# The physical properties and the evolution of the interacting system AM
1204-292

D. A. Rosa,1 I. Rodrigues,1 A. C. Krabbe, 1 A. C. Milone,2 and S. Carvalho 1
1Instituto de Pesquisa & Desenvolvimento (IP&D), Universidade do Vale do
Paraíba, Av. Shishima Hifumi 2911, 12244-000, São José dos Campos-SP, Brazil.
2Instituto Nacional de Pesquisas Espaciais (INPE), Divisão de Astrofísica, Av.
dos Astronautas 1758, 12227-010, São José dos Campos-SP, Brazil E-mail:
<EMAIL_ADDRESS>
(Accepted 2020 December 07. Received 2020 December 06; in original form 2019
December 09)

###### Abstract

We investigate the interaction effects in the stellar and gas kinematics,
stellar population and ionized gas properties in the interacting galaxy pair
AM 1209-292, composed by NGC 4105 and NGC 4106. The data consist of long-slit
spectra in the range of $3000-7050$ Å. The massive E3 galaxy NGC 4105 presents
a flat stellar velocity profile, while the ionized gas is in strong rotation,
suggesting external origin. Its companion, NGC 4106, shows asymmetries in the
radial velocity field, likely due to the interaction. The dynamics of the
interacting pair was modelled using P-Gadget3 TreePM/SPH code, from which we
show that the system has just passed the first perigalacticum, which triggered
an outbreak of star formation, currently at full maximum. We characterized the
stellar population properties using the stellar population synthesis code
STARLIGHT and, on average, both galaxies are predominantly composed of old
stellar populations. NGC 4105 has a slightly negative age gradient, comparable
to that of the most massive elliptical galaxies, but a steeper metallicity
gradient. The SB0 galaxy NGC 4106 presents smaller radial variations in both
age and metallicity in comparison with intermediate mass early-type galaxies.
These gradients have not been disturbed by the interaction since the star
formation happened very recently and was not extensive in mass. Electron
density estimates for the pair are systematically higher than those obtained
in isolated galaxies. The central O/H abundances were obtained from
photoionization models in combination with emission line ratios, which
resulted in $12+{\rm log(O/H)}=9.03\pm 0.02$ and $12+{\rm log(O/H)}=8.69\pm
0.05$ for NGC 4105 and NGC 4106, respectively.

###### keywords:

galaxies: groups: general, galaxies: stellar content, galaxies: abundances,
galaxies: kinematics and dynamics, galaxies: formation.

††pubyear: 2020††pagerange: The physical properties and the evolution of the
interacting system AM 1204-292–The physical properties and the evolution of
the interacting system AM 1204-292

##  1 Introduction

Interacting pairs of galaxies are excellent laboratories to study different
aspects of the galaxy evolution, such as the morphological transformation, the
induced star formation, and the fueling of a central supermassive black hole
(lighting an active nucleus) (e.g., Fernandes de Mello, 1995; De Mello et al.,
1995, 1996; Longhetti et al., 2000; Focardi et al., 2008; Rogers et al.,
2009). Besides, the study of binary galaxies at different interaction stages
can provide important constraints about the hierarchical scenario of galaxy
formation and evolution. The relatively small velocities of a close encounter
in a gravitationally coupled pair of galaxies of comparable masses/sizes make
the mutual interaction strong enough inside each galaxy. Numerous studies have
reported that a close encounter between the two galaxies modify the mutual
gravitational fields (Toomre & Toomre, 1972). A close encounter can create new
substructures in the gas, dust, and star spatial distributions, such as warps
or bars, tidal tails, plumes, and bridges, pulled out by tidal forces during
the gravitational interaction (e.g., Mendes de Oliveira & Hickson, 1994;
Rodrigues et al., 1999; Barnes, 2002; Renaud, 2010; Presotto et al., 2010;
Drzazga et al., 2011).

Mixed pairs of galaxies have components spanning different morphologies, such
as an elliptical (early-type galaxy) and a spiral (late-type galaxy). This
kind of galaxy pair becomes particularly more interesting because they
represent good sites for investigating in detail those physical processes
induced by close interactions (e.g., Fernandes de Mello, 1995; De Mello et
al., 1995, 1996). The Catalogue of Southern Peculiar Galaxies and Associations
by Arp and Madore (Arp et al., 1987) became a pioneer compilation of galaxy
pairs, in which a lot of mixed pairs can be found.

The dynamics of galaxy interactions can be studied through numerical
simulations, including the stellar and gaseous components besides the dark
matter as primary ingredients. For instance, Moreno et al. (2015) embarked on
a systematic study of 75 simulations of galaxies in major mergers. They report
that star formation is more elevated in the center than the ones on the
outskirts of interacting galaxies, and concluded that these trends are most
prominent in the smaller companion galaxies with strongly aligned disc spin
orientations. Another example is the numerical simulations by Torrey et al.
(2012) that showed that the nuclear metallicity evolution is a perfect
competition between the metal-poor gas inflows and enrichment due to the star
formation (or feedback from star formation and active galactic nuclei (AGN)
activity).

Mutual interaction between galaxies may enhance star formation (e.g., Larson &
Tinsley, 1978; Woods & Geller, 2007; Krabbe et al., 2008; Patton et al., 2013;
Knapen et al., 2015). However, the environment might not play a determinant
role in the formation and evolution of (very) massive early-type galaxies
(e.g., Xu et al., 2010; Bait et al., 2017; Yadav & Chen, 2018).

The star formation enhancements are accompanied by other events such as
perturbations in the radial velocity field and dilution of the metallicity
gradient. This is consistent with results obtained in chemical abundances and
mass-metallicity relation (Sánchez et al., 2013), which show that interaction-
induced flows of low metallicity gas from the outer parts of the disk of a
galaxy can decrease the metallicity in the inner regions and modify the radial
abundance gradients (e.g., Kewley et al., 2010; Rosa et al., 2014; Sánchez et
al., 2014; Muñoz-Elgueta et al., 2018). We can not exclude the possibility
that the inflow of low-metallicity gas would be considered as the principal
source that affects the nuclear metallicity of interacting galaxies. Although
there is evidence for a dilution in the central gas metallicity produced by
interactions, a few interactions may also be able to enhance it (e.g. Barrera-
Ballesteros et al., 2015), observationally seen by redder optical colours
(e.g., Darg et al., 2010; Patton et al., 2011), for instance.

Strong interactions in close pairs may produce an in-homogeneity in the
interstellar medium shown as a wide variation of the electron density (e.g.,
Krabbe et al., 2014; Roche et al., 2015). Formation of bars induced by the
perturbation from the companion is also observed (e.g., Méndez-Abreu et al.,
2012; Alonso et al., 2018).

Active galactic nuclei can also be produced, fueled by the infall of gas from
the galaxy outskirts towards the central region, or even from a late-type
galaxy to its companion (e.g., Eliche-Moral et al., 2011; Ellison et al.,
2011; Satyapal et al., 2014; Donley et al., 2018). The gravitational
interactions, as well as the tidal effects, can provide fuel to a supermassive
black hole in the center of the galaxy (nucleus), leading to a connection
between star formation and black hole activity (Trouille et al., 2011;
Silverman et al., 2011). The activity of the nucleus provides energy to the
interstellar medium and can even smooth the star formation (Croton et al.,
2006; Di Matteo et al., 2008; Henriques et al., 2019). However, Weaver et al.
(2018) who investigated the effects of galaxy merger throughout the
interaction sequence in NGC 7252 (nearest major-merger galaxy remnants),
revealing the extent of ongoing star formation and recent star formation
history. These authors also found a large ionized gas cloud previously
discovered $\sim 5$ kpc south of the nucleus, which may be associated with a
low-ionization nuclear emission-line region (LINER). Therefore, it is of great
significance to study the properties of ionized gas in the circumnuclear
region and AGN connected with interacting galaxies in order to evidence the
relationship between the active galactic nucleus and gravitational
interaction.

This paper presents a detailed study for the interacting system AM 1204-292,
where the physical properties of the ionized gas, the characteristics of the
stellar population, and the nuclear activity are linked with the process of
the interaction between the galaxies and the destiny of this system is
uncovered.

Figure 1: Slit positions for AM 1204-292 system superimposed on the optical
image of the pair in $I$-band from the Carnegie-Irvine Galaxy Survey. The
center of each galaxy is marked in red. The white section of each slit shows
the section of the slits from where useful data was extracted (Ho et al.,
2011; Li et al., 2011). Table 1: Main characteristics of the AM 1204-292
galaxy pair members: identification, morphology, total B-band luminosity,
colour index (B-V), effective radius Re, apparent major axis direction MA, and
inclination angle $i$.

ID | Morphology [1] | log $L_{B}\,~{}^{[6]}$ | (B-V)$\,{}^{[2]}$ | Re [3] | MA [2,4,5] |  i [2,5] | others designations |
---|---|---|---|---|---|---|---|---
AM 1204-292 |  | ($L_{\odot}$) | (mag) | (kpc) | (∘) | (∘) |  |
NGC 4105 | E3 | 10.86 | 0.99 | 10.26 | 136.4 | 54.7 | ESO440-054 |
NGC 4106 | SB(s)0+ | 10.75 | 1.01 | 3.47 | 91 | 50 | ESO440-056 |

References: [1] De Vaucouleurs et al. (1991); [2] Hyperleda (Makarov et al.,
2014); [3] Salo et al. (2015); [4] Capaccioli et al. (2015); [5] Herrera-
Endoqui et al. (2015); [6] Trinchieri & Rampazzo (2001).

