [1113 images]

  * •

vasc: Vascular skin lesions in the dataset range from cherry angiomas to
angiokeratomas and pyogenic granulomas. [142 images]

More than 50% of lesions are confirmed through histopathology (histo), the
ground truth for the rest of the cases is either follow-up examination
(followup), expert consensus (consensus), or confirmation by in-vivo confocal
microscopy (confocal). Other features in the individual dataset include age,
gender and body-site of lesion (localization)[17].

##  4 Model Architecture

Since their first appearance in Le Cun et al. publication [18], Convolutional
Neural Networks (CNN) have been widely applied to data that have a known,
grid-like structure. Possible set of interests are time-series data, which can
be modeled as a 1D grid taking samples at regular time intervals, and image
data, which can be thought of as a 2D grid of pixels. The foundational layer
of a convolutional network consists of three stages. In the first stage, the
layer performs several parallel convolutions to produce a set of linear
activations. In the second stage, each convolution output is run through a
nonlinear activation function, such as the rectified linear activation
function (ReLU). In the third stage, a pooling function is used to modify the
output of the layer further [19]. The max pooling operation, used in this
work, reports the maximum output within a rectangular neighborhood. Since the
objective images of skin lesions present great variation in size, we decided
to use a network made by inception modules, which make use of filters of
different size operating at the same level. This usage of wide modules with
multiple cheap convolutional operations entails a reduced computational
complexity with respect to a deep network with large convolutional layers. In
the specific model we used, based on Inception-ResNet-v2, another point of
speed improvement is the introduction of residual connections, which replace
pooling operations within the main inception modules. However, the previously
mentioned max pooling operations are still present in the reduction blocks.
The structure of the network used in this work is shown on Fig.1

Figure 1: General schema for the used network, an Inception-ResNet-v2
architecture with the addition of a flattening layer, 2 fully-connected layers
and a final softmax activation.

The original Inception-ResNet-v2 architecture [20] has a stem block consisting
of the concatenation of multiple convolutional and pooling layers, while
Inception-ResNet blocks (A, B and C) contain a set of convolutional filters
with an average pooling layer. As prevously mentioned, reduction blocks (A, B)
replace the average pooling operation with a max pooling one. This structure
has been extended with a final module consisting of a flattening step, two
fully-connected layers of 64 units each, and the softmax classifier. The
overall module is trainable on a single GPU with reduced memory consumption.

##  5 Training process and experimental results

This work consists of two training rounds, after a step of data processing in
order to deal with the strong imbalance of the dataset:

  * •

A first classification training process using class weights.

  * •

Rollback of previous obtained best model to improve classification performance
with a second training phase.

###  5.1 Data Processing

In the first stage of data processing, after the creation of a new column with
a more readable definition of labels, each class was translated into a
numerical code using pandas.Categorical.codes. Afterwards, missing values in
”age” column was filled with column mean value. Fig.2 and Fig.3 show the
HAM10000 data distribution.

Figure 2: Plotting the frequency of each class, the imbalance between
Melanocytic Nevi and the rest of the possible categories is manifest. Figure
3: Plot of samples’ body locations.

Finally, images are loaded and resized from $450\times$600 to $299\times$299
in order to be correctly processed by the network. After a normalization step
on RGB arrays, we split the dataset into a training and validation set with
80:20 ratio.

Figure 4: Some HAM10000 images resized with OpenCV.

In order to re-balance the dataset, we chose to shrink the amount of images
for each class to an equal maximum dimension of 450 samples. This significant
decrease of available images is then mitigated by applying a step of data
augmentation. Training set expansion is made by altering images with small
transformations to reproduce some variations, such as horizontal flips,
vertical flips, translations, rotations and shearing transformations.

###  5.2 Baseline

Due to the limited number of samples for the training process, we decided to
take advantage of transfer learning, utilizing Inception-ResNet-v2 pre-trained
on ImageNet[21] and Tensorflow, a deep learning framework developed by Google,
for fine-tuning of the last 40 layers. Keras library offers a wide range of
optimizers: Adaptive optimization methods such as AdaGrad, RMSProp, and Adam
are widely used for deep neural networks training due to their fast
convergence times. However, as described in [22], when the number of
parameters exceeds the number of data points these optimizers often determine
a worse generalization capability compared with non-adaptive methods. In this
work we used a stochastic gradient descent optimizer (SGD), with learning rate
set to 0.0006 and usage of momentum and Nesterov Accelerated Gradient in order
to adapt updates to the slope of the loss function (categorical crossentropy)
and speed up the training process. The total number of epochs was set to 100,
using a small batch size of 10. A set of class weight was introduced in the
training process to get more emphasis on minority class recognition. A maximum
patience of 15 epochs was set to the early stopping callback in order to
mitigate the overfitting visible in Fig.5, which shows the history of training
and validation process.

Figure 5: Accuracy and Loss for each epoch.

Finally, the model achieves an accuracy of 73.4% on the validation set, using
weights from the best epoch. Fig.6 shows the confusion matrix for the model on
the validation set.

Figure 6: Two of the minority classes, Actinic Keratoses (akiec) and
Dermatofibroma (df), are not properly recognized. Melanoma (mel) is often
mistaken for generic keratoses (bkl), as already mentioned in 3

###  5.3 Resuming training from the best epoch:

In order to improve classification performance, specially on minority classes,
we loaded the best model obtained in the first round to extend the training
phase and explore other potential local minimum points of the loss function,
by using an additional amount of 20 epochs. This second step led to an
enhancement in overall predictions, reaching the maximum accuracy value of
78.9%.

Figure 7: Final confusion matrix. Figure 8: Normalized final confusion matrix.

Fig.7 shows the normalized confusion matrix on the validation set for the
final fine-tuned model. In this case, 6 out of 7 categories are classified
with a total ratio of True Positives higher than 75%, even in presence of
extremely limited sample set, as vascular lesions (vasc), 30 samples, and
dermatofibroma (df), 16 samples. The whole process of training has required
less than four hours on Google Colab cloud’s GPU, for an overall RAM
utilization below 20 GB.

##  6 Conclusions and future works

In conclusion, in this paper we investigate the possibility of obtaining
improved performances in the classification of 7 significantly unbalanced
different types of skin diseases, with a small amount of available images.
With use of a fine-tuned deep inception network, data augmentation and class
weights, the model can achieve a good final diagnostic accuracy. The described
training process has a light resource usage, requiring less than 20 GB of RAM
space, and it can be executed in a Google Colab notebook. For future
improvements larger datasets of dermatoscopic images are needed. The model
shown in this paper can be regarded as a starting point to implement a
lightweight diagnostic support system for dermatologists, for example in the
Web as well as through a mobile application.

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