Table 2: Galaxy ID, slit position, exposure time, and wavelength range of the spectra on galaxy pair. galaxy ID | PA (∘) | Exposure time (s) |  $\Delta\lambda$ (Å)
---|---|---|---
NGC 4105 | 46.4 | 3$\times$1200 | 3000-7050
NGC 4106 | 161 | 3$\times$1200 | 3000-7050
NGC 4105/06 | 118 | 2$\times$1200 | 3000-7050

AM 1204-292 is pair of galaxies with mixed morphology components, containing
the very luminous elliptical galaxy NGC 4105 (according to the Third Reference
Catalogue of Bright Galaxies De Vaucouleurs et al., 1991, hereafter RC3), but
classified as S0 in the Revised Shapley Ames Catalog of Bright Galaxies
(Sandage & Tammann (1981), hereafter RSA) and NGC 4106, a (diffuse) lenticular
barred galaxy with extended tidal arms produced by the interaction (Koprolin &
Zeilinger, 2000). It was classified in RC3 as SB0(s)+, and as SB0/a in RSA
(tides).

The galaxies are separated by a projected distance of 9.6 Kpc and a difference
in the radial velocities of 274 km s-1. There is visual evidence for one tidal
arm in NGC 4106 that expands towards the primary galaxy in the south
direction, and its counterpart tidal arm/tail heading to the opposite
direction. On the other side, NGC 4105 does not show clear signs of
perturbation. The spectra of both galaxies are dominated by prominent
absorption features (Donzelli & Pastoriza, 2000) and some emission lines, such
as [O II]$\lambda$3727 Å (Longhetti et al., 1998) and [N II]$\lambda$6584 Å
(Caon et al., 2000). Moreover, Longhetti et al. (1998) found evidence from the
line-strength indices that these galaxies experienced recent star formation
episodes. Both galaxies are also fairly red with (B$-$V) color index very
similar to $\sim+0.9$ (Reduzzi & Rampazzo, 1996). The X-ray luminosity of NGC
4105 is L${}_{x}\sim 3.98\times 10^{39}$ erg s-1 (0.5 – 2.0 keV), and NGC 4106
is L${}_{x}\sim 1.26\times 10^{39}$ erg s-1 (Grützbauch et al., 2007).
Besides, Grützbauch et al. (2009) found dusty features in the central region
and modeled the brightness distribution, concluding that it presents a disky
outer structure.

The paper is organized as follows. In Section 2, we describe the observations
and data reduction. The stellar and gas kinematics in each individual galaxy
are presented in Section 3. The N-Body + SPH simulations built to reproduce
the interaction between the two galaxies and to uncover the destiny of this
system are explored in Section 4. We present in Section 5 the procedure
employed to perform the stellar population synthesis and the respective
results to estimate the age distribution of the stellar populations. In
Section 6, we analyze the results about the ionized gas properties, centering
on electron density and oxygen abundance. Finally, we summarize our
conclusions in Section 7.

##  2 Observations and data reduction

In Table 1 some general characteristics of the galaxy pair are listed, such
as: identification, morphology, absolute $B$-band magnitude, effective radius
Re (isophotal radius that contains half of the total flux of the component),
major axis direction (MA), and the inclination angle (i) of each galaxy. The
projected linear separation between the galaxy nuclei on the sky plane is
about 9.6 kpc (adopting $H_{0}=73\,\rm km\>s^{-1}Mpc^{-1}$ (Wright, 2006), and
the average distance of the galaxy pair of 28.6 Mpc).

The current study is based on long slit spectroscopic observations carried out
on 2017 March 04th with the Goodman High Throughput Spectrograph (GTHS)
attached to the 4.10m SOAR telescope (Cerro Pachon, Chile). Spectra with the
single long slit mask were acquired in the range $\lambda\lambda$3000-7050 Å
with a grating of 400 lines mm-1 (M1 mode) and a slit of 1.03 arcsec width,
providing an average spectral resolution of 5.5 Å in the region
$\lambda\lambda$3800-6800 Å, a spectral sampling of 1 Å px-1, and a spatial
scale of 0.15 ″ px-1. The estimate of the spectral resolution was based on the
broadening of He-Ar lines in the wavelength calibration spectra and cross
correlations between the stellar template spectra (representing
$\sigma_{v,inst}$ = 132 km s-1). The atmospheric seeing was about 1 ″, and we
also noted some cirrus clouds along the observations.

The long slit spectra of the galaxy pair AM 1204-292 were taken at three
different position angles (PA) in order to observe the central regions and the
principal-axes. PA$=118^{\circ}$ is crossing the disc of NGC 4105, nearby the
semi-major axis, and NGC 4106, but not in their centers. The minor photometric
axis of the elliptical galaxy NGC 4105 was observed with PA = 46.4∘. The slit
at PA$=161^{\circ}$ crossed the center of NGC 4106 and the southern tidal arm
that extends toward NGC 4105.

Multiple spectra were taken at the same slit position to increase the signal
to noise ratio, and the exposure times were limited to 1200 seconds to
minimize the effects of spurious cosmic rays. The slit positions are shown in
Fig. 1, superimposed on the optical image of the pair. The journal of
observations is listed in Table 2.

The spectroscopic data have been reduced with the  IRAF111Image Reduction and
Analysis Facility, published by National Optical Astronomical Observatory (
NOAO), operated by AURA, Inc, under agreement with NFS. package. The standard
spectroscopic reduction procedure has been applied: image trimming, bias
subtraction, flat field correction, sky background subtraction, cosmic rays
removal, one-dimensional spectrum extraction, and wavelength and flux
calibrations. Each one-dimensional spectrum comprises the flux contained in an
aperture of 1.03 $\times$ 0.9 ″, which at a distance to the par corresponds to
nearly 142 $\times$ 127.5 pc. The galactocentric radial coordinates to the
positions of each aperture extracted from the slits were corrected taking into
account the inclination angles of each galaxy (given in Table 1), in order to
make radial plots of quantities of interest.

Figure 2: Spatial profiles of observed $\lambda\,5735$ flux, in units of
$10^{-17}$ ergs cm-2 s-1 (top), radial velocities after subtraction of the
systemic velocities (middle) and velocity dispersion (bottom) for NGC 4105 and
NGC 4106 along the different position angles, as indicated.The velocity scale
was not corrected by the disk inclination.

##  3 Kinematics

###  3.1 Stellar Kinematics

The stellar radial velocity and line-of-sight velocity dispersion
$\sigma^{*}_{\rm v}$ for each aperture were estimated from the cross
correlation method of its absorption line spectrum against a set of stellar
template spectra. The radial velocity was reduced to a heliocentric frame of
reference $cz_{\rm Helio}$, as usual. The rotation curve and the
$\sigma^{*}_{\rm v}$ radial profile on the slit directions have been estimated
for each galaxy to characterize the dynamical system.

We adopted the Radial Velocity Analysis package of Smithsonian Astrophysical
Observatory ( RVSAO Kurtz & Mink, 1998), from  IRAF package to measure the
radial velocity and velocity dispersion. Spectra of K giant stars (collected
under the same instrumental setup of the galaxy pair observations) were used
as reference for the cross correlations since optical spectra of evolved ETGs
are dominated by them. The observed K giants are HD 120452, HD 132345, and HD
136028. Their heliocentric radial velocities are from Duflot et al. (1995)
catalogue. The best stellar template from the cross correlation of nuclear
aperture spectrum is assumed as a single template for all aperture spectra of
each galaxy. The spectral range of cross correlations is $\lambda\lambda$3800
– 6800Å, and the Fourier filters are those suggested by Tonry & Davis (1979).
A second order polynomial function is employed to fit half height of the cross
correlation peak for estimating its full width at half maximum (FWHM).

The observed radial velocity is simply read from the position of correlation
peak, and its correspondent error is directly given by
(3/8)$\times$(FWHM/(1+$r$)), where $r$ is the height of the correlation peak
divided by the amplitude of a sinusoidal noise for the correlation function
(Kurtz & Mink, 1998). The velocity dispersion of each galaxy aperture spectrum
has been estimated as a function of FWHM through a third order polynomial fit.
To establish the relation between $\sigma_{\rm v}$ and FWHM, we convoluted all
three template spectra took at March 4th night by different Gaussians,
representing $\sigma_{\rm v}$ from 50 up to 500 km s-1 with steps of 50 km
s-1. Cross correlations between the broadened template spectra and the
respective observed stellar spectrum provide the $\sigma_{\rm v}$ and the
FWHM. Errors in velocity dispersions were estimated artificially adding
Poisson noise in the broadened spectra of a stellar template.

The radial velocities after subtraction of the systemic velocities, the
velocity dispersion $\sigma^{*}_{\rm v}$, and the spatial profiles of the
$\lambda$5735 continuum flux across the observed slit positions are shown in
Fig. 2 (no geometric correction was applied due to the inclination of the
galaxy plane relative to the plane of the sky). These galaxies are separated
by a 1.15 arcmin in projection, corresponding to 9.6 kpc and a difference in
radial velocities of $\sim 274$ km s-1. These values are very similar to those
found by Kim et al. (2012).

NGC 4105. No signs of rotation were detected for the stellar component over
both slit directions observed for this elliptical galaxy (see Fig. 2). On the
PA$=46.4^{\circ}$ we can observe a U-shaped symmetry in the radial velocity,
with values reaching $209$ and $238$ km s-1 at $\sim 3.36$ kpc (SW) and $\sim
3.24$ kpc (NE), respectively, and no tendency to decline. In Section 4, we
present the velocity map of a simulation that describes quite well the
dynamics of this pair. We will then come back to this point. For
PA$=118^{\circ}$ the radial velocity and the velocity dispersion are
approximately homogeneous along the radius, with a slight decline towards the
SE side, reaching a maximum value of $52$ km s-1 at $\sim 1.5$ kpc, probably
due to the interaction. We considered the heliocentric velocity of NGC 4105
being a radial velocity measured in the continuum peak along the slit position
at PA$=46.4^{\circ}$, that is, v${}_{r}=1949\pm 18$ km s-1.

Samurović & Danziger (2005) investigated the dark matter in early-type
galaxies from dynamical modelling and found that the velocities are not anti-
symmetric due to the interaction with the companion. These authors also found
that the velocity dispersion decreases from the central values of $320-200$ km
s-1, in agreement with our results.

NGC 4106.  This galaxy shows a nearly symmetrical rotation velocity curve
across both slit directions. For the slit position crossing the nucleus,
PA$=161^{\circ}$, a heliocentric velocity of $2223\pm 15$ km s-1 and a maximum
value of the rotation velocity of $\sim 50\pm 21$ km s-1, was found. One side
of the curve at $\sim 0.7$ kpc towards the southeast of slit position
PA$=161^{\circ}$ presents oscillations in the flat region of the curve of the
order of 30 km s-1, where originates the tidal arm. The most symmetrical,
less-scattered, and smooth curve was derived using the slit position at
PA$=118^{\circ}$. Both sides of the curve display regions with fluctuations of
about 20 km s-1. One $\sim 0.5-1.5$ kpc towards the northwest and another at
$\sim 1-2$ kpc towards the SE. The maximum rotation velocity is $128\pm 22$ km
s-1 at $\sim 1.82$ kpc.

The radially symmetric velocity dispersion profile obtained by us along both
directions (PA$=161^{\circ}$ and PA$=118^{\circ}$) are in good agreement with
the results previously obtained by Longhetti et al. (1998).

Hence, we predict that NGC 4106 galaxy shows a fairly significant rotational
symmetry. For this reason, we adopted a very simple approximation for the
observed velocity distribution. Specifically, assuming circular orbits in a
plane $P(i,\psi_{0})$ that are characterized by its inclination relative to
the plane of the sky ($i$) and the position angle (PA) of the line of nodes
$\psi_{0}$. It results in an observed radial circular velocity at a position
$v$(R,$\psi$) in the plane of the sky given by Bertola et al. (1991):

$v({\rm R},\psi)={\rm V_{s}}+\tfrac{{\rm V_{0}}{\rm
R}cos(\psi-\psi_{0})sin(i)cos(i)}{\sqrt{{\rm
R^{2}}[sin^{2}(\psi-\psi_{0})+cos^{2}(i)cos^{2}(\psi-\psi_{0})]+{\rm
R_{c}^{2}}cos^{2}(i)}},$ (1)

where R is the radius in the plane of the galaxy, Vs is the systemic velocity,
and V0 and Rc are defined as amplitude parameters and shape of the curve.

The inclination of NGC 4106 with respect to the plane of the sky, estimated as
$i\sim 50^{\circ}$, was determined by the observed axial ratio taken from
Herrera-Endoqui et al. (2015). The position angle of the line of nodes,
$\psi_{0}$ = 91∘, was also taken from the same reference. From the fitting of
the radial velocity observed for NGC 4106 (PA$=118^{\circ}$), shown in the
bottom panel of Fig. 2, we obtain a deprojected velocity amplitude of
$V_{0}=138\pm 6$ km s-1. Supposing that the mass in a certain radius is given
by $M(R)=RV_{0}^{2}/G$, using the amplitude of this fit, the dynamical mass of
the S0 galaxy is about $1.6\times 10^{10}\,M_{\odot}$ up to a radius of 3.64
kpc.

Figure 3: Radial velocity and velocity dispersion profiles for the ionized gas
in NGC 4105 and NGC 4106 derived from [O II]3727Å and [N II]6583.5Å. Radial
velocities were obtained after subtraction of the velocity in the central
aperture, defined by the position of the flux peak in the spatial profile. As
in Fig. 2, the velocity scale was not corrected for galaxy inclination.

###  3.2 Gaseous Kinematics

The gaseous radial velocity in NGC 4105 and NGC 4106 was determined using the
task  EMSAO (Mink & Wyatt, 1995) implemented in the  RVSAO package (Kurtz &
Mink, 1998). This program finds emission lines in a spectrum and computes the
observed centre, yielding individual radial velocity and error for each line
and combining them into a single radial velocity. The program was successful
at finding the emission lines [O II]3727Å and [N II]6584Å. The errors are
estimated from the uncertainty in the dispersion function in Angstroms and in
fit to the center of the Gaussian. The velocity dispersion $\sigma^{\rm
gas}_{\rm v}$ and their error were calculated as a function of FWHM, measured
using the  SPLOT task from  IRAF. The FHWM uncertainty is estimated from the
Poisson error. The velocity dispersion was corrected by the instrumental
broadening $\sigma_{\rm v,inst}$ ($132\pm 5\,{\rm km\,s}^{-1}$), where
$\sigma_{\rm line}$ $\sim$ ($c$/2.352) $\times\,{\rm FWHM}_{\rm
line}$/$\lambda_{\rm central}$ and $\sigma_{\rm line}^{2}$ = $\sigma_{\rm
v}^{2}$ \+ $\sigma_{\rm v,inst}^{2}$. We assumed the radial velocity of the
central aperture of each galaxy as the heliocentric velocity.

The radial velocity and velocity dispersion profiles of each observed slit
positions are shown in Fig. 3. The number of kinematic measurements of the
gaseous component is smaller than the stellar one, reaching only the central
kpc of galaxies.

As can be seen in the top panel of Fig. 3, NGC 4105 does not show a U-shaped
radial velocity profile along PA = 46.4∘, like that observed for the stellar
kinematics. Gas velocity dispersions on this slit are lower than that of the
stars, but the profile is similar, with a peak in the nuclear region and
descending wings for both sides. The radial velocity profile across PA = 118∘
shows a clear rotational pattern, very different from the almost flat profile
of the stellar component, considering our error bars (see Fig. 2). It should
be noted that the slit did not cross the center of the galaxy. The behavior of
ionized gas velocity profiles in both directions is very different from that
of the stellar profiles, i.e., ionized gas and stars do not share the same
dynamics.

The gaseous and stellar kinematics in NGC 4105 was previously studied by Caon
et al. (2000), using long-slit spectroscopy and a similar methodology to ours.
They observed with the slit positioned at PA = 156∘, different than our slit
position and also not coincident with the photometric major axis position
angle (see Tab.1, PA = 136.4∘). Caon et al. (2000) also verified that the gas
velocity curve shows a strong rotation pattern with peak velocities above 200
km s-1. Their stellar velocities show a quite symmetric profile with positive
velocities towards the NW direction and negative to the SE ($\leq 40$ km s-1
for both sides), which they interpret as rotation. Based on it, they argue
that the ionized gas in NGC 4105 is counter-rotating with respect to the
stars, is of external origin, is not in equilibrium, and was acquired
recently, also affecting stellar kinematics. In the stellar velocity profile
of our PA$=118^{\circ}$ slit position, we also observed counter-rotation (with
a slightly lower velocity range). But our simulation fails to reproduce that.
We show in Section 4 that the origin of this gas is not the current companion
galaxy, as can be inferred from our simulation, in which no gas is transferred
from NGC 4106 to NGC 4105.

We argue that the stellar velocity profile obtained by Caon et al. (2000) may
reflect the transfer of orbital angular momentum to the outermost stars of NGC
4105 during the close encounter, in contrast to a pre-existing rotation.

In NGC 4106, the radially symmetric velocity profile across the slits reach a
maximum velocity of $206\pm 33$ km s-1 at $\sim-0.65$ kpc over
PA$=118^{\circ}$. The velocity dispersion profiles at both slits are very
steeper than that of the stellar profiles, and nearly symmetrical up to
$\sim$1.2 kpc, with a central value of $187\pm 21$ km s-1 for PA$=161^{\circ}$
and $178\pm 14$ km s-1 in the central aperture of PA$=118^{\circ}$.

The HI gas masses for the galaxies were measured by Bottinelli & Gouguenheim
(1979), who found $M_{\textrm{H}{\scriptsize\textrm{I}}}=5.2\times 10^{8}$ M⊙
for NGC 4105 and $M_{\textrm{H}{\scriptsize I}}=3.4\times 10^{8}$ M⊙ for NGC
4106.

##  4 N-Body + SPH simulations

Numerical simulations were carried out in order to reconstruct the interaction
history between the galaxies NGC 4105 and NGC 4106. The simulations were
performed using the TreePM/SPH code P-GADGET3, an updated and significantly
extended version of GADGET-2 (Springel, 2005), where the dark matter and stars
are followed by collision less particles and gravity is calculated in a tree-
code. The gas is treated through Smoothed Particle Hydrodynamics (SPH,
Springel & Hernquist, 2002), with star formation process, radiative cooling
and feedback, and sub-grid star formation model as defined in Springel &
Hernquist (2003), from which we used the same parameters.

###  4.1 Initial conditions for the simulation

Based on the observational data presented here and elsewhere, we began to
construct the initial conditions for our simulations. In the case of AM
1204-292, the collision is between an elliptical galaxy (NGC 4105) and a S0
galaxy (NGC 4106).

The total stellar mass of both galaxies were derived through the mass-to-light
ratios resulting from stellar population synthesis (see Sect. 5). We correct
the slit stellar mass obtained from  STARLIGHT to total stellar mass by
scaling the total and slit luminosity in the B-band. The total B-band
luminosity was taken from the ESO-LV catalogue and are L${}_{B}\sim
7.24\times\,10^{10}$ L⊙ and L${}_{B}\sim 5.62\times\,10^{10}$ L⊙ for NGC 4105
and NGC 4106, respectively. Then, the stellar masses are
$1.81\times\,10^{11}\,M_{\odot}$ and $8.30\times\,10^{10}\,M_{\odot}$, for NGC
4105 and NGC 4106 respectively.

Table 3: NGC 4105 (E with gas) model parameters. |  Phys. Units∗
---|---
NGC 4105 model (Elliptical)
Number of particles in halo | 700,000
$R_{200}$ | 155 kpc
$M_{200}$ | $86.59\times 10^{10}\,\mathrm{M}_{\odot}$
$V_{200}$ |  155 km s-1
$r_{h}$ | 23.4 kpc
Halo mass $(M_{\rm{dm}})$ | $71.69\times 10^{10}\,\mathrm{M}_{\odot}$
Number of particles in stellar disk | 200,000
Disk mass $(M_{\rm{d,s}})$ | $0.086\times 10^{10}\,\mathrm{M}_{\odot}$
Disk radial scale length $(R_{\rm{d,s}})$ | 3.2 kpc
Disk vertical scale thickness $(z_{\rm{0,s}})$ | 0.63 kpc
Number of particles in gas disk | 200,000
Gas disk mass $(M_{\rm{d,g}})$ | $0.064\times 10^{10}\,\mathrm{M}_{\odot}$
Gas disk radial scale length $(R_{\rm{d,g}})$ | 3.2 kpc
Gas disk vertical scale thickness $(z_{\rm{0,g}})$ | 0.035 kpc
Number of particles in spherical stellar component | 300,000
Spherical component mass $(M)$ | $14.72\times 10^{10}\,\mathrm{M}_{\odot}$
Spherical component radial scale length $(r_{h})$ | 4.77 kpc
Total mass of the model | $86.59\times 10^{10}\,\mathrm{M}_{\odot}$
Total barionic mass of the model | $15.70\times 10^{10}\,\mathrm{M}_{\odot}$

###  4.2 NGC 4105 model

The elliptical galaxy was modeled by two spheroidal Hernquist (1990)
components, one for the stars, other for the dark matter halo:

$\rho(r)=\dfrac{M}{2\pi}\dfrac{r_{h}}{r(r+r_{h})^{3}}$ (2)

where $M$ is the total mass and $a$ is the radial scale length of each
component. Two models have were constructed: (i) The first one has a spherical
stellar component within a concentric spherical dark matter halo to test the
hypothesis that the gas present in this galaxy was stolen from NGC 4106 during
the collision, as will be discussed in section 4.4; (ii) Stellar and gaseous
disks were added to the second model, with exponential density profiles:

$\rho(d)=\dfrac{M_{d}}{4\pi
R_{d}^{2}z_{0}}\exp(-\dfrac{R}{R_{d}})\rm{sech}^{2}\left(\dfrac{z}{z_{0}}\right),$
(3)

where $M_{d}$, $R_{d}$ and $z_{o}$ are the stellar or gaseous disk mass,
radial scale length and vertical scale length, respectively.

We start setting up our galaxy model by the values of the mass. The stellar
mass is presented in the previous section. For the halo model, we took typical
parameters for elliptical galaxies. The first guess for the total mass (dark
plus barionic) was obtained using equation 4.249b from Binney & Tremaine
(2008), based on the scalar Virial theorem. The velocity dispersion is
presented in Section 3.1 (Fig. 2), from the PA $=46.4^{\circ}$, which gave us
a mass of $M\sim 5\times 10^{11}$M⊙. The radial velocity curves are also used
as observational beacons. During the construction of the model galaxy, we
should find a good set of parameters that are committed to observational data,
and still produce a dynamically stable model. The final value of the total
mass is a little higher: $M_{200}\sim 8.7\times 10^{11}$M⊙, as the stellar
mass.

All galaxy models in the present paper were set following the prescription
given by Springel (2005). Spheroidal halo and bulge follow the Hernquist
(1990) profile (eq. 2), while the gas and stellar disks have an exponential
density profile (eq. 3). The parameters of the final model are given in Table
LABEL:tab:model_par_4105.

###  4.3 NGC 4106 model

For NGC 4106, the S0 type galaxy, we started with the stellar mass derived
from the M/L ratio obtained from the stellar population synthesis (see Sect.
4.1, and 5. For each model generated, we compare the rotation curve with the
stellar radial velocity curves obtained from longslit spectroscopy (Fig. 2).
The slit position PA $=161^{\circ}$ crosses the galaxy through its nucleus, in
a direction close to the galaxy minor axis (at PA $=181^{\circ}$), so that no
rotation is detected. The other one, at PA $=118^{\circ}$, is close to the
major axis direction (at PA $=91^{\circ}$), but does not cross the galactic
center. Both radial velocity curves were corrected to estimate the rotation
curve of the galaxy, as explained in Section 3.1. Then we generated a set of
S0-like galaxy models, constructed with a bulge and a halo given by Hernquist
(1990) spheroids (Eq. 2) and an exponential disk (Eq. 3). We plotted the
corresponding rotation curve over the observed one and choose the one that
best reproduces the mean observed rotation curve and is dynamically stable
(i.e., it survives with its mass distribution nearly unaltered in a 1 Gyr
run). It was not a fit procedure. The parameters of the final NGC 4106 model
are also given in Table LABEL:tab:model_par_4106.

Table 4: NGC 4106 (S0) model parameters. |  Phys. Units∗
---|---
NGC 4106 model (S0)
Number of particles in halo | 500,000
$R_{200}$ | 160 kpc
$M_{200}$ | $95.24\times 10^{10}\,\mathrm{M}_{\odot}$
$V_{200}$ |  160 km s-1
$r_{h}$ | 32.35 kpc
Halo mass $(M_{\rm{dm}})$ | $90.94\times 10^{10}\,\mathrm{M}_{\odot}$
Number of particles in stellar disk | 450,000
Disk mass $(M_{\rm{d,s}})$ | $3.8\times 10^{10}\,\mathrm{M}_{\odot}$
Disk radial scale length $(R_{\rm{d,s}})$ | 2.8 kpc
Disk vertical scale thickness $(z_{\rm{0,s}})$ | 0.56 kpc
Number of particles in gas disk | 200,000
Gas disk mass $(M_{\rm{d,g}})$ | $0.064\times 10^{10}\,\mathrm{M}_{\odot}$
Gas disk radial scale length $(R_{\rm{d,g}})$ | 2.8 kpc
Gas disk vertical scale thickness $(z_{\rm{0,g}})$ | 0.56 kpc
Number of particles in bulge | 50,000
Bulge mass $(M)$ | $0.48\times 10^{10}\,\mathrm{M}_{\odot}$
Bulge radial scale length $(r_{h})$ | 0.56 kpc
Total mass of the model | $95.52\times 10^{10}\,\mathrm{M}_{\odot}$
Total barionic mass of the model | $4.35\times 10^{10}\,\mathrm{M}_{\odot}$

Each galaxy model was evolved in isolation for 3 Gyr to test stability and to
allow its numerical relaxation.

Table 5: Orbital parameters from the best model.

|  Phys. Units∗
---|---
NGC 4106 model (S0)
Eccentricity | 0.9
Pericenter distance | 5.7 kpc
Present distance | 17.7 kpc
$V_{sys}$ |  -329 km s-1
Pericenter vector direction∗ | (0.004, 0.961, -0.275)
Orbital plane normal vector direction∗∗ | (0.356, -0.871, -0.338)
Present day position vector∗ | (-8.8, -4.2, -14.8) kpc
Present day velocity vector∗ |  (-792, -91, -329) km s-1
Initial galaxy separation | 1.0 Mpc

Notes:

∗Vector components defined in a Cartesian reference frame (X, Y, Z) centered
on NGC 4105, with X to the North, Y to the West, and Z in the line of sight
direction, with positive values towards the observer.

∗∗The orbital plane normal vector is the direction of the orbital angular
momentum (reference frame defined as in ∗).

Figure 4: Selected views of the snapshot that best reproduces the observed
kinematics and morphology of AM 1204-292, 14.2 Myr after perigalacticum. Top
left panel: Only stellar particles are shown. Top wright: Gas density
distribution. Bottom left panel: stars formed during the simulation are drawn
in green over the gas distribution. Bottom right panel: The gas distribution
vied from a different view point to show that the galaxies are completely
detached from each other, with no connecting bridges. No gas is transferred
between galaxies. The square grid at the center of each panel is 20 kpc wide.

###  4.4 The orbit

In order to find the orbit followed by both galaxies, we wrote a code that
takes as input parameters the distance to the galaxy pair, the positions of
the galaxies projected in the sky plane, the radial velocities of their mass
centers. Then the algorithm calculates orbits that may lead to the present
state of the system. Many thousands of orbits are presented. Then we identify
families of orbits and select those that, by morphological and kinematic
constraints, are suitable to take the system to its present state. Then, a
large series of low resolution test simulations were run, out of which, 6 best
cases were chosen and ran in higher resolution. Orbital parameters are
presented in Table LABEL:tab:model_par.

The simulation starts 1 Gyr before perigalacticum, with the galaxies at an
initial distance of 1 Mpc. In this initial configuration, dark matter halos
are far enough from each other, so the dynamical damage caused by the sudden
presence of the other galaxy model is negligible. Galaxy models were evolved
in isolation for 1 Gyr, before being stacked together for the collision run.

A whole set of simulations was performed using a simple elliptical galaxy
model for NGC 4105 (spherical model with dark matter and stars), with the
gaseous content of the pair entirely placed in the NGC 4106 model. Remember
that both galaxies have a similar amount of H I gas, as stated at the end of
Section 3.2. Despite being very successful in several aspects (reproduction of
morphology, kinematics, consistent evolution of the star formation rate, etc
…), the model failed in one respect: there was no gas transfer from NGC 4106
to NGC 4105. From this result, we can be sure that, as already suggested by
Caon et al. (2000), the gas present in NGC 4105 is pre-existing, and must have
been captured in a previous event.

Figure 5:  Stellar radial velocity map of the snapshot that best reproduces
the observed kinematics and morphology of AM 1204-292, 14.2 Myr after
perigalacticum. Only disk components of both galaxies are shown. Red crosses
mark the center of each model galaxy.

Then, the second set of simulations was run with stellar and gaseous disks
placed added to the NGC 4105 model. It makes sense in view of the results by
Grützbauch et al. (2009), who found dusty features in the central region and
modeled the brightness distribution, concluding that it presents a disky outer
structure. The subtraction of their model showed a central bar-like structure.
From the best fit simulation out of this set, we present in Fig. 4 different
renderings of the snapshots that reproduces the current situation of the
system, 14.2 Myr after perigalacticum. The main morphological features, such
as the general shape and the tidal arms, are well reproduced by the
simulation. In the last panel of Fig. 4 we show the gas disks from a viewpoint
different than our line of sight. The disks are completely disconnected, with
no bridges, connecting tails or stream between them, showing that no gas is
transferred from one galaxy to the other.

The velocity distribution, measured by long slit spectroscopy, can also be
compared directly to the velocity map extracted from the simulation at the
current system state. It is presented in Fig. 5, where only the stellar disks
are shown.

Fig. 6 shows the time evolution around perigalacticum (at 1.311 Gyr after the
beginning of the simulation) of the star formation rate (SFR) during the
simulation. Note that just after perigalacticum, a sudden peak in SFR occurs,
as expected. We plotted in orange the SFR evolution of the isolated models to
show that the peak is due to the collision. At the moment, the system is
undergoing an outbreak of star formation. In Section 5 we present a stellar
population synthesis, from which we detect, in some apertures, and for both
galaxies, the presence of a few percent of the young stellar population. The
star formation rate peaks near 60 Myr after perigalacticum, with an SFR of
0.12 M${}_{\odot}\,Myr^{-1}$.

Figure 6:  Star formation rate evolution. Time is given in Gyr from the
beginning of the simulation. The blue line shows the SFR in the encounter
simulation. In orange we plot the sum of the SFR of the S0 and E galaxies when
evolved isolated. Perigalacticum is at 1.311 Gyr (vertical solid red line),
where a sudden star formation episode begins. The dotted vertical line at
1.325 Gyr marks the present state of the system, indicating that the SFR is
rising due to the collision and will peak about 60 Myr after perigalacticum.

Regarding the radial velocity distribution (Fig. 5), it also corresponds
nicely with the observed velocities over the slit positions. If we take the
velocity profile along PA$=46.4^{\circ}$ from the velocity map, we will have a
beautiful U-shaped profile, unlike what happens along the line that connects
the nuclei of the galaxies.

The velocity profile along the PA=$118^{\circ}$ is presented in Fig. 7. The
overall shape of the velocity profile taken from the simulation is in good
agreement with the observed one. At the left end of the graph, which
corresponds to the SE side of the slit, the velocity in the simulation keeps
rising to values above 300 km s-1, while the observed profile becomes flat at
${\sim}250$ km s-1. This is due to the details on that part of the simulated
galaxy that develops a prominent bar, stronger than that on NGC 4106. On the
other side, the velocity profile in the NGC 4105 section is quite strange,
with a low velocity tail in the direction of its companion galaxy. It is clear
that a simulation can hardly reproduce the observations in a very high degree
of detail. Both simulations and observations have limiting factors that affect
the data. The objective here is to study the dynamics in general lines.

Figure 7:  Velocity profile along the PA =$118^{\circ}$. The observed data
points are plotted in blue. In orange is a cut on the velocity map of the
simulation (Fig. 5), over the same direction. The center of each galaxy is
indicated by the dotted vertical red lines.

##  5 Stellar populations synthesis

To compute the stellar population contribution on the galaxy pair, we utilized
the  STARLIGHT stellar population synthesis code (Cid Fernandes et al., 2005).
Full details on the code and methodology are discussed in Cid Fernandes et al.
(2004, 2005); Mateus et al. (2006); Cid Fernandes et al. (2007); Asari et al.
(2007). Briefly, the code fits the observed spectrum (Oλ) in terms of the
overlap of simple stellar population synthesis models (SSPs) with a wide range
of ages and metallicities. In this code, we adopt the empirical stellar
spectral library by Bruzual & Charlot (2003). The library includes spectra
with a spectral resolution of 3 Å (FWHM) over a wavelength range
$\lambda\lambda 3200\,-\,9500$ Å under a sampling of $1$ Å. We also adopted
the Padova 1994 tracks, and an initial mass function (IMF) by Chabrier (2003)
for stars of masses between 0.1 and 100 M⊙.

The basic equation of the synthetic spectrum solved by the  STARLIGHT is given
by:

$\centering
M_{\lambda}/M_{\lambda_{0}}=\left[\sum_{j=1}^{N_{\star}}\vec{x}_{j}b_{j},_{\lambda}r_{\lambda}\right]\otimes
G(\nu_{\star},\sigma_{\star}),\@add@centering$ (4)

where M${}_{\lambda_{0}}$ is the synthetic flux at that rest-frame wavelength;
$\vec{x}_{j}$ is the population vector; $b_{j}$,λ is the reddened spectrum of
the $j^{\rm th}$ SSP model that is flux normalized at $\lambda_{0}=5750$ Å;
$r_{\lambda}$ $\equiv$ 10${}^{-0.4(A_{\lambda}\,-\,A_{\lambda_{0}})}$ is the
extinction term; $\otimes$ represents the convolution operator;
G$(\nu_{\star},\sigma_{\star})$ denotes a Gaussian distribution along the line
of sight centered at velocity $\nu_{\star}$ and with a star velocity
dispersion $\sigma_{\star}$; The population vector $\vec{x}$ denote the
percentage contribution of the SSPs at $\lambda_{0}$ weighted by flux, being
expressed in terms of age and metallicity ($t_{j}$, $Z_{j}$). The population
vector can also be expressed as a function of the SSP mass fractional
contribution and is designed by the vector $\vec{m}$.

The goodness-of-fit between observed and synthetic spectra is derived from an
algorithm that finds the results for the minimum value of ($\chi^{2}$). The
intrinsic reddening is modeled by the code as due to the light scattering by
dust, adopting the extinction law of Cardelli et al. (1989). The SSPs
considered in this work take into account 15 ages ($t=0.001$, $0.003$,
$0.005$, $0.01$, $0.025$, $0.04$, $0.1$, $0.3$, $0.6$, $0.9$, $1.4$, $2.5$,
$5$, $11$ and $13$ Gyr), three metallicities ($Z$ = $0.2$, $1$, and
$2.5Z_{\odot}$), summing it up N${}_{\star}=45$ SSP components.

We combined the individual components in age bins following the prescription
of Cid Fernandes et al. (2005) as young stellar population, $x_{y}$ or $m_{y}$
($t\,<\,1\times 10^{8}$ yr); intermediate stellar population, $x_{i}$ or
$m_{i}$ (1$\times 10^{8}\leq\,t\,\leq 1\times 10^{9}$ yr), and old stellar
population $x_{o}$ or $m_{o}$ ($t\,>1\times 10^{9}$ yr). These components are
employed to construct the SSPs with their flux ($\vec{x}$) and mass
($\vec{m}$) fractional contributions.

Another important point is that the  STARLIGHT code usually not provide the
uncertainties of the resulting parameters. However, we estimated the
uncertainties of the average parameters as a function of the spectrum quality
indicated by signal to noise ratio (SNR) variations. For this, we applied 20
levels of random flux perturbations over the central spectrum of each slit
position observed, and calculated the percentage differences between the new
estimates and standard ones. These percentage differences were scaled to the
other apertures of the slit according to the SNR of the spectrum considered.
We considered only spectra with a good quality signal-to-noise of the
continuum (at least 10).

Figure 8: Stellar population synthesis for different regions of NGC 4105 (top)
and NGC 4106 (bottom). The observed spectrum (black), the synthesized spectrum
(red), and the residual spectrum, with its pure emission line structure
(blue), are shown. The main emission and absorption lines are identified, and
a zoom-in of the region between 6500 to 6800 Å is shown.

Figure 9: Flux and mass fractions (black and red bars, respectively) of SSP
contributions from the best solution of  STARLIGHT population synthesis.

In Fig. 8 we present some examples of the results of the spectral synthesis
fitting for the central and outskirt regions of NGC 4105 and NGC 4106. The
results of stellar population synthesis for the individual spatial bins across
the slit positions are shown in Fig. 9, stated as the fractional contribution
of each base element weighted by flux and mass. It can be noted that the
spatial distributions of stellar populations are homogeneous across all slit
positions. NGC 4106 is a kind of lenticular galaxy, so we would expect to find
a light-weighted stellar age a few billion years younger than in the
elliptical galaxy NGC 4105. From the simulation, the maximum fraction of young
stellar population ($\leq 30$ Myr) in NGC 4105 would be 3.8%. What we actually
found for both galaxies was that the stellar population in both galaxies is
predominantly composed by the old component in light, as well as in mass, and
a very small fraction (up to 5 percent) of young and intermediate population,
very close to the detection threshold of the stellar population synthesis
method.

Figure 10: The mean stellar age and metallicity (both in logarithmic scales)
as weighted by the flux contribution of SSP models addressed by the stellar
population synthesis as a function of the linear nuclear distance R (in kpc)
and its logarithm, for NGC 4105 and NGC 4106.

We show, in Fig. 10, the light-weighted means of stellar age and metallicity
as a function of the linear distance towards the nucleus in each observed
galaxy (expressed in kpc). The age (in Gyr) is presented in the logarithm
scale and the metallicity Z in the logarithm scale normalized to the solar
value, i.e. log ($Z/Z_{\odot}$) = [Z], for which $Z_{\odot}\,=\,0.02$ to be
consistent with the adopted SSPs grid. The stellar population synthesis has
extracted light-weighted stellar age and metallicity on two distinct projected
directions of each galaxy, across which we have also measured the stellar
kinematics (see Subsection 3.1). The stellar population properties have been
derived from a set of aperture spectra along radial distances up to almost 0.5
Re in the case of NGC 4105, and up to around 1 Re for NGC 4106. NGC 4105 is
morphologically classified as a E3 galaxy and NGC 4106 as a SB(s)0+ galaxy,
i.e. a lenticular barred galaxy. Specifically for NGC 4106, our kinematics
observations have shown that the slit direction at PA = 118o is more aligned
to a stellar rotating disc than one at PA = 161o, on which the maximum line-
of-sight rotational velocity is $128\pm 22$ km s-1 and $50\pm 21$ km s-1
respectively (see Fig. 2). The other galaxy, the elliptical NGC 4105, does not
exhibit any stellar rotating disc. Since the radial variations of both stellar
properties are indistinguishable across both directions in each observed
galaxy, we have aggregated them all for each object into a single dataset in
order to estimate representative radial gradients of light-weighted stellar
age and metallicity. The stellar properties derived for different apertures at
each side of the radial profile across two distinct projected directions were
averaged in the case they correspond to the same nuclear distance within 1
arcsec, otherwise, the apertures are individually considered. The radial
distances of all apertures towards the galaxy nucleus were corrected by the
inclination angle of the reference plane of each correspondent galaxy.

In the case of NGC 4105 (a massive E3 galaxy), we have measured, up to a
nuclear distance of about 4.6 kpc (i.e., up to around 0.45 Re), a slightly
negative gradient in age ($\Delta{\log}(\rm age\,(\rm Gyr))$/$\Delta{{\rm
R(kpc)}}$ = $-0.018\pm 0.002$ dex kpc-1 and $\Delta{\log}(\rm age\,(\rm
Gyr))$/$\Delta{\log}({\rm R(kpc)})$ = $-0.051\pm 0.012$) and a very sharp
negative metallicity gradient ($\Delta{\log}(\rm Z/Z_{\odot})$/$\Delta{{\rm
R(kpc)}}$ = $-0.070\pm 0.006$ dex kpc-1 and $\Delta{\log}(\rm
Z/Z_{\odot})$/$\Delta{\log}({\rm R(kpc)})$ = $-0.210\pm 0.023$). Whilst the
variation in stellar age is very small (i.e., from nearly 13 Gyr in the
nucleus down to around 11 Gyr at about 0.5 Re), the metallicity deeply
decreases from about [Z] = +0.1 dex in the nucleus down to nearly [Z] = -0.2
dex at about 0.5 Re.

In the case of NGC 4106 (an intermediate mass lenticular galaxy), we have
measured nearly up to 3.4 kpc along the nuclear distance (i.e., up to about 1
Re) a null gradient in age ($\Delta{\log}(\rm age\,(\rm Gyr))$/$\Delta{{\rm
R(kpc)}}$ = $-0.010\pm 0.005$ dex kpc-1, $\Delta{\log}(\rm age\,(\rm
Gyr))$/$\Delta{\log}({\rm R(kpc)})$ = $-0.028\pm 0.015$) and a negative
metallicity gradient ($\Delta{\log}(\rm Z/Z_{\odot})$/$\Delta{{\rm R(kpc)}}$ =
$-0.045\pm 0.008$ dex kpc-1 and $\Delta{\log}(\rm
Z/Z_{\odot})$/$\Delta{\log}({\rm R(kpc)})$ = $-0.114\pm 0.027$). Whilst the
stellar age is constant around 12.6 Gyr inside 1 Re ($\pm$0.6 Gyr), the
metallicity decreases from around the solar value in the nucleus (+0.03 dex
indeed) down to nearly [Z] = -0.1 dex at about 1 Re (-0.12 dex indeed) that
closely corresponds to one third of the whole metallicity variation on a two
times greater normalized radial distance in comparison against NGC 4105.

Negative gradients in stellar age and metallicity with different intensities
are observed in massive early-type galaxies (especially ellipticals) (Sánchez
et al., 2012; González Delgado et al., 2015; Zibetti et al., 2020), such that
older metal-richer stars are found within the innermost regions of the galaxy
and the relatively younger and metal-poorer stars in the outskirt ones, as a
result of a long two-phase process (Oser et al., 2010). The initial phase
occurs at high redshift and builds up the main body of the elliptical with
stellar populations passively evolving afterwards. In the second phase lasting
billions of years, minor/major dry mergers make the main body progressively
bigger and redder. An extended relatively younger stellar envelope with
relatively smaller metallicity than the nuclear region is also formed (Obreja
et al., 2013; Huang et al., 2016). Very recently, Zibetti et al. (2020)
investigated and compiled the spatial distribution of the light-weighted mean
stellar age and metallicity inside extended galactic regions (i.e., up to 2
Re) in a wide variety of nearby ETGs (48 E’s and 21 S0’s extracted from the
CALIFA integral field spectroscopic survey). They found that the age profiles
typically follow a U-shaped variation in the plane log(age(Gyr))-Re with a
minimum around 0.4 Re, asymptotically increasing outwards beyond 1.5 Re and
increasing towards the nucleus. The greater the $\sigma_{\rm v}$ within 1 Re
($\sigma_{\rm e}$) is, the smaller the depth of the minimum and the central
increment are, i.e., flatter the age gradient is. The metallicity gradients
are found universally negative and strong such that the metallicity flattens
out moving towards larger radial distances (about -0.3 dex per 1 Re within 1
Re). They analysed the metallicity profiles over the plane ${\log}(\rm
Z/Z_{\odot})$-${\log}({\rm R_{e}})$. They state that a possible qualitative
interpretation for their observations is a two-phase scenario for the
formation of ETGs.

In comparison with the age and metallicity radial gradients investigated by
Zibetti et al. (2020) in the most massive elliptical galaxies (i.e.,
$\sigma_{\rm e}$ $\geq$ 210 km s-1), NGC 4105 exhibits, within the central
region up to 0.5 Re, a comparable change in age and a greater variation in
metallicity. In the case of NGC 4106, this SB0 galaxy presents, within the
central region up to 1 Re, smaller variations in both age and metallicity in
comparison against ETGs with intermediate mass (i.e., 170 $\leq$ $\sigma_{e}$
$<$ 210 km s-1). According to E’s against S0’s, Zibetti et al. (2020) found
that they all have U-shaped radial profiles in age, which are dislocated to
smaller ages for S0’s by around 1.6 Gyr, and comparable metallicity gradients
within 0.5 Re that become distinct between 0.5 and 1 Re (steeper in E’s).

Since the nearest approximation between the galaxies seems to have happened
only 30 Myr ago, we suppose the gradients of age and metallicity in both
galaxies have not been yet substantially modified by the interaction in
course. Both galaxies might have preserved their intrinsic stellar population
spatial distributions according to their particular mass assembly histories.
In fact, Kobayashi (2004) stated that the merging histories of elliptical
galaxies could, in principle, be extracted from the observed metallicity
gradients in the local Universe galaxies. Kobayashi (2004) obtained that there
is no correlation between metallicity gradient and galaxy mass based on a
variety of chemodynamical simulations of elliptical galaxies (GRAPE-SPH code).
An average gradient in metallicity that Kobayashi (2004) compiled was
$\Delta{\log}(\rm Z)$/$\Delta{\log}({\rm R/R_{e}})$ = -0.3 ($\pm$0.2 as
dispersion). Kobayashi (2004) also concluded that E galaxies that
hypothetically would have been formed through a monolithic process would have
steeper gradients, while E galaxies that undergo major mergers would have
shallower gradients.

The stellar population synthesis has provided small flux contributions of
young and intermediate age populations (respectively, with age $<100$ Myr and
100 Myr $\leq$ age $\leq 1$ Gyr) in both galaxies, i.e., up to about 6 per
cent, which is very close to the detection limit of the stellar population
synthesis applied through the  STARLIGHT code. Therefore, these very small
contributions did not represent any significant perturbation on the age and
metallicity radial gradients. However, Grützbauch et al. (2007) argue that NGC
4106 shows a value for the H+K (Ca II) line-strength index larger than 1.3,
i.e., larger than the maximum value attainable in post star-burst models (with
both solar and supersolar metallicity), which suggests the presence of
H$\epsilon$ in emission, considered a good indicator of recent star formation.
The detection of [O II]$\lambda$3727, 3729 emission in the nucleus of NGC 4105
is also suggestive of a recent star formation episode.

On the one hand, in strongly interacting galaxies, in which there was some
reservoir of cold gas to form new stars during a long period of time, the
metallicity gradient becomes significantly flatter than the ones observed in
isolated galaxies, as shown for instance by Krabbe et al. (2008); Kewley et
al. (2010); Krabbe et al. (2011); Bresolin et al. (2009); Rosa et al. (2014).
The shallow metallicity gradients found in strongly interacting galaxies could
be explained as the interaction-induced gas flows from the outer parts to the
centre of each component (Toomre & Toomre, 1972; Dalcanton, 2007). On the
other hand, the observed galaxy pair under a strong interaction too, taking
into account that the galaxies likely had a single encounter in a few tens of
million years ago and there was no great available amount of cold gas, none
significant star formation was actually induced inside both components by the
mutual interaction.

Figure 11:  [S II]$\lambda\lambda$6716/6731Å line ratio and electron density
($N_{\rm e}$) as a function of linear radial distance in kpc.

##  6 Ionized gas properties

The presence of emission lines are not unique to star forming galaxies, but
may also appear in early-type galaxies (e.g., Macchetto et al., 1996; Caon et
al., 2000; Rosa et al., 2018; Diniz et al., 2018). Several studies have shown
that line emission is prevalent in more than 50% of passive red galaxies
(Phillips et al., 1986; Goudfrooij et al., 1994; Yan et al., 2006; Capetti &
Baldi, 2011), and many of the early-type emission line galaxies show
characteristic spectra of a LINER (Heckman, 1980). As can be seen in Fig. 8,
after the subtraction of the modelled stellar spectrum from the observed one,
the pure nebular spectra in both galaxies present pronounced emission lines,
that are: [O II]$\lambda$3727, H$\beta$, [O II]$\lambda\lambda$4959,5007, [N
II]$\lambda\lambda$6548,6584, H$\alpha$, and [S II]$\lambda\lambda$6716,6731.
In fact, previous studies have already detected some emission lines the
elliptical galaxy NGC 4105. For example, for NGC 4105 Longhetti et al. (1998)
detected [O II]$\lambda$3727, and Caon et al. (2000) measured [N
II]$\lambda$6584.

The line intensities were determined from fitting Gaussian profiles on the
pure emission spectra, using the  SPLOT task in the NOAO.ONEDSPEC package of
IRAF. The associated error with each line flux was given as
$\sigma^{2}=\sigma_{\rm cont}^{2}+\sigma_{\rm line}^{2}$, where the
$\sigma_{\rm cont}$ and $\sigma_{\rm line}$ are the continuum $rms$ and
Poisson error on the line flux, respectively. We corrected the observed line
intensities for the effect of interstellar extinction. This was performed
comparing the observed H$\alpha$/H$\beta$ ratio with the theoretical value of
$2.86$ from Pengelly (1964), for an electron temperature of 10 000 K. The
starburst extinction law of Calzetti et al. (1994) was adopted.

###  6.1 Electron density

We have determined the electron density ($N_{\rm e}$) from the observed [S
II]$\lambda 6716/\lambda 6731$Å ratio, using  TEMDEN routine of the  NEBULAR
package of the  STSDAS/IRAF, assuming an electron temperature of $10^{4}$ K,
since the lines sensitive to temperature were not observed in both galaxies.
It is important to note that some regions in both galaxies present [S II]
ratio near the low-density limit, and therefore, the electron densities for
these areas cannot be estimated.

The radial profiles of [S II] line ratio and $N_{\rm e}$ are plotted in Fig.
11 as a function of the linear radial distance for NGC 4105 and NGC 4106. The
values of electron density obtained for both galaxies are in the range of
$N_{\rm e}=33-445\,\rm cm^{-3}$ . These estimations are in agreement with
those obtained by Krabbe et al. (2014) and Mora et al. (2019) for H II regions
of interacting galaxies, which are systematically higher than those derived
for isolated galaxies of $N_{\rm e}=40-137\,\rm cm^{-3}$, as previously
estimated by Krabbe et al. (2014).

###  6.2 Oxygen Abundance

The identification of the dominant ionization source for the emitting gas
across the galaxy is essential to determine the chemical abundance correctly.
The elliptical galaxy, NGC 4105, is classified as a LINER galaxy by Mauch &
Sadler (2007) and Véron-Cetty & Véron (2010), and NGC 4106 does not have a
classification in the literature. Therefore, with the goal of confirming the
ionization source of NGC 4105 and to classify NGC 4106, we used the
$[\text{O\,{iii}}]/\rm H\beta$ versus $[\text{N\,{ii}}]/\rm H\alpha$
diagnostic diagram proposed by Baldwin et al. (1981), and commonly known as
BPT diagram. This diagram is used to distinguish objects ionized by massive
stars (SF), AGN, and LINER natures. In our case, we can not use the
$[\text{O\,{iii}}]/\rm H\beta$ versus $[\text{S\,{ii}}]/\rm H\alpha$ because
the [S II]$\lambda\lambda$6716,6731 emission lines presented a very low S/N in
the central regions of the galaxies.

Fig. 12 present the BPT diagram for the regions measured along the different
slit positions of AM 1204-292. We can note that all regions of NGC 4105
present emission like LINER, in agreement with the findings of (De Vaucouleurs
et al., 1991). NGC 4106 shows extended LINER-type emission in their disc, as
well as their nuclei.

Figure 12: The [O III]/H$\beta$ versus [N II]$\lambda$/H$\alpha$ diagnostic
diagram. The central region of each galaxy is marked by red symbols. The solid
black line represents the theoretical upper limit for SF galaxies from Kewley
et al. (2001) (Ke01), and the black dashed curve pure star-formation line from
Kauffmann et al. (2003) (Ka03). The region between Ke01 and Ka03 is
denominated composite region. The dividing line between Seyferts and LINERs
(long dashed, green) was set by Ho et al. (1997).

The accurate determination of the metallicity is critically dependent on the
emission lines sensitive to the electron temperature, which are not detected
in the spectra of the objects. Therefore, to estimate the oxygen abundance of
the gas phase in the nuclear regions of each galaxy, indirect methods that use
relatively strong emission lines easily observed in AGNs or SFs and/or
photoionization models should be applied. The most common method used for
estimating the oxygen abundance use the $R_{23}$=([O II]$\lambda$3727+[O
III]$\lambda\lambda$4959,5007)/H$\beta$ parameter initially proposed Pagel et
al. (1979). We used the $R_{23}$ indicator to estimate the metallicity in the
nuclear regions of both galaxies, comparing the observed values with a grid of
photoionization models from the code  CLOUDY code (Ferland et al., 2017).

The grids were built following the same methodology presented in Dors et al.
(2014, 2017). We created theoretical models based on power laws of the form
$f_{\nu}\propto\nu^{\alpha}$, where $\alpha$ was assumed to be equal to
$\alpha=-1.4$, which is representative value for AGNs (Zamorani et al., 1981;
Miller et al., 2011), with metallicities of
$Z=0.2,0.5,1.0,2.0,3.0\,$and$\,4.0\,Z_{\odot}$, ionization parameter ${\rm
log}\,U=-3.46,-2.46,1.46,0.46,0.54,1.54\,$and$\,2.54$, and a fixed electron
density value of 500 cm-3. The solar oxygen abundance of $12+{\rm
log(O/H)}_{\odot}=8.69\pm 0.05$ is taken from Allende Prieto et al. (2001).

Fig. 13 shows the $R_{23}$ vs. [O III]/[O II] diagram, with the observed
values superposed in the computed models. Black and red squares correspond to
the nuclear aperture in NGC 4105 and NGC 4106, respectively. The central O/H
abundances to each object were obtained by linear interpolation from the model
grid, which resulted in $12+{\rm log(O/H)}=9.03\pm 0.02$ and $12+{\rm
log(O/H)}=8.69\pm 0.05$ for NGC 4105 and NGC 4106, respectively. We compared
the metallicities obtained from the gas and stellar phase and verified that
for NGC 4106 the $Z/Z_{\odot}$ is the same, that is, $Z/Z_{\odot}=1$, and for
NGC 4105, the metallicity in the stellar phase ($Z/Z_{\odot}=1.6$) is slightly
lower than that obtained from the gas $Z/Z_{\odot}=2.2$, about 30%.

Figure 13: log(R23) vs. log([O III]/[O II]) index. The dashed lines connect
the photoionization model curves with different values of the logarithm
ionization parameters, while the solid lines to different gas-phase
metallicity. Black and red squares correspond to the nuclear region in NGC
4105 and NGC 4106, respectively.

##  7 Conclusions

With the present paper, we aimed at understanding the effects of the
interaction in the stellar and gas kinematics, stellar populations and ionized
gas properties in the strongly interacting system pair AM 1209-292. Long-slit
spectra in the range $3000-7050$ Å were obtained with the Goodman High
Throughput Spectrograph attached to the $4.1-$m SOAR telescope. The main
results are the following:

  1. 1.

No signs of rotation were detected in the stellar components over both slit
directions observed for NGC 4105. As for gas, the kinematics show a clear
pattern of rotation for PA=$118^{\circ}$, that is, gas and stars do not share
the same kinematics. We can deduce from this that the ionized gas in NGC 4105
is of external origin and that it must have been incorporated into the galaxy
recently. However, it was not the result of the ongoing interaction with NGC
4106, as the modeling of the system dynamics indicates. In contrast, NGC 4106
shows nearly symmetrical rotation curves across both slit directions, and a
dynamical mass of $1.6\times 10^{10}\,M_{\odot}$ was calculated from the V0 =
138$\pm$6 km s-1 at a radius of 3.64 kpc.

  2. 2.

We also perform a set of N-body numerical simulations of the encounter between
both components using P-Gadget3 TreePM/SPH code. It was found that the
perigalacticum would have occurred about 14.2 Myr ago. At this point, the
system is undergoing an outbreak of star formation. In addition, our models
successfully reproduce some important observational features of the system.

  3. 3.

The contribution of the stellar components in relation to the optical flux at
$\lambda 5870$ Å for both galaxies is dominated by old ($t\,>1\times 10^{9}$
yr) population, with a not negligible contribution of an intermediate
population and a small amount of young population.

  4. 4.

The electron density estimates for the pair AM 1204-292 are systematically
higher than those derived for isolated galaxies in the literature. Some
regions of NGC 4106 and NGC 4105 galaxies show an increment of $N_{\rm e}$
towards the outskirt and in the close central region, respectively.

  5. 5.

We observe a LINER-like line ratio in all regions of NGC 4105. While NGC 4106
shows extended LINER-type emission in the disc, as well as their nuclei.

  6. 6.

The central O/H abundances to NGC 4105 and NGC 4106 were obtained by linear
interpolation from a model grid and intensity of emission lines, which
resulted in $12+{\rm log(O/H)}=9.03\pm 0.02$ and $12+{\rm log(O/H)}=8.69\pm
0.05$ for NGC 4105 and NGC 4106, respectively.

  7. 7.

We compared the metallicities obtained from the gas and stellar phase and
verified that for NGC 4106 the $Z/Z_{\odot}$ is the same, that is,
$Z/Z_{\odot}=1$, and for NGC 4105, the metallicity in the stellar phase
($Z/Z_{\odot}=1.6$) is slightly lower than that obtained from the gas
$Z/Z_{\odot}=2.2$, about 30%.

In summary, the current work has contributed to the understanding of how the
mutual dynamic perturbation in a strongly interacting galaxy pair can disturb
the global morphology of each galaxy, the stellar and gas kinematics, the
physic-chemical properties of the gaseous component, and the stellar formation
inside each one of the galaxies. The AGN phenomenon can also be intensified by
encounters between galaxies containing gas in some amount, having been
investigated in this work as well. Curiously, the system contains galaxies of
different morphological types (and with distinct masses too), such that one
object could have been wrongly classified as a spiral galaxy due to the
formation of a double tidal stellar tail during the interaction as proposed by
us, which has easily been misinterpreted as a spiral arm. This kind of
observational study on a detailed characterization of both stellar and gaseous
components, when confronted with a set of N-body numerical simulations for the
encounter, has provided a lot of interesting results about the formation and
evolution of interacting galaxies in very low density environments like nearby
galaxy pairs and sparse groups.

## Acknowledgements

D. A. Rosa is grateful for the scholarship from CAPES and CNPq foundations for
the PCI/MCTIC/INPE posdoc fellowship (process 300082/2016-9). I. Rodrigues and
A. Krabbe thank the Brazilian foundation CNPq/MCTIC (grants number
313489/2018-1 and 311331/2017-3, respectively). A. Krabbe thanks FAPESP (grant
number 2016/21532-9). Numerical simulations were run on the Hypercubo HPC
cluster at IP&D-UNIVAP (FINEP 01.10.0661-00, FAPESP 2011/13250-0, FAPESP
2013/17247-9 and FAPESP 2014/10489-0). We all acknowledge the usage of the
HyperLeda database (http://leda.univ-lyon1.fr). This research made use of the
NASA/IPAC Extragalactic Database (NED), which is operated by the Jet
Propulsion Laboratory, California Institute of Technology, under contract with
the National Aeronautics and Space Administration.  IRAF is written and
supported by the National Optical Astronomy Observatories (NOAO) in Tucson,
Arizona. NOAO is operated by the Association of Universities for Research in
Astronomy (AURA), Inc. under cooperative agreement with the National Science
Foundation. We would like to thank Louis Ho, Zhao-Yu Li, and the CGS team for
kindly providing the image in the $I$ band of AM 1204-292.

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

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##  Appendix A Vertical chemical and cloud profiles

Figure 16: Vertical abundances profiles of radiatively active species computed
with Exo-REM. Solid lines assume solar metallicity, and dashed lines have 10x
solar metallicity. Disequilibrium chemistry is assumed in both cases. The
upper atmosphere is dominated by CO, with a strong abundance of H2O. TiO and
VO are negligible at these temperatures even though they are included in the
computation.

To derive the radiative data used in our 3D simulations, we computed vertical
abundances of typical absorbers using the 1D radiative convective code Exo-REM
(Baudino et al. 2015; Charnay et al. 2018; Blain et al. 2021) with
disequilibrium chemistry. For each planet, we assumed that H2O, CO, CH4, CO2,
FeH, HCN, H2S, TiO, VO, Na, K, PH3, and NH3 will be radiatively active. The
vertical profiles of the absorbers are presented in Fig.16 for a solar and a
10x solar metallicity. In both cases, CO and H2O dominates the atmosphere
below 10 bars. TiO and VO are negligible for WASP-43 b.
Figure 17 displays the initial condensable vapour mass mixing ratio used to
initial the 3D spherical shell of the generic PCM. These profiles are based on
analytical derivations from (Visscher et al. 2010) for Mg2SiO4 and (Morley et
al. 2012) for MnS.

Figure 17: Initial condensable vapour profile for MnS and Mg2SiO4 clouds.
These profiles are computed using analytical profiles from Visscher et al.
(2010) and Morley et al. (2012), with a planetary averaged temperature profile
from the cloudless simulation.

##  Appendix B Axial angular momentum transport induced by 1 $\mathbf{\mu}$m
clouds

Here are plotted the horizontal and vertical transport of angular momentum
contribution as wall as the net transport, as described in section 3.2.

Figure 18: Axial angular momentum horizontal transport for the cloudless
simulation of WASP-43 b (_left_) and 1$\rm\mu$m clouds of MnS (_middle_) and
Mg2SiO4 (_right_). Each row displays a different contribution: mean
circulation contribution (a) stationary wave contribution (b), transient wave
contribution (c), and net transport (d). Each row is plotted with a different
range to show the amplitude of each contribution to the net transport. Figure
19: Axial angular momentum vertical transport for the cloudless simulation of
WASP-43 b (_left_) and 1$\rm\mu$m clouds of MnS (_middle_) and Mg2SiO4
(_right_). Each row displays a different contribution: mean circulation
contribution (a), stationary wave contribution (b), transient wave
contribution (c), and net transport (d). Each row is plotted with a different
range to show the amplitude of each contribution to the net transport.

##  Appendix C Solar and super-solar simulation: Dynamical and thermal
structure of the atmosphere

Here are plotted the broad dynamical and thermal pattern of the atmosphere.
Figure 20 is for the solar metallicity simulations with Mg2SiO4 clouds. Figure
21 is the same for super-solar metallicity, with the first column being the
cloudless simulation and the three other panels cloudy simulations with
different particle sizes. In all plots where clouds are included, clouds are
composed of Mg2SiO4 .

Figure 20: Comparison of wind patterns, temperature maps at 10 mbar, cloud
latitudinal distributions, and jet acceleration for simulations with Mg2SiO4
clouds and particle sizes of $0.1$, $1$ and $10$ $\mathrm{\mu}$m. Top row:
Zonal mean and time-averaged zonal wind. Black contours are the mass stream
function, with solid lines indicating a clockwise circulation and dashed lines
an anti-clockwise circulation. Second row: Time-averaged temperature map at
the 10 mbar isobaric level. Black contours denote the location of clouds.
White vertical dashed lines are the terminators. Third row: Latitude-pressure
map of the cloud distribution, averaged in time and longitude. Bottom panel:
Vertically integrated jet zonal wind acceleration in zonal and time mean
between 5 and 20 mbar. We show the decomposition into residual mean
circulation, the eddy contributions, and the net acceleration. From left to
right, cloud particles increase in size.  Figure 21: Comparison of wind
patterns, temperature maps at 10 mbar, cloud latitudinal distributions, and
jet acceleration for simulations with Mg2SiO4 clouds at super-solar
metallicity and particle sizes of $0.1$, $1,$ and $10$ $\mathrm{\mu}$m. The
first column is the cloudless simulation, and the following are the cloudy
ones. Top row: Zonal mean and time-averaged zonal wind. Black contours are the
mass stream function, with solid lines indicating a clockwise circulation and
dashed lines an anti-clockwise circulation. Second row: Time-averaged
temperature map at the 10 mbar isobaric level. Black contours denote the
location of clouds. White vertical dashed lines are the terminators. Third
row: Latitude-pressure map of the cloud distribution, averaged in time and
longitude. Bottom panel: Vertically integrated jet zonal wind acceleration in
zonal and time mean between 5 and 20 mbar. We show the decomposition into
residual mean circulation, the eddy contributions, and the net acceleration.

We also plot the cloud radiative forcing for the super-solar metallicity
simulations with Mg2SiO4 clouds, in Fig. 22

Figure 22: Cloud radiative forcing (normalised by the stellar irradiance) for
super-solar metallicity atmospheres and Mg2SiO4 clouds. See Section 3.4 for a
definition.